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
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)`	-

Eigenvalue decomposition

eigenenergies: return eigenenergies (eigenvalues)
eigenstates: return EigsolveResult (contains eigenvalues and eigenvectors)
eigvals: return eigenvalues
eigen: using dense eigen solver and 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]   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=[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]   size=(5, 5)   ishermitian=true
5×5 Matrix{ComplexF64}:
 0.367911+0.0im   0.367744+0.0im  …  0.146207+0.0im   0.088267+0.0im
 0.367744+0.0im   0.367577+0.0im      0.14614+0.0im  0.0882269+0.0im
 0.261054+0.0im   0.260936+0.0im     0.103742+0.0im  0.0626306+0.0im
 0.146207+0.0im    0.14614+0.0im     0.058102+0.0im   0.035077+0.0im
 0.088267+0.0im  0.0882269+0.0im     0.035077+0.0im  0.0211765+0.0im

diag(ρ)

5-element Vector{ComplexF64}:
   0.3679111729923387 + 0.0im
   0.3675770456232403 + 0.0im
  0.18523331233838003 + 0.0im
  0.05810197190350208 + 0.0im
 0.021176497142538265 + 0.0im

get_data(ρ)

5×5 Matrix{ComplexF64}:
 0.367911+0.0im   0.367744+0.0im  …  0.146207+0.0im   0.088267+0.0im
 0.367744+0.0im   0.367577+0.0im      0.14614+0.0im  0.0882269+0.0im
 0.261054+0.0im   0.260936+0.0im     0.103742+0.0im  0.0626306+0.0im
 0.146207+0.0im    0.14614+0.0im     0.058102+0.0im   0.035077+0.0im
 0.088267+0.0im  0.0882269+0.0im     0.035077+0.0im  0.0211765+0.0im

norm(ρ)

0.9999999999999996

sqrtm(ρ)

Quantum Object:   type=Operator   dims=[5]   size=(5, 5)   ishermitian=true
5×5 Matrix{ComplexF64}:
 0.367911+0.0im   0.367744+0.0im  …  0.146207+0.0im   0.088267+0.0im
 0.367744-0.0im   0.367577+0.0im      0.14614+0.0im  0.0882269+0.0im
 0.261054-0.0im   0.260936-0.0im     0.103742+0.0im  0.0626306+0.0im
 0.146207-0.0im    0.14614-0.0im     0.058102+0.0im   0.035077+0.0im
 0.088267-0.0im  0.0882269-0.0im     0.035077-0.0im  0.0211765+0.0im

tr(ρ)

0.9999999999999993 + 0.0im

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]
values:
5-element Vector{ComplexF64}:
 -2.8177392874225097e-17 + 0.0im
   3.416070845000482e-17 + 0.0im
   7.728341127110703e-17 + 0.0im
   8.881784197001252e-16 + 0.0im
      0.9999999999999994 + 0.0im
vectors:
5×5 Matrix{ComplexF64}:
  0.349081+0.0im   0.699533+0.0im  …  -0.0892167+0.0im  -0.606557+0.0im
 -0.698381+0.0im  -0.132457+0.0im     -0.0891762+0.0im  -0.606281+0.0im
  0.596321+0.0im  -0.648729+0.0im     -0.0633045+0.0im  -0.430387+0.0im
 -0.186568+0.0im  -0.268811+0.0im     -0.0354544+0.0im  -0.241044+0.0im
       0.0+0.0im        0.0+0.0im       0.989355+0.0im  -0.145521-0.0im

λ, ψ = result
λ # eigenvalues

5-element Vector{ComplexF64}:
 -2.8177392874225097e-17 + 0.0im
   3.416070845000482e-17 + 0.0im
   7.728341127110703e-17 + 0.0im
   8.881784197001252e-16 + 0.0im
      0.9999999999999994 + 0.0im

ψ # eigenvectors

5-element Vector{QuantumObject{Vector{ComplexF64}, KetQuantumObject, 1}}:
 Quantum Object:   type=Ket   dims=[5]   size=(5,)
5-element Vector{ComplexF64}:
  0.34908090726365426 + 0.0im
  -0.6983812275352181 + 0.0im
   0.5963209517322131 + 0.0im
 -0.18656769210014237 + 0.0im
                  0.0 + 0.0im
 Quantum Object:   type=Ket   dims=[5]   size=(5,)
5-element Vector{ComplexF64}:
  0.6995328227558701 + 0.0im
 -0.1324572834510804 + 0.0im
 -0.6487290623177562 + 0.0im
 -0.2688112751584079 + 0.0im
                 0.0 + 0.0im
 Quantum Object:   type=Ket   dims=[5]   size=(5,)
5-element Vector{ComplexF64}:
  0.11369058957774669 + 0.0im
 -0.34523801249281383 + 0.0im
 -0.18523271117272722 + 0.0im
   0.9130027422100534 + 0.0im
                  0.0 + 0.0im
 Quantum Object:   type=Ket   dims=[5]   size=(5,)
5-element Vector{ComplexF64}:
  -0.08921674125585327 + 0.0im
   -0.0891762198903574 + 0.0im
  -0.06330447537681073 + 0.0im
 -0.035454413343867425 + 0.0im
    0.9893550944213416 + 0.0im
 Quantum Object:   type=Ket   dims=[5]   size=(5,)
5-element Vector{ComplexF64}:
  -0.6065568176126116 + 0.0im
   -0.606281325477901 + 0.0im
 -0.43038739797812414 + 0.0im
 -0.24104350624628368 + 0.0im
 -0.14552146626026788 - 0.0im

λ, ψ, T = result
T # transformation matrix

5×5 Matrix{ComplexF64}:
  0.349081+0.0im   0.699533+0.0im  …  -0.0892167+0.0im  -0.606557+0.0im
 -0.698381+0.0im  -0.132457+0.0im     -0.0891762+0.0im  -0.606281+0.0im
  0.596321+0.0im  -0.648729+0.0im     -0.0633045+0.0im  -0.430387+0.0im
 -0.186568+0.0im  -0.268811+0.0im     -0.0354544+0.0im  -0.241044+0.0im
       0.0+0.0im        0.0+0.0im       0.989355+0.0im  -0.145521-0.0im