Journal Article

DISCRIMINATING QUANTUM STATES: THE MULTIPLE CHERNOFF DISTANCE

Li, Ke

2016

Overview

We consider the problem of testing multiple quantum hypotheses $\\left\\{ {\\rho _1^{ \\otimes n},...,\\rho _r^{ \\otimes n}} \\right\\}$, 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 Pe decays exponentially to zero, that is, Pe = exp{–ξn + 0(n)}. However, this error exponent ξ 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 Szkola's conjecture that ξ = mini≠j C(ρi, ρj). The right-hand side of this equality is called the multiple quantum Chernoff distance, and $C\\left( {{\\rho _i},{\\rho _j}} \\right): = \\max {}_{0 \\leqslant s \\leqslant 1}\\left\\{ { - \\log Tr\\rho _i^s\\rho _j^{1 - s}} \\right\\}$ 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 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.

• Publisher: Institute of Mathematical Statistics
• Subject: Codes / Eigenvalues / Error analysis / Error rates / Hilbert spaces / Hypotheses / Hypothesis testing / Information theory / Mathematical lattices / Mathematical problems / Mathematical theorems / Matrices / Probability distribution / Proof theory / Quantum physics / Quantum states / Statistical analysis / Studies / Unit vectors