Add quantum state discrimination tutorial

[odow]

x-ref #2348 (comment)

Also fixes #3245 (comment)

Still a WIP, because I don't really know enough physics to write about the problem we're solving, or what the solution represents.

Preview: https://jump.dev/JuMP.jl/previews/PR3250/tutorials/conic/quantum_discrimination/

[codecov]

Codecov Report

Patch and project coverage have no change.

Comparison is base (6329762) 98.09% compared to head (b2dd210) 98.09%.

Additional details and impacted files

@@           Coverage Diff           @@
##           master    #3250   +/-   ##
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  Coverage   98.09%   98.09%           
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  Files          34       34           
  Lines        4663     4664    +1     
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+ Hits         4574     4575    +1     
  Misses         89       89           

Impacted Files	Coverage Δ
src/macros.jl	98.57% <0.00%> (+<0.01%)	⬆️

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[jd-foster]

A POVM is the set of PSD matrices over the states 1:N, not the individual matrices.

[jd-foster]

This all looks great to me. My only last question is whether you wanted to seed the random generation?

[odow]

My only last question is whether you wanted to seed the random generation?

Nah. It'll be okay.

[araujoms]

The task is the following: you are given an unknown quantum state, promised to come from a known set of n states (sampled with uniform probability, for simplicity). You need to find out which quantum state it is. The objective of the SDP is the probability of success, \sum_i p( guess i| state i)/n. The matrices E[i] are the POVM elements, encoding the optimal measurement you can do.

[araujoms]

More specific comments: one is interested in the value of the objective, not on the POVM elements, so I'd print the objective instead. Also, in the problem formulation I'd write the objective as \sum_i \operatorname{tr}(\rho_i E_i)/N instead of \mathbb{E}_i[ \operatorname{tr}(\rho_i E_i)].

Also worth remarking is that for N=2 the problem is trivial, with solution given by 0.5 + 0.25*sum(svdvals(rho[1]-rho[2])).

[ericphanson]

In case it’s helpful, I have an example written with Convex.jl here, where I compare to the analytic solution for a simple example

[odow]

Updated with @araujoms suggestions. I don't know if we need to go into detail about the theory behind the model. People interested in HermitianPSDCone will probably already have the necessary background.

[odow]

Merging since we seem to have converged on this. Any other changes can happen in a new PR.