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Faithful Squashed Entanglement

Brandão, Fernando G. S. L. and Christandl, Matthias and Yard, Jon (2011) Faithful Squashed Entanglement. Communications in Mathematical Physics, 306 (3). pp. 805-830. ISSN 0010-3616. doi:10.1007/s00220-011-1302-1.

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Squashed entanglement is a measure for the entanglement of bipartite quantum states. In this paper we present a lower bound for squashed entanglement in terms of a distance to the set of separable states. This implies that squashed entanglement is faithful, that is, it is strictly positive if and only if the state is entangled. We derive the lower bound on squashed entanglement from a lower bound on the quantum conditional mutual information which is used to define squashed entanglement. The quantum conditional mutual information corresponds to the amount by which strong subadditivity of von Neumann entropy fails to be saturated. Our result therefore sheds light on the structure of states that almost satisfy strong subadditivity with equality. The proof is based on two recent results from quantum information theory: the operational interpretation of the quantum mutual information as the optimal rate for state redistribution and the interpretation of the regularised relative entropy of entanglement as an error exponent in hypothesis testing. The distance to the set of separable states is measured in terms of the LOCC norm, an operationally motivated norm giving the optimal probability of distinguishing two bipartite quantum states, each shared by two parties, using any protocol formed by local quantum operations and classical communication (LOCC) between the parties. A similar result for the Frobenius or Euclidean norm follows as an immediate consequence. The result has two applications in complexity theory. The first application is a quasipolynomial-time algorithm solving the weak membership problem for the set of separable states in LOCC or Euclidean norm. The second application concerns quantum Merlin-Arthur games. Here we show that multiple provers are not more powerful than a single prover when the verifier is restricted to LOCC operations thereby providing a new characterisation of the complexity class QMA.

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Brandão, Fernando G. S. L.0000-0003-3866-9378
Additional Information:© 2011 Springer-Verlag. Received: 8 October 2010; Accepted: 14 June 2011; Published online: 2 August 2011. We thank Mario Berta, Aram Harrow and Andreas Winter for helpful discussions and in particular David Reeb for pointing out that our LOCC norm bound on squashed entanglement would extend to a bound on the conditional mutual information. FB is supported by a “Conhecimento Novo” fellowship from the Brazilian agency Fundacão de Amparo a Pesquisa do Estado de Minas Gerais (FAPEMIG). MC is supported by the Swiss National Science Foundation (grant PP00P2-128455) and the German Science Foundation (grants CH 843/1-1 and CH 843/2-1). JY is supported by a grant through the LDRD program of the United States Department of Energy. FB and JY thank the Institute Mittag Leffler, where part of this work was done, for their hospitality.
Funding AgencyGrant Number
Fundacão de Amparo a Pesquisa do Estado de Minas Gerais (FAPEMIG)UNSPECIFIED
Swiss National Science Foundation (SNSF)PP00P2-128455
Deutsche Forschungsgemeinschaft (DFG)CH 843/1-1
Deutsche Forschungsgemeinschaft (DFG)CH 843/2-1
Department of Energy (DOE)UNSPECIFIED
Issue or Number:3
Record Number:CaltechAUTHORS:20160607-124125598
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:67735
Deposited By: Tony Diaz
Deposited On:07 Jun 2016 19:49
Last Modified:11 Nov 2021 03:53

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