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International Journal of Fundamental Physical Sciences

Von Neumann Entropy by Logarithmic Method

Bijan Nikouravan

Published Dec 1, 2019• 1 citation
Publisher

Abstract

The Von Neumann entropy plays a central role in the quantum information theory and is a concave function and following the property . In this paper, we introduce a new proof for the linearity of Von Neumann entropy in the rate without using the above inequality. Here the Von Neumann entropy is concave; that is, given weights and density matrices . Roughly speaking, we will show that in the rate case, the Von Neumann entropy is linear without using Fannes inequality. https://orcid.org/0000-0003-4308-1632

DOI
10.14331/ijfps.2019.330132

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Abstract Chapter 20 provides a discussion of entropies and entropic quantities in quantum information theory, briefly considering the generalizations of the Rényi entropy to the quantum regime, but mainly focusing on the von Neumann entropy and its properties such as concavity, the Araki-Lieb inequality, and subadditivity. We introduce the quantum relative entropy, quantum joint entropy, quantum conditional entropy, and quantum mutual information, and prove key properties and results for these quantities, including Klein’s inequality, joint convexity, additivity, and monotonicity under completely positive and trace-preserving maps of the relative entropy, as well as weak monotonicity and strong subadditivity of the von Neumann entropy. We then analyse the relation of (negative) conditional entropy and entanglement. In this context we discuss the conditional amplitude operator and the mutual amplitude operator, as well as conditional Rényi entropies and their role for entanglement detection.
Von Neumann defined the informational entropy of a density matrix ρ by the expression S(ρ)=−tr(ρ ln ρ). [J. Von Neumann, Die Mathematischen Grundlagen der Quantummechanik (Springer-Verlag, Berlin, 1932)]. Here, starting from the definitions of Shannon entropy and of Renyi entropy of random variables, and using some (reasonable) rules of inference, two expressions are obtained for the quantum entropy of a given square (deterministic) matrix, irrespective of any probabilistic framework and/or any randomization technique. These definitions apply directly to linear operators on Hilbert spaces even when they are not of positive trace class, and they contain Von Neumann entropy as a special case. These new measures of uncertainty could provide new approaches to some problems related to structure complexity.
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.
We exhibit infinitely many new, constrained inequalities for the von Neumann entropy, and show that they are independent of each other and the known inequalities obeyed by the von Neumann entropy (basically strong subadditivity). The new inequalities were proved originally by Makarychevfor the Shannon entropy, using properties of probability distributions. Our approach extends the proof of the inequalities to the quantum domain, and includes their independence for the quantum and also the classical cases.