This is a package that provides basic routines to perform quantum tomography of finite dimensional systems.
Given the estimated means and variances of the observations, tomography proceeds by finding the state, process, or measurements that optimize some figure of merit -- either the likelihood or the χ² statistic -- often with the additional constraint the reconstruction be physical (e.g., correspond to a valid quantum mechanical object).
Namely, states are constrained to be given by positive semidefinite matrices with unit trace, processes are constrained to be given by the Choi matrix of a completely positive, trace preserving map, and measurements are constrained to be given by POVM elements.
Using the Julia package manager:
(v1.0) pkg> add QuantumTomography
Each different tomography method is associated with a type. Objects of
this type must be instantiated before reconstruction can be performed
with a call to fit(). These objects are also needed to make
predictions about tomography experiments with predict(). Currently available
tomography methods are
-
FreeLSStateTomo: Unconstrained least-squares state tomography. -
LSStateTomo: Least-squares state tomography constrained to yield physical states. -
MLStateTomo: Maximum-likelihood state tomography (including the option for entropy maximization or hedging).
In order to perform quantum state tomography, we need an informationally complete set of measurement effects. In the case of a single qubit, that can be given by the eigenstates of the 3 Pauli operators.
julia> using Cliffords, QuantumTomography
julia> import QuantumInfo.eye
julia> obs = Matrix[ (complex(Pauli(i))+eye(2))/2 for i in 1:3 ];
julia> append!(obs, Matrix[ (-complex(Pauli(i))+eye(2))/2 for i in 1:3 ]);
julia> tomo = LSStateTomo(obs);
We choose some random pure state to generate the fictitious experiment
julia> using RandomQuantum, QuantumInfo
julia> ψ = rand(FubiniStudyPureState(2));
julia> normalize!(ψ)
2-element Array{Complex{Float64},1}:
0.264298+0.850605im
0.449897-0.0648884im
julia> ρ = projector(ψ)
2x2 Array{Complex{Float64},2}:
0.793382+0.0im 0.0637123+0.399834im
0.0637123-0.399834im 0.206618+0.0im
Predict the expectation values of the observations for some hypothesized ρ
ideal_means = predict(tomo, ρ) |> real
With these in hand, we can finally reconstruct ρ from the observed expectation values and variances.
julia> fit(tomo, ideal_means, ones(6))
(
2x2 Array{Complex{Float64},2}:
0.793382-9.39512e-25im 0.0637123+0.399834im
0.0637123-0.399834im 0.206618+7.98287e-25im ,
3.730685819507896e-11,:Optimal)
Using the data generated above, we can instead choose to reconstruct the state by maximizing the likelihood function for some set of binomial observations.
julia> using Distributions
julia> ml_tomo = MLStateTomo(obs)
julia> freqs = Float64[rand(Binomial(10_000, μ))/10_000 for μ in ideal_means[1:3]]
julia> append!(freqs,1-freqs)
julia> fit(ml_tomo, freqs)
(
2x2 Array{Complex{Float64},2}:
0.789799-7.92259e-17im 0.0641298+0.401799im
0.0641298-0.401799im 0.210201-4.28412e-17im ,
-1.5215022154657007,:Optimal)
If the observations are incomplete (in the sense that they do not uniquely specify the quantum state), one can still perform reconstruction by maximizing a mixture of the likelihood and the entropy of the resulting state (see PRL 107 020404 2011). In this package, this would correspond to
julia> fit(ml_tomo, freqs, λ=1e-3)
(
2x2 Array{Complex{Float64},2}:
0.789155-2.68147e-17im 0.0639152+0.401322im
0.0639152-0.401322im 0.210845-9.18039e-18im ,
-1.5215005466837999,:Optimal)
Constrained maximum-likelihood also suffers from biasing towards low
rank states. This can be avoided by hedging (see Blume-Kohout, PRL
105, 200504 2010), which essentially follows a modification of
Laplace's rule to penalize low rank estimates. Hedging can be enabled
by using the experimental fitting routine fitA with MLStateTomo:
julia> QuantumTomography.fitA(ml_tomo, freqs)
(
2x2 Array{Complex{Float64},2}:
0.789155-2.68147e-17im 0.0639152+0.401322im
0.0639152-0.401322im 0.210845-9.18039e-18im ,
-1.5215005466837999,:Optimal)
- Implement least-squares and ML process tomography
- Implement compressed sensing state and process tomography
Raytheon BBN Technologies.
Apache Lincense 2.0 (summary)
Marcus Silva (@marcusps on GitHub)