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The problem of discriminating between many quantum channels with certainty is analyzed under the assumption of prior knowledge of algebraic relations among possible channels. It is shown, by explicit construction of a novel family of quantum algorithms, that when the set of possible channels faithfully represents a finite subgroup of SU(2) (e.g., $C_n, D_{2n}, A_4, S_4, A_5$) the recently-developed techniques of quantum signal processing can be modified to constitute subroutines for quantum hypothesis testing. These algorithms, for group quantum hypothesis testing (G-QHT), intuitively encode discrete properties of the channel set in SU(2) and improve query complexity at least quadratically in $n$, the size of the channel set and group, compared to naive repetition of binary hypothesis testing. Intriguingly, performance is completely defined by explicit group homomorphisms; these in turn inform simple constraints on polynomials embedded in unitary matrices. These constructions demonstrate a flexible technique for mapping questions in quantum inference to the well-understood subfields of functional approximation and discrete algebra. Extensions to larger groups and noisy settings are discussed, as well as paths by which improved protocols for quantum hypothesis testing against structured channel sets have application in the transmission of reference frames, proofs of security in quantum cryptography, and algorithms for property testing.
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For any pair of quantum states (the hypotheses), the task of binary quantum hypotheses testing is to derive the tradeoff relation between the probability $p_{01}$ of rejecting the null hypothesis and $p_{10}$ of accepting the alternative hypothesis. The case when both hypotheses are explicitly given was solved in the pioneering work by Helstrom. Here, instead, for any given null hypothesis as a pure state, we consider the worst-case alternative hypothesis that maximizes $p_{10}$ under a constraint on the distinguishability of such hypotheses. Additionally, we restrict the optimization to separable measurements, in order to describe tests that are performed locally. The case $p_{01}=0$ has been recently studied under the name of quantum state verification. We show that the problem can be cast as a semi-definite program (SDP). Then we study in detail the two-qubit case. A comprehensive study in parameter space is done by solving the SDP numerically. We also obtain analytical solutions in the case of commuting hypotheses, and in the case where the two hypotheses can be orthogonal (in the latter case, we prove that the restriction to separable measurements generically prevents perfect distinguishability). In regards to quantum state verification, our work shows the existence of more efficient strategies for noisy measurement scenarios.
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We consider sequential hypothesis testing between two quantum states using adaptive and non-adaptive strategies. In this setting, samples of an unknown state are requested sequentially and a decision to either continue or to accept one of the two hypotheses is made after each test. Under the constraint that the number of samples is bounded, either in expectation or with high probability, we exhibit adaptive strategies that minimize both types of misidentification errors. Namely, we show that these errors decrease exponentially (in the stopping time) with decay rates given by the measured relative entropies between the two states. Moreover, if we allow joint measurements on multiple samples, the rates are increased to the respective quantum relative entropies. We also fully characterize the achievable error exponents for non-adaptive strategies and provide numerical evidence showing that adaptive measurements are necessary to achieve our bounds under some additional assumptions.
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