TY - JOUR

T1 - From Log-Determinant Inequalities to Gaussian Entanglement via Recoverability Theory

AU - Lami, Ludovico

AU - Hirche, Christoph

AU - Adesso, Gerardo

AU - Winter, Andreas

N1 - Funding Information: Manuscript received March 17, 2017; accepted July 22, 2017. Date of publication August 8, 2017; date of current version October 18, 2017. L. Lami and A. Winter were supported in part by the Spanish MINECO through the FEDER funds under Project FIS2013-40627-P and Project FIS2016-86681-P, in part by the Generalitat de Catalunya, CIRIT, under Project 2014-SGR-966, and in part by the European Research Council, Advanced Grant IRQUAT, under Grant 2010-AdG-267386. C. Hirche was supported in part by the Spanish MINECO through the FEDER funds under Project FIS2013-40627-P and Project FIS2016-86681-P, in part by the Generalitat de Catalunya, CIRIT, under Project 2014-SGR-966, in part by the European Research Council, Advanced Grant IRQUAT, under Grant 2010-AdG-267386, and in part by FPI Scholarship under Grant BES-2014-068888. G. Adesso was supported in part by The European Research Council, Starting Grant GQCOP, under Grant 637352 and in part by the Foundational Questions Institute (FQXi) Physics of the Observer Programme under Grant FQXi-RFP-1601.

PY - 2017/11

Y1 - 2017/11

N2 - Many determinantal inequalities for positive definite block matrices are consequences of general entropy inequalities, specialized to Gaussian distributed vectors with prescribed covariances. In particular, strong subadditivity (SSA) yields VAC + \ln \det VBC - \ln \det V ABC V C 0 for all 3× 3 block matrices V ABC , where subscripts identify principal submatrices. We shall refer to the above-mentioned inequality as SSA of log-det entropy. In this paper, we develop further insights on the properties of the above-mentioned inequality and its applications to classical and quantum information theory. In the first part of this paper, we show how to find known and new necessary and sufficient conditions under which saturation with equality occurs. Subsequently, we discuss the role of the classical transpose channel (also known as Petz recovery map) in this problem and find its action explicitly. We then prove some extensions of the saturation theorem, by finding faithful lower bounds on a log-det conditional mutual information. In the second part, we focus on quantum Gaussian states, whose covariance matrices are not only positive but obey additional constraints due to the uncertainty relation. For Gaussian states, the log-det entropy is equivalent to the Rényi entropy of order 2. We provide a strengthening of log-det SSA for quantum covariance matrices that involves the so-called Gaussian Rényi-2 entanglement of formation, a well-behaved entanglement measure defined via a Gaussian convex roof construction. We then employ this result to define a log-det entropy equivalent of the squashed entanglement measure, which is remarkably shown to coincide with the Gaussian Rényi-2 entanglement of formation. This allows us to establish useful properties of such measure(s), such as monogamy, faithfulness, and additivity on Gaussian states.

AB - Many determinantal inequalities for positive definite block matrices are consequences of general entropy inequalities, specialized to Gaussian distributed vectors with prescribed covariances. In particular, strong subadditivity (SSA) yields VAC + \ln \det VBC - \ln \det V ABC V C 0 for all 3× 3 block matrices V ABC , where subscripts identify principal submatrices. We shall refer to the above-mentioned inequality as SSA of log-det entropy. In this paper, we develop further insights on the properties of the above-mentioned inequality and its applications to classical and quantum information theory. In the first part of this paper, we show how to find known and new necessary and sufficient conditions under which saturation with equality occurs. Subsequently, we discuss the role of the classical transpose channel (also known as Petz recovery map) in this problem and find its action explicitly. We then prove some extensions of the saturation theorem, by finding faithful lower bounds on a log-det conditional mutual information. In the second part, we focus on quantum Gaussian states, whose covariance matrices are not only positive but obey additional constraints due to the uncertainty relation. For Gaussian states, the log-det entropy is equivalent to the Rényi entropy of order 2. We provide a strengthening of log-det SSA for quantum covariance matrices that involves the so-called Gaussian Rényi-2 entanglement of formation, a well-behaved entanglement measure defined via a Gaussian convex roof construction. We then employ this result to define a log-det entropy equivalent of the squashed entanglement measure, which is remarkably shown to coincide with the Gaussian Rényi-2 entanglement of formation. This allows us to establish useful properties of such measure(s), such as monogamy, faithfulness, and additivity on Gaussian states.

KW - linear matrix inequalities

KW - mutual information

KW - quantum entanglement

KW - Quantum mechanics

UR - http://www.scopus.com/inward/record.url?scp=85029184120&partnerID=8YFLogxK

U2 - 10.1109/TIT.2017.2737546

DO - 10.1109/TIT.2017.2737546

M3 - Article

AN - SCOPUS:85029184120

VL - 63

SP - 7553

EP - 7568

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

SN - 0018-9448

IS - 11

M1 - 8004445

ER -