From log-determinant inequalities to Gaussian entanglement via recoverability theory

Lami, Ludovico; Hirche, Christoph; Adesso, Gerardo; Winter, Andreas

Abstract

Many 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 × 3-block matrices VABC , where subscripts identify principal submatrices. We shall refer to the above inequality as SSA of log-det 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 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 Renyi entropy of order 2. We provide a strengthening of log-det SSA for quantum covariance matrices that involves the so-called Gaussian Renyi-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 Renyi-2 entanglement of formation. This allows us to establish useful properties of such measure(s), like monogamy, faithfulness, and additivity on Gaussian states.

Citation

Lami, L., Hirche, C., Adesso, G., & Winter, A. (in press). From log-determinant inequalities to Gaussian entanglement via recoverability theory. IEEE Transactions on Information Theory, 63(11), https://doi.org/10.1109/TIT.2017.2737546

• Journal Article Type: Article
• Acceptance Date: Aug 9, 2017
• Online Publication Date: Aug 9, 2017
• Deposit Date: Oct 6, 2017
• Publicly Available Date: Oct 6, 2017
• Journal: IEEE Transactions on Information Theory
• Print ISSN: 0018-9448
• Electronic ISSN: 0018-9448
• Publisher: Institute of Electrical and Electronics Engineers
• Peer Reviewed: Peer Reviewed
• Volume: 63
• Issue: 11
• DOI: https://doi.org/10.1109/TIT.2017.2737546
• Public URL: https://nottingham-repository.worktribe.com/output/877290
• Publisher URL: http://ieeexplore.ieee.org/document/8004445/
• Additional Information: (c) 2017 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.
• Contract Date: Oct 6, 2017