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arXiv:quant-ph/9805055 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Information measures and classicality in quantum mechanics

Charis Anastopoulos

Published Dec 30, 1998• 14 citations
arXivHTMLPDF

Abstract

We study information measures in quantum mechanics, with the particular emphasis on providing a quantification of the notion of predictability and classicality. Our primary tool is the Shannon-Wehrl entropy I. We give a precise criterion for phase space classicality of states and argue that in view of this (a) I provides a good measure for the degree of deviation from classicality in closed systems and (b) $I\ensuremath{-}S (S$ the von Neumann entropy) plays the same role in open quantum system. We examine particular examples in nonrelativistic quantum mechanics. Finally we generalize the discussion into the field theory case, and (this being one of our main motivations) we comment on the field classicalization in early universe cosmology.

DOI
10.1103/physrevd.59.045001

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We study information measures in quantum mechanics, with the particular emphasis on providing a quantification of the notion of predictability and classicality. Our primary tool is the Shannon-Wehrl entropy I. We give a precise criterion for phase space classicality of states and argue that in view of this (a) I provides a good measure for the degree of deviation from classicality in closed systems and (b) I\ensuremath−S(SI\ensuremath{-}S (SI\ensuremath−S(S the von Neumann entropy) plays the same role in open quantum system. We examine particular examples in nonrelativistic quantum mechanics. Finally we generalize the discussion into the field theory case, and (this being one of our main motivations) we comment on the field classicalization in early universe cosmology.
Information plays an important role in our understanding of the physical world. Hence we propose an entropic measure of information for any physical theory that admits systems, states and measurements. In the quantum and classical worlds, our measure reduces to the von Neumann and Shannon entropies, respectively. It can even be used in a quantum or classical setting where we are only allowed to perform a limited set of operations. In a world that admits superstrong correlations in the form of non-local boxes, our measure can be used to analyze protocols such as superstrong random access encodings and the violation of 'information causality'. However, we also show that in such a world no entropic measure can exhibit all the properties we commonly accept in a quantum setting. For example, there exists no 'reasonable' measure of conditional entropy that is subadditive. Finally, we prove a coding theorem for some theories that is analogous to the quantum and classical settings, providing us with an appealing operational interpretation.
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.
It is well known that a Shannon-based definition of information entropy leads in the classical case to the Boltzmann entropy. It is tempting to regard the von Neumann entropy as the corresponding quantum-mechanical definition. But the latter is problematic from the quantum information point of view. Consequently, we introduce a new definition of entropy that reflects the inherent uncertainty of quantum-mechanical states. We derive for it an explicit expression, and discuss some of its general properties. We distinguish between the minimum uncertainty entropy of pure states, and the excess statistical entropy of mixtures.
We present an entropic description of quantum-mechanical states based on an operational approach to a phase-space measurement. We give a simple phase-space interpretation of sampling entropies in terms of which we derive very general entropic uncertainty relations reflecting the degree of the phase-space uncertainty of the quantum-mechanical state in the given measurement (i.e., for a given ``quantum-ruler'' state). We relate the sampling entropy to the von Neumann and Shannon entropy and show that the Wehrl entropy represents a particular example of a sampling entropy when the quantum ruler is represented by coherent states.
The notion of Shannon entropy is crucial for the theory of classical information. In quantum information theory, an analogous key role is played by the von Neumann entropy: quantum information processing is closely related to entropy dynamics. This reveals a direct link with the theory of quantum chaotic systems, which can be characterized by a positive entropy production. Furthermore, noise, which inevitably affects any quantum system, can be modeled by a random quantum operation or by coupling to an environment in a generic chaotic state. In this contribution, we emphasize the universality of quantum chaotic dynamics and discuss its implications for quantum information processing.
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 Neumman entropy. It is the quantum analog of the Shannon entropy, but it captures both classical and quantum uncertainty in a quantum state. The von Neumann entropy gives meaning to a notion of the information qubit . This notion is different from that of the physical qubit, which is the description of a quantum state in an electron or a photon. The information qubit is the fundamental quantum informational unit of measure, determining how much quantum information is in a quantum system. The beginning 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 a special meaning in quantum information theory. Pure quantum states that are entangled have stronger-than-classical spatial 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.