Unambiguous state discrimination

[atg205]

Sometimes Hypatia is not able to solve the SDP (even though it should be). In that case we return the failure state and a warning message is shown .

[araujoms]

This is not Hypatia's fault; this SDP is not strictly feasible, so you'll always run into numerical problems. I'll have to think about whether it's possible to reformulate it in a strictly feasible way.

Comments about the code:

1. Don't do something arbitrary in case of error, no trying the dual or returning some specific error thing. Just throw an exception with the code JuMP.is_solved_and_feasible(model) || throw(error(JuMP.raw_status(model))), like we do in other SDPs.
2. I see lots of concrete types: Float64, Float32, ComplexF64, etc. Don't use them, the code must be generically typed. Use T or _solver_type(T).
3. tr(A' * B) is a rather inefficient representation for the Hilbert-Schmidt inner product, because it includes an unnecessary matrix multiplication. Use instead dot(A, B).
4. Don't create a default empty q just to overwrite it later, you can create the correct q already in the default argument. And please create it with the correct type.
5. Don't use the constraint sum(E) == I. Instead create only the first N-1 POVM elements as variables and define the last as identity minus their sum, like I did in my state discrimination tutorial.
6. The expression "positive operator-valued measure" is only of historical interest. Never use it. The name of the thing is POVM.
7. The name of the function should be descriptive, something like state_discrimination_unambiguous.

[araujoms]

It's possible to make it strictly feasible (and thus reliable). It's a bit of work, though. I don't know how much time you want to spend on this.

First consider the case where the input is a set of pure states $\{|\psi_i\rangle\}_{i=0}^{N-1}$. If they are linearly dependent, throw an exception. Else, let $L$ be the vector space spanned by them. For every $k$ there exists a unique vector $|\phi_k\rangle \in L$ such that $\langle \phi_k | \psi_i \rangle = 0$ for all $i \neq k$. We can take w.l.o.g. $E_k = \alpha_k |\phi_k\rangle \langle \phi_k|$. Then the SDP is strictly feasible, and we only have $N$ real variables $\alpha_k$ to worry about.

Now consider the case where the input is a set of (possibly) mixed states $\{\rho_i\}_{i=0}^{N-1}$. You need $\mathrm{tr}(E_k \rho_i) = 0$ for all $i \neq k$. Let then $L_k$ be the vector space spanned be the support of these states. $E_k$ must then be supported on its orthogonal complement $L_k^\perp$. Let then $\{|\phi^i_k \rangle\} _{i=0}^{d_k-1}$ be a basis for $L_k^\perp$, and define the isometry $V_k = \sum _{i=0}^{d_k-1} |\phi^i_k \rangle \langle i|$. Then $E_k = V_k e_k V_k^\dagger$ for some $d_k \times d_k$ positive semidefinite matrix $e_k$. If you then declare the $e_k$ as variables and write the program in terms of them the SDP will be strictly feasible and will run reliably.

[atg205]

Thank you very much for the suggestion, I will see if I manage to implement it next week. I will let you know.

[atg205]

I still have some problems understanding why it is unfeasible. Isn't it that in the case where it is impossible to distinguish unambiguously between the two states, the failure state M_n is returned (=I)?

[araujoms]

It is feasible. It is not strictly feasible, which means it can be solved, but not reliably. Strict feasibility means that there must exist a point which is in the interior of the cones and respects all the equality constraints. In the case of this SDP, it means that there must exist full rank $E_i$ that respect the constraints $\mathrm{tr}(E_i \rho_j) = 0$. But this is not possible, because these constraints imply that for all $|\psi\rangle$ in the support of $\rho_j$ we must have $E_i |\psi\rangle = 0$, that is, $|\psi\rangle$ is a null eigenvector of $E_i$, and thus $E_i$ can't be full rank.

See https://arxiv.org/abs/2302.03529 and https://arxiv.org/abs/1706.03705