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)	-
identity-like matrix	one(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)	-

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{KetQuantumObject, Dimensions{1, Tuple{Space}}, Vector{ComplexF64}}}:
 
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