The entropy of an ensemble of systems is defined in classical statistical mechanics by the relation where k is the Boltzmann constant and p n is the probability of finding the system in state n. It is possible to generalize this definition of entropy to a quantum state ψ(x) with Fourier transform ϕ(p) by defining p n as1 The corresponding entropies for the two quadratures x ^ and p ^ are defined as For the coherent state S ( x ) = S ( p ) and has the minimum value. For the x quadrature squeezed state, S ( x ) decreases at the expense of S ( p ) and vice versa. This definition of entropy has applications in information theory and optical communication using squeezed states.
The entropy of an ensemble of systems is defined in classical statistical mechanics by the relation where k is the Boltzmann constant and p n is the probability of finding the system in state n. It is possible to generalize this definition of entropy to a quantum state ψ(x) with Fourier transform ϕ(p) by defining p n as1 The corresponding entropies for the two quadratures x ^ and p ^ are defined as For the coherent state S ( x ) = S ( p ) and has the minimum value. For the x quadrature squeezed state, S ( x ) decreases at the expense of S ( p ) and vice versa. This definition of entropy has applications in information theory and optical communication using squeezed states.
It is well known that a Shannon based definition of information entropy leads in the classical case to the Boltzmann definition of entropy. It is tempting to regard the Von Neumann entropy as the corresponding quantum mechanical definition. But the latter has little to do with information theory. 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.
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.
This paper introduces a novel definition of non-classical information entropy, rooted in the principles of statistical physics, particularly quantum mechanics. The core premise is that traditional information entropy, derived from classical probability theory, is insufficient to capture the essence of information within non-classical systems exhibiting phenomena such as superposition and entanglement. We propose a definition based on the concept of accessible states and their associated probabilities, extending the notion of entropy beyond the realm of purely deterministic systems. The paper details the mathematical formulation of this non-classical information entropy, denoted as *Hnc*, and presents a method for its calculation. The method relies on identifying the set of accessible states for a given system and then applying a generalized form of Shannon's entropy formula, modified to account for the inherent uncertainties associated with quantum mechanics. The resulting *Hnc* provides a quantitative measure of the information content within non-classical systems, offering a potential framework for studying information processing and dynamics in quantum environments. The key contribution lies in bridging the gap between information theory and non-classical statistical physics, offering a new lens through which to understand information in quantum systems.
The Gibbs entropy of a macroscopic classical system is a function of a probability distribution over phase space, i.e., of an ensemble. In contrast, the Boltzmann entropy is a function on phase space, and is thus defined for an individual system. Our aim is to discuss and compare these two notions of entropy, along with the associated ensemblist and individualist views of thermal equilibrium. Using the Gibbsian ensembles for the computation of the Gibbs entropy, the two notions yield the same (leading order) values for the entropy of a macroscopic system in thermal equilibrium. The two approaches do not, however, necessarily agree for non-equilibrium systems. For those, we argue that the Boltzmann entropy is the one that corresponds to thermodynamic entropy, in particular in connection with the second law of thermodynamics. Moreover, we describe the quantum analog of the Boltzmann entropy, and we argue that the individualist (Boltzmannian) concept of equilibrium is supported by the recent works on thermalization of closed quantum systems.
The Gibbs entropy of a macroscopic classical system is a function of a probability distribution over phase space, i.e., of an ensemble. In contrast, the Boltzmann entropy is a function on phase space, and is thus defined for an individual system. Our aim is to discuss and compare these two notions of entropy, along with the associated ensemblist and individualist views of thermal equilibrium. Using the Gibbsian ensembles for the computation of the Gibbs entropy, the two notions yield the same (leading order) values for the entropy of a macroscopic system in thermal equilibrium. The two approaches do not, however, necessarily agree for non-equilibrium systems. For those, we argue that the Boltzmann entropy is the one that corresponds to thermodynamic entropy, in particular, in connection with the second law of thermodynamics. Moreover, we describe the quantum analog of the Boltzmann entropy, and we argue that the individualist (Boltzmannian) concept of equilibrium is supported by the recent works on thermalization of closed quantum systems.
In thermodynamics and statistical mechanics entropy is the fundamental physical quantity to describe the evolution of a statistical ensemble. Its microscopic definition was provided by Boltzmann through the celebrated expression S = kB ln Γ, where kB is the Boltzmann constant and Γ is the number of microstates exploiting the given macroscopic properties. In this context, it is well known that in classical mechanics the entropy: i) allows the violation of the uncertainty principle [1], [2]; ii) can be considered as a special case of the so-called Boltzmann-Gibbs-Shannon entropy that enables one to apply results of information theory to physics [1], [3]. In particular, by introducing the principle of maximum entropy (MEP) it was found possible to derive rigorous hydrodynamic (HD) models based on the moments of the distribution function to all orders of a power expansion and including appropriate closure conditions [4], [5]. Accordingly, making use of the Lagrange multipliers technique, it was found possible to construct the set of evolution equations for the macro-variables of interest. Apart from some partial attempts [3], [6], this is no longer the case in quantum mechanics. Here, the main difficulties concern with: i) the definition of a proper quantum entropy that includes particle indistinguishability; ii) the formulation of a global quantum MEP (QMEP) that allows one to obtain a quantum distribution function both for thermodynamic equilibrium and nonequilibrium configurations. From one hand, in the framework of a nonlocal quantum theory, the generalization of the corresponding Lagrange multipliers is also an open problem. Fron another hand, a rigorous formulation of quantum HD (QHD) closed models is a demanding issue for many kinds of problems in quantum systems like, interacting fermionic and bosonic gases, quantum turbulence, quantum fluids, quantized vortices, nuclear physics, confined carrier transport in semiconductor heterostrucures, phonon and electron transport in nanostructures, nanowires and thin layers. The aim of this talk is to address the above drawbacks by defining a quantum entropy in terms of the reduced density matrix, thus explicitly incorporating quantum statistics into problems involving a system of identical particles. As a further step, with respect to the uncertainty principle, starting from the Wigner representation we formulate a quantum maximum entropy principle which, in the framework of information theory, allows us to obtain a non-local theory for the system under study. As a final step, we determine a closed quantum hydrodynamic model for the macroscopic variables used as constraints in the QMEP approach.