Phase-space features of the Wigner flow are examined so to provide a set of continuity equations that describe the flux of quantum information in the phase-space. The reported results suggest that the non-classicality profile of anharmonic (periodic) quantum systems can be consistently obtained in terms of the fluxes of probability, purity and von Neumann entropy. Extensions of the such phase-space quantifiers are also investigated in the context of the so-called Renyi entropy.
Phase-space features of the Wigner flow are examined so to provide a set of continuity equations that describe the flux of quantum information in the phase-space. The reported results suggest that the non-classicality profile of anharmonic (periodic) quantum systems can be consistently obtained in terms of the fluxes of probability, purity and von Neumann entropy. Extensions of the such phase-space quantifiers are also investigated in the context of the so-called Renyi entropy.
A fluid analog of the information flux in the phase-space associated to purity and von Neumann entropy is identified in the Weyl-Wigner formalism of quantum mechanics. Once constrained by symmetry and positiveness, the encountered continuity equations provide novel quantifiers for non-classicality (non-Liouvillian fluidity) given in terms of quantum decoherence, purity and von Neumann entropy fluxes. Through definitions in the Weyl-Wigner formalism, one can identify the quantum fluctuations that distort the classical-quantum coincidence regime, and the corresponding quantum information profile, whenever some bounded x − p volume of the phase-space is specified. The dynamics of anharmonic systems is investigated in order to illustrate such a novel paradigm for describing quantumness and classicality through the flux of quantum information in the phase-space.
Phase-space features of the Wigner flow for an anharmonic quantum system driven by the harmonic oscillator potential modified by the addition of an inverse square (one-dimensional Coulomb-like) contribution are analytically described in terms of Wigner functions and Wigner currents. Reporting about three correlated continuity equations which quantify the flux of quantum information in the phase space, the nonclassicality profile of such an anharmonic system can be consistently obtained in terms of the fluxes of probability, purity, and von Neumann--like entropy. Considering that quantum fluctuations can be identified from distortions over the classical regime, they can be quantified through the above-mentioned information fluxes whenever some classically bounded volume of the phase space is selected. Our results suggest that the Wigner flow approach works as a probe of quantumness and classicality for a large set of anharmonic quantum systems driven by quantum wells.
Phase-space versions of quantum mechanics -- from Wigner's original distribution to modern discrete-qudit constructions -- represent some states with negative quasi-probabilities. Conventional Shannon and Rényi entropies become complex-valued in this setting and lose their operational meaning. Building on the axiomatic treatments of Rényi (1961) and Daróczy (1963), we develop a conservative extension that applies to signed finite phase spaces and identify a single admissible entropy family, which we call signed Rényi α-entropy (for a free parameter α≥0). The obvious signed Shannon candidate is ruled out because it violates extensivity. We prove four results that bolster the usefulness of the new measure. (i) It serves as a witness of the presence of cancellation, detecting the coexistence of positive and negative weight in a signed measure. (ii) For α>1, it is Schur-concave, delivering the intuitive property that mixing increases entropy (iii) The same parametric family obeys a quantum H-theorem, namely, that under de-phasing dynamics entropy cannot decrease. (iv) The 2-entropy is conserved under discrete Moyal-bracket dynamics, mirroring conservation of von Neumann entropy under unitary evolution on Hilbert space. We also comment on interpreting the Rényi order parameter as an inverse temperature. Overall, we believe that our investigation provides good evidence that our axiomatically derived signed Rényi entropy may be a useful addition to existing entropy measures employed in quantum information, foundations, and thermodynamics.
Quantifiers of stationarity, classicality, purity and vorticity are derived from phase-space differential geometrical structures within the Weyl-Wigner framework, after which they are related to the hyperbolic stability of classical and quantum-modified Hamiltonian (non-linear) equations of motion. By examining the equilibrium regime produced by such an autonomous system of ordinary differential equations, a correspondence between Wigner flow properties and hyperbolic stability boundaries in the phase-space is identified. Explicit analytical expressions for equilibrium-stability parameters are obtained for quantum Gaussian ensembles, wherein information quantifiers driven by Wigner currents are identified. Illustrated by an application to a Harper-like system, the results provide a self-contained analysis for identifying the influence of quantum fluctuations associated to the emergence of phase-space vorticity in order to quantify equilibrium and stability properties of Hamiltonian non-linear dynamics.
Abstract Given its well known spectral decomposition profile, the 1-dim harmonic oscillator potential modified by an inverse square (1-dim angular momentum-like) contribution works as an efficient platform for probing classical and quantum information quantifiers in the context of the phase-space Weyl-Wigner formalism. In particular, the phase-space informational content related to the canonical ensemble driven by such a singular oscillator can be quantified in terms of well established analytical structures. Considering that, on one hand, the singular oscillator produce a spectral decomposition profile equivalent to that one of the unmodified harmonic system – in the sense that they result into identical thermodynamic statistics, even for different statistical mixtures – on the other hand, a more complete scrutinization of their phase-space information content can capture some different aspects of the encoded information for the related quantum ensembles. Besides the identification of decoherence effects, the Wigner flow analysis is presumedly useful in identifying stable quantum configurations, according to finite temperature and interaction parameter values. Unexpectedly, our results show that the equivalence between the statistical (quantum) mechanics of the anharmonic singular oscillator and an ordinary harmonic oscillator can also be extended to the phase-space quantum purity quantifier, which is analytically computed and reproduces exactly the same quantum ensemble statistical mixture profile, which does not depend on interaction parameter values.
We obtain a classical analog of the quantum covariance matrix by performing its classical approximation for any continuous quantum state, and we illustrate this approach with the anharmonic oscillator. Using this classical covariance matrix, we propose classical analogs of the purity, linear quantum entropy, and von Neumann entropy for classical integrable systems, when the quantum counterpart of the system under consideration is in a Gaussian state. As is well known, this matrix completely characterizes the purity, linear quantum entropy, and von Neumann entropy for Gaussian states. These classical analogs can be interpreted as quantities that reveal how much information from the complete system remains in the considered subsystem. To illustrate our approach, we calculate these classical analogs for three coupled harmonic oscillators and two linearly coupled oscillators. We find that they exactly reproduce the results of their quantum counterparts. In this sense, it is remarkable that we can calculate these quantities from the classical viewpoint.