Brandão, Fernando G. S. L. and Harrow, Aram W. and Lee, James R. and Peres, Yuval
(2020)
*Adversarial Hypothesis Testing and a Quantum Stein’s Lemma for Restricted Measurements.*
IEEE Transactions on Information Theory, 66
(8).
pp. 5037-5054.
ISSN 0018-9448.
https://resolver.caltech.edu/CaltechAUTHORS:20200730-074100066

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## Abstract

Recall the classical hypothesis testing setting with two sets of probability distributions P and Q. One receives either n i.i.d. samples from a distribution p ∈ P or from a distribution q ∈ Q and wants to decide from which set the points were sampled. It is known that the optimal exponential rate at which errors decrease can be achieved by a simple maximum-likelihood ratio test which does not depend on p or q, but only on the sets P and Q . We consider an adaptive generalization of this model where the choice of p ∈ P and q ∈ Q can change in each sample in some way that depends arbitrarily on the previous samples. In other words, in the kth round, an adversary, having observed all the previous samples in rounds 1,…,k−1 , chooses p_k ∈ P and q_k ∈ Q , with the goal of confusing the hypothesis test. We prove that even in this case, the optimal exponential error rate can be achieved by a simple maximum-likelihood test that depends only on P and Q. We then show that the adversarial model has applications in hypothesis testing for quantum states using restricted measurements. For example, it can be used to study the problem of distinguishing entangled states from the set of all separable states using only measurements that can be implemented with local operations and classical communication (LOCC). The basic idea is that in our setup, the deleterious effects of entanglement can be simulated by an adaptive classical adversary. We prove a quantum Stein’s Lemma in this setting: In many circumstances, the optimal hypothesis testing rate is equal to an appropriate notion of quantum relative entropy between two states. In particular, our arguments yield an alternate proof of Li and Winter’s recent strengthening of strong subadditivity for von Neumann entropy.

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Additional Information: | © 2020 IEEE. Manuscript received March 11, 2019; revised December 22, 2019; accepted January 20, 2020. Date of publication March 10, 2020; date of current version July 14, 2020. The work of Fernando G. S. L. Brandão was supported by the Engineering and Physical Sciences Research Council (EPSRC). The work of Aram W. Harrow was supported in part by the NSF under Grant CCF-1111382, Grant CCF-1452616, Grant CCF-1729369, and Grant PHY-1818914 and in part by the Army Research Office (ARO) under Contract W911NF-12-1-0486. The work of James R. Lee was supported by the NSF under Grant CCF-1217256 and Grant CCF-0905626. This article was presented at the 5th Conference on Innovations in Theoretical Computer Science (ITCS 2014). | ||||||||||||||||||

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Subject Keywords: | Hypothesis testing, quantum information theory, quantum Stein’s Lemma, subadditivity, von Neumann entropy | ||||||||||||||||||

Issue or Number: | 8 | ||||||||||||||||||

Record Number: | CaltechAUTHORS:20200730-074100066 | ||||||||||||||||||

Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20200730-074100066 | ||||||||||||||||||

Official Citation: | F. G. S. L. Brandão, A. W. Harrow, J. R. Lee and Y. Peres, "Adversarial Hypothesis Testing and a Quantum Stein’s Lemma for Restricted Measurements," in IEEE Transactions on Information Theory, vol. 66, no. 8, pp. 5037-5054, Aug. 2020, doi: 10.1109/TIT.2020.2979704 | ||||||||||||||||||

Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||||||||||

ID Code: | 104652 | ||||||||||||||||||

Collection: | CaltechAUTHORS | ||||||||||||||||||

Deposited By: | Tony Diaz | ||||||||||||||||||

Deposited On: | 30 Jul 2020 15:45 | ||||||||||||||||||

Last Modified: | 30 Jul 2020 15:45 |

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