Quantum Information Processing

, Volume 15, Issue 11, pp 4563–4580 | Cite as

Squashed entanglement and approximate private states

  • Mark M. WildeEmail author


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 \(\log _{2}K\)) in an \(\varepsilon \)-approximate private state is bounded from above by the squashed entanglement of that state plus a term that depends only \(\varepsilon \) and \(\log _{2}K\). 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.


Private states Distillable secret key Distillable entanglement 



I am grateful to Koji Azuma and Stefan Baeuml for pointing out the main issue discussed in this paper. I am as well thankful to Koji Azuma, Stefan Baeuml, Saikat Guha, Ryan Gregory James, Masahiro Takeoka, and Stephanie Wehner for discussions related to the topic of this paper. I thank the anonymous referees for helpful comments that improved the readability of the paper. I acknowledge support from the NSF under Award No. CCF-1350397 and thank Stefano Mancini for hosting me at University of Camerino during late June 2016, where this result was developed.


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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Center for Computation and TechnologyLouisiana State UniversityBaton RougeUSA

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