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Fundamental properties of Tsallis relative entropy

Shigeru Furuichi, 研二郎 柳, 憲 栗山

Published Dec 1, 2004• 0 citations
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Abstract

Fundamental propenies for the Tsallis relative entropy in both classical and quantum systems are studied.As one of our main results, we give the parametric extension of the trace inequality between the quantum relative entropy and the minus of the trace of the relative operator entropy given by Hiai and Petz.The monotonicity of the quantum Tsallis relative entropy for the trace preserving completely positive linear map is also shown without the assumption that the density operators are invertible.The generalized Tsallis relative entropy is defined and its subadditivity is shown by its joint convexity.Moreover, the generalized Peierls-Bogoliubov inequality is also proven.

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Fundamental propenies for the Tsallis relative entropy in both classical and quantum systems are studied.As one of our main results, we give the parametric extension of the trace inequality between the quantum relative entropy and the minus of the trace of the relative operator entropy given by Hiai and Petz.The monotonicity of the quantum Tsallis relative entropy for the trace preserving completely positive linear map is also shown without the assumption that the density operators are invertible.The generalized Tsallis relative entropy is defined and its subadditivity is shown by its joint convexity.Moreover, the generalized Peierls-Bogoliubov inequality is also proven.
Fundamental properties for the Tsallis relative entropy in both classical and quantum systems are studied. As one of our main results, we give the parametric extension of the trace inequality between the quantum relative entropy and the minus of the trace of the relative operator entropy given by Hiai and Petz. The monotonicity of the quantum Tsallis relative entropy for the trace preserving completely positive linear map is also shown without the assumption that the density operators are invertible. The generalized Tsallis relative entropy is defined and its subadditivity is shown by its joint convexity. Moreover, the generalized Peierls–Bogoliubov inequality is also proven.
Quantum entropy and skew information play important roles in quantum information science. They are defined by the trace of the positive operators so that the trace inequalities often have important roles to develop the mathematical theory in quantum information science. In this paper, we study some properties for information quantities in quantum system through trace inequalities. Especially, we give upper bounds and lower bounds of Tsallis relative entropy, which is a one-parameter extension of the relative entropy in quantum system. In addition, we compare the known bounds and the new bounds, for both upper and lower bounds, respectively. We also give an inequality for generalized skew information by introducing a generalized correlation measure.
Quantum entropy and skew information play important roles in quantum information science. They are defined by traces of positive operators, hence trace inequalities often have important roles to develop the mathematical theory in quantum information science. In this article, we study some properties for information quantities in quantum systems through trace inequalities. Especially, we give upper bounds and lower bounds on the Tsallis relative entropy, which is a one-parameter extension of the relative entropy in quantum systems. In addition, we compare the known bounds and the new bounds, for both upper and lower bounds, respectively. We also give an inequality for generalized skew information by introducing a generalized correlation measure.
The relative entropy is a principal measure of distinguishability in quantum information theory, with its most important property being that it is non-increasing with respect to noisy quantum operations. Here, we establish a remainder term for this inequality that quantifies how well one can recover from a loss of information by employing a rotated Petz recovery map. The main approach for proving this refinement is to combine the methods of [Fawzi and Renner, 2014] with the notion of a relative typical subspace from [Bjelakovic and Siegmund-Schultze, 2003]. Our paper constitutes partial progress towards a remainder term which features just the Petz recovery map (not a rotated Petz map), a conjecture which would have many consequences in quantum information theory. A well known result states that the monotonicity of relative entropy with respect to quantum operations is equivalent to each of the following inequalities: strong subadditivity of entropy, concavity of conditional entropy, joint convexity of relative entropy, and monotonicity of relative entropy with respect to partial trace. We show that this equivalence holds true for refinements of all these inequalities in terms of the Petz recovery map. So either all of these refinements are true or all are false.
The relative entropy is a principal measure of distinguishability in quantum information theory, with its most important property being that it is non-increasing with respect to noisy quantum operations. Here, we establish a remainder term for this inequality that quantifies how well one can recover from a loss of information by employing a rotated Petz recovery map. The main approach for proving this refinement is to combine the methods of [Fawzi and Renner, arXiv:1410.0664] with the notion of a relative typical subspace from [Bjelakovic and Siegmund-Schultze, arXiv:quant-ph/0307170]. Our paper constitutes partial progress towards a remainder term which features just the Petz recovery map (not a rotated Petz map), a conjecture which would have many consequences in quantum information theory. A well known result states that the monotonicity of relative entropy with respect to quantum operations is equivalent to each of the following inequalities: strong subadditivity of entropy, concavity of conditional entropy, joint convexity of relative entropy, and monotonicity of relative entropy with respect to partial trace. We show that this equivalence holds true for refinements of all these inequalities in terms of the Petz recovery map. So either all of these refinements are true or all are false.