From logdeterminant inequalities to Gaussian entanglement via recoverability theoryTools Lami, Ludovico and Hirche, Christoph and Adesso, Gerardo and Winter, Andreas (2017) From logdeterminant inequalities to Gaussian entanglement via recoverability theory. IEEE Transactions on Information Theory, 63 (11). pp. 75537568. ISSN 00189448
Official URL: http://ieeexplore.ieee.org/document/8004445/
AbstractMany determinantal inequalities for positive definite block matrices are consequences of general entropy inequalities, specialised to Gaussian distributed vectors with prescribed covariances. In particular, strong subadditivity (SSA) yields ln det VAC + ln det VBC − ln det VABC − ln det VC ≥ 0 for all 3 × 3block matrices VABC , where subscripts identify principal submatrices. We shall refer to the above inequality as SSA of logdet entropy. In this paper we develop further insights on the properties of the above inequality and its applications to classical and quantum information theory. In the first part of the 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 logdet 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 logdet entropy is equivalent to the Renyi entropy of order 2. We provide a strengthening of logdet SSA for quantum covariance matrices that involves the socalled Gaussian Renyi2 entanglement of formation, a wellbehaved entanglement measure defined via a Gaussian convex roof construction. We then employ this result to define a logdet entropy equivalent of the squashed entanglement measure, which is remarkably shown to coincide with the Gaussian Renyi2 entanglement of formation. This allows us to establish useful properties of such measure(s), like monogamy, faithfulness, and additivity on Gaussian states.
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