Functions operating on Qobj

QuantumToolbox also provide functions (methods) that operates on QuantumObject.

You can click the function links and see the corresponding docstring for more information.

Linear algebra and attributes

Here is a table that summarizes all the supported linear algebra functions and attribute functions operating on a given QuantumObject Q:

Description	Function call	Synonyms
zero-like array	zero(Q)	qzero_like(Q)
identity-like matrix	one(Q)	qeye_like(Q)
conjugate	conj(Q)	-
transpose	transpose(Q)	trans(Q)
conjugate transposition	adjoint(Q)	Q', dag(Q)
partial transpose	partial_transpose(Q, mask)	-
dot product	dot(Q1, Q2)	-
generalized dot product	dot(Q1, Q2, Q3)	matrix_element(Q1, Q2, Q3)
trace	tr(Q)	-
partial trace	ptrace(Q, sel)	-
singular values	svdvals(Q)	-
standard vector p-norm or Schatten p-norm	norm(Q, p)	-
normalization	normalize(Q, p)	unit(Q, p)
normalization (in-place)	normalize!(Q, p)	-
matrix inverse	inv(Q)	-
matrix square root	sqrt(Q)	√(Q), sqrtm(Q)
matrix logarithm	log(Q)	logm(Q)
matrix exponential	exp(Q)	expm(Q)
matrix sine	sin(Q)	sinm(Q)
matrix cosine	cos(Q)	cosm(Q)
diagonal elements	diag(Q)	-
projector	proj(Q)	-
purity	purity(Q)	-
permute	permute(Q, order)	-
remove small elements	tidyup(Q, tol)	-
remove small elements (in-place)	tidyup!(Q, tol)	-
get data	get_data(Q)	-
get coherence	get_coherence(Q)	-

Eigen decomposition

LinearAlgebra.jl eigen solvers

• eigvals: return eigenenergies (eigenvalues)

• eigen: return EigsolveResult (contains eigenvalues and eigenvectors)

Only work for dense matrices

The eigen decomposition methods provided by LinearAlgebra.jl only works for dense matrices. If you pass-in a sparse matrix, the function automatically convert it to dense matrix.

Extra eigen solvers

• eigenenergies: return eigenenergies (eigenvalues)

• eigenstates: return EigsolveResult (contains eigenvalues and eigenvectors)

• eigsolve: using sparse eigen solver and return EigsolveResult (contains eigenvalues and eigenvectors)

• eigsolve_al: using the Arnoldi-Lindblad eigen solver and return EigsolveResult (contains eigenvalues and eigenvectors)

Examples

ψ = normalize(basis(4, 1) + basis(4, 2))

Quantum Object:   type=Ket()   dims=([4], [1])   size=(4,)
4-element Vector{ComplexF64}:
                0.0 + 0.0im
 0.7071067811865475 + 0.0im
 0.7071067811865475 + 0.0im
                0.0 + 0.0im

ψ'

Quantum Object:   type=Bra()   dims=([1], [4])   size=(1, 4)
1×4 adjoint(::Vector{ComplexF64}) with eltype ComplexF64:
 0.0-0.0im  0.707107-0.0im  0.707107-0.0im  0.0-0.0im

ρ = coherent_dm(5, 1)

Quantum Object:   type=Operator()   dims=([5], [5])   size=(5, 5)   ishermitian=true
5×5 Matrix{Float64}:
 0.367911  0.367744   0.261054   0.146207  0.088267
 0.367744  0.367577   0.260936   0.14614   0.0882269
 0.261054  0.260936   0.185233   0.103742  0.0626306
 0.146207  0.14614    0.103742   0.058102  0.035077
 0.088267  0.0882269  0.0626306  0.035077  0.0211765

diag(ρ)

5-element Vector{Float64}:
 0.3679111729923387
 0.3675770456232403
 0.18523331233838003
 0.05810197190350208
 0.021176497142538265

get_data(ρ)

5×5 Matrix{Float64}:
 0.367911  0.367744   0.261054   0.146207  0.088267
 0.367744  0.367577   0.260936   0.14614   0.0882269
 0.261054  0.260936   0.185233   0.103742  0.0626306
 0.146207  0.14614    0.103742   0.058102  0.035077
 0.088267  0.0882269  0.0626306  0.035077  0.0211765

norm(ρ)

0.9999999999999996

sqrtm(ρ)

Quantum Object:   type=Operator()   dims=([5], [5])   size=(5, 5)   ishermitian=true
5×5 Matrix{Float64}:
 0.367911  0.367744   0.261054   0.146207  0.088267
 0.367744  0.367577   0.260936   0.14614   0.0882269
 0.261054  0.260936   0.185233   0.103742  0.0626306
 0.146207  0.14614    0.103742   0.058102  0.035077
 0.088267  0.0882269  0.0626306  0.035077  0.0211765

tr(ρ)

0.9999999999999993

eigenenergies(ρ)

5-element Vector{Float64}:
 -3.188562758433179e-17
 -7.70881124130806e-18
  2.0684351320241445e-17
  1.0235138990670176e-16
  0.9999999999999996

result = eigenstates(ρ)

EigsolveResult:   type=Operator()   dims=([5], [5])
values:
5-element Vector{Float64}:
 -2.8177392874225097e-17
  3.416070845000482e-17
  7.728341127110703e-17
  8.881784197001252e-16
  0.9999999999999994
vectors:
5×5 Matrix{Float64}:
  0.349081   0.699533   0.113691  -0.0892167  -0.606557
 -0.698381  -0.132457  -0.345238  -0.0891762  -0.606281
  0.596321  -0.648729  -0.185233  -0.0633045  -0.430387
 -0.186568  -0.268811   0.913003  -0.0354544  -0.241044
  0.0        0.0        0.0        0.989355   -0.145521

λ, ψ = result
λ # eigenvalues

5-element Vector{Float64}:
 -2.8177392874225097e-17
  3.416070845000482e-17
  7.728341127110703e-17
  8.881784197001252e-16
  0.9999999999999994

ψ # eigenvectors

5-element Vector{QuantumObject{Ket, Dimensions{Space, Space}, Vector{Float64}}}:
 
Quantum Object:   type=Ket()   dims=([5], [1])   size=(5,)
5-element Vector{Float64}:
  0.34908090726365426
 -0.6983812275352181
  0.5963209517322131
 -0.18656769210014237
  0.0
 
Quantum Object:   type=Ket()   dims=([5], [1])   size=(5,)
5-element Vector{Float64}:
  0.6995328227558701
 -0.1324572834510804
 -0.6487290623177562
 -0.2688112751584079
  0.0
 
Quantum Object:   type=Ket()   dims=([5], [1])   size=(5,)
5-element Vector{Float64}:
  0.11369058957774669
 -0.34523801249281383
 -0.18523271117272722
  0.9130027422100534
  0.0
 
Quantum Object:   type=Ket()   dims=([5], [1])   size=(5,)
5-element Vector{Float64}:
 -0.08921674125585327
 -0.0891762198903574
 -0.06330447537681073
 -0.035454413343867425
  0.9893550944213416
 
Quantum Object:   type=Ket()   dims=([5], [1])   size=(5,)
5-element Vector{Float64}:
 -0.6065568176126116
 -0.606281325477901
 -0.43038739797812414
 -0.24104350624628368
 -0.14552146626026788

λ, ψ, T = result
T # transformation matrix

5×5 Matrix{Float64}:
  0.349081   0.699533   0.113691  -0.0892167  -0.606557
 -0.698381  -0.132457  -0.345238  -0.0891762  -0.606281
  0.596321  -0.648729  -0.185233  -0.0633045  -0.430387
 -0.186568  -0.268811   0.913003  -0.0354544  -0.241044
  0.0        0.0        0.0        0.989355   -0.145521