Abstract
We are interested in the properties and relations of entanglement measures. Especially, we focus on the squashed entanglement and relative entropy of entanglement, as well as their analogues and variants.
Our first result is a monogamy-like inequality involving the relative entropy of entanglement and its one-way LOCC variant. The proof is accomplished by exploring the properties of relative entropy in the context of hypothesis testing via one-way LOCC operations, and by making use of an argument resembling that by Piani on the faithfulness of regularized relative entropy of entanglement.
Following this, we obtain a commensurate and faithful lower bound for squashed entanglement, in the form of one-way LOCC relative entropy of entanglement. This gives a strengthening to the strong subadditivity of von Neumann entropy. Our result improves the trace-distance-type bound derived in Brandão et al. (Commun Math Phys, 306:805–830, 2011), where faithfulness of squashed entanglement was first proved. Applying Pinsker’s inequality, we are able to recover the trace-distance-type bound, even with slightly better constant factor. However, the main improvement is that our new lower bound can be much larger than the old one and it is almost a genuine entanglement measure.
We evaluate exactly the relative entropy of entanglement under various restricted measurement classes, for maximally entangled states. Then, by proving asymptotic continuity, we extend the exact evaluation to their regularized versions for all pure states. Finally, we consider comparisons and separations between some important entanglement measures and obtain several new results on these, too.
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Horodecki R., Horodecki P., Horodecki M., Horodecki K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)
Christandl M., Winter A.: ‘Squashed entanglement’—an additive entanglement measure. J. Math. Phys. 45, 829 (2004)
Tucci, R.R.: Quantum entanglement and conditional information transmission. http://arXiv.org/abs/quant-ph/9909041v1, 1999
Tucci, R.R.: Entanglement of distillation and conditional mutual information. http://arXiv.org/abs/quant-ph/0202144v2, 2002
Maurer U.M., Wolf S.: Unconditionally secure key agreement and the intrinsic conditional information. IEEE Trans. Inf. Theory 45, 499 (1999)
Alicki R., Fannes M.: Continuity of quantum conditional information. J. Phys. A: Math. Gen. 37, L55 (2004)
Koashi M., Winter A.: Monogamy of entanglement and other correlations. Phys. Rev. A 69, 022309 (2004)
Devetak I., Yard J.: Exact cost of redistributing quantum states. Phys. Rev. Lett. 100, 230501 (2008)
Yard J., Devetak I.: Optimal quantum source coding with quantum information at the encoder and decoder. IEEE Trans. Inf. Theory 55, 5339 (2009)
Ye M.-Y., Bai Y.-K., Wang Z.D.: Quantum state redistribution based on a generalized decoupling. Phys. Rev. A 78, 030302(R) (2008)
Oppenheim, J.: A paradigm for entanglement theory based on quantum communication. http://arXiv.org/abs/0801.0458v1 [quant-ph], 2008
Hiai F., Petz D.: The proper formula for relative entropy and its asymptotics in quantum probability. Commun. Math. Phys. 143, 99 (1991)
Ogawa T., Nagaoka H.: Strong converse and Stein’s lemma in quantum hypothesis testing. IEEE. Tran. Inf. Theory 46, 2428 (2000)
Vedral V., Plenio M.B., Rippin M.A., Knight P.L.: Quantifying entanglement. Phys. Rev. Lett. 78, 2275 (1997)
Vedral V., Plenio M.B.: Entanglement measures and purification procedures. Phys. Rev. A 57, 1619 (1998)
Vollbrecht K.G.H., Werner R.F.: Entanglement measures under symmetry. Phys. Rev. A 64, 062307 (2001)
Brandão F.G.S.L., Plenio M.B.: Entanglement theory and the second law of thermodynamics. Nat. Phys. 4, 873 (2008)
Brandão F.G.S.L., Plenio M.B.: A reversible theory of entanglement and its relation to the second law. Commun. Math. Phys. 295, 829 (2010)
Brandão F.G.S.L., Plenio M.B.: A generalization of quantum Stein’s lemma. Commun. Math. Phys. 295, 791 (2010)
Lieb E.H., Ruskai M.B.: Proof of the strong subadditivity of quantum-mechanical entropy. J. Math. Phys. 14, 1938 (1973)
Brandão F.G.S.L., Christandl, M., Yard, J.: Faithful squashed entanglement. Commun. Math. Phys. 306, 805–830 (2011); Erratum to: Faithful Squashed Entanglement. Commun. Math. Phys. 316, 287–288 (2012)
Matthews W., Wehner S., Winter A.: Distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding. Commun. Math. Phys. 291, 813 (2009)
Piani M.: Relative entropy of entanglement and restricted measurements. Phys. Rev. Lett. 103, 160504 (2009)
Christandl M., Schuch N., Winter A.: Entanglement of the antisymmetric state. Commun. Math. Phys. 311, 397 (2012)
Christandl M., Schuch N., Winter A.: Highly entangled states with almost no secrecy. Phys. Rev. Lett. 104, 240405 (2010)
Fuchs C.A., van de Graaf J.: Cryptographic distinguishability measures for quantum-mechanical states. IEEE. Tran. Inf. Theory 45, 1216 (1999)
Cover, T.M., Thomas, J.A.: Elements of Information Theory. New York: Wiley, 1991
Matthews W., Winter A.: On the Chernoff distance for asymptotic LOCC discrimination of bipartite quantum states. Commun. Math. Phys. 285, 161 (2008)
Owari, M., Hayashi, M.: Asymptotic local hypothesis testing between a pure bipartite state and the completely mixed state. http://arXiv.org/abs/1105.3789v1 [quant-ph], 2011
Winter A.: Coding theorem and strong converse for quantum channels. IEEE. Tran. Inf. Theory 45, 2481 (1999)
Lindblad G.: Completely positive maps and entropy inequalities. Commun. Math. Phys. 40, 147 (1975)
Uhlmann A.: Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory. Commun. Math. Phys. 54, 21 (1977)
Donald M., Horodecki M.: Continuity of relative entropy of entanglement. Phys. Lett. A 264, 257 (1999)
Horodecki M., Horodecki P.: Reduction criterion of separability and limits for a class of distillation protocols. Phys. Rev. A 59, 4206 (1999)
Yang D., Horodecki M., Wang Z.: An additive and operational entanglement measure: conditional entanglement of mutual information. Phys. Rev. Lett. 101, 140501 (2008)
Yang D., Horodecki K., Horodecki M., Horodecki P., Oppenheim J., Song W.: Squashed entanglement for multipartite states and entanglement measures based on the mixed convex roof. IEEE Trans. Inf. Theory 55(7), 3375 (2009)
Yang, D., Horodecki, M., Wang, Z.: Conditional entanglement. http://arXiv.org/abs/quant-ph/0701149v3, 2007
Christandl, M.: The structure of bipartite quantum states—insights from group theory and cryptography. Ph. D. thesis, University of Cambridge. http://arXiv.org/abs/quant-ph/0604183v1, 2006
Horodecki K., Horodecki M., Horodecki P., Oppenheim J.: Secure key from bound entanglement. Phys. Rev. Lett. 94, 160502 (2005)
Horodecki K., Horodecki M., Horodecki P., Oppenheim J.: Locking entanglement measures with a single qubit. Phys. Rev. Lett. 94, 200501 (2005)
Christandl M., Winter A.: Uncertainty, monogamy, and locking of quantum correlations. IEEE Trans. Inf. Theory 51(9), 3159 (2005)
Horodecki, M., Horodecki, P., Horodecki, R.: Mixed-state entanglement and distillation: Is there a ‘bound’ entanglement in nature? Phys. Rev. Lett. 80, 5239 (1998)
Hastings M.B.: Superadditivity of communication capacity using entangled inputs. Nat. Phys. 5, 255 (2009)
Shor P.W.: Equivalence of additivity questions in quantum information theory. Commun. Math. Phys. 246, 453 (2003)
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Communicated by M. B. Ruskai
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Li, K., Winter, A. Relative Entropy and Squashed Entanglement. Commun. Math. Phys. 326, 63–80 (2014). https://doi.org/10.1007/s00220-013-1871-2
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DOI: https://doi.org/10.1007/s00220-013-1871-2