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arXiv:cond-mat/0101479 · Physical Review A

Quantum entanglement and entropy

Filippo Giraldi, Paolo Grigolini A

Published Aug 20, 2001• 29 citations
arXivHTMLPDF

Abstract

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 $\mathfrak{I}.$ 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 $\mathfrak{I},$ 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.

DOI
10.1103/physreva.64.032310

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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.
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.
In this chapter, we discuss several information measures that are important for quantifying the amount of information and correlations that are present in quantum systems. The first fundamental measure that we introduce is the von Neumann entropy (or simply quantum entropy ). It is the quantum generalization of the Shannon entropy, but it captures both classical and quantum uncertainty in a quantum state.1 The quantum entropy gives meaning to the notion of an information qubit . This notion is different from that of the physical qubit, which is the description of a quantum state of an electron or a photon. The information qubit is the fundamental quantum informational unit of measure, determining how much quantum information is present in a quantum system. The initial definitions here are analogous to the classical definitions of entropy, but we soon discover a radical departure from the intuitive classical notions from the previous chapter: the conditional quantum entropy can be negative for certain quantum states. In the classical world, this negativity simply does not occur, but it takes on a special meaning in quantum information theory. Pure quantum states that are entangled have stronger-than-classical correlations and are examples of states that have negative conditional entropy. The negative of the conditional quantum entropy is so important in quantum information theory that we even have a special name for it: the coherent information. We discover that the coherent information obeys a quantum data-processing inequality, placing it on a firm footing as a particular informational measure of quantum correlations. We then define several other quantum information measures, such as quantum mutual information, that bear similar definitions as in the classical world, but with Shannon entropies replaced with quantum entropies. This replacement may seem to make quantum entropy somewhat trivial on the surface, but a simple calculation reveals that a maximally entangled state on two qubits registers two bits of quantum mutual information (recall that the largest the mutual information can be in the classical world is one bit for the case of two maximally correlated bits). We then discuss several entropy inequalities that play an important role in quantum information processing: the monotonicity of quantum relative entropy, strong subadditivity, the quantum data-processing inequalities, and continuity of quantum entropy.
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 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 $\stackrel{\ensuremath{\rightarrow}}{q}\ensuremath{\infty}$ is also shown.
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.
We discuss the notion of hidden correlations in classical and quantum indivisible systems along with such characteristics of the correlations as the mutual information and conditional information corresponding to the entropic subadditivity condition and the entropic strong subadditivity condition. We present an analog of the Bayes formula for systems without subsystems, study entropic inequality for von Neumann entropy and Tsallis entropy of the single-qudit state, and discuss the inequalities for qubit and qutrit states as an example.