Squashed entanglement, k-extendibility, quantum Markov chains, and recovery maps
arXiv:1410.4184 · doi:10.1007/s10701-018-0143-6
Abstract
Squashed entanglement [Christandl and Winter, J. Math. Phys. 45(3):829-840 (2004)] is a monogamous entanglement measure, which implies that highly extendible states have small value of the squashed entanglement. Here, invoking a recent inequality for the quantum conditional mutual information [Fawzi and Renner, Commun. Math. Phys. 340(2):575-611 (2015)] greatly extended and simplified in various work since, we show the converse, that a small value of squashed entanglement implies that the state is close to a highly extendible state. As a corollary, we establish an alternative proof of the faithfulness of squashed entanglement [Brandao, Christandl and Yard, Commun. Math. Phys. 306:805-830 (2011)]. We briefly discuss the previous and subsequent history of the Fawzi-Renner bound and related conjectures, and close by advertising a potentially far-reaching generalization to universal and functorial recovery maps for the monotonicity of the relative entropy.
14 pages, Springer journal style. In v2 we have removed the claim about multi-party squashed entanglement (see new Appendix B for why) and corrected other smaller mistakes; v3 takes into account the developments since the original posting and has a new structure; v4 final accepted version
References in corpus (18)
- Structure of states which satisfy strong subadditivity of quantum entropy with equality
- Aspects of generic entanglement
- Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints
- Quantum conditional mutual information and approximate Markov chains
- Is Entanglement Monogamous?
- One-and-a-half quantum de Finetti theorems
- Recoverability in quantum information theory
- Should Entanglement Measures be Monogamous or Faithful?
- Renyi generalizations of the conditional quantum mutual information
- Multivariate Trace Inequalities
- An additive and operational entanglement measure: conditional entanglement of mutual information
- Strengthened Monotonicity of Relative Entropy via Pinched Petz Recovery Map
- Rényi squashed entanglement, discord, and relative entropy differences
- Multipartite quantum correlations and local recoverability
- Dynamical maps, quantum detailed balance and Petz recovery map
- Approximate reversal of quantum Gaussian dynamics
- Highly Entangled States With Almost No Secrecy
- Swiveled Rényi entropies
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