A Generalization of Quantum Stein's Lemma
Abstract
Given many independent and identicallydistributed (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 noni.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 nonzero. Therefore, the operational definition of multipartite entanglement is equivalent to its mathematical definition
Additional Information
© 2010 SpringerVerlag. 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 McGillBellairs 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 QIPIRC 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.Attached Files
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Additional details
 Eprint ID
 67389
 DOI
 10.1007/s002200101005z
 Resolver ID
 CaltechAUTHORS:20160526105804802
 Engineering and Physical Sciences Research Council (EPSRC)
 GR/S82176/0
 IST directorate
 015848
 Fundaçã de Amparo a Pesquisa de Minas Gerais (FAPEMIG)
 Royal Society
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20160526Created from EPrint's datestamp field
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20211111Created from EPrint's last_modified field