Squashed entanglement, k-extendibility, quantum Markov chains, and recovery maps

Squashed entanglement [Christandl/AW, JMP 45:829 (2004)] is a seemingly blessed measure of bipartite entanglement: it is and upper bound both on the distillable entanglement and the distillable key in a state, it is convex in the state, additive and monogamous. The latter property endears it to many but it implies that it is very small for highly extendible states, such as O(1/d) for the dxd fully antisymmetric state. Indeed, it was only a few years ago that faithfulness was proved, i.e. positivity on all entangled (=non-separable) states [Brandao et al., CMP 306:805 (2011)].

Using a new result of Fawzi and Renner [arXiv:1410.0664], which proves a version of a conjecture of the speaker from 2008 (later refined by Isaac Kim and Berta et al. [arXiv:1403.6102]), we show the converse of the above observation: small squashed entanglement of a state implies that it is close to a highly extendable one. This should be contrasted with entanglement of formation, which is small if and only if the state is close to a separable one. As is well-known, k-extendibility approximates separability as k grows, and in this way the faithfulness statement of Brandao et al. follows.

Fawzi/Renner's result is a characterization of approximate quantum Markov chains, i.e. states $rho_{ABC}$ with $I(A:C|B) < epsilon$, showing that there exists a cptp map $R:B-->BC$ with $-log F[R(rho_{AB}),rho_{ABC}] < epsilon$, where $F[]$ is the fidelity between two states. This map is a variation of a "recovery" map introduced in the quantum setting by Petz [CMP 105:123 (1986)]. I will show a much stronger and more general characterization in the classical case using the Petz recovery map (aka "transpose channel") and ask whether it has a quantum analogue.