Squashed Entanglement for Multipartite States and Entanglement Measures Based on the Mixed Convex Roof

• Published in: IEEE transactions on information theory 2009-07, Vol.55 (7), p.3375-3387
• Main Authors: Dong Yang, Horodecki, K., Horodecki, M., Horodecki, P., Oppenheim, J., Wei Song
• Format: Article
• Language: English
• Subjects: Applied sciences; C-squashed entanglement; Cryptography; Decomposition; Density; Density measurement; Entanglement; Entropy; Exact sciences and technology; Extraterrestrial measurements; Infimum; Information processing; Information theory; Information, signal and communications theory; Mathematics; Matrix decomposition; Measurement; Measurement standards; mixed convex roof; multipartite distillable key; Mutual information; Optimization; Optimization methods; Optimization techniques; Physics; Roofs; Signal and communications theory; squashed entanglement; Telecommunications and information theory; Upper bound; Upper bounds
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2009.2021373

New measures of multipartite entanglement are constructed based on two definitions of multipartite information and different methods of optimizing over extensions of the states. One is a generalization of the squashed entanglement where one takes the mutual information of parties conditioned on the state's extension and takes the infimum over such extensions. Additivity of the multipartite squashed entanglement is proved for both versions of the multipartite information which turn out to be related. The second one is based on taking classical extensions. This scheme is generalized, which enables to construct measures of entanglement based on the mixed convex roof of a quantity, which in contrast to the standard convex roof method involves optimization over all decompositions of a density matrix rather than just the decompositions into pure states. As one of the possible applications of these results we prove that any multipartite monotone is an upper bound on the amount of multipartite distillable key. The findings are finally related to analogous results in classical key agreement.