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A Generalization of Quantum Stein’s Lemma

Brandão, Fernando G. S. L. and Plenio, Martin B. (2010) A Generalization of Quantum Stein’s Lemma. Communications in Mathematical Physics, 295 (3). pp. 791-828. ISSN 0010-3616. https://resolver.caltech.edu/CaltechAUTHORS:20160526-105804802

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Abstract

Given many independent and identically-distributed (i.i.d.) copies of a quantum system described either by the state ρ or σ (called null and alternative hypotheses, respectively), what is the optimal measurement to learn the identity of the true state? In asymmetric hypothesis testing one is interested in minimizing the probability of mistakenly identifying ρ instead of σ, while requiring that the probability that σ is identified in the place of ρ is bounded by a small fixed number. Quantum Stein’s Lemma identifies the asymptotic exponential rate at which the specified error probability tends to zero as the quantum relative entropy of ρ and σ. We present a generalization of quantum Stein’s Lemma to the situation in which the alternative hypothesis is formed by a family of states, which can moreover be non-i.i.d. We consider sets of states which satisfy a few natural properties, the most important being the closedness under permutations of the copies. We then determine the error rate function in a very similar fashion to quantum Stein’s Lemma, in terms of the quantum relative entropy. Our result has two applications to entanglement theory. First it gives an operational meaning to an entanglement measure known as regularized relative entropy of entanglement. Second, it shows that this measure is faithful, being strictly positive on every entangled state. This implies, in particular, that whenever a multipartite state can be asymptotically converted into another entangled state by local operations and classical communication, the rate of conversion must be non-zero. Therefore, the operational definition of multipartite entanglement is equivalent to its mathematical definition


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1007/s00220-010-1005-zDOIArticle
http://link.springer.com/article/10.1007%2Fs00220-010-1005-zPublisherArticle
http://arxiv.org/abs/0904.0281arXivDiscussion Paper
ORCID:
AuthorORCID
Brandão, Fernando G. S. L.0000-0003-3866-9378
Additional Information:© 2010 Springer-Verlag. Received: 5 March 2009; Accepted: 20 November 2009. We gratefully thank Koenraad Audenaert, Nilanjana Datta, Jens Eisert, Andrzej Grudka, Masahito Hayashi, Michał and Ryszard Horodecki, Renato Renner, Shashank Virmani, Reinhard Werner, Andreas Winter and the participants in the 2009 McGill-Bellairs workshop for many interesting discussions, and an anonymous referee for filling in gaps in the proofs of Lemma III.6 and Proposition III.1, for pointing out that our main result could be extended to cover the original quantum Stein’s Lemma and for many other extremely useful comments on the manuscript. This work is part of the QIP-IRC supported by EPSRC (GR/S82176/0) as well as the Integrated Project Qubit Applications (QAP) supported by the IST directorate as Contract Number 015848’ and was supported by the Brazilian agency Fundao de Amparo Pesquisa do Estado de Minas Gerais (FAPEMIG), an EPSRC Postdoctoral Fellowship for Theoretical Physics and a Royal Society Wolfson Research Merit Award.
Funders:
Funding AgencyGrant Number
Engineering and Physical Sciences Research Council (EPSRC)GR/S82176/0
IST directorate015848
Fundaçã de Amparo a Pesquisa de Minas Gerais (FAPEMIG)UNSPECIFIED
Royal SocietyUNSPECIFIED
Issue or Number:3
Record Number:CaltechAUTHORS:20160526-105804802
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20160526-105804802
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:67389
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:26 May 2016 19:09
Last Modified:03 Oct 2019 10:05

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