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Laser Physics Letters

On the relationship between: L Boltzmann entropy, M Planck distribution, C Shannon information and the principle of particles indistinguishability

S. N. Molotkov

Published Sep 22, 2020• 1 citation
Publisher

Abstract

Abstract The relationship between the fundamental formulas of statistical physics and information theory is considered. It is shown that the M Plank distribution for blackbody radiation and the quantum analog of C Shannon’s capacity for a frequency band-limited classical-quantum channel have a common basis – the principle of particles indistinguishability.

DOI
10.1088/1612-202x/abb093

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Abstract The relationship between the fundamental formulas of statistical physics and information theory is considered. It is shown that the M Plank distribution for blackbody radiation and the quantum analog of C Shannon’s capacity for a frequency band-limited classical-quantum channel have a common basis – the principle of particles indistinguishability.
Abstract The aim of the paper is the derivation of formulas for the capacities of a classical–quantum communication channel with Gaussian noise, which are analogs of the classical capacities and arise in many problems of quantum information theory, on a simple, heuristic and physically intuitively clear level. The classical capacity of a classical communication channel with limited frequency band and Gaussian information states diverges as noise tends to zero. In a consistent quantum consideration, the divergence is removed, which is a consequence of the fundamental physical principle of indistinguishability of particles—bosons in information states.
Information theory, introduced by Claude Shannon in 1948, provides a mathematical model, predominantly probabilistic, for addressing the challenges associated with measuring the amount of information in a message and efficiently and reliably storing the message (compression and protection). This theory has always been intricately connected to communication theory. Information theory not only relies on communication theory but also draws from fields such as statistics, statistical physics, and inference theory. In this chapter, a concise overview is presented, covering the mathematical foundations of classical and quantum information, their manipulation and associated qualifications. The fundamental concepts of classical information theory, including Shannon entropy, joint entropy, conditional entropy, and mutual information, are introduced. Additionally, basic concepts related to quantum information and quantum correlation measurements are presented.
Abstract This chapter discusses the principles of quantum information. The quantum mechanical equivalent of the Shannon noisy-channel communication theorem is obtained, along with some profound statements about the behavior of quantum information during generalised quantum measurements. The latter are very important in studies of quantum entanglement, but also in exploring the connections between thermodynamics, information theory, and quantum physics. This chapter also discusses equalities and inequalities related to entropy, the Holevo bound, capacity of a bosonic channel, information gained through measurements, relative entropy and thermodynamics, entropy increase due to erasure, and Landauer's erasure and data compression.
Abstract This book has discussed the foundations of quantum information science as well as the relationship between physics and information theory in general. It has considered the quantum equivalents of the Shannon coding and channel capacity theorems. The von Neumann entropy plays a role analogous to the Shannon entropy, and the Holevo bound is the analogue of Shannon's mutual information used to quantify the capacity of a classical channel. Quantum systems can process information more efficiently than classical systems in a number of different ways. Quantum teleportation and quantum dense coding can be performed using quantum entanglement. Entanglement is an excess of correlations that can exist in quantum physics and is impossible to reproduce classically (with what is termed “separable” states). The book has also demonstrated how to discriminate entangled from separable states using entanglement witnesses, as well as how to quantify entanglement, and looked at quantum computation and quantum algorithms.
The science of information theory begins with the observation that there is a fundamental link between probabilities and information. As early as the mid eighteenth century, Bayes recognized that probabilities depend on what we know; if we acquire additional information then this modifies the probability. For example, the probability that it is raining when I leave for work in the morning is about 0.2, but if I look out of the window ten minutes before leaving and see that it is raining then this additional information adjusts the probability to in excess of 0.9. Information is a function of probabilities: it is the entropy associated with the probability distribution. This conclusion grew out of investigations into the physical nature of entropy by Boltzmann and his followers. The full power of entropy as the quantity of information was revealed by Shannon in his mathematical theory of communication. This work laid the foundations for the development of information and communications theory by proving two powerful theorems which limit our ability to communicate information. The link between probability and information has far wider application than just communications. Indeed, we can expect the ideas of information theory to be applicable to any statistical or probabilistic problem. Quantum mechanics is a probabilistic theory and so it was inevitable that a quantum information theory would be developed. In quantum theory, probabilities are secondary quantities calculated by taking the squared modulus of probability amplitudes and this gives rise to interference effects. Consider, for example, the famous two-slit experiment depicted in Fig. 1.1. A single particle launched at the slits can arrive at a point P on the screen by passing either through slit 1 or through slit 2. In classical statistical mechanics this leads to a probability . . . P = P1 + P2, (1.1) . . . where P1 and P2 are, respectively, the probabilities that the particle passed through slit 1 or slit 2 and went on to arrive at the point P.