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Theory of Probability and Its Applications

Conditional Entropy for Infinite-Dimensional Quantum Systems

A. A. Kuznetsova

Published Nov 1, 2011• 33 citations
Publisher

Abstract

In this paper a general definition of conditional entropy for infinite-dimensional quantum systems is given based on the recent work of Holevo and Shirokov [Problems Inform. Transmission, 46 (2010), pp. 201–218] devoted to quantum mutual and coherent information in the infinite-dimensional case. Properties of the conditional entropy such as monotonicity, concavity, and subadditivity are also generalized to the infinite-dimensional case.

DOI
10.1137/s0040585x97985121

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In this paper a general definition of conditional entropy for infinite-dimensional quantum systems is given based on the recent work of Holevo and Shirokov [Problems Inform. Transmission, 46 (2010), pp. 201–218] devoted to quantum mutual and coherent information in the infinite-dimensional case. Properties of the conditional entropy such as monotonicity, concavity, and subadditivity are also generalized to the infinite-dimensional case.
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.
The work is devoted to study of quantum mutual information and coherent information -- the two important characteristics of quantum communication channel. Appropriate definitions of these quantities in the infinite-dimensional case are given and their properties are studied in detail. Basic identities relating quantum mutual information and coherent information of a pair of complementary channels are proved. An unexpected continuity property of quantum mutual information and coherent information following from the above identities is observed. An upper bound for the coherent information is obtained.
The paper is devoted to the study of quantum mutual information and coherent information, two important characteristics of a quantum communication channel. Appropriate definitions of these quantities in the infinite-dimensional case are given, and their properties are studied in detail. Basic identities relating the quantum mutual information and coherent information of a pair of complementary channels are proved. An unexpected continuity property of the quantum mutual information and coherent information, following from the above identities, is observed. An upper bound for the coherent information is obtained.
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.
What would be a reasonable definition of the conditional entropy of bipartite quantum processes, and what novel insight would it provide? We develop this notion using four information-theoretic axioms and define the corresponding quantitative formulas. Our definitions of the conditional entropies of channels are based on the generalized state and channel divergences, for instance, quantum relative entropy. We find that the conditional entropy of quantum channels has potential to reveal insights for quantum processes that aren't already captured by the existing entropic functions, entropy or conditional entropy, of the states and channels. The von Neumann conditional entropy S[A∣B]NS[A|B]_{\mathcal{N}}S[A∣B]N​ of the channel NA′B′→AB\mathcal{N}_{A'B'\to AB}NA′B′→AB​ is based on the quantum relative entropy, with system pairs A′,AA',AA′,A and B′,BB',BB′,B being nonconditioning and conditioning systems, respectively. We identify a connection between the underlying causal structure of a bipartite channel and its conditional entropy. In particular, we provide a necessary and sufficient condition for a bipartite quantum channel NA′B′→AB\mathcal{N}_{A'B'\to AB}NA′B′→AB​ in terms of its von Neumann conditional entropy S[A∣B]NS[A|B]_{\mathcal{N}}S[A∣B]N​, to have no causal influence from A′A'A′ to BBB. As a consequence, if S[A∣B]N<−log⁡∣A∣S[A|B]_{\mathcal{N}}< -\log|A|S[A∣B]N​<−log∣A∣ then the channel necessarily has causal influence (signaling) from A′A'A′ to BBB. Our definition of the conditional entropy establishes the strong subadditivity of the entropy for quantum channels. We also study the total amount of correlations possible due to quantum processes by defining the multipartite mutual information of quantum channels.