2015

Squashed entanglement in infinite dimensions

Shirokov, M. E.

Understand

We analyse two possible definitions of the squashed entanglement in an infinite-dimensional bipartite system: direct translation of the finite-dimensional definition and its universal extension.

It is shown that the both definitions produce the same lower semicontinuous entanglement measure possessing all basis properties of the squashed entanglement on the set of states having at least one finite marginal entropy.
Is also shown that the second definition gives an adequate extension of this measure to the set of all states of infinite-dimensional bipartite system.
A general condition relating continuity of the squashed entanglement to continuity of the quantum mutual information is proved and its corollaries are considered.

Built on

E.H.Lieb, M.B.Ruskai, ”Proof of the strong suadditivity of quantum mechanical entropy”, J.Math.Phys. V.14. 1938 (1973)
1973
Earlier work this paper cites.
G.Lindblad ”Entropy, information and quantum measurements”, Comm. Math. Phys. V.33. 305-322 (1973)
1973
Earlier work this paper cites.
G.Lindblad ”Expectation and Entropy Inequalities for Finite Quantum Systems”, Comm. Math. Phys. V.39. N.2. 111-119 (1974)
1974
Earlier work this paper cites.
A.Wehrl, ”General properties of entropy”, Rev. Mod. Phys. 50, 221-250, (1978)
1978
Earlier work this paper cites.
L.P.Hughston, R.Jozsa, W.K.Wootters, ”A complete classification of quantum ensembles having a given density matrix”, Phys. Lett. A 183, N.1, 14-18 (1993)
1993
Earlier work this paper cites.
C.H.Bennett, D.P.DiVincenzo, J.A.Smolin, W.K.Wootters, ”Mixed State Entanglement and Quantum Error Correction”, Phys. Rev. A, 54, 3824-3851, (1996)
1996
Earlier work this paper cites.
M.A.Nielsen, I.L.Chuang ”Quantum Computation and Quantum Information”, Cambridge University Press, 2000
2000
Earlier work this paper cites.
M.A.Nielsen, ”Continuity bounds for entanglement”, Physical Review A 61 (6): Art. No. 064301 (2000); arXiv:quant-ph/9908086;
2000
Earlier work this paper cites.
G.Vidal, ”Entanglement monotones”, J. Modern Opt. V.47 N.2-3, 355-376 (2000)
2000
Earlier work this paper cites.

Similar

J.Eisert, Ch.Simon, M.B.Plenio ”On the quantification of entanglement in infinite-dimensional quantum systems” J. Phys. A V.35, N.17, 3911-3923 (2002)
2002
Cited alongside, same era.
M.Christandl, A.Winter, ”Squashed entanglements - an additive entanglement measure”, J. Math. Phys., V.45, 829-840 (2003)
2003
Cited alongside, same era.
R.Alicki, M.Fannes, ”Continuity of quantum conditional information”, Journal of Physics A: Mathematical and General, V.37, N.5, L55-L57 (2004); arXiv: quant-ph/0312081
2004
Cited alongside, same era.
P.Hayden, R.Jozsa, D.Petz, A.Winter, ”Structure of states which satisfy strong subadditivity of quantum entropy with equality”, Commun. Math. Phys. V.246, N.2, 359-374 (2004); arXiv:quant-ph/0304007
2004
Cited alongside, same era.
M.Koashi, A.Winter, ”Monogamy of entanglement and other correlations”, Phys. Rev. A 69(2):022309 (2004); arXiv:quant-ph/0310037
2004
Cited alongside, same era.
A.S.Holevo, M.E.Shirokov, R.F.Werner ”On the notion of entanglement in Hilbert space”, Russian Math. Surveys, V.60, N.2, 153-154 (2005); arXiv:quant-ph/0504204;
2005
Cited alongside, same era.
M.E.Shirokov, ”Entropic characteristics of subsets of states I”, Izvestiya: Mathematics, V.70, N.6, 1265-1292 (2006); arXiv:quant-ph/0510073
2006
Cited alongside, same era.
M.B.Plenio, S.Virmani, ”An introduction to entanglement measures”, Quantum Inf. Comput., V.7 N.1-2, 1-51 (2007)
2007
Cited alongside, same era.
I.Devetak, J.Yard, ”The operational meaning of quantum conditional information”, Phys. Rev. Lett. 100, 230501 (2008); arXiv:quant-ph/0612050
2008
Cited alongside, same era.
M.Christandl, ”The Structure of Bipartite Quantum States - Insights from Group Theory and Cryptography”, arXiv:quant-ph/0604183
Cited in the paper.
J.Eisert, ”Entanglement in quantum information theory”, arxiv:quant-ph/0610253
Cited in the paper.
K.Li, A.Winter, ”Squashed entanglement, k-extendibility, quantum Markov chains, and recovery maps”, arXiv:1410.4184
Original
Cited in the paper.

Then

A.S.Holevo, M.E.Shirokov ”On approximation of infinite dimensional quantum channels”, Problems of Information Transmission. V.44. N.2. 3-22 (2008); arXiv:0711.2245
Original
2008
Later among the works it cites.
R.Horodecki, P.Horodecki, M.Horodecki, K.Horodecki, ”Quantum entanglement”, Rev.Mod.Phys. V.81. 865-942 (2009); arXiv:quant-ph/0702225
2009
Later among the works it cites.
M.E.Shirokov ”On properties of the space of quantum states and their application to construction of entanglement monotones”, Izvestiya: Mathematics, V.74, N.4, 849-882 (2010); arXiv:0804.1515
Original
2010
Later among the works it cites.
F.G.S.L.Brandao, M.Christandl, J.Yard, ”Faithful squashed entanglement”, Comm. Math. Phys., V.306, N.3, 805-830 (2011); arXiv:1010.1750
Original
2011
Later among the works it cites.
A.A.Kuznetsova, ”Quantum conditional entropy for infinite-dimensional systems”, Theory of Probability and its Applications, V.55, N.4, 709-717 (2011)
2011
Later among the works it cites.
N.Li, S.Luo, ”Classical and quantum correlative capacities of quantum systems”, Phys. Rev. A 84, 042124 (2011)
2011
Later among the works it cites.
A.S.Holevo ”Quantum systems, channels, information. A mathematical introduction”, Berlin, DeGruyter, 2012
2012
Later among the works it cites.
K.Li, A.Winter, ”Relative entropy and squashed entanglement”, Comm. Math. Phys., V.326, N.1, 6380 (2014); arXiv:1210.3181
Original
2014
Later among the works it cites.

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

Reading the bibliography…