QuantumTomography.jl

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.

Installation

Using the Julia package manager:

(v1.0) pkg> add QuantumTomography

API

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).

Examples

Constrained least-squares tomography

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)

Constrained maximum-likelihood tomography (with maximum entropy and hedging options)

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)

TODO

• Implement least-squares and ML process tomography
• Implement compressed sensing state and process tomography

Copyright

Raytheon BBN Technologies.

License

Apache Lincense 2.0 (summary)

Authors

Marcus Silva (@marcusps on GitHub)