API

Table of contents

• Quantum object (Qobj) and type
• Qobj boolean functions
• Qobj arithmetic and attributes
• Qobj eigenvalues and eigenvectors
• Qobj manipulation
• Generate states and operators
• Synonyms of functions for Qobj
• Time evolution
  • Steady State Solvers
  • Dynamical Shifted Fock method
  • Low-rank time evolution
• Correlations and Spectrum
• Entropy and Metrics
• Spin Lattice
• Symmetries and Block Diagonalization
• Miscellaneous
• Linear Maps
• Utility functions
• Visualization

Quantum object (Qobj) and type

QuantumToolbox.Space Type

struct Space <: AbstractSpace
    size::Int
end

A structure that describes a single Hilbert space with size = size.

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QuantumToolbox.Dimensions Type

struct Dimensions{N,T<:Tuple} <: AbstractDimensions{N}
    to::T
end

A structure that describes the Hilbert Space of each subsystems.

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QuantumToolbox.GeneralDimensions Type

struct GeneralDimensions{N,T1<:Tuple,T2<:Tuple} <: AbstractDimensions{N}
    to::T1
    from::T2
end

A structure that describes the left-hand side (to) and right-hand side (from) Hilbert Space of an Operator.

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QuantumToolbox.AbstractQuantumObject Type

abstract type AbstractQuantumObject{ObjType,DimType,DataType}

Abstract type for all quantum objects like QuantumObject and QuantumObjectEvolution.

Example

julia> sigmax() isa AbstractQuantumObject
true

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QuantumToolbox.BraQuantumObject Type

BraQuantumObject <: QuantumObjectType

Constructor representing a bra state $⟨\psi |$.

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QuantumToolbox.Bra Constant

const Bra = BraQuantumObject()

A constant representing the type of BraQuantumObject: a bra state $⟨\psi |$

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QuantumToolbox.KetQuantumObject Type

KetQuantumObject <: QuantumObjectType

Constructor representing a ket state $|\psi ⟩$.

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QuantumToolbox.Ket Constant

const Ket = KetQuantumObject()

A constant representing the type of KetQuantumObject: a ket state $|\psi ⟩$

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QuantumToolbox.OperatorQuantumObject Type

OperatorQuantumObject <: QuantumObjectType

Constructor representing an operator $\hat{O}$.

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QuantumToolbox.Operator Constant

const Operator = OperatorQuantumObject()

A constant representing the type of OperatorQuantumObject: an operator $\hat{O}$

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QuantumToolbox.OperatorBraQuantumObject Type

OperatorBraQuantumObject <: QuantumObjectType

Constructor representing a bra state in the SuperOperator formalism $⟨⟨\rho |$.

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QuantumToolbox.OperatorBra Constant

const OperatorBra = OperatorBraQuantumObject()

A constant representing the type of OperatorBraQuantumObject: a bra state in the SuperOperator formalism $⟨⟨\rho |$.

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QuantumToolbox.OperatorKetQuantumObject Type

OperatorKetQuantumObject <: QuantumObjectType

Constructor representing a ket state in the SuperOperator formalism $|\rho ⟩⟩$.

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QuantumToolbox.OperatorKet Constant

const OperatorKet = OperatorKetQuantumObject()

A constant representing the type of OperatorKetQuantumObject: a ket state in the SuperOperator formalism $|\rho ⟩⟩$

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QuantumToolbox.SuperOperatorQuantumObject Type

SuperOperatorQuantumObject <: QuantumObjectType

Constructor representing a super-operator $\hat{\mathcal{O}}$ acting on vectorized density operator matrices.

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QuantumToolbox.SuperOperator Constant

const SuperOperator = SuperOperatorQuantumObject()

A constant representing the type of SuperOperatorQuantumObject: a super-operator $\hat{\mathcal{O}}$ acting on vectorized density operator matrices

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QuantumToolbox.QuantumObject Type

struct QuantumObject{ObjType<:QuantumObjectType,DimType<:AbstractDimensions,DataType<:AbstractArray} <: AbstractQuantumObject{ObjType,DimType,DataType}
    data::DataType
    type::ObjType
    dimensions::DimType
end

Julia structure representing any time-independent quantum objects. For time-dependent cases, see QuantumObjectEvolution.

dims property

For a given H::QuantumObject, H.dims or getproperty(H, :dims) returns its dimensions in the type of integer-vector.

Examples

julia> a = destroy(20)

Quantum Object:   type=Operator   dims=[20]   size=(20, 20)   ishermitian=false
20×20 SparseMatrixCSC{ComplexF64, Int64} with 19 stored entries:
⎡⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⎤
⎢⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⎥
⎣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⎦

julia> a isa QuantumObject
true

julia> a.dims
1-element SVector{1, Int64} with indices SOneTo(1):
 20

julia> a.dimensions
Dimensions{1, Tuple{Space}}((Space(20),))

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QuantumToolbox.QuantumObjectEvolution Type

struct QuantumObjectEvolution{ObjType<:QuantumObjectType,DimType<:AbstractDimensions,DataType<:AbstractSciMLOperator} <: AbstractQuantumObject{ObjType,DimType,DataType}
    data::DataType
    type::ObjType
    dimensions::DimType
end

Julia struct representing any time-dependent quantum object. The data field is a AbstractSciMLOperator object that represents the time-dependent quantum object. It can be seen as

$$\hat{O}(t)=\sum _i{c}_i(p,t){\hat{O}}_i$$

where $c_i(p,t)$ is a function that depends on the parameters p and time t, and ${\hat{O}}_i$ are the operators that form the quantum object.

For time-independent cases, see QuantumObject, and for more information about AbstractSciMLOperator, see the SciML documentation.

dims property

For a given H::QuantumObjectEvolution, H.dims or getproperty(H, :dims) returns its dimensions in the type of integer-vector.

Examples

This operator can be initialized in the same way as the QuTiP QobjEvo object. For example

julia> a = tensor(destroy(10), qeye(2))

Quantum Object:   type=Operator   dims=[10, 2]   size=(20, 20)   ishermitian=false
20×20 SparseMatrixCSC{ComplexF64, Int64} with 18 stored entries:
⎡⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⎤
⎢⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⎥
⎣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⎦

julia> coef1(p, t) = exp(-1im * t)
coef1 (generic function with 1 method)

julia> op = QobjEvo(a, coef1)

Quantum Object Evo.:   type=Operator   dims=[10, 2]   size=(20, 20)   ishermitian=true   isconstant=false
ScalarOperator(0.0 + 0.0im) * MatrixOperator(20 × 20)

If there are more than 2 operators, we need to put each set of operator and coefficient function into a two-element Tuple, and put all these Tuples together in a larger Tuple:

julia> σm = tensor(qeye(10), sigmam())

Quantum Object:   type=Operator   dims=[10, 2]   size=(20, 20)   ishermitian=false
20×20 SparseMatrixCSC{ComplexF64, Int64} with 10 stored entries:
⎡⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⎤
⎢⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⎥
⎣⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⎦

julia> coef2(p, t) = sin(t)
coef2 (generic function with 1 method)

julia> op1 = QobjEvo(((a, coef1), (σm, coef2)))

Quantum Object Evo.:   type=Operator   dims=[10, 2]   size=(20, 20)   ishermitian=true   isconstant=false
(ScalarOperator(0.0 + 0.0im) * MatrixOperator(20 × 20) + ScalarOperator(0.0 + 0.0im) * MatrixOperator(20 × 20))

We can also concretize the operator at a specific time t

julia> op1(0.1)

Quantum Object:   type=Operator   dims=[10, 2]   size=(20, 20)   ishermitian=false
20×20 SparseMatrixCSC{ComplexF64, Int64} with 28 stored entries:
⎡⠂⡑⢄⠀⠀⠀⠀⠀⠀⠀⎤
⎢⠀⠀⠂⡑⢄⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠂⡑⢄⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠂⡑⢄⠀⎥
⎣⠀⠀⠀⠀⠀⠀⠀⠀⠂⡑⎦

It also supports parameter-dependent time evolution

julia> coef1(p, t) = exp(-1im * p.ω1 * t)
coef1 (generic function with 1 method)

julia> coef2(p, t) = sin(p.ω2 * t)
coef2 (generic function with 1 method)

julia> op1 = QobjEvo(((a, coef1), (σm, coef2)))

Quantum Object Evo.:   type=Operator   dims=[10, 2]   size=(20, 20)   ishermitian=true   isconstant=false
(ScalarOperator(0.0 + 0.0im) * MatrixOperator(20 × 20) + ScalarOperator(0.0 + 0.0im) * MatrixOperator(20 × 20))

julia> p = (ω1 = 1.0, ω2 = 0.5)
(ω1 = 1.0, ω2 = 0.5)

julia> op1(p, 0.1)

Quantum Object:   type=Operator   dims=[10, 2]   size=(20, 20)   ishermitian=false
20×20 SparseMatrixCSC{ComplexF64, Int64} with 28 stored entries:
⎡⠂⡑⢄⠀⠀⠀⠀⠀⠀⠀⎤
⎢⠀⠀⠂⡑⢄⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠂⡑⢄⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠂⡑⢄⠀⎥
⎣⠀⠀⠀⠀⠀⠀⠀⠀⠂⡑⎦

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Base.size Function

size(A::AbstractQuantumObject)
size(A::AbstractQuantumObject, idx::Int)
shape(A::AbstractQuantumObject)
shape(A::AbstractQuantumObject, idx::Int)

Returns a tuple containing each dimensions of the array in the AbstractQuantumObject.

Optionally, you can specify an index (idx) to just get the corresponding dimension of the array.

Note

shape is a synonym of size.

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Base.eltype Function

eltype(A::AbstractQuantumObject)

Returns the elements type of the matrix or vector corresponding to the AbstractQuantumObject A.

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Base.length Function

length(A::AbstractQuantumObject)

Returns the length of the matrix or vector corresponding to the AbstractQuantumObject A.

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SciMLOperators.cache_operator Function

SciMLOperators.cached_operator(L::AbstractQuantumObject, u)

Allocate caches for AbstractQuantumObject L for in-place evaluation with u-like input vectors.

Here, u can be in either the following types:

• AbstractVector

• Ket-type QuantumObject (if L is an Operator)

• OperatorKet-type QuantumObject (if L is a SuperOperator)

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Qobj boolean functions

QuantumToolbox.isbra Function

isbra(A)

Checks if the QuantumObject A is a BraQuantumObject. Default case returns false for any other inputs.

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QuantumToolbox.isket Function

isket(A)

Checks if the QuantumObject A is a KetQuantumObject. Default case returns false for any other inputs.

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QuantumToolbox.isoper Function

isoper(A)

Checks if the AbstractQuantumObject A is a OperatorQuantumObject. Default case returns false for any other inputs.

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QuantumToolbox.isoperbra Function

isoperbra(A)

Checks if the QuantumObject A is a OperatorBraQuantumObject. Default case returns false for any other inputs.

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QuantumToolbox.isoperket Function

isoperket(A)

Checks if the QuantumObject A is a OperatorKetQuantumObject. Default case returns false for any other inputs.

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QuantumToolbox.issuper Function

issuper(A)

Checks if the AbstractQuantumObject A is a SuperOperatorQuantumObject. Default case returns false for any other inputs.

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LinearAlgebra.ishermitian Function

ishermitian(A::AbstractQuantumObject)
isherm(A::AbstractQuantumObject)

Test whether the AbstractQuantumObject is Hermitian.

Note

isherm is a synonym of ishermitian.

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LinearAlgebra.issymmetric Function

issymmetric(A::AbstractQuantumObject)

Test whether the AbstractQuantumObject is symmetric.

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LinearAlgebra.isposdef Function

isposdef(A::AbstractQuantumObject)

Test whether the AbstractQuantumObject is positive definite (and Hermitian) by trying to perform a Cholesky factorization of A.

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QuantumToolbox.isunitary Function

isunitary(U::QuantumObject; kwargs...)

Test whether the QuantumObject $U$ is unitary operator. This function calls Base.isapprox to test whether $U{U}^{†}$ is approximately equal to identity operator.

Note that all the keyword arguments will be passed to Base.isapprox.

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SciMLOperators.iscached Function

SciMLOperators.iscached(A::AbstractQuantumObject)

Test whether the AbstractQuantumObject A has preallocated caches for inplace evaluations.

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SciMLOperators.isconstant Function

SciMLOperators.isconstant(A::AbstractQuantumObject)

Test whether the AbstractQuantumObject A is constant in time. For a QuantumObject, this function returns true, while for a QuantumObjectEvolution, this function returns true if the operator is constant in time.

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Qobj arithmetic and attributes

Base.zero Function

zero(A::AbstractQuantumObject)

Return a similar AbstractQuantumObject with dims and type are same as A, but data is a zero-array.

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Base.one Function

one(A::AbstractQuantumObject)

Return a similar AbstractQuantumObject with dims and type are same as A, but data is an identity matrix.

Note that A must be Operator or SuperOperator.

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Base.conj Function

conj(A::AbstractQuantumObject)

Return the element-wise complex conjugation of the AbstractQuantumObject.

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Base.transpose Function

transpose(A::AbstractQuantumObject)

Lazy matrix transpose of the AbstractQuantumObject.

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Base.adjoint Function

A'
adjoint(A::AbstractQuantumObject)
dag(A::AbstractQuantumObject)

Lazy adjoint (conjugate transposition) of the AbstractQuantumObject

Note

A' and dag(A) are synonyms of adjoint(A).

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LinearAlgebra.dot Function

A ⋅ B
dot(A::QuantumObject, B::QuantumObject)

Compute the dot product between two QuantumObject: $⟨A|B⟩$

Note that A and B should be Ket or OperatorKet

Note

A ⋅ B (where ⋅ can be typed by tab-completing \cdot in the REPL) is a synonym of dot(A, B).

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dot(i::QuantumObject, A::AbstractQuantumObject j::QuantumObject)
matrix_element(i::QuantumObject, A::AbstractQuantumObject j::QuantumObject)

Compute the generalized dot product dot(i, A*j) between a AbstractQuantumObject and two QuantumObject (i and j), namely $⟨i|\hat{A}|j⟩$.

Supports the following inputs:

• A is in the type of Operator, with i and j are both Ket.

• A is in the type of SuperOperator, with i and j are both OperatorKet

Note

matrix_element(i, A, j) is a synonym of dot(i, A, j).

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Base.sqrt Function

√(A)
sqrt(A::QuantumObject)

Matrix square root of QuantumObject

Note

√(A) (where √ can be typed by tab-completing \sqrt in the REPL) is a synonym of sqrt(A).

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Base.log Function

log(A::QuantumObject)

Matrix logarithm of QuantumObject

Note that this function only supports for Operator and SuperOperator

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Base.exp Function

exp(A::QuantumObject)

Matrix exponential of QuantumObject

Note that this function only supports for Operator and SuperOperator

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Base.sin Function

sin(A::QuantumObject)

Matrix sine of QuantumObject, defined as

$\sin \left(\hat{A}\right)=\frac{e^{i\hat{A}}-e^{-i\hat{A}}}{2i}$

Note that this function only supports for Operator and SuperOperator

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Base.cos Function

cos(A::QuantumObject)

Matrix cosine of QuantumObject, defined as

$\cos \left(\hat{A}\right)=\frac{e^{i\hat{A}}+e^{-i\hat{A}}}{2}$

Note that this function only supports for Operator and SuperOperator

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LinearAlgebra.tr Function

tr(A::QuantumObject)

Returns the trace of QuantumObject.

Note that this function only supports for Operator and SuperOperator

Examples

julia> a = destroy(20)

Quantum Object:   type=Operator   dims=[20]   size=(20, 20)   ishermitian=false
20×20 SparseMatrixCSC{ComplexF64, Int64} with 19 stored entries:
⎡⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⎤
⎢⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⎥
⎣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⎦

julia> tr(a' * a)
190.0 + 0.0im

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LinearAlgebra.svdvals Function

svdvals(A::QuantumObject)

Return the singular values of a QuantumObject in descending order

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LinearAlgebra.norm Function

norm(A::QuantumObject, p::Real)

Return the standard vector p-norm or Schatten p-norm of a QuantumObject depending on the type of A:

• If A is either Ket, Bra, OperatorKet, or OperatorBra, returns the standard vector p-norm (default p=2) of A.

• If A is either Operator or SuperOperator, returns Schatten p-norm (default p=1) of A.

Examples

julia> ψ = fock(10, 2)

Quantum Object:   type=Ket   dims=[10]   size=(10,)
10-element Vector{ComplexF64}:
 0.0 + 0.0im
 0.0 + 0.0im
 1.0 + 0.0im
 0.0 + 0.0im
 0.0 + 0.0im
 0.0 + 0.0im
 0.0 + 0.0im
 0.0 + 0.0im
 0.0 + 0.0im
 0.0 + 0.0im

julia> norm(ψ)
1.0

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LinearAlgebra.normalize Function

normalize(A::QuantumObject, p::Real)
unit(A::QuantumObject, p::Real)

Return normalized QuantumObject so that its p-norm equals to unity, i.e. norm(A, p) == 1.

Support for the following types of QuantumObject:

• If A is Ket or Bra, default p = 2

• If A is Operator, default p = 1

Note

unit is a synonym of normalize.

Also, see norm about its definition for different types of QuantumObject.

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LinearAlgebra.normalize! Function

normalize!(A::QuantumObject, p::Real)

Normalize QuantumObject in-place so that its p-norm equals to unity, i.e. norm(A, p) == 1.

Support for the following types of QuantumObject:

• If A is Ket or Bra, default p = 2

• If A is Operator, default p = 1

Also, see norm about its definition for different types of QuantumObject.

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Base.inv Function

inv(A::AbstractQuantumObject)

Matrix inverse of the AbstractQuantumObject. If A is a QuantumObjectEvolution, the inverse is computed at the last computed time.

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LinearAlgebra.diag Function

diag(A::QuantumObject, k::Int=0)

Return the k-th diagonal elements of a matrix-type QuantumObject

Note that this function only supports for Operator and SuperOperator

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QuantumToolbox.proj Function

proj(ψ::QuantumObject)

Return the projector for a Ket or Bra type of QuantumObject

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QuantumToolbox.ptrace Function

ptrace(QO::QuantumObject, sel)

Partial trace of a quantum state QO leaving only the dimensions with the indices present in the sel vector.

Note that this function will always return Operator. No matter the input QuantumObject is a Ket, Bra, or Operator.

Examples

Two qubits in the state $|\psi ⟩=|e,g⟩$:

julia> ψ = kron(fock(2,0), fock(2,1))

Quantum Object:   type=Ket   dims=[2, 2]   size=(4,)
4-element Vector{ComplexF64}:
 0.0 + 0.0im
 1.0 + 0.0im
 0.0 + 0.0im
 0.0 + 0.0im

julia> ptrace(ψ, 2)

Quantum Object:   type=Operator   dims=[2]   size=(2, 2)   ishermitian=true
2×2 Matrix{ComplexF64}:
 0.0+0.0im  0.0+0.0im
 0.0+0.0im  1.0+0.0im

or in an entangled state $|\psi ⟩=\frac{1}{\sqrt{2}}(|e,e⟩+|g,g⟩)$:

julia> ψ = 1 / √2 * (kron(fock(2,0), fock(2,0)) + kron(fock(2,1), fock(2,1)))

Quantum Object:   type=Ket   dims=[2, 2]   size=(4,)
4-element Vector{ComplexF64}:
 0.7071067811865475 + 0.0im
                0.0 + 0.0im
                0.0 + 0.0im
 0.7071067811865475 + 0.0im

julia> ptrace(ψ, 1)

Quantum Object:   type=Operator   dims=[2]   size=(2, 2)   ishermitian=true
2×2 Matrix{ComplexF64}:
 0.5+0.0im  0.0+0.0im
 0.0+0.0im  0.5+0.0im

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QuantumToolbox.purity Function

purity(ρ::QuantumObject)

Calculate the purity of a QuantumObject: $\text{Tr}({\rho }^2)$

Note that this function only supports for Ket, Bra, and Operator

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SparseArrays.permute Function

permute(A::QuantumObject, order::Union{AbstractVector{Int},Tuple})

Permute the tensor structure of a QuantumObject A according to the specified order list

Note that this method currently works for Ket, Bra, and Operator types of QuantumObject.

Examples

If order = [2, 1, 3], the Hilbert space structure will be re-arranged: ${\mathcal{H}}_1\otimes {\mathcal{H}}_2\otimes {\mathcal{H}}_3\to {\mathcal{H}}_2\otimes {\mathcal{H}}_1\otimes {\mathcal{H}}_3$.

julia> ψ1 = fock(2, 0);

julia> ψ2 = fock(3, 1);

julia> ψ3 = fock(4, 2);

julia> ψ_123 = tensor(ψ1, ψ2, ψ3);

julia> permute(ψ_123, (2, 1, 3)) ≈ tensor(ψ2, ψ1, ψ3)
true

Beware of type-stability!

It is highly recommended to use permute(A, order) with order as Tuple or SVector to keep type stability. See the related Section about type stability for more details.

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QuantumToolbox.tidyup Function

tidyup(A::QuantumObject, tol::Real=1e-14)

Given a QuantumObject A, check the real and imaginary parts of each element separately. Remove the real or imaginary value if its absolute value is less than tol.

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QuantumToolbox.tidyup! Function

tidyup!(A::QuantumObject, tol::Real=1e-14)

Given a QuantumObject A, check the real and imaginary parts of each element separately. Remove the real or imaginary value if its absolute value is less than tol.

Note that this function is an in-place version of tidyup.

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QuantumToolbox.get_data Function

get_data(A::AbstractQuantumObject)

Returns the data of a AbstractQuantumObject.

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QuantumToolbox.get_coherence Function

get_coherence(ψ::QuantumObject)

Get the coherence value $\alpha$ by measuring the expectation value of the destruction operator $\hat{a}$ on a state $|\psi ⟩$ or a density matrix $\hat{\rho }$.

It returns both $\alpha$ and the corresponding state with the coherence removed: $|{\delta }_{\alpha }⟩=\exp ({\alpha }^{\ast }\hat{a}-\alpha {\hat{a}}^{†})|\psi ⟩$ for a pure state, and $\widehat{{\rho }_{\alpha }}=\exp ({\alpha }^{\ast }\hat{a}-\alpha {\hat{a}}^{†})\hat{\rho }\exp (-\overline{\alpha }\hat{a}+\alpha {\hat{a}}^{†})$ for a density matrix. These states correspond to the quantum fluctuations around the coherent state $|\alpha ⟩$ or $|\alpha ⟩⟨\alpha |$.

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QuantumToolbox.partial_transpose Function

partial_transpose(ρ::QuantumObject, mask::Vector{Bool})

Return the partial transpose of a density matrix $\rho$, where mask is an array/vector with length that equals the length of ρ.dims. The elements in mask are boolean (true or false) which indicates whether or not the corresponding subsystem should be transposed.

Arguments

• ρ::QuantumObject: The density matrix (ρ.type must be OperatorQuantumObject).

• mask::Vector{Bool}: A boolean vector selects which subsystems should be transposed.

Returns

• ρ_pt::QuantumObject: The density matrix with the selected subsystems transposed.

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Qobj eigenvalues and eigenvectors

QuantumToolbox.EigsolveResult Type

struct EigsolveResult

A struct containing the eigenvalues, the eigenvectors, and some information from the solver

Fields (Attributes)

• values::AbstractVector: the eigenvalues

• vectors::AbstractMatrix: the transformation matrix (eigenvectors)

• type::Union{Nothing,QuantumObjectType}: the type of QuantumObject, or nothing means solving eigen equation for general matrix

• dimensions::Union{Nothing,AbstractDimensions}: the dimensions of QuantumObject, or nothing means solving eigen equation for general matrix

• iter::Int: the number of iteration during the solving process

• numops::Int : number of times the linear map was applied in krylov methods

• converged::Bool: Whether the result is converged

dims property

For a given res::EigsolveResult, res.dims or getproperty(res, :dims) returns its dimensions in the type of integer-vector.

Examples

One can obtain the eigenvalues and the corresponding QuantumObject-type eigenvectors by:

julia> result = eigenstates(sigmax())
EigsolveResult:   type=Operator   dims=[2]
values:
2-element Vector{ComplexF64}:
 -1.0 + 0.0im
  1.0 + 0.0im
vectors:
2×2 Matrix{ComplexF64}:
 -0.707107+0.0im  0.707107+0.0im
  0.707107+0.0im  0.707107+0.0im

julia> λ, ψ, U = result;

julia> λ
2-element Vector{ComplexF64}:
 -1.0 + 0.0im
  1.0 + 0.0im

julia> ψ
2-element Vector{QuantumObject{KetQuantumObject, Dimensions{1, Tuple{Space}}, Vector{ComplexF64}}}:

Quantum Object:   type=Ket   dims=[2]   size=(2,)
2-element Vector{ComplexF64}:
 -0.7071067811865475 + 0.0im
  0.7071067811865475 + 0.0im

Quantum Object:   type=Ket   dims=[2]   size=(2,)
2-element Vector{ComplexF64}:
 0.7071067811865475 + 0.0im
 0.7071067811865475 + 0.0im

julia> U
2×2 Matrix{ComplexF64}:
 -0.707107+0.0im  0.707107+0.0im
  0.707107+0.0im  0.707107+0.0im

source

QuantumToolbox.eigenenergies Function

eigenenergies(A::QuantumObject; sparse::Bool=false, kwargs...)

Calculate the eigenenergies

Arguments

• A::QuantumObject: the QuantumObject to solve eigenvalues

• sparse::Bool: if false call eigvals(A::QuantumObject; kwargs...), otherwise call eigsolve. Default to false.

• kwargs: Additional keyword arguments passed to the solver. If sparse=true, the keyword arguments are passed to eigsolve, otherwise to eigen.

Returns

• ::Vector{<:Number}: a list of eigenvalues

source

QuantumToolbox.eigenstates Function

eigenstates(A::QuantumObject; sparse::Bool=false, kwargs...)

Calculate the eigenvalues and corresponding eigenvectors

Arguments

• A::QuantumObject: the QuantumObject to solve eigenvalues and eigenvectors

• sparse::Bool: if false call eigen(A::QuantumObject; kwargs...), otherwise call eigsolve. Default to false.

• kwargs: Additional keyword arguments passed to the solver. If sparse=true, the keyword arguments are passed to eigsolve, otherwise to eigen.

Returns

• ::EigsolveResult: containing the eigenvalues, the eigenvectors, and some information from the solver. see also EigsolveResult

source

LinearAlgebra.eigen Function

LinearAlgebra.eigen(A::QuantumObject; kwargs...)

Calculates the eigenvalues and eigenvectors of the QuantumObject A using the Julia LinearAlgebra package.

julia> a = destroy(5);

julia> H = a + a';

julia> E, ψ, U = eigen(H)
EigsolveResult:   type=Operator   dims=[5]
values:
5-element Vector{ComplexF64}:
       -2.8569700138728 + 0.0im
    -1.3556261799742608 + 0.0im
 1.3322676295501878e-15 + 0.0im
     1.3556261799742677 + 0.0im
     2.8569700138728056 + 0.0im
vectors:
5×5 Matrix{ComplexF64}:
  0.106101+0.0im  -0.471249-0.0im  …   0.471249-0.0im  0.106101-0.0im
 -0.303127-0.0im   0.638838+0.0im      0.638838+0.0im  0.303127-0.0im
  0.537348+0.0im  -0.279149-0.0im      0.279149-0.0im  0.537348-0.0im
 -0.638838-0.0im  -0.303127-0.0im     -0.303127-0.0im  0.638838+0.0im
  0.447214+0.0im   0.447214+0.0im     -0.447214-0.0im  0.447214-0.0im

julia> expect(H, ψ[1]) ≈ E[1]
true

source

LinearAlgebra.eigvals Function

LinearAlgebra.eigvals(A::QuantumObject; kwargs...)

Same as eigen(A::QuantumObject; kwargs...) but for only the eigenvalues.

source

QuantumToolbox.eigsolve Function

eigsolve(A::QuantumObject; 
    v0::Union{Nothing,AbstractVector}=nothing, 
    sigma::Union{Nothing, Real}=nothing,
    eigvals::Int = 1,
    krylovdim::Int = max(20, 2*k+1),
    tol::Real = 1e-8,
    maxiter::Int = 200,
    solver::Union{Nothing, SciMLLinearSolveAlgorithm} = nothing,
    kwargs...)

Solve for the eigenvalues and eigenvectors of a matrix A using the Arnoldi method.

Arguments

• A::QuantumObject: the QuantumObject to solve eigenvalues and eigenvectors.

• v0::Union{Nothing,AbstractVector}: the initial vector for the Arnoldi method. Default is a random vector.

• sigma::Union{Nothing, Real}: the shift for the eigenvalue problem. Default is nothing.

• eigvals::Int: the number of eigenvalues to compute. Default is 1.

• krylovdim::Int: the dimension of the Krylov subspace. Default is max(20, 2*k+1).

• tol::Real: the tolerance for the Arnoldi method. Default is 1e-8.

• maxiter::Int: the maximum number of iterations for the Arnoldi method. Default is 200.

• solver::Union{Nothing, SciMLLinearSolveAlgorithm}: the linear solver algorithm. Default is nothing.

• kwargs: Additional keyword arguments passed to the solver.

Notes

• For more details about solver and extra kwargs, please refer to LinearSolve.jl

Returns

• EigsolveResult: A struct containing the eigenvalues, the eigenvectors, and some information about the eigsolver

source

QuantumToolbox.eigsolve_al Function

eigsolve_al(
    H::Union{AbstractQuantumObject{HOpType},Tuple},
    T::Real,
    c_ops::Union{Nothing,AbstractVector,Tuple} = nothing;
    alg::OrdinaryDiffEqAlgorithm = Tsit5(),
    params::NamedTuple = NamedTuple(),
    ρ0::AbstractMatrix = rand_dm(prod(H.dimensions)).data,
    eigvals::Int = 1,
    krylovdim::Int = min(10, size(H, 1)),
    maxiter::Int = 200,
    eigstol::Real = 1e-6,
    kwargs...,
)

Solve the eigenvalue problem for a Liouvillian superoperator L using the Arnoldi-Lindblad method.

Arguments

• H: The Hamiltonian (or directly the Liouvillian) of the system. It can be a QuantumObject, a QuantumObjectEvolution, or a tuple of the form supported by mesolve.

• T: The time at which to evaluate the time evolution.

• c_ops: A vector of collapse operators. Default is nothing meaning the system is closed.

• alg: The differential equation solver algorithm. Default is Tsit5().

• params: A NamedTuple containing the parameters of the system.

• ρ0: The initial density matrix. If not specified, a random density matrix is used.

• eigvals: The number of eigenvalues to compute.

• krylovdim: The dimension of the Krylov subspace.

• maxiter: The maximum number of iterations for the eigsolver.

• eigstol: The tolerance for the eigsolver.

• kwargs: Additional keyword arguments passed to the differential equation solver.

Notes

• For more details about alg please refer to DifferentialEquations.jl (ODE Solvers)

• For more details about kwargs please refer to DifferentialEquations.jl (Keyword Arguments)

Returns

• EigsolveResult: A struct containing the eigenvalues, the eigenvectors, and some information about the eigsolver

References

• [1] Minganti, F., & Huybrechts, D. (2022). Arnoldi-Lindblad time evolution: Faster-than-the-clock algorithm for the spectrum of time-independent and Floquet open quantum systems. Quantum, 6, 649.

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Qobj manipulation

QuantumToolbox.ket2dm Function

ket2dm(ψ::QuantumObject)

Transform the ket state $|\psi ⟩$ into a pure density matrix $\hat{\rho }=|\psi ⟩⟨\psi |$.

source

QuantumToolbox.expect Function

expect(O::Union{AbstractQuantumObject,Vector{AbstractQuantumObject}}, ψ::Union{QuantumObject,Vector{QuantumObject}})

Expectation value of the Operator O with the state ψ. The state can be a Ket, Bra or Operator.

If ψ is a Ket or Bra, the function calculates $⟨\psi |\hat{O}|\psi ⟩$.

If ψ is a density matrix (Operator), the function calculates $\text{Tr}\left[\hat{O}\hat{\psi }\right]$

The function returns a real number if O is of Hermitian type or Symmetric type, and returns a complex number otherwise. You can make an operator O hermitian by using Hermitian(O).

List of observables and states

The observable O and state ψ can be given as a list of QuantumObject, it returns a list of expectation values. If both of them are given as a list, it returns a Matrix of expectation values.

Examples

julia> ψ1 = 1 / √2 * (fock(10,2) + fock(10,4));

julia> ψ2 = coherent(10, 0.6 + 0.8im);

julia> a = destroy(10);

julia> expect(a' * a, ψ1) |> round
3.0 + 0.0im

julia> expect(Hermitian(a' * a), ψ1) |> round
3.0

julia> round.(expect([a' * a, a' + a, a], [ψ1, ψ2]), digits = 1)
3×2 Matrix{ComplexF64}:
 3.0+0.0im  1.0+0.0im
 0.0+0.0im  1.2-0.0im
 0.0+0.0im  0.6+0.8im

source

QuantumToolbox.variance Function

variance(O::QuantumObject, ψ::Union{QuantumObject,Vector{QuantumObject}})

Variance of the Operator O: $⟨{\hat{O}}^2⟩-⟨\hat{O}{⟩}^2$,

where $⟨\hat{O}⟩$ is the expectation value of O with the state ψ (see also expect), and the state ψ can be a Ket, Bra or Operator.

The function returns a real number if O is hermitian, and returns a complex number otherwise.

Note that ψ can also be given as a list of QuantumObject, it returns a list of expectation values.

source

Base.kron Function

kron(A::AbstractQuantumObject, B::AbstractQuantumObject, ...)
tensor(A::AbstractQuantumObject, B::AbstractQuantumObject, ...)
⊗(A::AbstractQuantumObject, B::AbstractQuantumObject, ...)
A ⊗ B

Returns the Kronecker product $\hat{A}\otimes \hat{B}\otimes \cdots$.

Note

tensor and ⊗ (where ⊗ can be typed by tab-completing \otimes in the REPL) are synonyms of kron.

Examples

julia> a = destroy(20)

Quantum Object:   type=Operator   dims=[20]   size=(20, 20)   ishermitian=false
20×20 SparseMatrixCSC{ComplexF64, Int64} with 19 stored entries:
⎡⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⎤
⎢⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⎥
⎣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⎦

julia> O = kron(a, a);

julia> size(a), size(O)
((20, 20), (400, 400))

julia> a.dims, O.dims
([20], [20, 20])

source

QuantumToolbox.to_dense Function

to_dense(A::QuantumObject)

Converts a sparse QuantumObject to a dense QuantumObject.

source

QuantumToolbox.to_sparse Function

to_sparse(A::QuantumObject)

Converts a dense QuantumObject to a sparse QuantumObject.

source

QuantumToolbox.vec2mat Function

vec2mat(A::AbstractVector)

Converts a vector to a matrix.

source

vec2mat(A::QuantumObject)
vector_to_operator(A::QuantumObject)

Convert a quantum object from vector (OperatorKetQuantumObject-type) to matrix (OperatorQuantumObject-type)

Note

vector_to_operator is a synonym of vec2mat.

source

QuantumToolbox.mat2vec Function

mat2vec(A::QuantumObject)
operator_to_vector(A::QuantumObject)

Convert a quantum object from matrix (OperatorQuantumObject-type) to vector (OperatorKetQuantumObject-type)

Note

operator_to_vector is a synonym of mat2vec.

source

mat2vec(A::AbstractMatrix)

Converts a matrix to a vector.

source

Generate states and operators

QuantumToolbox.zero_ket Function

zero_ket(dimensions)

Returns a zero Ket vector with given argument dimensions.

The dimensions can be either the following types:

• dimensions::Int: Number of basis states in the Hilbert space.

• dimensions::Union{Dimensions,AbstractVector{Int}, Tuple}: list of dimensions representing the each number of basis in the subsystems.

Beware of type-stability!

It is highly recommended to use zero_ket(dimensions) with dimensions as Tuple or SVector to keep type stability. See the related Section about type stability for more details.

source

QuantumToolbox.fock Function

fock(N::Int, j::Int=0; dims::Union{Int,AbstractVector{Int},Tuple}=N, sparse::Union{Bool,Val}=Val(false))

Generates a fock state $|\psi ⟩$ of dimension N.

It is also possible to specify the list of dimensions dims if different subsystems are present.

Beware of type-stability!

If you want to keep type stability, it is recommended to use fock(N, j, dims=dims, sparse=Val(sparse)) instead of fock(N, j, dims=dims, sparse=sparse). Consider also to use dims as a Tuple or SVector instead of Vector. See this link and the related Section about type stability for more details.

source

QuantumToolbox.basis Function

basis(N::Int, j::Int = 0; dims::Union{Int,AbstractVector{Int},Tuple}=N)

Generates a fock state like fock.

It is also possible to specify the list of dimensions dims if different subsystems are present.

Beware of type-stability!

If you want to keep type stability, it is recommended to use basis(N, j, dims=dims) with dims as a Tuple or SVector instead of Vector. See this link and the related Section about type stability for more details.

source

QuantumToolbox.coherent Function

coherent(N::Int, α::Number)

Generates a coherent state $|\alpha ⟩$, which is defined as an eigenvector of the bosonic annihilation operator $\hat{a}|\alpha ⟩=\alpha |\alpha ⟩$.

This state is constructed via the displacement operator displace and zero-fock state fock: $|\alpha ⟩=\hat{D}(\alpha )|0⟩$

source

QuantumToolbox.rand_ket Function

rand_ket(dimensions)

Generate a random normalized Ket vector with given argument dimensions.

The dimensions can be either the following types:

• dimensions::Int: Number of basis states in the Hilbert space.

• dimensions::Union{Dimensions,AbstractVector{Int},Tuple}: list of dimensions representing the each number of basis in the subsystems.

Beware of type-stability!

If you want to keep type stability, it is recommended to use rand_ket(dimensions) with dimensions as Tuple or SVector to keep type stability. See the related Section about type stability for more details.

source

QuantumToolbox.fock_dm Function

fock_dm(N::Int, j::Int=0; dims::Union{Int,AbstractVector{Int},Tuple}=N, sparse::Union{Bool,Val}=Val(false))

Density matrix representation of a Fock state.

Constructed via outer product of fock.

Beware of type-stability!

If you want to keep type stability, it is recommended to use fock_dm(N, j, dims=dims, sparse=Val(sparse)) instead of fock_dm(N, j, dims=dims, sparse=sparse). Consider also to use dims as a Tuple or SVector instead of Vector. See this link and the related Section about type stability for more details.

source

QuantumToolbox.coherent_dm Function

coherent_dm(N::Int, α::Number)

Density matrix representation of a coherent state.

Constructed via outer product of coherent.

source

QuantumToolbox.thermal_dm Function

thermal_dm(N::Int, n::Real; sparse::Union{Bool,Val}=Val(false))

Density matrix for a thermal state (generating thermal state probabilities) with the following arguments:

• N::Int: Number of basis states in the Hilbert space

• n::Real: Expectation value for number of particles in the thermal state.

• sparse::Union{Bool,Val}: If true, return a sparse matrix representation.

Beware of type-stability!

If you want to keep type stability, it is recommended to use thermal_dm(N, n, sparse=Val(sparse)) instead of thermal_dm(N, n, sparse=sparse). See this link and the related Section about type stability for more details.

source

QuantumToolbox.maximally_mixed_dm Function

maximally_mixed_dm(dimensions)

Returns the maximally mixed density matrix with given argument dimensions.

The dimensions can be either the following types:

• dimensions::Int: Number of basis states in the Hilbert space.

• dimensions::Union{Dimensions,AbstractVector{Int},Tuple}: list of dimensions representing the each number of basis in the subsystems.

Beware of type-stability!

If you want to keep type stability, it is recommended to use maximally_mixed_dm(dimensions) with dimensions as Tuple or SVector to keep type stability. See the related Section about type stability for more details.

source

QuantumToolbox.rand_dm Function

rand_dm(dimensions; rank::Int=prod(dimensions))

Generate a random density matrix from Ginibre ensemble with given argument dimensions and rank, ensuring that it is positive semi-definite and trace equals to 1.

The dimensions can be either the following types:

• dimensions::Int: Number of basis states in the Hilbert space.

• dimensions::Union{Dimensions,AbstractVector{Int},Tuple}: list of dimensions representing the each number of basis in the subsystems.

The default keyword argument rank = prod(dimensions) (full rank).

Beware of type-stability!

If you want to keep type stability, it is recommended to use rand_dm(dimensions; rank=rank) with dimensions as Tuple or SVector instead of Vector. See this link and the related Section about type stability for more details.

References

• J. Ginibre, Statistical ensembles of complex, quaternion, and real matrices, Journal of Mathematical Physics 6.3 (1965): 440-449

• K. Życzkowski, et al., Generating random density matrices, Journal of Mathematical Physics 52, 062201 (2011)

source

QuantumToolbox.spin_state Function

spin_state(j::Real, m::Real)

Generate the spin state: $|j,m⟩$

The eigenstate of the Spin-j ${\hat{S}}_z$ operator with eigenvalue m, where where j is the spin quantum number and can be a non-negative integer or half-integer

See also jmat.

source

QuantumToolbox.spin_coherent Function

spin_coherent(j::Real, θ::Real, ϕ::Real)

Generate the coherent spin state (rotation of the $|j,j⟩$ state), namely

$$|\theta ,\varphi ⟩=\hat{R}(\theta ,\varphi )|j,j⟩$$

where the rotation operator is defined as

$$\hat{R}(\theta ,\varphi )=\exp (\frac{\theta }{2}({\hat{S}}_{-}{e}^{i\varphi }-{\hat{S}}_{+}{e}^{-i\varphi }))$$

and ${\hat{S}}_{\pm }$ are plus and minus Spin-j operators, respectively.

Arguments

• j::Real: The spin quantum number and can be a non-negative integer or half-integer

• θ::Real: rotation angle from z-axis

• ϕ::Real: rotation angle from x-axis

See also jmat and spin_state.

Reference

• Robert Jones, Spin Coherent States and Statistical Physics

source

QuantumToolbox.bell_state Function

bell_state(x::Union{Int}, z::Union{Int})

Return the Bell state depending on the arguments (x, z):

• (0, 0): $|{\mathrm{\Phi }}^{+}⟩=(|00⟩+|11⟩)/\sqrt{2}$

• (0, 1): $|{\mathrm{\Phi }}^{-}⟩=(|00⟩-|11⟩)/\sqrt{2}$

• (1, 0): $|{\mathrm{\Psi }}^{+}⟩=(|01⟩+|10⟩)/\sqrt{2}$

• (1, 1): $|{\mathrm{\Psi }}^{-}⟩=(|01⟩-|10⟩)/\sqrt{2}$

Here, x = 1 (z = 1) means applying Pauli-$X$ ( Pauli-$Z$) unitary transformation on $|{\mathrm{\Phi }}^{+}⟩$.

Example

julia> bell_state(0, 0)

Quantum Object:   type=Ket   dims=[2, 2]   size=(4,)
4-element Vector{ComplexF64}:
 0.7071067811865475 + 0.0im
                0.0 + 0.0im
                0.0 + 0.0im
 0.7071067811865475 + 0.0im

julia> bell_state(Val(1), Val(0))

Quantum Object:   type=Ket   dims=[2, 2]   size=(4,)
4-element Vector{ComplexF64}:
                0.0 + 0.0im
 0.7071067811865475 + 0.0im
 0.7071067811865475 + 0.0im
                0.0 + 0.0im

Beware of type-stability!

If you want to keep type stability, it is recommended to use bell_state(Val(x), Val(z)) instead of bell_state(x, z). See this link and the related Section for more details.

source

QuantumToolbox.singlet_state Function

singlet_state()

Return the two particle singlet state: $\frac{1}{\sqrt{2}}(|01⟩-|10⟩)$

source

QuantumToolbox.triplet_states Function

triplet_states()

Return a list of the two particle triplet states:

• $|11⟩$

• $(|01⟩+|10⟩)/\sqrt{2}$

• $|00⟩$

source

QuantumToolbox.w_state Function

w_state(n::Union{Int,Val})

Returns the n-qubit W-state:

$$\frac{1}{\sqrt{n}}(|100...0⟩+|010...0⟩+\cdots +|00...01⟩)$$

Beware of type-stability!

If you want to keep type stability, it is recommended to use w_state(Val(n)) instead of w_state(n). See this link and the related Section for more details.

source

QuantumToolbox.ghz_state Function

ghz_state(n::Union{Int,Val}; d::Int=2)

Returns the generalized n-qudit Greenberger–Horne–Zeilinger (GHZ) state:

$$\frac{1}{\sqrt{d}}\sum _{i=0}^{d-1}|i⟩\otimes \cdots \otimes |i⟩$$

Here, d specifies the dimension of each qudit. Default to d=2 (qubit).

Beware of type-stability!

If you want to keep type stability, it is recommended to use ghz_state(Val(n)) instead of ghz_state(n). See this link and the related Section for more details.

source

QuantumToolbox.rand_unitary Function

rand_unitary(dimensions, distribution=Val(:haar))

Returns a random unitary QuantumObject.

The dimensions can be either the following types:

• dimensions::Int: Number of basis states in the Hilbert space.

• dimensions::Union{Dimensions,AbstractVector{Int},Tuple}: list of dimensions representing the each number of basis in the subsystems.

The distribution specifies which of the method used to obtain the unitary matrix:

• :haar: Haar random unitary matrix using the algorithm from reference 1

• :exp: Uses $\exp (-i\hat{H})$, where $\hat{H}$ is a randomly generated Hermitian operator.

References

1. F. Mezzadri, How to generate random matrices from the classical compact groups, arXiv:math-ph/0609050 (2007)

Beware of type-stability!

If you want to keep type stability, it is recommended to use rand_unitary(dimensions, Val(distribution)) instead of rand_unitary(dimensions, distribution). Also, put dimensions as Tuple or SVector. See this link and the related Section about type stability for more details.

source

QuantumToolbox.sigmap Function

sigmap()

Pauli ladder operator ${\hat{\sigma }}_{+}=({\hat{\sigma }}_x+i{\hat{\sigma }}_y)/2$.

See also jmat.

source

QuantumToolbox.sigmam Function

sigmam()

Pauli ladder operator ${\hat{\sigma }}_{-}=({\hat{\sigma }}_x-i{\hat{\sigma }}_y)/2$.

See also jmat.

source

QuantumToolbox.sigmax Function

sigmax()

Pauli operator ${\hat{\sigma }}_x={\hat{\sigma }}_{-}+{\hat{\sigma }}_{+}$.

See also jmat.

source

QuantumToolbox.sigmay Function

sigmay()

Pauli operator ${\hat{\sigma }}_y=i({\hat{\sigma }}_{-}-{\hat{\sigma }}_{+})$.

See also jmat.

source

QuantumToolbox.sigmaz Function

sigmaz()

Pauli operator ${\hat{\sigma }}_z=[{\hat{\sigma }}_{+},{\hat{\sigma }}_{-}]$.

See also jmat.

source

QuantumToolbox.jmat Function

jmat(j::Real, which::Union{Symbol,Val})

Generate higher-order Spin-j operators, where j is the spin quantum number and can be a non-negative integer or half-integer

The parameter which specifies which of the following operator to return.

• :x: ${\hat{S}}_x$

• :y: ${\hat{S}}_y$

• :z: ${\hat{S}}_z$

• :+: ${\hat{S}}_{+}$

• :-: ${\hat{S}}_{-}$

Note that if the parameter which is not specified, returns a set of Spin-j operators: $({\hat{S}}_x,{\hat{S}}_y,{\hat{S}}_z)$

Examples

julia> jmat(0.5, :x)

Quantum Object:   type=Operator   dims=[2]   size=(2, 2)   ishermitian=true
2×2 SparseMatrixCSC{ComplexF64, Int64} with 2 stored entries:
     ⋅      0.5+0.0im
 0.5+0.0im      ⋅

julia> jmat(0.5, Val(:-))

Quantum Object:   type=Operator   dims=[2]   size=(2, 2)   ishermitian=false
2×2 SparseMatrixCSC{ComplexF64, Int64} with 1 stored entry:
     ⋅          ⋅    
 1.0+0.0im      ⋅

julia> jmat(1.5, Val(:z))

Quantum Object:   type=Operator   dims=[4]   size=(4, 4)   ishermitian=true
4×4 SparseMatrixCSC{ComplexF64, Int64} with 4 stored entries:
 1.5+0.0im      ⋅           ⋅           ⋅    
     ⋅      0.5+0.0im       ⋅           ⋅    
     ⋅          ⋅      -0.5+0.0im       ⋅    
     ⋅          ⋅           ⋅      -1.5+0.0im

Beware of type-stability!

If you want to keep type stability, it is recommended to use jmat(j, Val(which)) instead of jmat(j, which). See this link and the related Section about type stability for more details.

source

QuantumToolbox.spin_Jx Function

spin_Jx(j::Real)

${\hat{S}}_x$ operator for Spin-j, where j is the spin quantum number and can be a non-negative integer or half-integer.

See also jmat.

source

QuantumToolbox.spin_Jy Function

spin_Jy(j::Real)

${\hat{S}}_y$ operator for Spin-j, where j is the spin quantum number and can be a non-negative integer or half-integer.

See also jmat.

source

QuantumToolbox.spin_Jz Function

spin_Jz(j::Real)

${\hat{S}}_z$ operator for Spin-j, where j is the spin quantum number and can be a non-negative integer or half-integer.

See also jmat.

source

QuantumToolbox.spin_Jm Function

spin_Jm(j::Real)

${\hat{S}}_{-}$ operator for Spin-j, where j is the spin quantum number and can be a non-negative integer or half-integer.

See also jmat.

source

QuantumToolbox.spin_Jp Function

spin_Jp(j::Real)

${\hat{S}}_{+}$ operator for Spin-j, where j is the spin quantum number and can be a non-negative integer or half-integer.

See also jmat.

source

QuantumToolbox.spin_J_set Function

spin_J_set(j::Real)

A set of Spin-j operators $({\hat{S}}_x,{\hat{S}}_y,{\hat{S}}_z)$, where j is the spin quantum number and can be a non-negative integer or half-integer.

Note that this functions is same as jmat(j). See also jmat.

source

QuantumToolbox.destroy Function

destroy(N::Int)

Bosonic annihilation operator with Hilbert space cutoff N.

This operator acts on a fock state as $\hat{a}|n⟩=\sqrt{n}|n-1⟩$.

Examples

julia> a = destroy(20)

Quantum Object:   type=Operator   dims=[20]   size=(20, 20)   ishermitian=false
20×20 SparseMatrixCSC{ComplexF64, Int64} with 19 stored entries:
⎡⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⎤
⎢⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⎥
⎣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⎦

julia> fock(20, 3)' * a * fock(20, 4)
2.0 + 0.0im

source

QuantumToolbox.create Function

create(N::Int)

Bosonic creation operator with Hilbert space cutoff N.

This operator acts on a fock state as ${\hat{a}}^{†}|n⟩=\sqrt{n+1}|n+1⟩$.

Examples

julia> a_d = create(20)

Quantum Object:   type=Operator   dims=[20]   size=(20, 20)   ishermitian=false
20×20 SparseMatrixCSC{ComplexF64, Int64} with 19 stored entries:
⎡⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⎤
⎢⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⎥
⎣⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⎦

julia> fock(20, 4)' * a_d * fock(20, 3)
2.0 + 0.0im

source

QuantumToolbox.displace Function

displace(N::Int, α::Number)

Generate a displacement operator:

$$\hat{D}(\alpha )=\exp (\alpha {\hat{a}}^{†}-{\alpha }^{\ast }\hat{a}),$$

where $\hat{a}$ is the bosonic annihilation operator, and $\alpha$ is the amount of displacement in optical phase space.

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QuantumToolbox.squeeze Function

squeeze(N::Int, z::Number)

Generate a single-mode squeeze operator:

$$\hat{S}(z)=\exp (\frac{1}{2}(z^{\ast }{\hat{a}}^2-z({\hat{a}}^{†}{)}^2)),$$

where $\hat{a}$ is the bosonic annihilation operator.

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QuantumToolbox.num Function

num(N::Int)

Bosonic number operator with Hilbert space cutoff N.

This operator is defined as $\hat{N}={\hat{a}}^{†}\hat{a}$, where $\hat{a}$ is the bosonic annihilation operator.

source

QuantumToolbox.position Function

position(N::Int)

Position operator with Hilbert space cutoff N.

This operator is defined as $\hat{x}=\frac{1}{\sqrt{2}}({\hat{a}}^{†}+\hat{a})$, where $\hat{a}$ is the bosonic annihilation operator.

source

QuantumToolbox.momentum Function

momentum(N::Int)

Momentum operator with Hilbert space cutoff N.

This operator is defined as $\hat{p}=\frac{i}{\sqrt{2}}({\hat{a}}^{†}-\hat{a})$, where $\hat{a}$ is the bosonic annihilation operator.

source

QuantumToolbox.phase Function

phase(N::Int, ϕ0::Real=0)

Single-mode Pegg-Barnett phase operator with Hilbert space cutoff $N$ and the reference phase ${\varphi }_0$.

This operator is defined as

$$\hat{\varphi }=\sum _{m=0}^{N-1}{\varphi }_m|{\varphi }_m⟩⟨{\varphi }_m|,$$

where

$${\varphi }_m={\varphi }_0+\frac{2m\pi }{N},$$

and

$$|{\varphi }_m⟩=\frac{1}{\sqrt{N}}\sum _{n=0}^{N-1}\exp (in{\varphi }_m)|n⟩.$$

Reference

• Michael Martin Nieto, QUANTUM PHASE AND QUANTUM PHASE OPERATORS: Some Physics and Some History, arXiv:hep-th/9304036, Equation (30-32).

source

QuantumToolbox.fdestroy Function

fdestroy(N::Union{Int,Val}, j::Int)

Construct a fermionic destruction operator acting on the j-th site, where the fock space has totally N-sites:

Here, we use the Jordan-Wigner transformation, namely

$${\hat{d}}_j={\hat{\sigma }}_z^{\otimes j-1}\otimes {\hat{\sigma }}_{+}\otimes {\hat{\mathbb{1}}}^{\otimes N-j}$$

The site index j should satisfy: 1 ≤ j ≤ N.

Note that we put ${\hat{\sigma }}_{+}=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$ here because we consider $|0⟩=\left(\begin{array}{c}1\\ 0\end{array}\right)$ to be ground (vacant) state, and $|1⟩=\left(\begin{array}{c}0\\ 1\end{array}\right)$ to be excited (occupied) state.

Beware of type-stability!

If you want to keep type stability, it is recommended to use fdestroy(Val(N), j) instead of fdestroy(N, j). See this link and the related Section about type stability for more details.

source

QuantumToolbox.fcreate Function

fcreate(N::Union{Int,Val}, j::Int)

Construct a fermionic creation operator acting on the j-th site, where the fock space has totally N-sites:

Here, we use the Jordan-Wigner transformation, namely

$${\hat{d}}_j^{†}={\hat{\sigma }}_z^{\otimes j-1}\otimes {\hat{\sigma }}_{-}\otimes {\hat{\mathbb{1}}}^{\otimes N-j}$$

The site index j should satisfy: 1 ≤ j ≤ N.

Note that we put ${\hat{\sigma }}_{-}=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)$ here because we consider $|0⟩=\left(\begin{array}{c}1\\ 0\end{array}\right)$ to be ground (vacant) state, and $|1⟩=\left(\begin{array}{c}0\\ 1\end{array}\right)$ to be excited (occupied) state.

Beware of type-stability!

If you want to keep type stability, it is recommended to use fcreate(Val(N), j) instead of fcreate(N, j). See this link and the related Section about type stability for more details.

source

QuantumToolbox.tunneling Function

tunneling(N::Int, m::Int=1; sparse::Union{Bool,Val{<:Bool}}=Val(false))

Generate a tunneling operator defined as:

$$\sum _{n=0}^{N-m}|n⟩⟨n+m|+|n+m⟩⟨n|,$$

where $N$ is the number of basis states in the Hilbert space, and $m$ is the number of excitations in tunneling event.

If sparse=true, the operator is returned as a sparse matrix, otherwise a dense matrix is returned.

Beware of type-stability!

If you want to keep type stability, it is recommended to use tunneling(N, m, Val(sparse)) instead of tunneling(N, m, sparse). See this link and the related Section about type stability for more details.

source

QuantumToolbox.qft Function

qft(dimensions)

Generates a discrete Fourier transform matrix ${\hat{F}}_N$ for Quantum Fourier Transform (QFT) with given argument dimensions.

The dimensions can be either the following types:

• dimensions::Int: Number of basis states in the Hilbert space.

• dimensions::Union{Dimensions,AbstractVector{Int},Tuple}: list of dimensions representing the each number of basis in the subsystems.

$N$ represents the total dimension, and therefore the matrix is defined as

$${\hat{F}}_N=\frac{1}{\sqrt{N}}\left[\begin{array}{cccccc}1& 1& 1& 1& \cdots & 1\\ 1& \omega & {\omega }^2& {\omega }^3& \cdots & {\omega }^{N-1}\\ 1& {\omega }^2& {\omega }^4& {\omega }^6& \cdots & {\omega }^{2(N-1)}\\ 1& {\omega }^3& {\omega }^6& {\omega }^9& \cdots & {\omega }^{3(N-1)}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& {\omega }^{N-1}& {\omega }^{2(N-1)}& {\omega }^{3(N-1)}& \cdots & {\omega }^{(N-1)(N-1)}\end{array}\right],$$

where $\omega =\exp (\frac{2\pi i}{N})$.

Beware of type-stability!

It is highly recommended to use qft(dimensions) with dimensions as Tuple or SVector to keep type stability. See the related Section about type stability for more details.

source

QuantumToolbox.eye Function

eye(N::Int; type=Operator, dims=nothing)
qeye(N::Int; type=Operator, dims=nothing)

Identity operator $\hat{\mathbb{1}}$ with size N.

It is also possible to specify the list of Hilbert dimensions dims if different subsystems are present.

Note that type can only be either Operator or SuperOperator

Note

qeye is a synonym of eye.

source

QuantumToolbox.projection Function

projection(N::Int, i::Int, j::Int)

Generates the projection operator $\hat{O}=|i⟩⟨j|$ with Hilbert space dimension N.

source

QuantumToolbox.commutator Function

commutator(A::QuantumObject, B::QuantumObject; anti::Bool=false)

Return the commutator (or anti-commutator) of the two QuantumObject:

• commutator (anti=false): $\hat{A}\hat{B}-\hat{B}\hat{A}$

• anticommutator (anti=true): $\hat{A}\hat{B}+\hat{B}\hat{A}$

Note that A and B must be Operator

source

QuantumToolbox.spre Function

spre(A::AbstractQuantumObject, Id_cache=I(size(A,1)))

Returns the SuperOperator form of A acting on the left of the density matrix operator: $\mathcal{O}\left(\hat{A}\right)\left[\hat{\rho }\right]=\hat{A}\hat{\rho }$.

Since the density matrix is vectorized in OperatorKet form: $|\hat{\rho }⟩⟩$, this SuperOperator is always a matrix $\hat{\mathbb{1}}\otimes \hat{A}$, namely

$$\mathcal{O}\left(\hat{A}\right)\left[\hat{\rho }\right]=\hat{\mathbb{1}}\otimes \hat{A}|\hat{\rho }⟩⟩$$

(see the section in documentation: Superoperators and Vectorized Operators for more details)

The optional argument Id_cache can be used to pass a precomputed identity matrix. This can be useful when the same function is applied multiple times with a known Hilbert space dimension.

See also spost and sprepost.

source

QuantumToolbox.spost Function

spost(B::AbstractQuantumObject, Id_cache=I(size(B,1)))

Returns the SuperOperator form of B acting on the right of the density matrix operator: $\mathcal{O}\left(\hat{B}\right)\left[\hat{\rho }\right]=\hat{\rho }\hat{B}$.

Since the density matrix is vectorized in OperatorKet form: $|\hat{\rho }⟩⟩$, this SuperOperator is always a matrix ${\hat{B}}^T\otimes \hat{\mathbb{1}}$, namely

$$\mathcal{O}\left(\hat{B}\right)\left[\hat{\rho }\right]={\hat{B}}^T\otimes \hat{\mathbb{1}}|\hat{\rho }⟩⟩$$

(see the section in documentation: Superoperators and Vectorized Operators for more details)

The optional argument Id_cache can be used to pass a precomputed identity matrix. This can be useful when the same function is applied multiple times with a known Hilbert space dimension.

See also spre and sprepost.

source

QuantumToolbox.sprepost Function

sprepost(A::AbstractQuantumObject, B::AbstractQuantumObject)

Returns the SuperOperator form of A and B acting on the left and right of the density matrix operator, respectively: $\mathcal{O}(\hat{A},\hat{B})\left[\hat{\rho }\right]=\hat{A}\hat{\rho }\hat{B}$.

Since the density matrix is vectorized in OperatorKet form: $|\hat{\rho }⟩⟩$, this SuperOperator is always a matrix ${\hat{B}}^T\otimes \hat{A}$, namely

$$\mathcal{O}(\hat{A},\hat{B})\left[\hat{\rho }\right]={\hat{B}}^T\otimes \hat{A}|\hat{\rho }⟩⟩=\text{spre}(\hat{A})\ast \text{spost}(\hat{B})|\hat{\rho }⟩⟩$$

(see the section in documentation: Superoperators and Vectorized Operators for more details)

See also spre and spost.

source

QuantumToolbox.lindblad_dissipator Function

lindblad_dissipator(O::AbstractQuantumObject, Id_cache=I(size(O,1))

Returns the Lindblad SuperOperator defined as

$$\mathcal{D}\left(\hat{O}\right)\left[\hat{\rho }\right]=\frac{1}{2}(2\hat{O}\hat{\rho }{\hat{O}}^{†}-{\hat{O}}^{†}\hat{O}\hat{\rho }-\hat{\rho }{\hat{O}}^{†}\hat{O})$$

The optional argument Id_cache can be used to pass a precomputed identity matrix. This can be useful when the same function is applied multiple times with a known Hilbert space dimension.

See also spre, spost, and sprepost.

source

Synonyms of functions for Qobj

QuantumToolbox.Qobj Type

Qobj(A; kwargs...)

Note

Qobj is a synonym for generating QuantumObject. See the docstring of QuantumObject for more details.

source

QuantumToolbox.QobjEvo Type

QobjEvo(args...; kwargs...)

Note

QobjEvo is a synonym for generating QuantumObjectEvolution. See the docstrings of QuantumObjectEvolution for more details.

source

QuantumToolbox.shape Function

size(A::AbstractQuantumObject)
size(A::AbstractQuantumObject, idx::Int)
shape(A::AbstractQuantumObject)
shape(A::AbstractQuantumObject, idx::Int)

Returns a tuple containing each dimensions of the array in the AbstractQuantumObject.

Optionally, you can specify an index (idx) to just get the corresponding dimension of the array.

Note

shape is a synonym of size.

source

QuantumToolbox.isherm Function

ishermitian(A::AbstractQuantumObject)
isherm(A::AbstractQuantumObject)

Test whether the AbstractQuantumObject is Hermitian.

Note

isherm is a synonym of ishermitian.

source

QuantumToolbox.trans Function

transpose(A::AbstractQuantumObject)

Lazy matrix transpose of the AbstractQuantumObject.

source

QuantumToolbox.dag Function

A'
adjoint(A::AbstractQuantumObject)
dag(A::AbstractQuantumObject)

Lazy adjoint (conjugate transposition) of the AbstractQuantumObject

Note

A' and dag(A) are synonyms of adjoint(A).

source

QuantumToolbox.matrix_element Function

matrix_element(i::QuantumObject, A::QuantumObject j::QuantumObject)

Compute the generalized dot product dot(i, A*j) between three QuantumObject: $⟨i|\hat{A}|j⟩$

Note that this function is same as dot(i, A, j)

Supports the following inputs:

• A is in the type of Operator, with i and j are both Ket.

• A is in the type of SuperOperator, with i and j are both OperatorKet

source

QuantumToolbox.unit Function

normalize(A::QuantumObject, p::Real)
unit(A::QuantumObject, p::Real)

Return normalized QuantumObject so that its p-norm equals to unity, i.e. norm(A, p) == 1.

Support for the following types of QuantumObject:

• If A is Ket or Bra, default p = 2

• If A is Operator, default p = 1

Note

unit is a synonym of normalize.

Also, see norm about its definition for different types of QuantumObject.

source

QuantumToolbox.tensor Function

kron(A::AbstractQuantumObject, B::AbstractQuantumObject, ...)
tensor(A::AbstractQuantumObject, B::AbstractQuantumObject, ...)
⊗(A::AbstractQuantumObject, B::AbstractQuantumObject, ...)
A ⊗ B

Returns the Kronecker product $\hat{A}\otimes \hat{B}\otimes \cdots$.

Note

tensor and ⊗ (where ⊗ can be typed by tab-completing \otimes in the REPL) are synonyms of kron.

Examples

julia> a = destroy(20)

Quantum Object:   type=Operator   dims=[20]   size=(20, 20)   ishermitian=false
20×20 SparseMatrixCSC{ComplexF64, Int64} with 19 stored entries:
⎡⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⎤
⎢⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⎥
⎣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⎦

julia> O = kron(a, a);

julia> size(a), size(O)
((20, 20), (400, 400))

julia> a.dims, O.dims
([20], [20, 20])

source

QuantumToolbox.:⊗ Function

kron(A::AbstractQuantumObject, B::AbstractQuantumObject, ...)
tensor(A::AbstractQuantumObject, B::AbstractQuantumObject, ...)
⊗(A::AbstractQuantumObject, B::AbstractQuantumObject, ...)
A ⊗ B

Returns the Kronecker product $\hat{A}\otimes \hat{B}\otimes \cdots$.

Note

tensor and ⊗ (where ⊗ can be typed by tab-completing \otimes in the REPL) are synonyms of kron.

Examples

julia> a = destroy(20)

Quantum Object:   type=Operator   dims=[20]   size=(20, 20)   ishermitian=false
20×20 SparseMatrixCSC{ComplexF64, Int64} with 19 stored entries:
⎡⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⎤
⎢⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⎥
⎣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⎦

julia> O = kron(a, a);

julia> size(a), size(O)
((20, 20), (400, 400))

julia> a.dims, O.dims
([20], [20, 20])

source

QuantumToolbox.qeye Function

eye(N::Int; type=Operator, dims=nothing)
qeye(N::Int; type=Operator, dims=nothing)

Identity operator $\hat{\mathbb{1}}$ with size N.

It is also possible to specify the list of Hilbert dimensions dims if different subsystems are present.

Note that type can only be either Operator or SuperOperator

Note

qeye is a synonym of eye.

source

QuantumToolbox.vector_to_operator Function

vec2mat(A::AbstractVector)

Converts a vector to a matrix.

source

vec2mat(A::QuantumObject)
vector_to_operator(A::QuantumObject)

Convert a quantum object from vector (OperatorKetQuantumObject-type) to matrix (OperatorQuantumObject-type)

Note

vector_to_operator is a synonym of vec2mat.

source

QuantumToolbox.operator_to_vector Function

mat2vec(A::QuantumObject)
operator_to_vector(A::QuantumObject)

Convert a quantum object from matrix (OperatorQuantumObject-type) to vector (OperatorKetQuantumObject-type)

Note

operator_to_vector is a synonym of mat2vec.

source

mat2vec(A::AbstractMatrix)

Converts a matrix to a vector.

source

QuantumToolbox.sqrtm Function

sqrtm(A::QuantumObject)

Matrix square root of Operator type of QuantumObject

Note that for other types of QuantumObject use sprt(A) instead.

source

QuantumToolbox.logm Function

logm(A::QuantumObject)

Matrix logarithm of QuantumObject

Note that this function is same as log(A) and only supports for Operator and SuperOperator.

source

QuantumToolbox.expm Function

expm(A::QuantumObject)

Matrix exponential of QuantumObject

Note that this function is same as exp(A) and only supports for Operator and SuperOperator.

source

QuantumToolbox.sinm Function

sinm(A::QuantumObject)

Matrix sine of QuantumObject, defined as

$\sin \left(\hat{A}\right)=\frac{e^{i\hat{A}}-e^{-i\hat{A}}}{2i}$

Note that this function is same as sin(A) and only supports for Operator and SuperOperator.

source

QuantumToolbox.cosm Function

cosm(A::QuantumObject)

Matrix cosine of QuantumObject, defined as

$\cos \left(\hat{A}\right)=\frac{e^{i\hat{A}}+e^{-i\hat{A}}}{2}$

Note that this function is same as cos(A) and only supports for Operator and SuperOperator.

source

Time evolution

QuantumToolbox.TimeEvolutionProblem Type

struct TimeEvolutionProblem

A Julia constructor for handling the ODEProblem of the time evolution of quantum systems.

Fields (Attributes)

• prob::AbstractSciMLProblem: The ODEProblem of the time evolution.

• times::Abstractvector: The time list of the evolution.

• dimensions::AbstractDimensions: The dimensions of the Hilbert space.

• kwargs::KWT: Generic keyword arguments.

dims property

For a given prob::TimeEvolutionProblem, prob.dims or getproperty(prob, :dims) returns its dimensions in the type of integer-vector.

source

QuantumToolbox.TimeEvolutionSol Type

struct TimeEvolutionSol

A structure storing the results and some information from solving time evolution.

Fields (Attributes)

• times::AbstractVector: The time list of the evolution.

• states::Vector{QuantumObject}: The list of result states.

• expect::Union{AbstractMatrix,Nothing}: The expectation values corresponding to each time point in times.

• retcode: The return code from the solver.

• alg: The algorithm which is used during the solving process.

• abstol::Real: The absolute tolerance which is used during the solving process.

• reltol::Real: The relative tolerance which is used during the solving process.

source

QuantumToolbox.TimeEvolutionMCSol Type

struct TimeEvolutionMCSol

A structure storing the results and some information from solving quantum trajectories of the Monte Carlo wave function time evolution.

Fields (Attributes)

• ntraj::Int: Number of trajectories

• times::AbstractVector: The time list of the evolution.

• states::Vector{Vector{QuantumObject}}: The list of result states in each trajectory.

• expect::Union{AbstractMatrix,Nothing}: The expectation values (averaging all trajectories) corresponding to each time point in times.

• average_expect::Union{AbstractMatrix,Nothing}: The expectation values (averaging all trajectories) corresponding to each time point in times.

• runs_expect::Union{AbstractArray,Nothing}: The expectation values corresponding to each trajectory and each time point in times

• col_times::Vector{Vector{Real}}: The time records of every quantum jump occurred in each trajectory.

• col_which::Vector{Vector{Int}}: The indices of which collapse operator was responsible for each quantum jump in col_times.

• converged::Bool: Whether the solution is converged or not.

• alg: The algorithm which is used during the solving process.

• abstol::Real: The absolute tolerance which is used during the solving process.

• reltol::Real: The relative tolerance which is used during the solving process.

source

QuantumToolbox.TimeEvolutionStochasticSol Type

struct TimeEvolutionStochasticSol

A structure storing the results and some information from solving trajectories of the Stochastic time evolution.

Fields (Attributes)

• ntraj::Int: Number of trajectories

• times::AbstractVector: The time list of the evolution.

• states::Vector{Vector{QuantumObject}}: The list of result states in each trajectory.

• expect::Union{AbstractMatrix,Nothing}: The expectation values (averaging all trajectories) corresponding to each time point in times.

• average_expect::Union{AbstractMatrix,Nothing}: The expectation values (averaging all trajectories) corresponding to each time point in times.

• runs_expect::Union{AbstractArray,Nothing}: The expectation values corresponding to each trajectory and each time point in times

• converged::Bool: Whether the solution is converged or not.

• alg: The algorithm which is used during the solving process.

• abstol::Real: The absolute tolerance which is used during the solving process.

• reltol::Real: The relative tolerance which is used during the solving process.

source

QuantumToolbox.sesolveProblem Function

sesolveProblem(
    H::Union{AbstractQuantumObject{OperatorQuantumObject},Tuple},
    ψ0::QuantumObject{KetQuantumObject},
    tlist::AbstractVector;
    e_ops::Union{Nothing,AbstractVector,Tuple} = nothing,
    params = NullParameters(),
    progress_bar::Union{Val,Bool} = Val(true),
    inplace::Union{Val,Bool} = Val(true),
    kwargs...,
)

Generate the ODEProblem for the Schrödinger time evolution of a quantum system:

$$\frac{\partial }{\partial t}|\psi (t)⟩=-i\hat{H}|\psi (t)⟩$$

Arguments

• H: Hamiltonian of the system $\hat{H}$. It can be either a QuantumObject, a QuantumObjectEvolution, or a Tuple of operator-function pairs.

• ψ0: Initial state of the system $|\psi (0)⟩$.

• tlist: List of times at which to save either the state or the expectation values of the system.

• e_ops: List of operators for which to calculate expectation values. It can be either a Vector or a Tuple.

• params: Parameters to pass to the solver. This argument is usually expressed as a NamedTuple or AbstractVector of parameters. For more advanced usage, any custom struct can be used.

• progress_bar: Whether to show the progress bar. Using non-Val types might lead to type instabilities.

• inplace: Whether to use the inplace version of the ODEProblem. The default is Val(true). It is recommended to use Val(true) for better performance, but it is sometimes necessary to use Val(false), for example when performing automatic differentiation using Zygote.jl.

• kwargs: The keyword arguments for the ODEProblem.

Notes

• The states will be saved depend on the keyword argument saveat in kwargs.

• If e_ops is empty, the default value of saveat=tlist (saving the states corresponding to tlist), otherwise, saveat=[tlist[end]] (only save the final state). You can also specify e_ops and saveat separately.

• The default tolerances in kwargs are given as reltol=1e-6 and abstol=1e-8.

• For more details about kwargs please refer to DifferentialEquations.jl (Keyword Arguments)

Returns

• prob: The TimeEvolutionProblem containing the ODEProblem for the Schrödinger time evolution of the system.

source

QuantumToolbox.mesolveProblem Function

mesolveProblem(
    H::Union{AbstractQuantumObject,Tuple},
    ψ0::QuantumObject,
    tlist,
    c_ops::Union{Nothing,AbstractVector,Tuple} = nothing;
    e_ops::Union{Nothing,AbstractVector,Tuple} = nothing,
    params = NullParameters(),
    progress_bar::Union{Val,Bool} = Val(true),
    inplace::Union{Val,Bool} = Val(true),
    kwargs...,
)

Generate the ODEProblem for the master equation time evolution of an open quantum system:

$$\frac{\partial \hat{\rho }(t)}{\partial t}=-i[\hat{H},\hat{\rho }(t)]+\sum _n\mathcal{D}({\hat{C}}_n)[\hat{\rho }(t)]$$

where

$$\mathcal{D}({\hat{C}}_n)[\hat{\rho }(t)]={\hat{C}}_n\hat{\rho }(t){\hat{C}}_n^{†}-\frac{1}{2}{\hat{C}}_n^{†}{\hat{C}}_n\hat{\rho }(t)-\frac{1}{2}\hat{\rho }(t){\hat{C}}_n^{†}{\hat{C}}_n$$

Arguments

• H: Hamiltonian of the system $\hat{H}$. It can be either a QuantumObject, a QuantumObjectEvolution, or a Tuple of operator-function pairs.

• ψ0: Initial state of the system $|\psi (0)⟩$. It can be either a Ket or a Operator.

• tlist: List of times at which to save either the state or the expectation values of the system.

• c_ops: List of collapse operators $\{{\hat{C}}_n{\}}_n$. It can be either a Vector or a Tuple.

• e_ops: List of operators for which to calculate expectation values. It can be either a Vector or a Tuple.

• params: Parameters to pass to the solver. This argument is usually expressed as a NamedTuple or AbstractVector of parameters. For more advanced usage, any custom struct can be used.

• progress_bar: Whether to show the progress bar. Using non-Val types might lead to type instabilities.

• inplace: Whether to use the inplace version of the ODEProblem. The default is Val(true). It is recommended to use Val(true) for better performance, but it is sometimes necessary to use Val(false), for example when performing automatic differentiation using Zygote.jl.

• kwargs: The keyword arguments for the ODEProblem.

Notes

• The states will be saved depend on the keyword argument saveat in kwargs.

• If e_ops is empty, the default value of saveat=tlist (saving the states corresponding to tlist), otherwise, saveat=[tlist[end]] (only save the final state). You can also specify e_ops and saveat separately.

• The default tolerances in kwargs are given as reltol=1e-6 and abstol=1e-8.

• For more details about kwargs please refer to DifferentialEquations.jl (Keyword Arguments)

Returns

• prob::ODEProblem: The ODEProblem for the master equation time evolution.

source

QuantumToolbox.mcsolveProblem Function

mcsolveProblem(
    H::Union{AbstractQuantumObject{OperatorQuantumObject},Tuple},
    ψ0::QuantumObject{KetQuantumObject},
    tlist::AbstractVector,
    c_ops::Union{Nothing,AbstractVector,Tuple} = nothing;
    e_ops::Union{Nothing,AbstractVector,Tuple} = nothing,
    params = NullParameters(),
    rng::AbstractRNG = default_rng(),
    jump_callback::TJC = ContinuousLindbladJumpCallback(),
    kwargs...,
)

Generate the ODEProblem for a single trajectory of the Monte Carlo wave function time evolution of an open quantum system.

Given a system Hamiltonian $\hat{H}$ and a list of collapse (jump) operators $\{{\hat{C}}_n{\}}_n$, the evolution of the state $|\psi (t)⟩$ is governed by the Schrodinger equation:

$$\frac{\partial }{\partial t}|\psi (t)⟩=-i{\hat{H}}_{\text{eff}}|\psi (t)⟩$$

with a non-Hermitian effective Hamiltonian:

$${\hat{H}}_{\text{eff}}=\hat{H}-\frac{i}{2}\sum _n{\hat{C}}_n^{†}{\hat{C}}_n.$$

To the first-order of the wave function in a small time $\delta t$, the strictly negative non-Hermitian portion in ${\hat{H}}_{\text{eff}}$ gives rise to a reduction in the norm of the wave function, namely

$$⟨\psi (t+\delta t)|\psi (t+\delta t)⟩=1-\delta p,$$

where

$$\delta p=\delta t\sum _n⟨\psi (t)|{\hat{C}}_n^{†}{\hat{C}}_n|\psi (t)⟩$$

is the corresponding quantum jump probability.

If the environmental measurements register a quantum jump, the wave function undergoes a jump into a state defined by projecting $|\psi (t)⟩$ using the collapse operator ${\hat{C}}_n$ corresponding to the measurement, namely

$$|\psi (t+\delta t)⟩=\frac{{\hat{C}}_n|\psi (t)⟩}{\sqrt{⟨\psi (t)|{\hat{C}}_n^{†}{\hat{C}}_n|\psi (t)⟩}}$$

Arguments

• H: Hamiltonian of the system $\hat{H}$. It can be either a QuantumObject, a QuantumObjectEvolution, or a Tuple of operator-function pairs.

• ψ0: Initial state of the system $|\psi (0)⟩$.

• tlist: List of times at which to save either the state or the expectation values of the system.

• c_ops: List of collapse operators $\{{\hat{C}}_n{\}}_n$. It can be either a Vector or a Tuple.

• e_ops: List of operators for which to calculate expectation values. It can be either a Vector or a Tuple.

• params: Parameters to pass to the solver. This argument is usually expressed as a NamedTuple or AbstractVector of parameters. For more advanced usage, any custom struct can be used.

• rng: Random number generator for reproducibility.

• jump_callback: The Jump Callback type: Discrete or Continuous. The default is ContinuousLindbladJumpCallback(), which is more precise.

• kwargs: The keyword arguments for the ODEProblem.

Notes

• The states will be saved depend on the keyword argument saveat in kwargs.

• If e_ops is empty, the default value of saveat=tlist (saving the states corresponding to tlist), otherwise, saveat=[tlist[end]] (only save the final state). You can also specify e_ops and saveat separately.

• The default tolerances in kwargs are given as reltol=1e-6 and abstol=1e-8.

• For more details about kwargs please refer to DifferentialEquations.jl (Keyword Arguments)

Returns

• prob: The TimeEvolutionProblem containing the ODEProblem for the Monte Carlo wave function time evolution.

source

QuantumToolbox.mcsolveEnsembleProblem Function

mcsolveEnsembleProblem(
    H::Union{AbstractQuantumObject{OperatorQuantumObject},Tuple},
    ψ0::QuantumObject{KetQuantumObject},
    tlist::AbstractVector,
    c_ops::Union{Nothing,AbstractVector,Tuple} = nothing;
    e_ops::Union{Nothing,AbstractVector,Tuple} = nothing,
    params = NullParameters(),
    rng::AbstractRNG = default_rng(),
    ntraj::Int = 500,
    ensemblealg::EnsembleAlgorithm = EnsembleThreads(),
    jump_callback::TJC = ContinuousLindbladJumpCallback(),
    progress_bar::Union{Val,Bool} = Val(true),
    prob_func::Union{Function, Nothing} = nothing,
    output_func::Union{Tuple,Nothing} = nothing,
    kwargs...,
)

Generate the EnsembleProblem of ODEProblems for the ensemble of trajectories of the Monte Carlo wave function time evolution of an open quantum system.

Given a system Hamiltonian $\hat{H}$ and a list of collapse (jump) operators $\{{\hat{C}}_n{\}}_n$, the evolution of the state $|\psi (t)⟩$ is governed by the Schrodinger equation:

$$\frac{\partial }{\partial t}|\psi (t)⟩=-i{\hat{H}}_{\text{eff}}|\psi (t)⟩$$

with a non-Hermitian effective Hamiltonian:

$${\hat{H}}_{\text{eff}}=\hat{H}-\frac{i}{2}\sum _n{\hat{C}}_n^{†}{\hat{C}}_n.$$

To the first-order of the wave function in a small time $\delta t$, the strictly negative non-Hermitian portion in ${\hat{H}}_{\text{eff}}$ gives rise to a reduction in the norm of the wave function, namely

$$⟨\psi (t+\delta t)|\psi (t+\delta t)⟩=1-\delta p,$$

where

$$\delta p=\delta t\sum _n⟨\psi (t)|{\hat{C}}_n^{†}{\hat{C}}_n|\psi (t)⟩$$

is the corresponding quantum jump probability.

If the environmental measurements register a quantum jump, the wave function undergoes a jump into a state defined by projecting $|\psi (t)⟩$ using the collapse operator ${\hat{C}}_n$ corresponding to the measurement, namely

$$|\psi (t+\delta t)⟩=\frac{{\hat{C}}_n|\psi (t)⟩}{\sqrt{⟨\psi (t)|{\hat{C}}_n^{†}{\hat{C}}_n|\psi (t)⟩}}$$

Arguments

• H: Hamiltonian of the system $\hat{H}$. It can be either a QuantumObject, a QuantumObjectEvolution, or a Tuple of operator-function pairs.

• ψ0: Initial state of the system $|\psi (0)⟩$.

• tlist: List of times at which to save either the state or the expectation values of the system.

• c_ops: List of collapse operators $\{{\hat{C}}_n{\}}_n$. It can be either a Vector or a Tuple.

• e_ops: List of operators for which to calculate expectation values. It can be either a Vector or a Tuple.

• params: Parameters to pass to the solver. This argument is usually expressed as a NamedTuple or AbstractVector of parameters. For more advanced usage, any custom struct can be used.

• rng: Random number generator for reproducibility.

• ntraj: Number of trajectories to use.

• ensemblealg: Ensemble algorithm to use. Default to EnsembleThreads().

• jump_callback: The Jump Callback type: Discrete or Continuous. The default is ContinuousLindbladJumpCallback(), which is more precise.

• progress_bar: Whether to show the progress bar. Using non-Val types might lead to type instabilities.

• prob_func: Function to use for generating the ODEProblem.

• output_func: a Tuple containing the Function to use for generating the output of a single trajectory, the (optional) ProgressBar object, and the (optional) RemoteChannel object.

• kwargs: The keyword arguments for the ODEProblem.

Notes

• The states will be saved depend on the keyword argument saveat in kwargs.

• If e_ops is empty, the default value of saveat=tlist (saving the states corresponding to tlist), otherwise, saveat=[tlist[end]] (only save the final state). You can also specify e_ops and saveat separately.

• The default tolerances in kwargs are given as reltol=1e-6 and abstol=1e-8.

• For more details about kwargs please refer to DifferentialEquations.jl (Keyword Arguments)

Returns

• prob: The TimeEvolutionProblem containing the Ensemble ODEProblem for the Monte Carlo wave function time evolution.

source

QuantumToolbox.ssesolveProblem Function

ssesolveProblem(
    H::Union{AbstractQuantumObject{OperatorQuantumObject},Tuple},
    ψ0::QuantumObject{KetQuantumObject},
    tlist::AbstractVector,
    sc_ops::Union{Nothing,AbstractVector,Tuple,AbstractQuantumObject} = nothing;
    e_ops::Union{Nothing,AbstractVector,Tuple} = nothing,
    params = NullParameters(),
    rng::AbstractRNG = default_rng(),
    progress_bar::Union{Val,Bool} = Val(true),
    store_measurement::Union{Val, Bool} = Val(false),
    kwargs...,
)

Generate the SDEProblem for the Stochastic Schrödinger time evolution of a quantum system. This is defined by the following stochastic differential equation:

$$d|\psi (t)⟩=-i\hat{K}|\psi (t)⟩dt+\sum _n{\hat{M}}_n|\psi (t)⟩d{W}_n(t)$$

where

$$\hat{K}=\hat{H}+i\sum _n(\frac{e_n}{2}{\hat{S}}_n-\frac{1}{2}{\hat{S}}_n^{†}{\hat{S}}_n-\frac{e_n^2}{8}),$$$${\hat{M}}_n={\hat{S}}_n-\frac{e_n}{2},$$

and

$$e_n=⟨{\hat{S}}_n+{\hat{S}}_n^{†}⟩.$$

Above, ${\hat{S}}_n$ are the stochastic collapse operators and $d{W}_n(t)$ is the real Wiener increment associated to ${\hat{S}}_n$. See (Wiseman and Milburn, 2009) for more details.

Arguments

• H: Hamiltonian of the system $\hat{H}$. It can be either a QuantumObject, a QuantumObjectEvolution, or a Tuple of operator-function pairs.

• ψ0: Initial state of the system $|\psi (0)⟩$.

• tlist: List of times at which to save either the state or the expectation values of the system.

• sc_ops: List of stochastic collapse operators $\{{\hat{S}}_n{\}}_n$. It can be either a Vector, a Tuple or a AbstractQuantumObject. It is recommended to use the last case when only one operator is provided.

• e_ops: List of operators for which to calculate expectation values. It can be either a Vector or a Tuple.

• params: NullParameters of parameters to pass to the solver.

• rng: Random number generator for reproducibility.

• progress_bar: Whether to show the progress bar. Using non-Val types might lead to type instabilities.

• store_measurement: Whether to store the measurement results. Default is Val(false).

• kwargs: The keyword arguments for the ODEProblem.

Notes

• The states will be saved depend on the keyword argument saveat in kwargs.

• If e_ops is empty, the default value of saveat=tlist (saving the states corresponding to tlist), otherwise, saveat=[tlist[end]] (only save the final state). You can also specify e_ops and saveat separately.

• The default tolerances in kwargs are given as reltol=1e-2 and abstol=1e-2.

• For more details about kwargs please refer to DifferentialEquations.jl (Keyword Arguments)

Performance Tip

When sc_ops contains only a single operator, it is recommended to pass only that operator as the argument. This ensures that the stochastic noise is diagonal, making the simulation faster.

Returns

• prob: The SDEProblem for the Stochastic Schrödinger time evolution of the system.

source

QuantumToolbox.ssesolveEnsembleProblem Function

ssesolveEnsembleProblem(
    H::Union{AbstractQuantumObject{OperatorQuantumObject},Tuple},
    ψ0::QuantumObject{KetQuantumObject},
    tlist::AbstractVector,
    sc_ops::Union{Nothing,AbstractVector,Tuple,AbstractQuantumObject} = nothing;
    e_ops::Union{Nothing,AbstractVector,Tuple} = nothing,
    params = NullParameters(),
    rng::AbstractRNG = default_rng(),
    ntraj::Int = 500,
    ensemblealg::EnsembleAlgorithm = EnsembleThreads(),
    prob_func::Union{Function, Nothing} = nothing,
    output_func::Union{Tuple,Nothing} = nothing,
    progress_bar::Union{Val,Bool} = Val(true),
    store_measurement::Union{Val,Bool} = Val(false),
    kwargs...,
)

Generate the SDE EnsembleProblem for the Stochastic Schrödinger time evolution of a quantum system. This is defined by the following stochastic differential equation:

$$d|\psi (t)⟩=-i\hat{K}|\psi (t)⟩dt+\sum _n{\hat{M}}_n|\psi (t)⟩d{W}_n(t)$$

where

$$\hat{K}=\hat{H}+i\sum _n(\frac{e_n}{2}{\hat{S}}_n-\frac{1}{2}{\hat{S}}_n^{†}{\hat{S}}_n-\frac{e_n^2}{8}),$$$${\hat{M}}_n={\hat{S}}_n-\frac{e_n}{2},$$

and

$$e_n=⟨{\hat{S}}_n+{\hat{S}}_n^{†}⟩.$$

Above, ${\hat{S}}_n$ are the stochastic collapse operators and $d{W}_n(t)$ is the real Wiener increment associated to ${\hat{S}}_n$. See (Wiseman and Milburn, 2009) for more details.

Arguments

• H: Hamiltonian of the system $\hat{H}$. It can be either a QuantumObject, a QuantumObjectEvolution, or a Tuple of operator-function pairs.

• ψ0: Initial state of the system $|\psi (0)⟩$.

• tlist: List of times at which to save either the state or the expectation values of the system.

• sc_ops: List of stochastic collapse operators $\{{\hat{S}}_n{\}}_n$. It can be either a Vector, a Tuple or a AbstractQuantumObject. It is recommended to use the last case when only one operator is provided.

• e_ops: List of operators for which to calculate expectation values. It can be either a Vector or a Tuple.

• params: NullParameters of parameters to pass to the solver.

• rng: Random number generator for reproducibility.

• ntraj: Number of trajectories to use. Default is 500.

• ensemblealg: Ensemble method to use. Default to EnsembleThreads().

• jump_callback: The Jump Callback type: Discrete or Continuous. The default is ContinuousLindbladJumpCallback(), which is more precise.

• prob_func: Function to use for generating the SDEProblem.

• output_func: a Tuple containing the Function to use for generating the output of a single trajectory, the (optional) ProgressBar object, and the (optional) RemoteChannel object.

• progress_bar: Whether to show the progress bar. Using non-Val types might lead to type instabilities.

• store_measurement: Whether to store the measurement results. Default is Val(false).

• kwargs: The keyword arguments for the ODEProblem.

Notes

• The states will be saved depend on the keyword argument saveat in kwargs.

• If e_ops is empty, the default value of saveat=tlist (saving the states corresponding to tlist), otherwise, saveat=[tlist[end]] (only save the final state). You can also specify e_ops and saveat separately.

• The default tolerances in kwargs are given as reltol=1e-2 and abstol=1e-2.

• For more details about kwargs please refer to DifferentialEquations.jl (Keyword Arguments)

Performance Tip

When sc_ops contains only a single operator, it is recommended to pass only that operator as the argument. This ensures that the stochastic noise is diagonal, making the simulation faster.

Returns

• prob::EnsembleProblem with SDEProblem: The Ensemble SDEProblem for the Stochastic Shrödinger time evolution.

source

QuantumToolbox.smesolveProblem Function

smesolveProblem(
    H::Union{AbstractQuantumObject{OperatorQuantumObject},Tuple},
    ψ0::QuantumObject,
    tlist::AbstractVector,
    c_ops::Union{Nothing,AbstractVector,Tuple} = nothing,
    sc_ops::Union{Nothing,AbstractVector,Tuple,AbstractQuantumObject} = nothing;
    e_ops::Union{Nothing,AbstractVector,Tuple} = nothing,
    params = NullParameters(),
    rng::AbstractRNG = default_rng(),
    progress_bar::Union{Val,Bool} = Val(true),
    store_measurement::Union{Val, Bool} = Val(false),
    kwargs...,
)

Generate the SDEProblem for the Stochastic Master Equation time evolution of an open quantum system. This is defined by the following stochastic differential equation:

$$d\rho (t)=-i[\hat{H},\rho (t)]dt+\sum _i\mathcal{D}[{\hat{C}}_i]\rho (t)dt+\sum _n\mathcal{D}[{\hat{S}}_n]\rho (t)dt+\sum _n\mathcal{H}[{\hat{S}}_n]\rho (t)d{W}_n(t),$$

where

$$\mathcal{D}[\hat{O}]\rho =\hat{O}\rho {\hat{O}}^{†}-\frac{1}{2}\{{\hat{O}}^{†}\hat{O},\rho \},$$

is the Lindblad superoperator, and

$$\mathcal{H}[\hat{O}]\rho =\hat{O}\rho +\rho {\hat{O}}^{†}-\mathrm{Tr}[\hat{O}\rho +\rho {\hat{O}}^{†}]\rho ,$$

Above, ${\hat{C}}_i$ represent the collapse operators related to pure dissipation, while ${\hat{S}}_n$ are the stochastic collapse operators. The $d{W}_n(t)$ term is the real Wiener increment associated to ${\hat{S}}_n$. See (Wiseman and Milburn, 2009) for more details.

Arguments

• H: Hamiltonian of the system $\hat{H}$. It can be either a QuantumObject, a QuantumObjectEvolution, or a Tuple of operator-function pairs.

• ψ0: Initial state of the system $|\psi (0)⟩$. It can be either a Ket or a Operator.

• tlist: List of times at which to save either the state or the expectation values of the system.

• c_ops: List of collapse operators $\{{\hat{C}}_i{\}}_i$. It can be either a Vector or a Tuple.

• sc_ops: List of stochastic collapse operators $\{{\hat{S}}_n{\}}_n$. It can be either a Vector, a Tuple or a AbstractQuantumObject. It is recommended to use the last case when only one operator is provided.

• e_ops: List of operators for which to calculate expectation values. It can be either a Vector or a Tuple.

• params: NullParameters of parameters to pass to the solver.

• rng: Random number generator for reproducibility.

• progress_bar: Whether to show the progress bar. Using non-Val types might lead to type instabilities.

• store_measurement: Whether to store the measurement expectation values. Default is Val(false).

• kwargs: The keyword arguments for the ODEProblem.

Notes

• The states will be saved depend on the keyword argument saveat in kwargs.

• If e_ops is empty, the default value of saveat=tlist (saving the states corresponding to tlist), otherwise, saveat=[tlist[end]] (only save the final state). You can also specify e_ops and saveat separately.

• The default tolerances in kwargs are given as reltol=1e-2 and abstol=1e-2.

• For more details about kwargs please refer to DifferentialEquations.jl (Keyword Arguments)

Performance Tip

When sc_ops contains only a single operator, it is recommended to pass only that operator as the argument. This ensures that the stochastic noise is diagonal, making the simulation faster.

Returns

• prob: The TimeEvolutionProblem containing the SDEProblem for the Stochastic Master Equation time evolution.

source

QuantumToolbox.smesolveEnsembleProblem Function

smesolveEnsembleProblem(
    H::Union{AbstractQuantumObject{OperatorQuantumObject},Tuple},
    ψ0::QuantumObject,
    tlist::AbstractVector,
    c_ops::Union{Nothing,AbstractVector,Tuple} = nothing,
    sc_ops::Union{Nothing,AbstractVector,Tuple,AbstractQuantumObject} = nothing;
    e_ops::Union{Nothing,AbstractVector,Tuple} = nothing,
    params = NullParameters(),
    rng::AbstractRNG = default_rng(),
    ntraj::Int = 500,
    ensemblealg::EnsembleAlgorithm = EnsembleThreads(),
    prob_func::Union{Function, Nothing} = nothing,
    output_func::Union{Tuple,Nothing} = nothing,
    progress_bar::Union{Val,Bool} = Val(true),
    store_measurement::Union{Val,Bool} = Val(false),
    kwargs...,
)

Generate the SDEProblem for the Stochastic Master Equation time evolution of an open quantum system. This is defined by the following stochastic differential equation:

$$d\rho (t)=-i[\hat{H},\rho (t)]dt+\sum _i\mathcal{D}[{\hat{C}}_i]\rho (t)dt+\sum _n\mathcal{D}[{\hat{S}}_n]\rho (t)dt+\sum _n\mathcal{H}[{\hat{S}}_n]\rho (t)d{W}_n(t),$$

where

$$\mathcal{D}[\hat{O}]\rho =\hat{O}\rho {\hat{O}}^{†}-\frac{1}{2}\{{\hat{O}}^{†}\hat{O},\rho \},$$

is the Lindblad superoperator, and

$$\mathcal{H}[\hat{O}]\rho =\hat{O}\rho +\rho {\hat{O}}^{†}-\mathrm{Tr}[\hat{O}\rho +\rho {\hat{O}}^{†}]\rho ,$$

Above, ${\hat{C}}_i$ represent the collapse operators related to pure dissipation, while ${\hat{S}}_n$ are the stochastic collapse operators. The $d{W}_n(t)$ term is the real Wiener increment associated to ${\hat{S}}_n$. See (Wiseman and Milburn, 2009) for more details.

Arguments

• H: Hamiltonian of the system $\hat{H}$. It can be either a QuantumObject, a QuantumObjectEvolution, or a Tuple of operator-function pairs.

• ψ0: Initial state of the system $|\psi (0)⟩$. It can be either a Ket or a Operator.

• tlist: List of times at which to save either the state or the expectation values of the system.

• c_ops: List of collapse operators $\{{\hat{C}}_i{\}}_i$. It can be either a Vector or a Tuple.

• sc_ops: List of stochastic collapse operators $\{{\hat{S}}_n{\}}_n$. It can be either a Vector, a Tuple or a AbstractQuantumObject. It is recommended to use the last case when only one operator is provided.

• e_ops: List of operators for which to calculate expectation values. It can be either a Vector or a Tuple.

• params: NullParameters of parameters to pass to the solver.

• rng: Random number generator for reproducibility.

• ntraj: Number of trajectories to use. Default is 500.

• ensemblealg: Ensemble method to use. Default to EnsembleThreads().

• prob_func: Function to use for generating the SDEProblem.

• output_func: a Tuple containing the Function to use for generating the output of a single trajectory, the (optional) ProgressBar object, and the (optional) RemoteChannel object.

• progress_bar: Whether to show the progress bar. Using non-Val types might lead to type instabilities.

• store_measurement: Whether to store the measurement expectation values. Default is Val(false).

• kwargs: The keyword arguments for the ODEProblem.

Notes

• The states will be saved depend on the keyword argument saveat in kwargs.

• If e_ops is empty, the default value of saveat=tlist (saving the states corresponding to tlist), otherwise, saveat=[tlist[end]] (only save the final state). You can also specify e_ops and saveat separately.

• The default tolerances in kwargs are given as reltol=1e-2 and abstol=1e-2.

• For more details about kwargs please refer to DifferentialEquations.jl (Keyword Arguments)

Performance Tip

When sc_ops contains only a single operator, it is recommended to pass only that operator as the argument. This ensures that the stochastic noise is diagonal, making the simulation faster.

Returns

• prob: The TimeEvolutionProblem containing the Ensemble SDEProblem for the Stochastic Master Equation time evolution.

source

QuantumToolbox.sesolve Function

sesolve(
    H::Union{AbstractQuantumObject{OperatorQuantumObject},Tuple},
    ψ0::QuantumObject{KetQuantumObject},
    tlist::AbstractVector;
    alg::OrdinaryDiffEqAlgorithm = Tsit5(),
    e_ops::Union{Nothing,AbstractVector,Tuple} = nothing,
    params = NullParameters(),
    progress_bar::Union{Val,Bool} = Val(true),
    inplace::Union{Val,Bool} = Val(true),
    kwargs...,
)

Time evolution of a closed quantum system using the Schrödinger equation:

$$\frac{\partial }{\partial t}|\psi (t)⟩=-i\hat{H}|\psi (t)⟩$$

Arguments

• H: Hamiltonian of the system $\hat{H}$. It can be either a QuantumObject, a QuantumObjectEvolution, or a Tuple of operator-function pairs.

• ψ0: Initial state of the system $|\psi (0)⟩$.

• tlist: List of times at which to save either the state or the expectation values of the system.

• alg: The algorithm for the ODE solver. The default is Tsit5().

• e_ops: List of operators for which to calculate expectation values. It can be either a Vector or a Tuple.

• params: Parameters to pass to the solver. This argument is usually expressed as a NamedTuple or AbstractVector of parameters. For more advanced usage, any custom struct can be used.

• progress_bar: Whether to show the progress bar. Using non-Val types might lead to type instabilities.

• inplace: Whether to use the inplace version of the ODEProblem. The default is Val(true). It is recommended to use Val(true) for better performance, but it is sometimes necessary to use Val(false), for example when performing automatic differentiation using Zygote.jl.

• kwargs: The keyword arguments for the ODEProblem.

Notes

• The states will be saved depend on the keyword argument saveat in kwargs.

• If e_ops is empty, the default value of saveat=tlist (saving the states corresponding to tlist), otherwise, saveat=[tlist[end]] (only save the final state). You can also specify e_ops and saveat separately.

• The default tolerances in kwargs are given as reltol=1e-6 and abstol=1e-8.

• For more details about alg please refer to DifferentialEquations.jl (ODE Solvers)

• For more details about kwargs please refer to DifferentialEquations.jl (Keyword Arguments)

Returns

• sol::TimeEvolutionSol: The solution of the time evolution. See also TimeEvolutionSol

source

QuantumToolbox.mesolve Function

mesolve(
    H::Union{AbstractQuantumObject,Tuple},
    ψ0::QuantumObject,
    tlist::AbstractVector,
    c_ops::Union{Nothing,AbstractVector,Tuple} = nothing;
    alg::OrdinaryDiffEqAlgorithm = Tsit5(),
    e_ops::Union{Nothing,AbstractVector,Tuple} = nothing,
    params = NullParameters(),
    progress_bar::Union{Val,Bool} = Val(true),
    inplace::Union{Val,Bool} = Val(true),
    kwargs...,
)

Time evolution of an open quantum system using Lindblad master equation:

$$\frac{\partial \hat{\rho }(t)}{\partial t}=-i[\hat{H},\hat{\rho }(t)]+\sum _n\mathcal{D}({\hat{C}}_n)[\hat{\rho }(t)]$$

where

$$\mathcal{D}({\hat{C}}_n)[\hat{\rho }(t)]={\hat{C}}_n\hat{\rho }(t){\hat{C}}_n^{†}-\frac{1}{2}{\hat{C}}_n^{†}{\hat{C}}_n\hat{\rho }(t)-\frac{1}{2}\hat{\rho }(t){\hat{C}}_n^{†}{\hat{C}}_n$$

Arguments

• H: Hamiltonian of the system $\hat{H}$. It can be either a QuantumObject, a QuantumObjectEvolution, or a Tuple of operator-function pairs.

• ψ0: Initial state of the system $|\psi (0)⟩$. It can be either a Ket or a Operator.

• tlist: List of times at which to save either the state or the expectation values of the system.

• c_ops: List of collapse operators $\{{\hat{C}}_n{\}}_n$. It can be either a Vector or a Tuple.

• alg: The algorithm for the ODE solver. The default value is Tsit5().

• e_ops: List of operators for which to calculate expectation values. It can be either a Vector or a Tuple.

• params: Parameters to pass to the solver. This argument is usually expressed as a NamedTuple or AbstractVector of parameters. For more advanced usage, any custom struct can be used.

• progress_bar: Whether to show the progress bar. Using non-Val types might lead to type instabilities.

• inplace: Whether to use the inplace version of the ODEProblem. The default is Val(true). It is recommended to use Val(true) for better performance, but it is sometimes necessary to use Val(false), for example when performing automatic differentiation using Zygote.jl.

• kwargs: The keyword arguments for the ODEProblem.

Notes

• The states will be saved depend on the keyword argument saveat in kwargs.

• If e_ops is empty, the default value of saveat=tlist (saving the states corresponding to tlist), otherwise, saveat=[tlist[end]] (only save the final state). You can also specify e_ops and saveat separately.

• The default tolerances in kwargs are given as reltol=1e-6 and abstol=1e-8.

• For more details about alg please refer to DifferentialEquations.jl (ODE Solvers)

• For more details about kwargs please refer to DifferentialEquations.jl (Keyword Arguments)

Returns

• sol::TimeEvolutionSol: The solution of the time evolution. See also TimeEvolutionSol

source

QuantumToolbox.mcsolve Function

mcsolve(
    H::Union{AbstractQuantumObject{OperatorQuantumObject},Tuple},
    ψ0::QuantumObject{KetQuantumObject},
    tlist::AbstractVector,
    c_ops::Union{Nothing,AbstractVector,Tuple} = nothing;
    alg::OrdinaryDiffEqAlgorithm = Tsit5(),
    e_ops::Union{Nothing,AbstractVector,Tuple} = nothing,
    params = NullParameters(),
    rng::AbstractRNG = default_rng(),
    ntraj::Int = 500,
    ensemblealg::EnsembleAlgorithm = EnsembleThreads(),
    jump_callback::TJC = ContinuousLindbladJumpCallback(),
    progress_bar::Union{Val,Bool} = Val(true),
    prob_func::Union{Function, Nothing} = nothing,
    output_func::Union{Tuple,Nothing} = nothing,
    normalize_states::Union{Val,Bool} = Val(true),
    kwargs...,
)

Time evolution of an open quantum system using quantum trajectories.

Given a system Hamiltonian $\hat{H}$ and a list of collapse (jump) operators $\{{\hat{C}}_n{\}}_n$, the evolution of the state $|\psi (t)⟩$ is governed by the Schrodinger equation:

$$\frac{\partial }{\partial t}|\psi (t)⟩=-i{\hat{H}}_{\text{eff}}|\psi (t)⟩$$

with a non-Hermitian effective Hamiltonian:

$${\hat{H}}_{\text{eff}}=\hat{H}-\frac{i}{2}\sum _n{\hat{C}}_n^{†}{\hat{C}}_n.$$

To the first-order of the wave function in a small time $\delta t$, the strictly negative non-Hermitian portion in ${\hat{H}}_{\text{eff}}$ gives rise to a reduction in the norm of the wave function, namely

$$⟨\psi (t+\delta t)|\psi (t+\delta t)⟩=1-\delta p,$$

where

$$\delta p=\delta t\sum _n⟨\psi (t)|{\hat{C}}_n^{†}{\hat{C}}_n|\psi (t)⟩$$

is the corresponding quantum jump probability.

If the environmental measurements register a quantum jump, the wave function undergoes a jump into a state defined by projecting $|\psi (t)⟩$ using the collapse operator ${\hat{C}}_n$ corresponding to the measurement, namely

$$|\psi (t+\delta t)⟩=\frac{{\hat{C}}_n|\psi (t)⟩}{\sqrt{⟨\psi (t)|{\hat{C}}_n^{†}{\hat{C}}_n|\psi (t)⟩}}$$

Arguments

• H: Hamiltonian of the system $\hat{H}$. It can be either a QuantumObject, a QuantumObjectEvolution, or a Tuple of operator-function pairs.

• ψ0: Initial state of the system $|\psi (0)⟩$.

• tlist: List of times at which to save either the state or the expectation values of the system.

• c_ops: List of collapse operators $\{{\hat{C}}_n{\}}_n$. It can be either a Vector or a Tuple.

• alg: The algorithm to use for the ODE solver. Default to Tsit5().

• e_ops: List of operators for which to calculate expectation values. It can be either a Vector or a Tuple.

• params: Parameters to pass to the solver. This argument is usually expressed as a NamedTuple or AbstractVector of parameters. For more advanced usage, any custom struct can be used.

• rng: Random number generator for reproducibility.

• ntraj: Number of trajectories to use.

• ensemblealg: Ensemble algorithm to use. Default to EnsembleThreads().

• jump_callback: The Jump Callback type: Discrete or Continuous. The default is ContinuousLindbladJumpCallback(), which is more precise.

• progress_bar: Whether to show the progress bar. Using non-Val types might lead to type instabilities.

• prob_func: Function to use for generating the ODEProblem.

• output_func: a Tuple containing the Function to use for generating the output of a single trajectory, the (optional) ProgressBar object, and the (optional) RemoteChannel object.

• normalize_states: Whether to normalize the states. Default to Val(true).

• kwargs: The keyword arguments for the ODEProblem.

Notes

• ensemblealg can be one of EnsembleThreads(), EnsembleSerial(), EnsembleDistributed()

• The states will be saved depend on the keyword argument saveat in kwargs.

• If e_ops is empty, the default value of saveat=tlist (saving the states corresponding to tlist), otherwise, saveat=[tlist[end]] (only save the final state). You can also specify e_ops and saveat separately.

• The default tolerances in kwargs are given as reltol=1e-6 and abstol=1e-8.

• For more details about alg please refer to DifferentialEquations.jl (ODE Solvers)

• For more details about kwargs please refer to DifferentialEquations.jl (Keyword Arguments)

Returns

• sol::TimeEvolutionMCSol: The solution of the time evolution. See also TimeEvolutionMCSol.

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QuantumToolbox.ssesolve Function

ssesolve(
    H::Union{AbstractQuantumObject{OperatorQuantumObject},Tuple},
    ψ0::QuantumObject{KetQuantumObject},
    tlist::AbstractVector,
    sc_ops::Union{Nothing,AbstractVector,Tuple,AbstractQuantumObject} = nothing;
    alg::Union{Nothing,StochasticDiffEqAlgorithm} = nothing,
    e_ops::Union{Nothing,AbstractVector,Tuple} = nothing,
    params = NullParameters(),
    rng::AbstractRNG = default_rng(),
    ntraj::Int = 500,
    ensemblealg::EnsembleAlgorithm = EnsembleThreads(),
    prob_func::Union{Function, Nothing} = nothing,
    output_func::Union{Tuple,Nothing} = nothing,
    progress_bar::Union{Val,Bool} = Val(true),
    store_measurement::Union{Val,Bool} = Val(false),
    kwargs...,
)

Stochastic Schrödinger equation evolution of a quantum system given the system Hamiltonian $\hat{H}$ and a list of stochastic collapse (jump) operators $\{{\hat{S}}_n{\}}_n$. The stochastic evolution of the state $|\psi (t)⟩$ is defined by:

$$d|\psi (t)⟩=-i\hat{K}|\psi (t)⟩dt+\sum _n{\hat{M}}_n|\psi (t)⟩d{W}_n(t)$$

where

$$\hat{K}=\hat{H}+i\sum _n(\frac{e_n}{2}{\hat{S}}_n-\frac{1}{2}{\hat{S}}_n^{†}{\hat{S}}_n-\frac{e_n^2}{8}),$$$${\hat{M}}_n={\hat{S}}_n-\frac{e_n}{2},$$

and

$$e_n=⟨{\hat{S}}_n+{\hat{S}}_n^{†}⟩.$$

Above, ${\hat{S}}_n$ are the stochastic collapse operators and $d{W}_n(t)$ is the real Wiener increment associated to ${\hat{S}}_n$. See (Wiseman and Milburn, 2009) for more details.

Arguments

• H: Hamiltonian of the system $\hat{H}$. It can be either a QuantumObject, a QuantumObjectEvolution, or a Tuple of operator-function pairs.

• ψ0: Initial state of the system $|\psi (0)⟩$.

• tlist: List of times at which to save either the state or the expectation values of the system.

• sc_ops: List of stochastic collapse operators $\{{\hat{S}}_n{\}}_n$. It can be either a Vector, a Tuple or a AbstractQuantumObject. It is recommended to use the last case when only one operator is provided.

• alg: The algorithm to use for the stochastic differential equation. Default is SRIW1() if sc_ops is an AbstractQuantumObject (diagonal noise), and SRA2() otherwise (non-diagonal noise).

• e_ops: List of operators for which to calculate expectation values. It can be either a Vector or a Tuple.

• params: NullParameters of parameters to pass to the solver.

• rng: Random number generator for reproducibility.

• ntraj: Number of trajectories to use. Default is 500.

• ensemblealg: Ensemble method to use. Default to EnsembleThreads().

• prob_func: Function to use for generating the SDEProblem.

• output_func: a Tuple containing the Function to use for generating the output of a single trajectory, the (optional) ProgressBar object, and the (optional) RemoteChannel object.

• progress_bar: Whether to show the progress bar. Using non-Val types might lead to type instabilities.

• store_measurement: Whether to store the measurement results. Default is Val(false).

• kwargs: The keyword arguments for the ODEProblem.

Notes

• The states will be saved depend on the keyword argument saveat in kwargs.

• If e_ops is empty, the default value of saveat=tlist (saving the states corresponding to tlist), otherwise, saveat=[tlist[end]] (only save the final state). You can also specify e_ops and saveat separately.

• The default tolerances in kwargs are given as reltol=1e-2 and abstol=1e-2.

• For more details about alg please refer to DifferentialEquations.jl (SDE Solvers)

• For more details about kwargs please refer to DifferentialEquations.jl (Keyword Arguments)

Performance Tip

When sc_ops contains only a single operator, it is recommended to pass only that operator as the argument. This ensures that the stochastic noise is diagonal, making the simulation faster.

Returns

• sol::TimeEvolutionStochasticSol: The solution of the time evolution. See TimeEvolutionStochasticSol.

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QuantumToolbox.smesolve Function

smesolve(
    H::Union{AbstractQuantumObject{OperatorQuantumObject},Tuple},
    ψ0::QuantumObject,
    tlist::AbstractVector,
    c_ops::Union{Nothing,AbstractVector,Tuple} = nothing,
    sc_ops::Union{Nothing,AbstractVector,Tuple,AbstractQuantumObject} = nothing;
    alg::Union{Nothing,StochasticDiffEqAlgorithm} = nothing,
    e_ops::Union{Nothing,AbstractVector,Tuple} = nothing,
    params = NullParameters(),
    rng::AbstractRNG = default_rng(),
    ntraj::Int = 500,
    ensemblealg::EnsembleAlgorithm = EnsembleThreads(),
    prob_func::Union{Function, Nothing} = nothing,
    output_func::Union{Tuple,Nothing} = nothing,
    progress_bar::Union{Val,Bool} = Val(true),
    store_measurement::Union{Val,Bool} = Val(false),
    kwargs...,
)

Stochastic Master Equation time evolution of an open quantum system. This is defined by the following stochastic differential equation:

$$d\rho (t)=-i[\hat{H},\rho (t)]dt+\sum _i\mathcal{D}[{\hat{C}}_i]\rho (t)dt+\sum _n\mathcal{D}[{\hat{S}}_n]\rho (t)dt+\sum _n\mathcal{H}[{\hat{S}}_n]\rho (t)d{W}_n(t),$$

where

$$\mathcal{D}[\hat{O}]\rho =\hat{O}\rho {\hat{O}}^{†}-\frac{1}{2}\{{\hat{O}}^{†}\hat{O},\rho \},$$

is the Lindblad superoperator, and

$$\mathcal{H}[\hat{O}]\rho =\hat{O}\rho +\rho {\hat{O}}^{†}-\mathrm{Tr}[\hat{O}\rho +\rho {\hat{O}}^{†}]\rho ,$$

Above, ${\hat{C}}_i$ represent the collapse operators related to pure dissipation, while ${\hat{S}}_n$ are the stochastic co operators. The $d{W}_n(t)$ term is the real Wiener increment associated to ${\hat{S}}_n$. See (Wiseman and Milburn, 2009) for more details.

Arguments

• H: Hamiltonian of the system $\hat{H}$. It can be either a QuantumObject, a QuantumObjectEvolution, or a Tuple of operator-function pairs.

• ψ0: Initial state of the system $|\psi (0)⟩$. It can be either a Ket or a Operator.

• tlist: List of times at which to save either the state or the expectation values of the system.

• c_ops: List of collapse operators $\{{\hat{C}}_i{\}}_i$. It can be either a Vector or a Tuple.

• sc_ops: List of stochastic collapse operators $\{{\hat{S}}_n{\}}_n$. It can be either a Vector, a Tuple or a AbstractQuantumObject. It is recommended to use the last case when only one operator is provided.

• alg: The algorithm to use for the stochastic differential equation. Default is SRIW1() if sc_ops is an AbstractQuantumObject (diagonal noise), and SRA2() otherwise (non-diagonal noise).

• e_ops: List of operators for which to calculate expectation values. It can be either a Vector or a Tuple.

• params: NullParameters of parameters to pass to the solver.

• rng: Random number generator for reproducibility.

• ntraj: Number of trajectories to use. Default is 500.

• ensemblealg: Ensemble method to use. Default to EnsembleThreads().

• prob_func: Function to use for generating the SDEProblem.

• output_func: a Tuple containing the Function to use for generating the output of a single trajectory, the (optional) ProgressBar object, and the (optional) RemoteChannel object.

• progress_bar: Whether to show the progress bar. Using non-Val types might lead to type instabilities.

• store_measurement: Whether to store the measurement expectation values. Default is Val(false).

• kwargs: The keyword arguments for the ODEProblem.

Notes

• The states will be saved depend on the keyword argument saveat in kwargs.

• If e_ops is empty, the default value of saveat=tlist (saving the states corresponding to tlist), otherwise, saveat=[tlist[end]] (only save the final state). You can also specify e_ops and saveat separately.

• The default tolerances in kwargs are given as reltol=1e-2 and abstol=1e-2.

• For more details about kwargs please refer to DifferentialEquations.jl (Keyword Arguments)

Performance Tip

When sc_ops contains only a single operator, it is recommended to pass only that operator as the argument. This ensures that the stochastic noise is diagonal, making the simulation faster.

Returns

• sol::TimeEvolutionStochasticSol: The solution of the time evolution. See TimeEvolutionStochasticSol.

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QuantumToolbox.dfd_mesolve Function

dfd_mesolve(H::Function, ψ0::QuantumObject,
    tlist::AbstractVector, c_ops::Function, maxdims::AbstractVector,
    dfd_params::NamedTuple=NamedTuple();
    alg::OrdinaryDiffEqAlgorithm=Tsit5(),
    e_ops::Function=(dim_list) -> Vector{Vector{T1}}([]),
    params::NamedTuple=NamedTuple(),
    tol_list::Vector{<:Number}=fill(1e-8, length(maxdims)),
    kwargs...)

Time evolution of an open quantum system using master equation, dynamically changing the dimension of the Hilbert subspaces.

Notes

• For more details about alg please refer to DifferentialEquations.jl (ODE Solvers)

• For more details about kwargs please refer to DifferentialEquations.jl (Keyword Arguments)

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QuantumToolbox.liouvillian Function

liouvillian(H::AbstractQuantumObject, c_ops::Union{Nothing,AbstractVector,Tuple}=nothing, Id_cache=I(prod(H.dimensions)))

Construct the Liouvillian SuperOperator for a system Hamiltonian $\hat{H}$ and a set of collapse operators $\{{\hat{C}}_n{\}}_n$:

$$\mathcal{L}[\cdot ]=-i[\hat{H},\cdot ]+\sum _n\mathcal{D}({\hat{C}}_n)[\cdot ]$$

where

$$\mathcal{D}({\hat{C}}_n)[\cdot ]={\hat{C}}_n[\cdot ]{\hat{C}}_n^{†}-\frac{1}{2}{\hat{C}}_n^{†}{\hat{C}}_n[\cdot ]-\frac{1}{2}[\cdot ]{\hat{C}}_n^{†}{\hat{C}}_n$$

The optional argument Id_cache can be used to pass a precomputed identity matrix. This can be useful when the same function is applied multiple times with a known Hilbert space dimension.

See also spre, spost, and lindblad_dissipator.

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QuantumToolbox.liouvillian_generalized Function

liouvillian_generalized(H::QuantumObject, fields::Vector, 
T_list::Vector; N_trunc::Int=size(H,1), tol::Float64=0.0, σ_filter::Union{Nothing, Real}=nothing)

Constructs the generalized Liouvillian for a system coupled to a bath of harmonic oscillators.

See, e.g., Settineri, Alessio, et al. "Dissipation and thermal noise in hybrid quantum systems in the ultrastrong-coupling regime." Physical Review A 98.5 (2018): 053834.

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Steady State Solvers

QuantumToolbox.steadystate Function

steadystate(
    H::QuantumObject{OpType},
    c_ops::Union{Nothing,AbstractVector,Tuple} = nothing;
    solver::SteadyStateSolver = SteadyStateDirectSolver(),
    kwargs...,
)

Solve the stationary state based on different solvers.

Parameters

• H: The Hamiltonian or the Liouvillian of the system.

• c_ops: The list of the collapse operators.

• solver: see documentation Solving for Steady-State Solutions for different solvers.

• kwargs: The keyword arguments for the solver.

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QuantumToolbox.steadystate_fourier Function

steadystate_fourier(
    H_0::QuantumObject,
    H_p::QuantumObject,
    H_m::QuantumObject,
    ωd::Number,
    c_ops::Union{Nothing,AbstractVector,Tuple} = nothing;
    n_max::Integer = 2,
    tol::R = 1e-8,
    solver::FSolver = SteadyStateLinearSolver(),
    kwargs...,
)

Calculates the steady state of a periodically driven system. Here H_0 is the Hamiltonian or the Liouvillian of the undriven system. Considering a monochromatic drive at frequency ${\omega }_d$, we divide it into two parts, H_p and H_m, where H_p oscillates as $e^{i\omega t}$ and H_m oscillates as $e^{-i\omega t}$. There are two solvers available for this function:

• SteadyStateLinearSolver: Solves the linear system of equations.

• SSFloquetEffectiveLiouvillian: Solves the effective Liouvillian.

For both cases, n_max is the number of Fourier components to consider, and tol is the tolerance for the solver.

In the case of SteadyStateLinearSolver, the full linear system is solved at once:

$$({\mathcal{L}}_0-in{\omega }_d){\hat{\rho }}_n+{\mathcal{L}}_1{\hat{\rho }}_{n-1}+{\mathcal{L}}_{-1}{\hat{\rho }}_{n+1}=0$$

This is a tridiagonal linear system of the form

$$\mathbf{A}\cdot \mathbf{b}=0$$

where

$$\mathbf{A}=\left(\begin{array}{ccccc}{\mathcal{L}}_0-i(-n_{\text{max}}){\omega }_{\text{d}}& {\mathcal{L}}_{-1}& 0& \cdots & 0\\ {\mathcal{L}}_1& {\mathcal{L}}_0-i(-n_{\text{max}}+1){\omega }_{\text{d}}& {\mathcal{L}}_{-1}& \cdots & 0\\ 0& {\mathcal{L}}_1& {\mathcal{L}}_0-i(-n_{\text{max}}+2){\omega }_{\text{d}}& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & {\mathcal{L}}_0-i{n}_{\text{max}}{\omega }_{\text{d}}\end{array}\right)$$

and

$$\mathbf{b}=\left(\begin{array}{c}{\hat{\rho }}_{-n_{\text{max}}}\\ {\hat{\rho }}_{-n_{\text{max}}+1}\\ ⋮\\ {\hat{\rho }}_0\\ ⋮\\ {\hat{\rho }}_{n_{\text{max}}-1}\\ {\hat{\rho }}_{n_{\text{max}}}\end{array}\right)$$

This will allow to simultaneously obtain all the ${\hat{\rho }}_n$.

In the case of SSFloquetEffectiveLiouvillian, instead, the effective Liouvillian is calculated using the matrix continued fraction method.

different return

The two solvers returns different objects. The SteadyStateLinearSolver returns a list of QuantumObject, containing the density matrices for each Fourier component (${\hat{\rho }}_{-n}$, with $n$ from $0$ to $n_{\text{max}}$), while the SSFloquetEffectiveLiouvillian returns only ${\hat{\rho }}_0$.

Note

steadystate_floquet is a synonym of steadystate_fourier.

Arguments

• H_0::QuantumObject: The Hamiltonian or the Liouvillian of the undriven system.

• H_p::QuantumObject: The Hamiltonian or the Liouvillian of the part of the drive that oscillates as $e^{i\omega t}$.

• H_m::QuantumObject: The Hamiltonian or the Liouvillian of the part of the drive that oscillates as $e^{-i\omega t}$.

• ωd::Number: The frequency of the drive.

• c_ops::Union{Nothing,AbstractVector} = nothing: The optional collapse operators.

• n_max::Integer = 2: The number of Fourier components to consider.

• tol::R = 1e-8: The tolerance for the solver.

• solver::FSolver = SteadyStateLinearSolver: The solver to use.

• kwargs...: Additional keyword arguments to be passed to the solver.

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QuantumToolbox.SteadyStateDirectSolver Type

SteadyStateDirectSolver()

A solver which solves steadystate by finding the inverse of Liouvillian SuperOperator using the standard method given in LinearAlgebra.

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QuantumToolbox.SteadyStateEigenSolver Type

SteadyStateEigenSolver()

A solver which solves steadystate by finding the zero (or lowest) eigenvalue of Liouvillian SuperOperator using eigsolve.

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QuantumToolbox.SteadyStateLinearSolver Type

SteadyStateLinearSolver(alg = KrylovJL_GMRES(), Pl = nothing, Pr = nothing)

A solver which solves steadystate by finding the inverse of Liouvillian SuperOperator using the algorithms given in LinearSolve.jl.

Arguments

• alg::SciMLLinearSolveAlgorithm=KrylovJL_GMRES(): algorithms given in LinearSolve.jl

• Pl::Union{Function,Nothing}=nothing: left preconditioner, see documentation Solving for Steady-State Solutions for more details.

• Pr::Union{Function,Nothing}=nothing: right preconditioner, see documentation Solving for Steady-State Solutions for more details.

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QuantumToolbox.SteadyStateODESolver Type

SteadyStateODESolver(
    alg = Tsit5(),
    ψ0 = nothing,
    tmax = Inf,
    )

An ordinary differential equation (ODE) solver for solving steadystate.

Solve the stationary state based on time evolution (ordinary differential equations; OrdinaryDiffEq.jl) with a given initial state.

The termination condition of the stationary state $|\rho ⟩⟩$ is that either the following condition is true:

$$\Vert \frac{\partial |\hat{\rho }⟩⟩}{\partial t}\Vert \le \text{reltol}\times \Vert \frac{\partial |\hat{\rho }⟩⟩}{\partial t}+|\hat{\rho }⟩⟩\Vert$$

or

$$\Vert \frac{\partial |\hat{\rho }⟩⟩}{\partial t}\Vert \le \text{abstol}$$

Arguments

• alg::OrdinaryDiffEqAlgorithm=Tsit5(): The algorithm to solve the ODE.

• ψ0::Union{Nothing,QuantumObject}=nothing: The initial state of the system. If not specified, a random pure state will be generated.

• tmax::Real=Inf: The final time step for the steady state problem.

For more details about the solvers, please refer to OrdinaryDiffEq.jl.

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QuantumToolbox.SSFloquetEffectiveLiouvillian Type

SSFloquetEffectiveLiouvillian(steadystate_solver = SteadyStateDirectSolver())

A solver which solves steadystate_fourier by first extracting an effective time-independent Liouvillian and then using the steadystate_solver to extract the steadystate..

Parameters

• steadystate_solver::SteadyStateSolver=SteadyStateDirectSolver(): The solver to use for the effective Liouvillian.

Note

This solver is only available for steadystate_fourier.

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Dynamical Shifted Fock method

QuantumToolbox.dsf_mesolve Function

dsf_mesolve(H::Function,
    ψ0::QuantumObject,
    tlist::AbstractVector, c_ops::Function,
    op_list::Vector{TOl},
    α0_l::Vector{<:Number}=zeros(length(op_list)),
    dsf_params::NamedTuple=NamedTuple();
    alg::OrdinaryDiffEqAlgorithm=Tsit5(),
    e_ops::Function=(op_list,p) -> Vector{TOl}([]),
    params::NamedTuple=NamedTuple(),
    δα_list::Vector{<:Number}=fill(0.2, length(op_list)),
    krylov_dim::Int=max(6, min(10, cld(length(ket2dm(ψ0).data), 4))),
    kwargs...)

Time evolution of an open quantum system using master equation and the Dynamical Shifted Fock algorithm.

Notes

• For more details about alg please refer to DifferentialEquations.jl (ODE Solvers)

• For more details about kwargs please refer to DifferentialEquations.jl (Keyword Arguments)

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QuantumToolbox.dsf_mcsolve Function

dsf_mcsolve(H::Function,
    ψ0::QuantumObject,
    tlist::AbstractVector, c_ops::Function,
    op_list::Vector{TOl},
    α0_l::Vector{<:Number}=zeros(length(op_list)),
    dsf_params::NamedTuple=NamedTuple();
    alg::OrdinaryDiffEqAlgorithm=Tsit5(),
    e_ops::Function=(op_list,p) -> Vector{TOl}([]),
    params::NamedTuple=NamedTuple(),
    δα_list::Vector{<:Real}=fill(0.2, length(op_list)),
    ntraj::Int=500,
    ensemblealg::EnsembleAlgorithm=EnsembleThreads(),
    jump_callback::LindbladJumpCallbackType=ContinuousLindbladJumpCallback(),
    krylov_dim::Int=max(6, min(10, cld(length(ket2dm(ψ0).data), 4))),
    progress_bar::Union{Bool,Val} = Val(true)
    kwargs...)

Time evolution of a quantum system using the Monte Carlo wave function method and the Dynamical Shifted Fock algorithm.

Notes

• For more details about alg please refer to DifferentialEquations.jl (ODE Solvers)

• For more details about kwargs please refer to DifferentialEquations.jl (Keyword Arguments)

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Low-rank time evolution

QuantumToolbox.TimeEvolutionLRSol Type

struct TimeEvolutionLRSol

A structure storing the results and some information from solving low-rank master equation time evolution.

Fields (Attributes)

• times::AbstractVector: The time list of the evolution.

• states::Vector{QuantumObject}: The list of result states.

• expect::Matrix: The expectation values corresponding to each time point in times.

• fexpect::Matrix: The function values at each time point.

• retcode: The return code from the solver.

• alg: The algorithm which is used during the solving process.

• abstol::Real: The absolute tolerance which is used during the solving process.

• reltol::Real: The relative tolerance which is used during the solving process.

• z::Vector{QuantumObject}: The `z`` matrix of the low-rank algorithm at each time point.

• B::Vector{QuantumObject}: The B matrix of the low-rank algorithm at each time point.

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QuantumToolbox.lr_mesolveProblem Function

lr_mesolveProblem(
    H::QuantumObject{OperatorQuantumObject},
    z::AbstractArray{T,2},
    B::AbstractArray{T,2},
    tlist::AbstractVector,
    c_ops::Union{AbstractVector,Tuple}=();
    e_ops::Union{AbstractVector,Tuple}=(),
    f_ops::Union{AbstractVector,Tuple}=(),
    opt::NamedTuple = lr_mesolve_options_default,
    kwargs...,
)

Formulates the ODEproblem for the low-rank time evolution of the system. The function is called by lr_mesolve. For more information about the low-rank master equation, see (Gravina and Savona, 2024).

Arguments

• H::QuantumObject: The Hamiltonian of the system.

• z::AbstractArray: The initial z matrix of the low-rank algorithm.

• B::AbstractArray: The initial B matrix of the low-rank algorithm.

• tlist::AbstractVector: The time steps at which the expectation values and function values are calculated.

• c_ops::Union{AbstractVector,Tuple}: The list of the collapse operators.

• e_ops::Union{AbstractVector,Tuple}: The list of the operators for which the expectation values are calculated.

• f_ops::Union{AbstractVector,Tuple}: The list of the functions for which the function values are calculated.

• opt::NamedTuple: The options of the low-rank master equation.

• kwargs: Additional keyword arguments.

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QuantumToolbox.lr_mesolve Function

lr_mesolve(
    H::QuantumObject{OperatorQuantumObject},
    z::AbstractArray{T,2},
    B::AbstractArray{T,2},
    tlist::AbstractVector,
    c_ops::Union{AbstractVector,Tuple}=();
    e_ops::Union{AbstractVector,Tuple}=(),
    f_ops::Union{AbstractVector,Tuple}=(),
    opt::NamedTuple = lr_mesolve_options_default,
    kwargs...,
)

Time evolution of an open quantum system using the low-rank master equation. For more information about the low-rank master equation, see (Gravina and Savona, 2024).

Arguments

• H::QuantumObject: The Hamiltonian of the system.

• z::AbstractArray: The initial z matrix of the low-rank algorithm.

• B::AbstractArray: The initial B matrix of the low-rank algorithm.

• tlist::AbstractVector: The time steps at which the expectation values and function values are calculated.

• c_ops::Union{AbstractVector,Tuple}: The list of the collapse operators.

• e_ops::Union{AbstractVector,Tuple}: The list of the operators for which the expectation values are calculated.

• f_ops::Union{AbstractVector,Tuple}: The list of the functions for which the function values are calculated.

• opt::NamedTuple: The options of the low-rank master equation.

• kwargs: Additional keyword arguments.

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Correlations and Spectrum

QuantumToolbox.correlation_3op_2t Function

correlation_3op_2t(H::AbstractQuantumObject,
    ψ0::Union{Nothing,QuantumObject},
    tlist::AbstractVector,
    τlist::AbstractVector,
    c_ops::Union{Nothing,AbstractVector,Tuple},
    A::QuantumObject,
    B::QuantumObject,
    C::QuantumObject;
    kwargs...)

Returns the two-times correlation function of three operators $\hat{A}$, $\hat{B}$ and $\hat{C}$: $⟨\hat{A}(t)\hat{B}(t+\tau )\hat{C}(t)⟩$ for a given initial state $|{\psi }_0⟩$.

If the initial state ψ0 is given as nothing, then the steadystate will be used as the initial state. Note that this is only implemented if H is constant (QuantumObject).

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QuantumToolbox.correlation_3op_1t Function

correlation_3op_1t(H::AbstractQuantumObject,
    ψ0::Union{Nothing,QuantumObject},
    τlist::AbstractVector,
    c_ops::Union{Nothing,AbstractVector,Tuple},
    A::QuantumObject,
    B::QuantumObject,
    C::QuantumObject;
    kwargs...)

Returns the two-time correlation function (with only one time coordinate $\tau$) of three operators $\hat{A}$, $\hat{B}$ and $\hat{C}$: $⟨\hat{A}(0)\hat{B}(\tau )\hat{C}(0)⟩$ for a given initial state $|{\psi }_0⟩$.

If the initial state ψ0 is given as nothing, then the steadystate will be used as the initial state. Note that this is only implemented if H is constant (QuantumObject).

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QuantumToolbox.correlation_2op_2t Function

correlation_2op_2t(H::AbstractQuantumObject,
    ψ0::Union{Nothing,QuantumObject},
    tlist::AbstractVector,
    τlist::AbstractVector,
    c_ops::Union{Nothing,AbstractVector,Tuple},
    A::QuantumObject,
    B::QuantumObject;
    reverse::Bool=false,
    kwargs...)

Returns the two-times correlation function of two operators $\hat{A}$ and $\hat{B}$ : $⟨\hat{A}(t+\tau )\hat{B}(t)⟩$ for a given initial state $|{\psi }_0⟩$.

If the initial state ψ0 is given as nothing, then the steadystate will be used as the initial state. Note that this is only implemented if H is constant (QuantumObject).

When reverse=true, the correlation function is calculated as $⟨\hat{A}(t)\hat{B}(t+\tau )⟩$.

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QuantumToolbox.correlation_2op_1t Function

correlation_2op_1t(H::AbstractQuantumObject,
    ψ0::Union{Nothing,QuantumObject},
    τlist::AbstractVector,
    c_ops::Union{Nothing,AbstractVector,Tuple},
    A::QuantumObject,
    B::QuantumObject;
    reverse::Bool=false,
    kwargs...)

Returns the two-time correlation function (with only one time coordinate $\tau$) of two operators $\hat{A}$ and $\hat{B}$ : $⟨\hat{A}(\tau )\hat{B}(0)⟩$ for a given initial state $|{\psi }_0⟩$.

If the initial state ψ0 is given as nothing, then the steadystate will be used as the initial state. Note that this is only implemented if H is constant (QuantumObject).

When reverse=true, the correlation function is calculated as $⟨\hat{A}(0)\hat{B}(\tau )⟩$.

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QuantumToolbox.spectrum_correlation_fft Function

spectrum_correlation_fft(tlist, corr; inverse=false)

Calculate the power spectrum corresponding to a two-time correlation function using fast Fourier transform (FFT).

Parameters

• tlist::AbstractVector: List of times at which the two-time correlation function is given.

• corr::AbstractVector: List of two-time correlations corresponding to the given time point in tlist.

• inverse::Bool: Whether to use the inverse Fourier transform or not. Default to false.

Returns

• ωlist: the list of angular frequencies $\omega$.

• Slist: the list of the power spectrum corresponding to the angular frequencies in ωlist.

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QuantumToolbox.spectrum Function

spectrum(H::QuantumObject,
    ωlist::AbstractVector,
    c_ops::Union{Nothing,AbstractVector,Tuple},
    A::QuantumObject{OperatorQuantumObject},
    B::QuantumObject{OperatorQuantumObject};
    solver::SpectrumSolver=ExponentialSeries(),
    kwargs...)

Calculate the spectrum of the correlation function

$$S(\omega )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\underset{t\to \mathrm{\infty }}{lim}⟨\hat{A}(t+\tau )\hat{B}(t)⟩e^{-i\omega \tau }d\tau$$

See also the following list for SpectrumSolver docstrings:

• ExponentialSeries

• PseudoInverse

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QuantumToolbox.ExponentialSeries Type

ExponentialSeries(; tol = 1e-14, calc_steadystate = false)

A solver which solves spectrum by finding the eigen decomposition of the Liouvillian SuperOperator and calculate the exponential series.

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QuantumToolbox.PseudoInverse Type

PseudoInverse(; alg::SciMLLinearSolveAlgorithm = KrylovJL_GMRES())

A solver which solves spectrum by finding the inverse of Liouvillian SuperOperator using the algorithms given in LinearSolve.jl.

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Entropy and Metrics

QuantumToolbox.entropy_vn Function

entropy_vn(ρ::QuantumObject; base::Int=0, tol::Real=1e-15)

Calculates the Von Neumann entropy $S=-\text{Tr}[\hat{\rho }\log \left(\hat{\rho }\right)]$, where $\hat{\rho }$ is the density matrix of the system.

Notes

• ρ is the quantum state, can be either a Ket or an Operator.

• base specifies the base of the logarithm to use, and when using the default value 0, the natural logarithm is used.

• tol describes the absolute tolerance for detecting the zero-valued eigenvalues of the density matrix $\hat{\rho }$.

Examples

Pure state:

julia> ψ = fock(2,0)

Quantum Object:   type=Ket   dims=[2]   size=(2,)
2-element Vector{ComplexF64}:
 1.0 + 0.0im
 0.0 + 0.0im

julia> entropy_vn(ψ, base=2)
-0.0

Mixed state:

julia> ρ = maximally_mixed_dm(2)

Quantum Object:   type=Operator   dims=[2]   size=(2, 2)   ishermitian=true
2×2 Diagonal{ComplexF64, Vector{ComplexF64}}:
 0.5-0.0im      ⋅    
     ⋅      0.5-0.0im

julia> entropy_vn(ρ, base=2)
1.0

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QuantumToolbox.entropy_relative Function

entropy_relative(ρ::QuantumObject, σ::QuantumObject; base::Int=0, tol::Real=1e-15)

Calculates the quantum relative entropy of $\hat{\rho }$ with respect to $\hat{\sigma }$: $D(\hat{\rho }||\hat{\sigma })=\text{Tr}[\hat{\rho }\log \left(\hat{\rho }\right)]-\text{Tr}[\hat{\rho }\log \left(\hat{\sigma }\right)]$.

Notes

• ρ is a quantum state, can be either a Ket or an Operator.

• σ is a quantum state, can be either a Ket or an Operator.

• base specifies the base of the logarithm to use, and when using the default value 0, the natural logarithm is used.

• tol describes the absolute tolerance for detecting the zero-valued eigenvalues of the density matrix $\hat{\rho }$.

References

• Nielsen and Chuang (2012), section 11.3.1, page 511

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QuantumToolbox.entropy_linear Function

entropy_linear(ρ::QuantumObject)

Calculates the quantum linear entropy $S_L=1-\text{Tr}\left[{\hat{\rho }}^2\right]$, where $\hat{\rho }$ is the density matrix of the system.

Note that ρ can be either a Ket or an Operator.

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QuantumToolbox.entropy_mutual Function

entropy_mutual(ρAB::QuantumObject, selA, selB; kwargs...)

Calculates the quantum mutual information $I(A:B)=S({\hat{\rho }}_A)+S({\hat{\rho }}_B)-S({\hat{\rho }}_{AB})$ between subsystems $A$ and $B$.

Here, $S$ is the Von Neumann entropy, ${\hat{\rho }}_{AB}$ is the density matrix of the entire system, ${\hat{\rho }}_A={\text{Tr}}_B\left[{\hat{\rho }}_{AB}\right]$, ${\hat{\rho }}_B={\text{Tr}}_A\left[{\hat{\rho }}_{AB}\right]$.

Notes

• ρAB can be either a Ket or an Operator.

• selA specifies the indices of the sub-system A in ρAB.dimensions. See also ptrace.

• selB specifies the indices of the sub-system B in ρAB.dimensions. See also ptrace.

• kwargs are the keyword arguments for calculating Von Neumann entropy. See also entropy_vn.

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QuantumToolbox.entropy_conditional Function

entropy_conditional(ρAB::QuantumObject, selB; kwargs...)

Calculates the conditional quantum entropy with respect to sub-system $B$: $S(A|B)=S({\hat{\rho }}_{AB})-S({\hat{\rho }}_B)$.

Here, $S$ is the Von Neumann entropy, ${\hat{\rho }}_{AB}$ is the density matrix of the entire system, and ${\hat{\rho }}_B={\text{Tr}}_A\left[{\hat{\rho }}_{AB}\right]$.

Notes

• ρAB can be either a Ket or an Operator.

• selB specifies the indices of the sub-system B in ρAB.dimensions. See also ptrace.

• kwargs are the keyword arguments for calculating Von Neumann entropy. See also entropy_vn.

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QuantumToolbox.entanglement Function

entanglement(ρ::QuantumObject, sel; kwargs...)

Calculates the entanglement entropy by doing the partial trace of ρ, selecting only the dimensions with the indices contained in the sel vector, and then use the Von Neumann entropy entropy_vn.

Notes

• ρ can be either a Ket or an Operator. But should be a pure state.

• sel specifies the indices of the remaining sub-system. See also ptrace.

• kwargs are the keyword arguments for calculating Von Neumann entropy. See also entropy_vn.

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QuantumToolbox.concurrence Function

concurrence(ρ::QuantumObject)

Calculate the concurrence for a two-qubit state.

Notes

• ρ can be either a Ket or an Operator.

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QuantumToolbox.negativity Function

negativity(ρ::QuantumObject, subsys::Int; logarithmic::Bool=false)

Compute the negativity $N(\hat{\rho })=\frac{\Vert {\hat{\rho }}^{\mathrm{\Gamma }}{\Vert }_1-1}{2}$ where ${\hat{\rho }}^{\mathrm{\Gamma }}$ is the partial transpose of $\hat{\rho }$ with respect to the subsystem, and $\Vert \hat{X}{\Vert }_1=\text{Tr}\sqrt{{\hat{X}}^{†}\hat{X}}$ is the trace norm.

Arguments

• ρ::QuantumObject: The density matrix (ρ.type must be OperatorQuantumObject).

• subsys::Int: an index that indicates which subsystem to compute the negativity for.

• logarithmic::Bool: choose whether to calculate logarithmic negativity or not. Default as false

Returns

• N::Real: The value of negativity.

Examples

julia> Ψ = bell_state(0, 0)

Quantum Object:   type=Ket   dims=[2, 2]   size=(4,)
4-element Vector{ComplexF64}:
 0.7071067811865475 + 0.0im
                0.0 + 0.0im
                0.0 + 0.0im
 0.7071067811865475 + 0.0im

julia> ρ = ket2dm(Ψ)

Quantum Object:   type=Operator   dims=[2, 2]   size=(4, 4)   ishermitian=true
4×4 Matrix{ComplexF64}:
 0.5+0.0im  0.0+0.0im  0.0+0.0im  0.5+0.0im
 0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im
 0.0+0.0im  0.0+0.0im  0.0+0.0im  0.0+0.0im
 0.5+0.0im  0.0+0.0im  0.0+0.0im  0.5+0.0im

julia> round(negativity(ρ, 2), digits=2)
0.5

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QuantumToolbox.fidelity Function

fidelity(ρ::QuantumObject, σ::QuantumObject)

Calculate the fidelity of two QuantumObject: $F(\hat{\rho },\hat{\sigma })=\text{Tr}\sqrt{\sqrt{\hat{\rho }}\hat{\sigma }\sqrt{\hat{\rho }}}$

Here, the definition is from Nielsen and Chuang (2012). It is the square root of the fidelity defined in Jozsa (1994).

Note that ρ and σ must be either Ket or Operator.

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QuantumToolbox.tracedist Function

tracedist(ρ::QuantumObject, σ::QuantumObject)

Calculates the trace distance between two QuantumObject: $T(\hat{\rho },\hat{\sigma })=\frac{1}{2}\Vert \hat{\rho }-\hat{\sigma }{\Vert }_1$

Note that ρ and σ must be either Ket or Operator.

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QuantumToolbox.hilbert_dist Function

hilbert_dist(ρ::QuantumObject, σ::QuantumObject)

Calculates the Hilbert-Schmidt distance between two QuantumObject: $D_{HS}(\hat{\rho },\hat{\sigma })=\text{Tr}\left[{\hat{A}}^{†}\hat{A}\right]$, where $\hat{A}=\hat{\rho }-\hat{\sigma }$.

Note that ρ and σ must be either Ket or Operator.

References

• Vedral and Plenio (1998)

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QuantumToolbox.hellinger_dist Function

hellinger_dist(ρ::QuantumObject, σ::QuantumObject)

Calculates the Hellinger distance between two QuantumObject: $D_H(\hat{\rho },\hat{\sigma })=\sqrt{2-2\text{Tr}\left(\sqrt{\hat{\rho }}\sqrt{\hat{\sigma }}\right)}$

Note that ρ and σ must be either Ket or Operator.

References

• Spehner et al. (2017)

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QuantumToolbox.bures_dist Function

bures_dist(ρ::QuantumObject, σ::QuantumObject)

Calculate the Bures distance between two QuantumObject: $D_B(\hat{\rho },\hat{\sigma })=\sqrt{2(1-F(\hat{\rho },\hat{\sigma }))}$

Here, the definition of fidelity $F$ is from Nielsen and Chuang (2012). It is the square root of the fidelity defined in Jozsa (1994).

Note that ρ and σ must be either Ket or Operator.

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QuantumToolbox.bures_angle Function

bures_angle(ρ::QuantumObject, σ::QuantumObject)

Calculate the Bures angle between two QuantumObject: $D_A(\hat{\rho },\hat{\sigma })=\arccos (F(\hat{\rho },\hat{\sigma }))$

Here, the definition of fidelity $F$ is from Nielsen and Chuang (2012). It is the square root of the fidelity defined in Jozsa (1994).

Note that ρ and σ must be either Ket or Operator.

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Spin Lattice

QuantumToolbox.Lattice Type

Lattice

A Julia constructor for a lattice object. The lattice object is used to define the geometry of the lattice. Nx and Ny are the number of sites in the x and y directions, respectively. N is the total number of sites. lin_idx is a LinearIndices object and car_idx is a CartesianIndices object, and they are used to efficiently select sites on the lattice.

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QuantumToolbox.multisite_operator Function

multisite_operator(dims::Union{AbstractVector, Tuple}, pairs::Pair{<:Integer,<:QuantumObject}...)

A Julia function for generating a multi-site operator $$\begin{gathered} hatO= \\ hat{O}_i \\ hat{O}_j \\ cdots \\ hat{O}_k \end{gathered}$$. The function takes a vector of dimensions dims and a list of pairs pairs where the first element of the pair is the site index and the second element is the operator acting on that site.

Arguments

• dims::Union{AbstractVector, Tuple}: A vector of dimensions of the lattice.

• pairs::Pair{<:Integer,<:QuantumObject}...: A list of pairs where the first element of the pair is the site index and the second element is the operator acting on that site.

Returns

QuantumObject: A QuantumObject representing the multi-site operator.

Example

julia> op = multisite_operator(Val(8), 5=>sigmax(), 7=>sigmaz());

julia> op.dims
8-element SVector{8, Int64} with indices SOneTo(8):
 2
 2
 2
 2
 2
 2
 2
 2

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QuantumToolbox.DissipativeIsing Function

DissipativeIsing(Jx::Real, Jy::Real, Jz::Real, hx::Real, hy::Real, hz::Real, γ::Real, latt::Lattice; boundary_condition::Union{Symbol, Val} = Val(:periodic_bc), order::Integer = 1)

A Julia constructor for a dissipative Ising model. The function returns the Hamiltonian

$$\hat{H}=\frac{J_x}{2}\sum _{⟨i,j⟩}{\hat{\sigma }}_i^x{\hat{\sigma }}_j^x+\frac{J_y}{2}\sum _{⟨i,j⟩}{\hat{\sigma }}_i^y{\hat{\sigma }}_j^y+\frac{J_z}{2}\sum _{⟨i,j⟩}{\hat{\sigma }}_i^z{\hat{\sigma }}_j^z+h_x\sum _i{\hat{\sigma }}_i^x$$

and the collapse operators

$${\hat{c}}_i=\sqrt{\gamma }{\hat{\sigma }}_i^{-}$$

Arguments

• Jx::Real: The coupling constant in the x-direction.

• Jy::Real: The coupling constant in the y-direction.

• Jz::Real: The coupling constant in the z-direction.

• hx::Real: The magnetic field in the x-direction.

• hy::Real: The magnetic field in the y-direction.

• hz::Real: The magnetic field in the z-direction.

• γ::Real: The local dissipation rate.

• latt::Lattice: A Lattice object that defines the geometry of the lattice.

• boundary_condition::Union{Symbol, Val}: The boundary conditions of the lattice. The possible inputs are periodic_bc and open_bc, for periodic or open boundary conditions, respectively. The default value is Val(:periodic_bc).

• order::Integer: The order of the nearest-neighbour sites. The default value is 1.

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Symmetries and Block Diagonalization

QuantumToolbox.block_diagonal_form Function

block_diagonal_form(A::QuantumObject)

Return the block-diagonal form of a QuantumObject. This is very useful in the presence of symmetries.

Arguments

• A::QuantumObject: The quantum object.

Returns

The BlockDiagonalForm of A.

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QuantumToolbox.BlockDiagonalForm Type

struct BlockDiagonalForm

A type for storing a block-diagonal form of a matrix.

Fields

• B::DT: The block-diagonal matrix. It can be a sparse matrix or a QuantumObject.

• P::DT: The permutation matrix. It can be a sparse matrix or a QuantumObject.

• blocks::AbstractVector: The blocks of the block-diagonal matrix.

• block_sizes::AbstractVector: The sizes of the blocks.

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Miscellaneous

QuantumToolbox.wigner Function

wigner(
    state::QuantumObject{OpType},
    xvec::AbstractVector,
    yvec::AbstractVector;
    g::Real = √2,
    method::WignerSolver = WignerClenshaw(),
)

Generates the Wigner quasipropability distribution of state at points xvec + 1im * yvec in phase space. The g parameter is a scaling factor related to the value of $\hslash$ in the commutation relation $[x,y]=i\hslash$ via $\hslash =2/g^2$ giving the default value $\hslash =1$.

The method parameter can be either WignerLaguerre() or WignerClenshaw(). The former uses the Laguerre polynomial expansion of the Wigner function, while the latter uses the Clenshaw algorithm. The Laguerre expansion is faster for sparse matrices, while the Clenshaw algorithm is faster for dense matrices. The WignerLaguerre method has an optional parallel parameter which defaults to true and uses multithreading to speed up the calculation.

Arguments

• state::QuantumObject: The quantum state for which the Wigner function is calculated. It can be either a KetQuantumObject, BraQuantumObject, or OperatorQuantumObject.

• xvec::AbstractVector: The x-coordinates of the phase space grid.

• yvec::AbstractVector: The y-coordinates of the phase space grid.

• g::Real: The scaling factor related to the value of $\hslash$ in the commutation relation $[x,y]=i\hslash$ via $\hslash =2/g^2$.

• method::WignerSolver: The method used to calculate the Wigner function. It can be either WignerLaguerre() or WignerClenshaw(), with WignerClenshaw() as default. The WignerLaguerre method has the optional parallel and tol parameters, with default values true and 1e-14, respectively.

Returns

• W::Matrix: The Wigner function of the state at the points xvec + 1im * yvec in phase space.

Example

julia> ψ = fock(10, 0) + fock(10, 1) |> normalize

Quantum Object:   type=Ket   dims=[10]   size=(10,)
10-element Vector{ComplexF64}:
 0.7071067811865475 + 0.0im
 0.7071067811865475 + 0.0im
                0.0 + 0.0im
                0.0 + 0.0im
                0.0 + 0.0im
                0.0 + 0.0im
                0.0 + 0.0im
                0.0 + 0.0im
                0.0 + 0.0im
                0.0 + 0.0im

julia> xvec = range(-5, 5, 200)
-5.0:0.05025125628140704:5.0

julia> wig = wigner(ψ, xvec, xvec);

or taking advantage of the parallel computation of the WignerLaguerre method

julia> ρ = ket2dm(ψ) |> to_sparse;

julia> wig = wigner(ρ, xvec, xvec, method=WignerLaguerre(parallel=true));

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Linear Maps

QuantumToolbox.AbstractLinearMap Type

AbstractLinearMap{T, TS}

Represents a general linear map with element type T and size TS.

Overview

A linear map is a transformation L that satisfies:

• Additivity: math L(u + v) = L(u) + L(v)

• Homogeneity: math L(cu) = cL(u)

It is typically represented as a matrix with dimensions given by size, and this abstract type helps to define this map when the matrix is not explicitly available.

Methods

• Base.eltype(A): Returns the element type T.

• Base.size(A): Returns the size A.size.

• Base.size(A, i): Returns the i-th dimension.

Example

As an example, we now define the linear map used in the eigsolve_al function for Arnoldi-Lindblad eigenvalue solver:

struct ArnoldiLindbladIntegratorMap{T,TS,TI} <: AbstractLinearMap{T,TS}
    elty::Type{T}
    size::TS
    integrator::TI
end

function LinearAlgebra.mul!(y::AbstractVector, A::ArnoldiLindbladIntegratorMap, x::AbstractVector)
    reinit!(A.integrator, x)
    solve!(A.integrator)
    return copyto!(y, A.integrator.u)
end

where integrator is the ODE integrator for the time-evolution. In this way, we can diagonalize this linear map using the eigsolve function.

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Utility functions

QuantumToolbox.versioninfo Function

QuantumToolbox.versioninfo(io::IO=stdout)

Command line output of information on QuantumToolbox, dependencies, and system information, same as QuantumToolbox.about.

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QuantumToolbox.about Function

QuantumToolbox.about(io::IO=stdout)

Command line output of information on QuantumToolbox, dependencies, and system information, same as QuantumToolbox.versioninfo.

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QuantumToolbox.gaussian Function

gaussian(x::Number, μ::Number, σ::Number)

Returns the gaussian function $\exp [-\frac{(x-\mu {)}^2}{2{\sigma }^2}]$, where $\mu$ and ${\sigma }^2$ are the mean and the variance respectively.

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QuantumToolbox.n_thermal Function

n_thermal(ω::Real, ω_th::Real)

Return the number of photons in thermal equilibrium for an harmonic oscillator mode with frequency $\omega$, at the temperature described by ${\omega }_{\text{th}}\equiv k_{B}T/\hslash$:

$$n(\omega ,{\omega }_{\text{th}})=\frac{1}{e^{\omega /{\omega }_{\text{th}}}-1},$$

where $\hslash$ is the reduced Planck constant, and $k_B$ is the Boltzmann constant.

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QuantumToolbox.PhysicalConstants Constant

const PhysicalConstants

A NamedTuple which stores some constant values listed in CODATA recommended values of the fundamental physical constants: 2022

The current stored constants are:

• c : (exact) speed of light in vacuum with unit $[\text{m}\cdot {\text{s}}^{-1}]$

• G : Newtonian constant of gravitation with unit $[{\text{m}}^3\cdot {\text{kg}}^{-1}\cdot {\text{s}}^{-2}]$

• h : (exact) Planck constant with unit $[\text{J}\cdot \text{s}]$

• ħ : reduced Planck constant with unit $[\text{J}\cdot \text{s}]$

• e : (exact) elementary charge with unit $[\text{C}]$

• μ0 : vacuum magnetic permeability with unit $[\text{N}\cdot {\text{A}}^{-2}]$

• ϵ0 : vacuum electric permittivity with unit $[\text{F}\cdot {\text{m}}^{-1}]$

• k : (exact) Boltzmann constant with unit $[\text{J}\cdot {\text{K}}^{-1}]$

• NA : (exact) Avogadro constant with unit $[{\text{mol}}^{-1}]$

Examples

julia> PhysicalConstants.ħ
1.0545718176461565e-34

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QuantumToolbox.convert_unit Function

convert_unit(value::Real, unit1::Symbol, unit2::Symbol)

Convert the energy value from unit1 to unit2. The unit1 and unit2 can be either the following Symbol:

• :J : Joule

• :eV : electron volt

• :meV : milli-electron volt

• :MHz : Mega-Hertz multiplied by Planck constant $h$

• :GHz : Giga-Hertz multiplied by Planck constant $h$

• :K : Kelvin multiplied by Boltzmann constant $k$

• :mK : milli-Kelvin multiplied by Boltzmann constant $k$

Note that we use the values stored in PhysicalConstants to do the conversion.

Examples

julia> convert_unit(1, :eV, :J)
1.602176634e-19

julia> convert_unit(1, :GHz, :J)
6.62607015e-25

julia> round(convert_unit(1, :meV, :mK), digits=4)
11604.5181

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QuantumToolbox.row_major_reshape Function

row_major_reshape(Q::AbstractArray, shapes...)

Reshapes Q in the row-major order, as numpy.

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QuantumToolbox.meshgrid Function

meshgrid(x::AbstractVector, y::AbstractVector)

Equivalent to numpy meshgrid.

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Visualization

QuantumToolbox.plot_wigner Function

plot_wigner(
    state::QuantumObject{OpType}; 
    library::Union{Val,Symbol}=Val(:CairoMakie), 
    kwargs...
) where {OpType<:Union{BraQuantumObject,KetQuantumObject,OperatorQuantumObject}

Plot the Wigner quasipropability distribution of state using the wigner function.

The library keyword argument specifies the plotting library to use, defaulting to CairoMakie.

Arguments

• state::QuantumObject: The quantum state for which to plot the Wigner distribution.

• library::Union{Val,Symbol}: The plotting library to use. Default is Val(:CairoMakie).

• kwargs...: Additional keyword arguments to pass to the plotting function. See the documentation for the specific plotting library for more information.

Import library first

The plotting libraries must first be imported before using them with this function.

Beware of type-stability!

If you want to keep type stability, it is recommended to use Val(:CairoMakie) instead of :CairoMakie as the plotting library. See this link and the related Section about type stability for more details.

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plot_wigner(
    library::Val{:CairoMakie},
    state::QuantumObject{OpType};
    xvec::Union{Nothing,AbstractVector} = nothing,
    yvec::Union{Nothing,AbstractVector} = nothing,
    g::Real = √2,
    method::WignerSolver = WignerClenshaw(),
    projection::Union{Val,Symbol} = Val(:two_dim),
    location::Union{GridPosition,Nothing} = nothing,
    colorbar::Bool = false,
    kwargs...
) where {OpType}

Plot the Wigner quasipropability distribution of state using the CairoMakie plotting library.

Arguments

• library::Val{:CairoMakie}: The plotting library to use.

• state::QuantumObject: The quantum state for which the Wigner function is calculated. It can be either a Ket, Bra, or Operator.

• xvec::AbstractVector: The x-coordinates of the phase space grid. Defaults to a linear range from -7.5 to 7.5 with 200 points.

• yvec::AbstractVector: The y-coordinates of the phase space grid. Defaults to a linear range from -7.5 to 7.5 with 200 points.

• g::Real: The scaling factor related to the value of $\hslash$ in the commutation relation $[x,y]=i\hslash$ via $\hslash =2/g^2$.

• method::WignerSolver: The method used to calculate the Wigner function. It can be either WignerLaguerre() or WignerClenshaw(), with WignerClenshaw() as default. The WignerLaguerre method has the optional parallel and tol parameters, with default values true and 1e-14, respectively.

• projection::Union{Val,Symbol}: Whether to plot the Wigner function in 2D or 3D. It can be either Val(:two_dim) or Val(:three_dim), with Val(:two_dim) as default.

• location::Union{GridPosition,Nothing}: The location of the plot in the layout. If nothing, the plot is created in a new figure. Default is nothing.

• colorbar::Bool: Whether to include a colorbar in the plot. Default is false.

• kwargs...: Additional keyword arguments to pass to the plotting function.

Returns

• fig: The figure object.

• ax: The axis object.

• hm: Either the heatmap or surface object, depending on the projection.

Import library first

CairoMakie must first be imported before using this function.

Beware of type-stability!

If you want to keep type stability, it is recommended to use Val(:two_dim) and Val(:three_dim) instead of :two_dim and :three_dim, respectively. Also, specify the library as Val(:CairoMakie) See this link and the related Section about type stability for more details.

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