Squashed entanglement and approximate private states

The squashed entanglement is a fundamental entanglement measure in quantum information theory, finding application as an upper bound on the distillable secret key or distillable entanglement of a quantum state or a quantum channel. This paper simplifies proofs that the squashed entanglement is an upper bound on distillable key for finite-dimensional quantum systems and solidifies such proofs for infinite-dimensional quantum systems. More specifically, this paper establishes that the logarithm of the dimension of the key system (call it $$\log _{2}K$$log2K) in an $$\varepsilon $$ε-approximate private state is bounded from above by the squashed entanglement of that state plus a term that depends only $$\varepsilon $$ε and $$\log _{2}K$$log2K. Importantly, the extra term does not depend on the dimension of the shield systems of the private state. The result holds for the bipartite squashed entanglement, and an extension of this result is established for two different flavors of the multipartite squashed entanglement.

[1]  Koji Azuma,et al.  Aggregating quantum repeaters for the quantum internet , 2016, 1606.00135.

[2]  L. Banchi,et al.  Fundamental limits of repeaterless quantum communications , 2015, Nature Communications.

[3]  Michael Satosi Watanabe,et al.  Information Theoretical Analysis of Multivariate Correlation , 1960, IBM J. Res. Dev..

[4]  Kenneth Goodenough,et al.  Assessing the performance of quantum repeaters for all phase-insensitive Gaussian bosonic channels , 2015, 1511.08710.

[5]  J. Oppenheim,et al.  Secure key from bound entanglement. , 2003, Physical Review Letters.

[6]  M. E. Shirokov,et al.  Tight uniform continuity bounds for the quantum conditional mutual information, for the Holevo quantity, and for capacities of quantum channels , 2015, 1512.09047.

[7]  M. Fannes,et al.  Continuity of quantum conditional information , 2003, quant-ph/0312081.

[8]  Andreas J. Winter,et al.  Tight Uniform Continuity Bounds for Quantum Entropies: Conditional Entropy, Relative Entropy Distance and Energy Constraints , 2015, ArXiv.

[9]  Saikat Guha,et al.  The Squashed Entanglement of a Quantum Channel , 2013, IEEE Transactions on Information Theory.

[10]  Pawel Horodecki,et al.  Quantum states representing perfectly secure bits are always distillable , 2006 .

[11]  Michal Horodecki,et al.  Unifying Classical and Quantum Key Distillation , 2007, TCC.

[12]  Mark M. Wilde,et al.  Bounds on Entanglement Distillation and Secret Key Agreement for Quantum Broadcast Channels , 2015, IEEE Transactions on Information Theory.

[13]  E. Lieb,et al.  Proof of the strong subadditivity of quantum‐mechanical entropy , 1973 .

[14]  M. Christandl.  The Structure of Bipartite Quantum States - Insights from Group Theory and Cryptography , 2006, quant-ph/0604183.

[15]  A. Lichnerowicz.  Proof of the Strong Subadditivity of Quantum-Mechanical Entropy , 2018 .

[16]  S. Massar,et al.  Multipartite classical and quantum secrecy monotones , 2002, quant-ph/0202103.

[17]  David Avis,et al.  Distributed compression and multiparty squashed entanglement , 2007, ArXiv.

[18]  M. Lewenstein,et al.  Quantum Entanglement , 2020, Quantum Mechanics.

[19]  Dong Yang,et al.  Squashed Entanglement for Multipartite States and Entanglement Measures Based on the Mixed Convex Roof , 2007, IEEE Transactions on Information Theory.

[20]  Matthias Christandl,et al.  Entanglement of the Antisymmetric State , 2009, 0910.4151.

[21]  Mario Berta,et al.  Rényi squashed entanglement, discord, and relative entropy differences , 2014, ArXiv.

[22]  S. Guha,et al.  Fundamental rate-loss tradeoff for optical quantum key distribution , 2014, Nature Communications.

[23]  Mark M Wilde,et al.  Multipartite quantum correlations and local recoverability , 2014, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[24]  Robert R. Tucci.  Quantum Entanglement and Conditional Information Transmission , 1999 .

[25]  Robert R.Tucci.  Entanglement of Distillation and Conditional Mutual Information , 2002 .

[26]  A. Uhlmann.  The "transition probability" in the state space of a ∗-algebra , 1976 .

[27]  M. Shirokov.  Squashed entanglement in infinite dimensions , 2015, 1507.08964.

[28]  Mark M. Wilde,et al.  Fidelity of recovery, geometric squashed entanglement, and measurement recoverability , 2014, 1410.1441.

[29]  Marco Tomamichel,et al.  Quantum Information Processing with Finite Resources - Mathematical Foundations , 2015, ArXiv.

[30]  Michal Horodecki,et al.  General Paradigm for Distilling Classical Key From Quantum States , 2009, IEEE Transactions on Information Theory.

[31]  E. Lieb,et al.  A Fundamental Property of Quantum-Mechanical Entropy , 1973 .

[32]  C. Helstrom.  Quantum detection and estimation theory , 1969 .

[33]  P. Horodecki,et al.  No-local-broadcasting theorem for multipartite quantum correlations. , 2007, Physical review letters.

[34]  F. Brandão,et al.  Faithful Squashed Entanglement , 2010, 1010.1750.

[35]  Mario Berta,et al.  Converse Bounds for Private Communication Over Quantum Channels , 2016, IEEE Transactions on Information Theory.

[36]  Mark M. Wilde,et al.  From Classical to Quantum Shannon Theory , 2011, ArXiv.

[37]  Te Sun Han.  Nonnegative Entropy Measures of Multivariate Symmetric Correlations , 1978, Inf. Control..

[38]  Jeroen van de Graaf,et al.  Cryptographic Distinguishability Measures for Quantum-Mechanical States , 1997, IEEE Trans. Inf. Theory.

[39]  Hoi-Kwong Lo,et al.  Fundamental rate-loss trade-off for the quantum internet , 2016, Nature Communications.

[40]  A. Winter,et al.  “Squashed entanglement”: An additive entanglement measure , 2003, quant-ph/0308088.

[41]  A. Winter,et al.  Monogamy of quantum entanglement and other correlations , 2003, quant-ph/0310037.

[42]  Michał Horodecki,et al.  An additive and operational entanglement measure: conditional entanglement of mutual information. , 2008, Physical review letters.

[43]  Te Sun Han,et al.  Linear Dependence Structure of the Entropy Space , 1975, Inf. Control..