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arXiv:quant-ph/0104135 · Physical Review A

Nonadditive information measure and quantum entanglement in a class of mixed states of an Nn system

Sumiyoshi Abe

Published May 8, 2002• 38 citations
arXivHTMLPDF

Abstract

Generalizing Khinchin's classical axiomatic foundation, a basis is developed for nonadditive information theory with the Tsallis entropy indexed by q. The classical nonadditive conditional entropy is introduced and then translated into quantum theory. To examine if this theory has points superior to the ordinary additive information theory with the von Neumann entropy corresponding to the limit $\stackrel{\ensuremath{\rightarrow}}{q}1,$ separability of a one-parameter family of the Werner-Popescu states of the ${N}^{n}$ system (i.e., the n-partite N-level system) is discussed. The nonadditive information theory with $q>1$ is shown to yield a limitation on separability that is stronger than the one derived from the additive theory. How the strongest limitation can be obtained in the limit $\stackrel{\ensuremath{\rightarrow}}{q}\ensuremath{\infty}$ is also shown.

DOI
10.1103/physreva.65.052323

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Generalizing Khinchin's classical axiomatic foundation, a basis is developed for nonadditive information theory with the Tsallis entropy indexed by q. The classical nonadditive conditional entropy is introduced and then translated into quantum theory. To examine if this theory has points superior to the ordinary additive information theory with the von Neumann entropy corresponding to the limit q\ensuremath→1,\stackrel{\ensuremath{\rightarrow}}{q}1,q\ensuremath→​1, separability of a one-parameter family of the Werner-Popescu states of the Nn{N}^{n}Nn system (i.e., the n-partite N-level system) is discussed. The nonadditive information theory with q>1q>1q>1 is shown to yield a limitation on separability that is stronger than the one derived from the additive theory. How the strongest limitation can be obtained in the limit q\ensuremath→\ensuremath∞\stackrel{\ensuremath{\rightarrow}}{q}\ensuremath{\infty}q\ensuremath→​\ensuremath∞ is also shown.
Nonadditive classical information theory is developed in the axiomatic framework and then translated into quantum theory. The nonadditive conditional entropy associated with the Tsallis entropy indexed by q is given in accordance with the formalism of nonextensive statistical mechanics. The theory is applied to the problems of quantum entanglement and separability of the Werner-Popescu-type mixed state of a multipartite system, in order to examine if it has any points superior to the additive theory with the von Neumann entropy realized in the limit q → 1. It is shown that the nonadditive theory can lead to the necessary and sufficient condition for separability of the Werner-Popescu-type state, whereas the von Neumann theory can give only a much weaker condition…. Tsallis' nonextensive generalization of Boltzmann-Gibbs statistical mechanics [3, 15, 16] and its success in describing behaviors of a large class of complex systems naturally lead to the question of whether information theory can also admit an analogous generalization. If the answer is affirmative, then that will be of particular importance in connection with the problem of quantum entanglement and quantum theory of measurement [6, 8], in which necessities of a nonadditive information measure and an information content are suggested. One should also remember that there exists a conceptual similarity between a complex system and an entangled quantum system. In these systems, a "part" is indivisibly connected with the rest. An external operation on any part drastically influences the whole system, in general. Thus, the traditional reductionistic approach to an understanding of the nature of such a system may not work efficiently. In this chapter, we report a recent development in nonadditive quantum information theory based on the Tsallis entropy indexed by q [15] and its associated nonadditive conditional entropy [1]. This theory includes the ordinary additive theory with the von Neumann entropy in a special limiting case: q → To see if it has points superior to the additive theory, we apply it to the problems of separability and quantum entanglement.
Entanglement is the fundamental quantum property behind the now popular field of quantum transport of information. This quantum property is incompatible with the separation of a single system into two uncorrelated subsystems. Consequently, it does not require the use of an additive form of entropy. We discuss the problem of the choice of the most convenient entropy indicator, focusing our attention on a system of two qubits, and on a special set, denoted by I.\mathfrak{I}.I. This set contains both the maximally and partially entangled states that are described by density matrices diagonal in the Bell basis set. We select this set for the main purpose of making our work of analysis more straightforward. As a matter of fact, we find that in general the conventional von Neumann entropy is not a monotonic function of the entanglement strength. This means that the von Neumann entropy is not a reliable indicator of the departure from the condition of maximum entanglement. We study the behavior of a form of nonadditive entropy, made popular by the 1988 work by Tsallis [J. Stat. Phys. 52, 479 (1988)]. We show that in the set I,\mathfrak{I},I, implying the key condition of nonvanishing entanglement, this nonadditive entropy indicator turns out to be a strictly monotonic function of the strength of the entanglement, if entropy indexes q larger than a critical value Q are adopted. We argue that this might be a consequence of the nonadditive nature of the Tsallis entropy, implying that the world is quantum and that uncorrelated subsystems do not exist.
The problem of quantum-state inference and the concept of quantum entanglement are studied using a nonadditive measure in the form of the Tsallis entropy indexed by the positive parameter q. The maximum entropy principle associated with this entropy along with its thermodynamic interpretation are discussed in detail for the Einstein-Podolsky-Rosen pair of two spin-12\frac{1}{2}21​ particles. Given the data on the Bell-Clauser-Horne-Shimony-Holt observable, the analytic expression is presented for the inferred quantum entangled state. It is shown that for q>1,q>1,q>1, indicating the subadditive feature of the Tsallis entropy, the entangled region is small and enlarges as one goes into the superadditive regime where q<1.q<1.q<1. It is also shown that quantum entanglement can be quantified by the generalized (Kullback-Leibler) relative entropy.
We investigate the concept of entropy in probabilistic theories more general than quantum mechanics, with particular reference to the notion of information causality (IC) recently proposed by Pawlowski et al (2009 arXiv:0905.2292). We consider two entropic quantities, which we term measurement and mixing entropy. In the context of classical and quantum theory, these coincide, being given by the Shannon and von Neumann entropies, respectively; in general, however, they are very different. In particular, while measurement entropy is easily seen to be concave, mixing entropy need not be. In fact, as we show, mixing entropy is not concave whenever the state space is a non-simplicial polytope. Thus, the condition that measurement and mixing entropies coincide is a strong constraint on possible theories. We call theories with this property monoentropic . Measurement entropy is subadditive, but not in general strongly subadditive. Equivalently, if we define the mutual information between two systems A and B by the usual formula I ( A : B )= H ( A )+ H ( B )- H ( AB ), where H denotes the measurement entropy and AB is a non-signaling composite of A and B , then it can happen that I ( A : BC )< I ( A : B ). This is relevant to IC in the sense of Pawlowski et al : we show that any monoentropic non-signaling theory in which measurement entropy is strongly subadditive, and also satisfies a version of the Holevo bound, is informationally causal, and on the other hand we observe that Popescu–Rohrlich boxes, which violate IC, also violate strong subadditivity. We also explore the interplay between measurement and mixing entropy and various natural conditions on theories that arise in quantum axiomatics.
The quantum relative R\'enyi entropy of two noncommuting density matrices was recently defined from which its conditional entropy is deduced. This framework is here extended to the corresponding Tsallis relative entropy and to its conditional form. This expression of Tsallis conditional entropy is shown to witness entanglement beyond the method based on global and local spectra of composite quantum states. When the reduced density matrix happens to be a maximally mixed state, this conditional entropy coincides with the expression in terms of Tsallis entropies derived earlier by Abe and Rajagopal [Physica A 289, 157 (2001)]. The separability range in a noisy one-parameter family of WWW and Greenberger-Horne-Zeilinger states with three and four qubits is explored here and it is shown that the results inferred from negative Tsallis conditional entropy matches that obtained through Peres's partial transpose criterion for one-parameter family of WWW states, in one of its partitions. The criterion is shown to be nonspectral through its usefulness in identifying entanglement in isospectral density matrices.
We discuss the entropic criterion for separability of compound quantum systems for general nonadditive entropic forms based on arbitrary concave functions f. For any separable state, the generalized entropy of the whole system is shown to be not smaller than that of the subsystems, for any choice of f, providing thus a necessary criterion for separability. Nevertheless, the criterion is not sufficient and examples of entangled states with the same property are provided. This entails, in particular, that the conjecture about the positivity of the conditional Tsallis entropy for all q, a more stringent requirement than the positivity of the conditional von Neumann entropy, is actually a necessary but not sufficient condition for separability in general. The direct relation between the entropic criterion and the largest eigenvalues of the full and reduced density operators of the system is also discussed.