We analyze manifestations of so-called information entropy production and the role of information dynamics in the Schrödinger picture quantum evolution. PACS numbers: 03.65.-w, 02.50.-r In a number of recent papers, [1]-[5] c.f. also [6]-[13], various links were disclosed between the concept of Fisher information in statistical inference theory (plus related indeterminacy-type inequalities) and localization properties of Schrödinger wave functions. Except for a brief notice in Ref. [13], no attention has been paid to the full-fledged Schrödinger quantum evolution and to the fact that a central role in the formalism is in fact played not by the Fisher information, but by the (formally related) concept of (configurational) information entropy production, which appears to be inseparably tied with the dynamics of quantum probability densities and the persistence of steady (stationary) states. Information theory concepts are normally introduced in the context of equilibrium and nonequlibrium statistical mechanics as natural tools for quantifying phase-space disorder (dynamical origin of increasing entropy, irreversibility and time arrow issues). However, their range of applicability
We analyze manifestations of so-called information entropy production and the role of information dynamics in the Schrödinger picture quantum evolution. PACS numbers: 03.65.-w, 02.50.-r In a number of recent papers, [1]-[5] c.f. also [6]-[13], various links were disclosed between the concept of Fisher information in statistical inference theory (plus related indeterminacy-type inequalities) and localization properties of Schrödinger wave functions. Except for a brief notice in Ref. [13], no attention has been paid to the full-fledged Schrödinger quantum evolution and to the fact that a central role in the formalism is in fact played not by the Fisher information, but by the (formally related) concept of (configurational) information entropy production, which appears to be inseparably tied with the dynamics of quantum probability densities and the persistence of steady (stationary) states. Information theory concepts are normally introduced in the context of equilibrium and nonequlibrium statistical mechanics as natural tools for quantifying phase-space disorder (dynamical origin of increasing entropy, irreversibility and time arrow issues). However, their range of applicability
Shannon entropy and Fisher information functionals are known to quantify certain information-theoretic properties of continuous probability distributions of various origins. We carry out a systematic study of these functionals, while assuming that the pertinent probability density has a quantum mechanical appearance ρ≐∣ψ∣2, with ψ∈L2(R). Their behavior in time, due to the quantum Schrödinger picture evolution-induced dynamics of ρ(x,t) is investigated as well, with an emphasis on thermodynamical features of quantum motion.
Currently, ‘time’ does not play any essential role in quantum information theory. In this sense, quantum information theory is underdeveloped similarly to how quantum physics was underdeveloped before Erwin Schrödinger introduced his famous equation for the evolution of a quantum wave function. In this review article, we cope with the problem of time for one of the central quantities in quantum information theory: entropy. Recently, a replica trick formalism, the so-called ‘multiple parallel world’ formalism, has been proposed that revolutionizes entropy evaluation for quantum systems. This formalism is one of the first attempts to introduce ‘time’ in quantum information theory. With the total entropy being conserved in a closed system, entropy can flow internally between subsystems; however, we show that this flow is not limited only to physical correlations as the literature suggest. The nonlinear dependence of entropy on the density matrix introduces new types of correlations with no analogue in physical quantities. Evolving a number of replicas simultaneously makes it possible for them to exchange particles between different replicas. We will summarize some of the recent news about entropy in some example quantum devices. Moreover, we take a quick look at a new correspondence that was recently proposed that provides an interesting link between quantum information theory and quantum physics. The mere existence of such a correspondence allows for exploring new physical phenomena as the result of controlling entanglement in a quantum device.
Non-relativistic quantum theory is derived from information codified into an appropriate statistical model. The basic assumption is that there is an irreducible uncertainty in the location of particles: positions constitute a configuration space and the corresponding probability distributions constitute a statistical manifold. The dynamics follows from a principle of inference, the method of Maximum Entropy. The concept of time is introduced as a convenient way to keep track of change. A welcome feature is that the entropic dynamics notion of time incorporates a natural distinction between past and future. The statistical manifold is assumed to be a dynamical entity: its curved and evolving geometry determines the evolution of the particles which, in their turn, react back and determine the evolution of the geometry. Imposing that the dynamics conserve energy leads to the Schroedinger equation and to a natural explanation of its linearity, its unitarity, and of the role of complex numbers. The phase of the wave function is explained as a feature of purely statistical origin. There is a quantum analogue to the gravitational equivalence principle.
The importance of the Fisher information metrics and its quantum generalisations is testified by the number of applications that this has in very different fields, ranging from hypothesis testing to metrology, passing through thermodynamics. Still, from the rich range of possible quantum Fisher informations, only a handful are typically used and studied. This review aims at collecting a number of results scattered in the literature and provide a cohesive treatment to people who begin the study of Fisher information and to those who are already working on it to have a more organic understanding of the topic. Moreover, we complement the review with new results about the relation between Fisher information and physical evolutions. Extending previous works, we show that dynamical properties such as (complete) positivity, Markovianity, detailed balance, retrodictive power of evolution maps can be characterised in terms of their relation with respect to the Fisher information metrics. These results show a fact that was partially overseen in the literature, namely the inherently dynamical nature of Fisher information.
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.
Information-theoretic inequalities play a fundamental role in numerous scientific and technological areas (e.g., estimation and communication theories, signal and information processing, quantum physics, …) as they generally express the impossibility to have a complete description of a system via a finite number of information measures. In particular, they gave rise to the design of various quantifiers (statistical complexity measures) of the internal complexity of a (quantum) system. In this paper, we introduce a three-parametric Fisher–Rényi complexity, named ( p , β , λ ) -Fisher–Rényi complexity, based on both a two-parametic extension of the Fisher information and the Rényi entropies of a probability density function ρ characteristic of the system. This complexity measure quantifies the combined balance of the spreading and the gradient contents of ρ , and has the three main properties of a statistical complexity: the invariance under translation and scaling transformations, and a universal bounding from below. The latter is proved by generalizing the Stam inequality, which lowerbounds the product of the Shannon entropy power and the Fisher information of a probability density function. An extension of this inequality was already proposed by Bercher and Lutwak, a particular case of the general one, where the three parameters are linked, allowing to determine the sharp lower bound and the associated probability density with minimal complexity. Using the notion of differential-escort deformation, we are able to determine the sharp bound of the complexity measure even when the three parameters are decoupled (in a certain range). We determine as well the distribution that saturates the inequality: the ( p , β , λ ) -Gaussian distribution, which involves an inverse incomplete beta function. Finally, the complexity measure is calculated for various quantum-mechanical states of the harmonic and hydrogenic systems, which are the two main prototypes of physical systems subject to a central potential.