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Squashed entanglement, k-extendibility, quantum Markov chains, and recovery maps
Ke Li, Andreas Winter
TL;DR
The paper asks whether small squashed entanglement can certify proximity to highly extendible states, complementing the known implication from high extendibility to small squashed entanglement. Using a recovery-map inequality, it proves such proximity and derives faithfulness of squashed entanglement, while noting limitations of the resulting bounds.
Problem
The paper asks whether small squashed entanglement implies proximity to highly extendible states, complementing the implication from high extendibility to small squashed entanglement.
Source: p. 2Method
The proof uses the Fawzi–Renner recovery-map result to relate small squashed entanglement to proximity to k-extendible states, then invokes a quantitative de Finetti theorem.
Source: pp. 2–3Results
Small squashed entanglement implies proximity to a k-extendible state, and squashed entanglement is zero if and only if the state is separable.
Source: p. 3Takeaways & Limitations
Squashed entanglement provides a quantitative relation to distance from highly extendible states, yielding an alternative proof of its faithfulness.
Source: p. 2The trace-norm bound has a worse polynomial dependence on ε than the comparison discussed, and the corollary’s dimensional dependence involves only one of the two systems.
Source: p. 3
Abstract
from arXiv · showhide
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.
Appendix — Multi-party squashed entanglement.
The appendix explores whether the bipartite relationship between squashed entanglement and distance from highly extendible states extends to multiple parties. It outlines a recovery-map approach, identifies interference between recovery maps as an obstacle, and notes a related route using conditional entanglement of mutual information.
Multipartite extension approach
- If multipartite conditional mutual information is at most 2ε, then every bipartition’s conditional mutual information is also at most 2ε.The appendix derives this from the chain-rule identity for conditional multiinformation.Source: Appendix — Multi-party squashed… · p. 11
- Recovery maps are proposed to extract k approximate copies of each subsystem from the conditioning system E.The maps act from E to E joined with each Ai; tracing out E gives a state containing k copies of each Ai.Source: Appendix — Multi-party squashed… · p. 12
- The recovered state satisfies the desired bound only for tuples in which at most one index differs from 1.The different recovery maps may interfere, preventing the authors from establishing the bound for all tuples.Source: Appendix — Multi-party squashed… · p. 12
- A related argument applies to the generally larger multi-party conditional entanglement of mutual information (CEMI).The proposed bound is a trace-distance bound from a fully separable state, with dependence on k and the dimensions of A2 through An.Source: Appendix — Multi-party squashed… · p. 12
- The authors leave an extension of Theorem 1 to more than two parties as an open problem.For zero conditional multiinformation, the appendix describes sequential measurements whose outcomes condition the state to factorize across the parties.Source: Appendix — Multi-party squashed… · p. 12