Usefulness of adaptive strategies in asymptotic quantum channel discrimination

Farzin Salek, Masahito Hayashi, and Andreas Winter
Phys. Rev. A 105, 022419 – Published 14 February 2022

Abstract

Adaptiveness is a key principle in information processing including statistics and machine learning. We investigate the usefulness adaptive methods in the framework of asymptotic binary hypothesis testing, when each hypothesis represents asymptotically many independent instances of a quantum channel, and the tests are based on using the unknown channel and observing outputs. Unlike the familiar setting of quantum states as hypotheses, there is a fundamental distinction between adaptive and nonadaptive strategies with respect to the channel uses, and we introduce a number of further variants of the discrimination tasks by imposing different restrictions on the test strategies. The following results are obtained: (1) We prove that for classical-quantum channels, adaptive and nonadaptive strategies lead to the same error exponents both in the symmetric (Chernoff) and asymmetric (Hoeffding, Stein) settings. (2) The first separation between adaptive and nonadaptive symmetric hypothesis testing exponents for quantum channels, which we derive from a general lower bound on the error probability for nonadaptive strategies; the concrete example we analyze is a pair of entanglement-breaking channels. (3) We prove, in some sense generalizing the previous statement, that for general channels adaptive strategies restricted to classical feed-forward and product state channel inputs are not superior in the asymptotic limit to nonadaptive product state strategies. (4) As an application of our findings, we address the discrimination power of an arbitrary quantum channel and show that adaptive strategies with classical feedback and no quantum memory at the input do not increase the discrimination power of the channel beyond nonadaptive tensor product input strategies.

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  • Received 10 June 2021
  • Revised 30 November 2021
  • Accepted 25 January 2022

DOI:https://doi.org/10.1103/PhysRevA.105.022419

©2022 American Physical Society

Physics Subject Headings (PhySH)

Quantum Information, Science & Technology

Authors & Affiliations

Farzin Salek1,*, Masahito Hayashi2,3,4,†, and Andreas Winter5,6,‡

  • 1Zentrum Mathematik, Technische Universität München, 85748 München, Germany
  • 2Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Nanshan District, Shenzhen, 518055, China
  • 3International Quantum Academy (SIQA), Futian District, Shenzhen 518048, China
  • 4Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan
  • 5Institució Catalana de Recerca i Estudis Avançats (ICREA), Pg. Lluis Companys, 23, 08010 Barcelona, Spain
  • 6Grup d'Informació Quàntica, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain

  • *farzin.salek@gmail.com
  • hayashi@sustech.edu.cn
  • andreas.winter@uab.cat

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Issue

Vol. 105, Iss. 2 — February 2022

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