698:, and gives the additional number of bits above the classical limit that can be transmitted in a quantum dense coding protocol. Positive conditional entropy of a state thus means the state cannot reach even the classical limit, while the negative conditional entropy provides for additional information.
496:
657:
371:
694:, the conditional quantum entropy can be negative. This is true even though the (quantum) von Neumann entropy of single variable is never negative. The negative conditional entropy is also known as the
204:, who showed that quantum conditional entropies can be negative, something that is forbidden in classical physics. The negativity of quantum conditional entropy is a sufficient criterion for quantum
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111:
893:
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982:
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534:. The von Neumann entropy measures an observer's uncertainty about the value of the state, that is, how much the state is a
36:
28:
717:
491:{\displaystyle S(A)_{\rho }\ {\stackrel {\mathrm {def} }{=}}\ S(\rho ^{A})=S(\mathrm {tr} _{B}\rho ^{AB})}
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By analogy with the classical conditional entropy, one defines the conditional quantum entropy as
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Horodecki, Michał; Oppenheim, Jonathan; Winter, Andreas (2005). "Partial quantum information".
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An equivalent operational definition of the quantum conditional entropy (as a measure of the
944:
852:
799:
744:
652:{\displaystyle S(A|B)_{\rho }\ {\stackrel {\mathrm {def} }{=}}\ S(AB)_{\rho }-S(B)_{\rho }}
885:
205:
161:. The quantum conditional entropy was defined in terms of a conditional density operator
940:
848:
795:
740:
715:
Cerf, N. J.; Adami, C. (1997). "Negative
Entropy and Information in Quantum Mechanics".
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Cerf, N. J.; Adami, C. (1999-08-01). "Quantum extension of conditional probability".
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366:{\displaystyle S(AB)_{\rho }\ {\stackrel {\mathrm {def} }{=}}\ S(\rho ^{AB})}
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931:
16:
Measure of relative information in quantum information theory
921:
Wilde, Mark M. (2017), "Preface to the Second
Edition",
898:(2nd ed.). Cambridge: Cambridge University Press.
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925:, Cambridge University Press, pp. xi–xii,
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895:Quantum Computation and Quantum Information
373:, and the entropies of the subsystems are
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282:, the entropy of the joint system AB is
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211:In what follows, we use the notation
69:, the conditional entropy is written
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31:. It is a generalization of the
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252:Given a bipartite quantum state
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150:{\displaystyle H(A|B)_{\rho }}
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106:{\displaystyle S(A|B)_{\rho }}
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79:
1:
527:{\displaystyle S(B)_{\rho }}
37:classical information theory
749:10.1103/physrevlett.79.5194
189:{\displaystyle \rho _{A|B}}
21:conditional quantum entropy
999:
983:Quantum mechanical entropy
923:Quantum Information Theory
275:{\displaystyle \rho ^{AB}}
62:{\displaystyle \rho ^{AB}}
39:. For a bipartite state
29:quantum information theory
949:10.1017/9781316809976.001
233:{\displaystyle S(\cdot )}
804:10.1103/PhysRevA.60.893
718:Physical Review Letters
670:merging) was given by
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690:Unlike the classical
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886:Nielsen, Michael A.
857:10.1038/nature03909
849:2005Natur.436..673H
796:1999PhRvA..60..893C
741:1997PhRvL..79.5194C
692:conditional entropy
242:von Neumann entropy
159:von Neumann entropy
33:conditional entropy
676:Jonathan Oppenheim
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833:(7051): 673–676.
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251:
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198:Nicolas Cerf
20:
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536:mixed state
202:Chris Adami
702:References
686:Properties
248:Definition
932:1106.1445
914:844974180
812:119451904
645:ρ
628:−
623:ρ
571:ρ
520:ρ
474:ρ
437:ρ
395:ρ
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261:ρ
225:⋅
170:ρ
143:ρ
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977:Category
892:(2010).
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737:Bibcode
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827:Nature
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927:arXiv
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727:arXiv
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953:ISBN
910:OCLC
900:ISBN
861:PMID
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200:and
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