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On a Quantum Version of Shannon's Conditional Entropy
Author(s) -
Schrader R.
Publication year - 2000
Publication title -
fortschritte der physik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.469
H-Index - 71
eISSN - 1521-3978
pISSN - 0015-8208
DOI - 10.1002/1521-3978(200008)48:8<747::aid-prop747>3.0.co;2-t
Subject(s) - von neumann entropy , hilbert space , sigma , mathematics , quantum relative entropy , conditional entropy , joint quantum entropy , conditional quantum entropy , entropy (arrow of time) , quantum , min entropy , rényi entropy , mathematical physics , generalized relative entropy , physics , quantum mechanics , combinatorics , statistical physics , pure mathematics , quantum discord , principle of maximum entropy , quantum entanglement , statistics , maximum entropy thermodynamics
In this article we propose a quantum version of Shannon's conditional entropy. Given two density matrices ρ and σ on a finite dimensional Hilbert space and with S ( ρ ) = —Tr ρ ln ρ being the usual von Neumann entropy, this quantity S ( ρ | σ ) is concave in ρ and satisfies 0 ≤ S ( ρ | σ ) ≤ S ( ρ ), a quantum analogue of Shannon's famous inequality. Thus we view S ( ρ | σ ) as the entropy of ρ conditioned by σ . The second inequality is an equality if σ is a multiple of the identity. In contrast to the classical case, however, S ( ρ | ρ ) = 0 if and only if the non‐vanishing eigenvalues of ρ are all non‐degenerate. Also in general and again in contrast to the corresponding classical situation S ( ρ , σ ) = S ( σ ) + S ( ρ | σ ) is not symmetric in ρ and σ even if they commute. We also show that there is no quantum version of conditional entropy in terms of two density matrices, which shares more properties with the classical case and which in particular reduces to the classical case when the two density matrices commute. As an alternative we propose to use spectral resolutions of the unit matrix instead of density matrices. We briefly compare this with the algebraic approach of Connes and Størmer and Connes, Narnhofer and Thirring.