I'm interested in operational ways of demonstrating (with high probability of confidence, in an error-free setting) that a POVM operator on n-qubit states is a projector. Specifically, I'm interested in ways in which one can do so using product states. (I'm more interested in describing it as an analytic characterization of projectors, but I find it easier to describe roughly what sort of characterization I'm looking for in terms of a protocol.) I'm not interested in poly-time approaches per se; just ones which are much better than process tomography.


One way to characterize a projector is process tomography. Given a POVM P, represented as a CPTP map of the form $$ P(\rho) \;=\; \sum_{j = 1}^m \;\;\mathrm{Tr}(\rho E_j) \;\;|j\rangle\!\langle j|\;, $$ one may determine whether any particular Er is a projector by collecting statistics on measurement results, e.g. on all n-fold tensor products of the eigenstates of the Pauli X, Y, and Z operators, using these to compute matrix coefficients for Er, and then determine whether that matrix is idempotent. If we are given a promise that Er is either a projector or bounded away from being a projector in some suitable norm, the probability of correctly deducing whether it is a projector then amounts to the confidence in the estimates of the matrix coefficients.

However, if you know in advance that Er is a projector, and you would like to convince someone else that it is, you can certify this somewhat more efficiently if you are able to prepare a complete basis of eigenstates of Er. The outcome Er will never arise for 0-eigenstates, and always will for +1-eigenstates. These eigenstates aren't necessarily product states, of course, but having them as a resource makes it in principle less laborious to give evidence that Er is a projector: it suffices to collect enough statistics to show that it is quite likely that any one eigenstate is indeed an eigenstate (made easier by the extremality of the measurement probabilities). There are only 2n such eigenstates, and the confidence in your estimates converges (and quickly) despite "sampling" fewer states, because of the extremality of the eigenvalues.


Is there a characterization of projectors, which one may (statistically, as with the eigenstates) verify more succinctly than tomography — and which can be performed with only (input-dependent) product states? If this cannot be done for all projectors in a way which is clearly better than tomography, what are the classes of operators (apart from the class of product projectors) for which there are better approaches?

For instance, this is easy for rank-1 projectors in the case n = 2 (even if we consider qudits of arbitrary dimension rather than qubits), just by using the Schmidt decomposition; we can find a (not necessarily orthogonal) basis for the kernel, and then complete this basis to allow us to estimate the trace of Er and show that its other eigenvalue must be close to 1. But it's not clear how to extend even this simple case to n = 3.


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