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QuantumStateDistributions

Documentation for QuantumStateDistributions.

Installation

The package can be installed with the Julia package manager. From the Julia REPL, type ] to enter the Pkg REPL mode and run:

pkg> add QuantumStateDistributions

Quick start

Construct a squeezed state via QuantumStateBase.jl and declare its quantum state distribution.

julia> using QuantumStateDistributions, QuantumStateBase

julia> r, θ, dim = 0.8, π/2, 100;

julia> ρ = SqueezedState(r, θ, Matrix, dim=dim)
100×100 Matrix{ComplexF64}:
       0.7477+0.0im          0.0+0.0im  …  -1.77607e-24-5.91946e-10im  0.0+0.0im
          0.0+0.0im          0.0+0.0im              0.0+0.0im          0.0+0.0im
 -2.14974e-17+0.351079im     0.0+0.0im      2.77945e-10-8.16923e-25im  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          0.0+0.0im
 -1.77607e-24+5.91946e-10im  0.0+0.0im      4.68637e-19+0.0im          0.0+0.0im
          0.0+0.0im          0.0+0.0im              0.0+0.0im          0.0+0.0im

julia> d = GaussianStateBHD(ρ);

Sample a point from the quantum state distribution in intensity-to-measurement-phase quadrature coordinate measured by balanced homodyne detection:

julia> rand(d)
2-element Vector{Float64}:
 0.8420476666965236
 1.6008878775912423

Sample n points from the quantum state distribution:

julia> rand(d, 4096)
2×4096 Matrix{Float64}:
  0.0018714   0.0034182   0.00403972  0.00780472  …   6.27393  6.27811    6.27884
 -0.706334   -1.16179    -0.195581    0.174201       -0.60763  0.853457  -0.217017

Estemate density matrix of a quantum state via MLE

Generate some data from a squeezed state:

julia> data = rand(d, 81920);

Construct a PositivePreservingIterator:

julia> t = 50;

julia> ppit = PositivePreservingIterator(data, t, dim=35);

Estermate density matrix:

julia> run!(ppit)

julia> ρ_mle = ppit.ρ

Index

APIs

QuantumStateDistributions.GaussianStateBHD — Type
GaussianStateBHD

A Gaussian state distribution in intensity-to-measurement-phase quadrature coordinate measured by balanced homodyne detection.

Example

julia> using QuantumStateBase

julia> d = GaussianStateBHD(SqueezedState(0.8, π, Matrix, dim=100));

julia> points = rand(d, 4096);
source
QuantumStateDistributions.PositivePreservingIterator — Method
PositivePreservingIterator(data::Matrix, steps::Integer; dim::Integer)

The maximum likelihood estimator with the magical positive preserving iterator derived by variational calculus.

Example

julia> using QuantumStateBase

julia> ρ = SqueezedState(0.8, π/4, Matrix, dim=100);

julia> data = rand(GaussianStateBHD(ρ), 8192);

julia> ρ_mle = run!(PositivePreservingIterator(data, 50, dim=35));
source
QuantumStateDistributions.next! — Method
next!(ppit::PositivePreservingIterator)

Iterate one step with the magic positive preserving iterator.

$\rho^{t+1} = \mathcal{R} \rho^t \mathcal{R}$

source
QuantumStateDistributions.qpdf — Method
qpdf([T=Float64], d::GaussianStateBHD, θ::Real, x::Real)
qpdf([T=Float64], d::GaussianStateBHD, θs::AbstractRange, xs::AbstractRange)
qpdf([T=Float64], d::QuantumStateBHD, θ::Real, x::Real)
qpdf([T=Float64], d::QuantumStateBHD, θs::AbstractRange, xs::AbstractRange)
qpdf([T=Float64], ρ::AbstractArray, θ::Real, x::Real)
qpdf([T=Float64], ρ::AbstractArray, θs::AbstractRange, xs::AbstractRange)

Quadrature prabability in intensity-to-measurement-phase quadrature coordinate.

Arguments

  • T: Data type, default as Float64.
  • state: State can be GaussianStateBHD distribution, QuantumStateBHD distribution or density matrix.
  • θ: Measurement phase, can be Real or AbstractRange.
  • x: Intensity in quadrature coordinate, can be Real or AbstractRange.

$p(\rho, \theta, x) = tr(\hat{\Pi}(\theta, x) \rho)$

source
QuantumStateDistributions.𝛑̂ — Method
𝛑̂(θ::Real, x::Real; dim::Integer)

BHD measurement operator.

$\hat{\Pi}_{m, n}(\theta, x) = \langle m | \hat{\Pi}(\theta, x) | n \rangle = \langle m | \theta, x \rangle \langle \theta, x | n \rangle$

source