Abstract Entropic distance measures for quantum mechanical probability distributions, which are characterized by nodal structure and symmetry holes, are considered. We illustrate how the Kullback‐Leibler (KL) distance is not well defined in some instances and propose instead the use of the cumulative residual Kullback‐Leibler (CRKL) distance. The KL and CRKL measures are compared and contrasted for some representative quantum mechanical systems: The particle in an infinite well, the harmonic oscillator, and hydrogenic systems. We present cases where CRKL can be used to obtain distances whereas KL cannot be used, and also highlight examples where the KL and CRKL measures yield different behaviors and interpretations. An extension of the CRKL definition is obtained for application to harmonic oscillator systems defined over [−∞, ∞]. Distance measures for two‐variable (particle) distributions are also considered to address generalizations of the mutual information correlation measure. The use of CRKL in measuring distances between orbitals is also discussed.
Abstract Entropic distance measures for quantum mechanical probability distributions, which are characterized by nodal structure and symmetry holes, are considered. We illustrate how the Kullback‐Leibler (KL) distance is not well defined in some instances and propose instead the use of the cumulative residual Kullback‐Leibler (CRKL) distance. The KL and CRKL measures are compared and contrasted for some representative quantum mechanical systems: The particle in an infinite well, the harmonic oscillator, and hydrogenic systems. We present cases where CRKL can be used to obtain distances whereas KL cannot be used, and also highlight examples where the KL and CRKL measures yield different behaviors and interpretations. An extension of the CRKL definition is obtained for application to harmonic oscillator systems defined over [−∞, ∞]. Distance measures for two‐variable (particle) distributions are also considered to address generalizations of the mutual information correlation measure. The use of CRKL in measuring distances between orbitals is also discussed.
This work presents an upper-bound to value that the Kullback-Leibler (KL) divergence can reach for a class of probability distributions called quantum distributions (QD). The aim is to find a distribution U which maximizes the KL divergence from a given distribution P under the assumption that P and U have been generated by distributing a given discrete quantity, a quantum. Quantum distributions naturally represent a wide range of probability distributions that are used in practical applications. Moreover, such a class of distributions can be obtained as an approximation of any probability distribution. The retrieving of an upper-bound for the entropic divergence is here shown to be possible under the condition that the compared distributions are quantum distributions over the same quantum value, thus they become comparable. Thus, entropic divergence acquires a more powerful meaning when it is applied to comparable distributions. This aspect should be taken into account in future developments of divergences. The theoretical findings are used for proposing a notion of normalized KL divergence that is empirically shown to behave differently from already known measures.
Abstract Entropic uncertainty and statistical correlation measures, based on survival and cumulative densities, are explored in some representative quantum systems. We illustrate how the cumulative residual entropy in the quantum well system recovers the correct classical behavior for larger quantum numbers while the Shannon entropy does not. Two interacting and noninteracting oscillators are used to examine two‐particle entropies and their related correlation measures. The joint cumulative residual entropy does distinguish between symmetric and antisymmetric wave functions in interacting systems as the interaction is turned on. The joint Shannon entropy does not distinguish between the symmetries even in the presence of interaction. Conversely, the joint Shannon entropy and joint cumulative residual entropy are both unable to distinguish between the symmetries for certain states of the noninteracting oscillators. As measures of statistical correlation, the mutual information and the cross cumulative residual entropy yield similar behaviors as a function of the strength of the interparticle interaction.
The Kullback-Leibler entropy is defined as: - Integral dx P(x) ln { P(x)/R(x)} where P(x) and R(x) are two classical probability distributions i.e. real, positive functions with values between 0 and 1. In (1), it is shown that the Kullback-Leibler entropy is equivalent to Fisher’s information if R(x) = P(x+delta x) where delta x is a tiny constant. We argue that in this case, the Kullback-Leibler entropy represents either average momentum multiplied by spatial density in a quantum bound state, i.e. Integral dx W(x)W(x) {dW/dx / W} which is 0 because overall the particle does not move in one direction (odd function). This suggests forwards and backwards internal motion which yields 0 overall translation, similar to fluctuations (in x not t) . In addition, it represents the “fluctuations” or average momentum for half the classical probability distribution associated with a quantum free particle exp(ipx), namely C cos(px) cos(px). Again the Kullback-Leibler entropy for this distribution (or sin(px)sin(px)) is 0. (This is equivalent to the result of a particle in an infinite potential well if one considers pave in cos(pave x), the wavefunction.) Thus a statistical quantity, namely Kullback-Leibler entropy, seems to be associated with quantum mechanical motion (average momentum). This motion is linked with zero Kullback-Leibler entropy where P(x) is a distribution and R(x)=P(x+delta x). Furthermore, this seems to show, we argue, that d/dx in Fisher’s information is linked with physical momentum.
In this chapter we shall discuss the recent progresses of information theoretic tools in the context of free and confined harmonic oscillator. Confined quantum systems have provided appreciable interest in areas of physics, chemistry, biology,etc., since its inception. A particle under extreme pressure environment unfolds many fascinating, notable physical and chemical changes. The desired effect is achieved by reducing the spatial boundary from infinity to a finite region. Similarly, in the last decade, information measures were investigated extensively in diverse quantum problems, in both free and constrained situations. The most prominent amongst these are: Fisher information, Shannon entropy, Renyi entropy , Tsallis entropy, Onicescu energy and several complexities. Arguably, these are the most effective measures of uncertainty, as they do not make any reference to some specific points of a respective Hilbert space. These have been invoked to explain several physico-chemical properties of a system under investigation. Kullback-Leibler divergence or relative entropy describes how a given probability distribution shifts from a reference distribution function. This characterizes a measure of discrimination between two states. In other words, it extracts the change of information in going from one state to another.
Chapter 5 delves into divergences and distance measures, which are crucial for comparing quantum states. It begins with classical divergences such as the Kullback–Leibler and Jensen–Shannon, then advances to their quantum counterparts, discussing their optimal characteristics. Influenced by quantum resource theories, these quantum extensions provide foundational insights into the robust tools of resource theories. The chapter concentrates on particular divergences that serve as true metrics, including the trace distance and a variant of the fidelity, and explores the concept of distance between subnormalized states, which is essential in the context of quantum measurements. It emphasizes the purified distance, a useful tool for understanding the entanglement cost of quantum systems, setting the stage for further exploration in later chapters. The chapter offers a mathematically approachable survey of these measures, underscoring their practical importance in quantum information theory.
A definition of three-variable cumulative residual entropy is introduced, and then used to obtain expressions for higher order or triple-wise correlation measures, that are based on cumulative residual densities. These information measures are calculated in continuous variable quantum systems comprised of three oscillators, and their behaviour compared to the analogous measures from Shannon information theory. There is an overall consistency in the behaviour of the newly introduced measures as compared to the Shannon ones. There are, however, differences in interpretation, in the case of three uncoupled oscillators, where the correlation is due to wave function symmetry. In interacting systems, the cumulative based measures are shown in order to detect salient features, which are also present in the Shannon based ones.