Discriminating quantum states: the multiple Chernoff distance. (English) Zbl 1397.62611

Summary: We consider the problem of testing multiple quantum hypotheses \(\{\rho_{1}^{\otimes n},\ldots,\rho_{r}^{\otimes n}\}\), where an arbitrary prior distribution is given and each of the \(r\) hypotheses is \(n\) copies of a quantum state. It is known that the minimal average error probability \(P_{e}\) decays exponentially to zero, that is, \(P_{e}=\exp\{-\xi n+o(n)\}\). However, this error exponent \(\xi\) is generally unknown, except for the case that \(r=2\).
In this paper, we solve the long-standing open problem of identifying the above error exponent, by proving Nussbaum and Szkoła’s conjecture that \(\xi=\min_{i\neq j}C(\rho_{i},\rho_{j})\). The right-hand side of this equality is called the multiple quantum Chernoff distance, and \(C(\rho_{i},\rho_{j}):=\max_{0\leq s\leq1}\{-\log\operatorname{Tr}\rho_{i}^{s}\rho_{j}^{1-s}\}\) has been previously identified as the optimal error exponent for testing two hypotheses, \(\rho_{i}^{\otimes n}\) versus \(\rho_{j}^{\otimes n}\).
The main ingredient of our proof is a new upper bound for the average error probability, for testing an ensemble of finite-dimensional, but otherwise general, quantum states. This upper bound, up to a states-dependent factor, matches the multiple-state generalization of M. Nussbaum and A. Szkoła’s lower bound [ibid. 37, No. 2, 1040–1057 (2009; Zbl 1162.62100)]. Specialized to the case \(r=2\), we give an alternative proof to the achievability of the binary-hypothesis Chernoff distance, which was originally proved by K. M. R. Audenaert and M. Mosonyi [J. Math. Phys. 55, No. 10, 102201, 39 p. (2014; Zbl 1309.81042)].


62P35 Applications of statistics to physics
62G10 Nonparametric hypothesis testing
81P45 Quantum information, communication, networks (quantum-theoretic aspects)
Full Text: DOI arXiv Euclid


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