By Ke Li
Suppose we are given n copies of one of the quantum states {rho_1,. . . , rho_r}, with an arbitrary prior distribution that is independent of n. The multiple hypothesis Chernoff bound problem concerns the minimal average error probability P_e in detecting the true state. It is known that P_e=exp{-En+o(n)} decays exponentially to zero. However, this error exponent E is generally unknown, except for the case r=2. In this talk, I will give a solution to the long-standing open problem of identifying the above error exponent, by proving Nussbaum and Szkola's conjecture that E=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 = s = 1} {-log Tr (rho_i^s rho_j^(1-s))} has been previously identified as the optimal error exponent for testing two hypotheses, rho_i versus rho_j. 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 Nussbaum and Szkola's lower bound. 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 Audenaert et al.
By Anthony Leverrier
By Zbigniew Puchała
By Benoit Collins
By Satya Majumdar
By Motohisa Fukuda
By Anand Natarajan
By Nilanjana Datta
By Daniel Stilck Franca
By Ke Li
By Radosław Adamczak
By Madalin Guta
By Antonio Acín
By Ashley Montanaro
By Ping Zhong
By James Mingo
By Camille Male
By Clément Delcamp
By Claudio Dappiaggi
By Alexander Strohmaier
