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arXiv:1403.7062 · Mathematical Inequalities & Applications

Some inequalities for quantum Tsallis entropy related to the strong subadditivity

Dénes Petz, Dániel Virosztek

Published Jan 1, 2015• 13 citations
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Abstract

In this paper we investigate the inequality S q ( 123 ) + S q ( 2 ) S q ( 12 ) + S q ( 23 )( * ) where 123 is a state on a finite dimensional Hilbert space H 1 H 2 H 3 , and S q is the Tsallis entropy. It is well-known that the strong subadditivity of the von Neumnann entropy can be derived from the monotonicity of the Umegaki relative entropy. Now, we present an equivalent form of (*), which is an inequality of relative quasi-entropies. We derive an inequality of the form S q ( 123 ) + S q ( 2 ) S q ( 12 ) + S q ( 23 ) + f q ( 123 ) , where f 1 ( 123 ) = 0 . Such a result can be considered as a generalization of the strong subadditivity of the von Neumnann entropy. One can see that (*) does not hold in general (a picturesque example is included in this paper), but we give a sufficient condition for this inequality, as well.

DOI
10.7153/mia-18-41

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In this paper we investigate the inequality S q ( 123 ) + S q ( 2 ) S q ( 12 ) + S q ( 23 )( * ) where 123 is a state on a finite dimensional Hilbert space H 1 H 2 H 3 , and S q is the Tsallis entropy. It is well-known that the strong subadditivity of the von Neumnann entropy can be derived from the monotonicity of the Umegaki relative entropy. Now, we present an equivalent form of (), which is an inequality of relative quasi-entropies. We derive an inequality of the form S q ( 123 ) + S q ( 2 ) S q ( 12 ) + S q ( 23 ) + f q ( 123 ) , where f 1 ( 123 ) = 0 . Such a result can be considered as a generalization of the strong subadditivity of the von Neumnann entropy. One can see that () does not hold in general (a picturesque example is included in this paper), but we give a sufficient condition for this inequality, as well.
In this paper we investigate the inequality Sq(ρ123) + Sq(ρ2) ≤ Sq(ρ12) + Sq(ρ23) (∗) where ρ123 is a state on a finite dimensional Hilbert space H1 ⊗ H2 ⊗ H3, and Sq is the Tsallis entropy. It is well-known that the strong subadditivity of the von Neumnann entropy can be derived from the monotonicity of the Umegaki relative entropy. Now, we present an equivalent form of (*), which is an inequality of relative quasi-entropies. We derive an inequality of the form Sq(ρ123) + Sq(ρ2) ≤ Sq(ρ12) + Sq(ρ23) + fq(ρ123), where f1(ρ123) = 0. Such a result can be considered as a generalization of the strong subadditivity of the von Neumnann entropy. One can see that (*) does not hold in general (a picturesque example is included in this paper), but we give a sufficient condition for this inequality, as well.
We prove a generalization of the strong-subadditivity (SSA) inequality in the context of finite dimensional von Neumann algebras. Our generalized SSA inequality subsumes various entropic uncertainty relations, such as the Maassen-Uffink inequality in the presence of quantum memory. Motivated by the generalization of SSA, we introduce an analog of conditional mutual information and discuss superadditivity and monotonicity under certain operations. Our proof uses complex interpolation and noncommutative mixed LpL_pLp​ spaces.
This article presents self-contained proofs of the strong subadditivity inequality for von Neumann’s quantum entropy, S(ρ), and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps. Moreover, the approach presented here, which is based on Klein’s inequality and Lieb’s theorem that the function A→Tr eK+log A is concave, allows one to obtain conditions for equality. In the case of strong subadditivity, which states that S(ρ123)+S(ρ2)⩽S(ρ12)+S(ρ23) where the subscripts denote subsystems of a composite system, equality holds if and only if log ρ123=log ρ12−log ρ2+log ρ23. Using the fact that the Holevo bound on the accessible information in a quantum ensemble can be obtained as a consequence of the monotonicity of relative entropy, we show that equality can be attained for that bound only when the states in the ensemble commute. The article concludes with an Appendix giving a short description of Epstein’s elegant proof of Lieb’s theorem.
We give an explicit characterisation of the quantum states which saturate the strong subadditivity inequality for the von Neumann entropy. By combining a result of Petz characterising the equality case for the monotonicity of relative entropy with a recent theorem by Koashi and Imoto, we show that such states will have the form of a so-called short quantum Markov chain, which in turn implies that two of the systems are independent conditioned on the third, in a physically meaningful sense. This characterisation simultaneously generalises known necessary and sufficient entropic conditions for quantum error correction as well as the conditions for the achievability of the Holevo bound on accessible information.
We prove a generalization of the strong subadditivity of the von Neumann entropy for bosonic quantum Gaussian systems. Such generalization determines the minimum values of linear combinations of the entropies of subsystems associated to arbitrary linear functions of the quadratures, and holds for arbitrary quantum states including the scenario where the entropies are conditioned on a memory quantum system. We apply our result to prove new entropic uncertainty relations with quantum memory, a generalization of the quantum Entropy Power Inequality, and the linear time scaling of the entanglement entropy produced by quadratic Hamiltonians.