Phase-space continuity equations for quantum decoherence, purity, von Neumann and Rényi entropies
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 Rényi entropy.
Departamento de Física, Universidade Federal de São Carlos, PO Box 676, 13565-905, São Carlos, SP, Brasil.
Departamento de Física e Astronomia, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007, Porto, Portugal
The formulation of the phase-space dynamics of quantum systems according to the Weyl-Wigner quantum mechanics [1, 2] provides elementary tools to connect the predictive features of quantum mechanics with the investigation of quantum information. As it has been shown, the Weyl-Wigner phase-space formulation [3, 4, 5] can provide a wider understanding of the limits between classical and quantum regimes of physical systems.
The Wigner function, , encompasses the information from coordinate and momentum Schrödinger wave function representation of quantum mechanics in a convolutional representation of the wave function described by the Weyl transform of a quantum operator, ,
the Weyl transform of a density matrix, , is interpreted as the Fourier transform of the off-diagonal elements of .
In spite of being consistent with the Moyal’s picture of quantum mechanics , where the noncommutative nature of canonical operators supports the Moyal star-product definition, the above definition does not yield a real-valued probability distribution, given that it admits local negative values, which is inconsistent with the positive-valued probability interpretation. This provides a quasi-probability interpretation which admits an enhanced phase-space framework and eventually fixes this issue .
Keeping in mind that the Weyl transform and the Wigner function connect quantum observables, , with their expectation values, one notices that the trace of the product of the operators, and , can be evaluated from the integral of the product of their Weyl transforms [1, 2], \@footnotemark\@footnotetextIn general one has
for any product of two operators, and .
which allows for the probability distribution interpretation, with the normalization condition established by , . The Weyl-Wigner definitions also admit extensions from pure states to statistical mixtures. In particular, the purity,
is constrained by the conditions of for pure states. In the investigation of classical and quantum regimes of one-dimensional physical systems, the statistical aspects related to the nature of the density matrix quantum operators play a relevant role in the computation of entropy and information quantifiers.
Considering these theoretical foundations, a fluid-analog of the phase-space quantum entropy (information) flux in the phase-space can be recovered from the Weyl-Wigner formalism of quantum mechanics . In this contribution, phase-space continuity equations for quantum decoherence, purity and von Neumann entropy are recast in a dimensionless framework in order to be useful for discussing a broad set of Hamiltonian quantum systems. In particular, extensions of such phase-space quantifiers are also investigated in the context of the Rényi entropy definition. The conclusion is that, the set of phase-space continuity equations discussed below emphasizes the relevancy of the Wigner formalism in discussing the flux of quantum information in the phase-space.
2 Phase-space flow analysis and continuity equations
The dynamics of the Wigner function, can be described in terms of a vector flux, , which represents the flow of in the phase-space [4, 9, 10, 11]. In this case, the flow field, , is decomposed into coordinate and momentum components, , with , through which one can identify the quantum version of the Liouville equation, that is a continuity equation given by [2, 7, 4, 8],
and is the potential for a non-relativistic quantum system described by the Hamiltonian
Notice from Eq. (2) that the contributions from in the series expansion quantify the distortion due to the quantum corrections with respect to the classical Liouvillian phase-space trajectories.
In particular, for the generalized discussion of non-relativistic quantum systems where is simply a mass scale, generalized results can be obtained when is put into a dimensionless form, . The dimensionless variables, and , provide the identification of and , where is the time scale.
The corresponding dimensionless form of the Wigner function and of the corresponding Wigner current are thus obtained from
which can be recast in the form 
where is a dimensionless quantity\@footnotemark\@footnotetextNotice that is consistent with the normalization condition given by
and the Eq. (2) can be multiplied by so to yield the dimensionless continuity equation,
To implement a self-contained procedure for obtaining the continuity equations derived from the fluid-analog properties of the Wigner functions, one notices that locally and globally conservative properties associated to a point in the phase-space, , and to a phase-space volume integral bounded by a comoving closed surface, , are connected by a convective (or substantial) derivative [4, 8] operator defined by
with , and where corresponds to the classical phase-space vector velocity, . Assuming that a two-dimensional classical path, , can be identified, from the definition of , one sets
which implies into a conservation law, . In this case, the divergenceless behavior of the classical fluid-analog associated to the Wigner function is expressed by , which defines the Liouvillian behavior.
To implement the above properties onto a quantum scenario, one has to notice that can be cast in the form of product , where the now introduced Wigner phase-velocity, , satisfies the constraint equation given by
For Hamiltonians describing periodic motions (in particular, for those driven by symmetric quantum wells), the parameterization to the two-dimensional volume enclosed by the classical path, , results into
where corresponds to the phase-space volume integrated probability flux.
which allows for identifying the role of the quantum corrections given in terms of , which effectively drives the volume variation of in terms of a path integral given by
where the unitary vector is defined by and has been used in the last step. One thus has a parametric integral given by
where, in the first step, the line element, , is set as , and, in the second step, one has and as typical solutions of the classical Hamiltonian problem. In addition, is the dimensionless period of the classical motion, and is identified by the series expansion from Eq. (2) for .
The above result straightforwardly quantifies the loss of information related to quantum decoherence. The same analysis can be applied to the informational content related to the Wigner associated von Neumann entropy and purity expressed respectively by 
In these cases, some suitable mathematical manipulations involving Eq. (2)\@footnotemark\@footnotetextIn particular, if one notices that . result into
for von Neumann entropy and purity, respectively, where and, in both cases, it has been used that ).
Notice that the surface terms have been consistently suppressed in the limit where , which leads to the results from Refs. [8, 14]. However, when a classical surface, , encloses a finite phase-space volume, , the surface term must be taken into account. In fact, given the definitions from Eqs. (2)-(2), it is possible to verify that is proportional to
and for parity symmetric potentials, which lead to periodic anharmonic motions, the above obtained averaged terms vanish. Therefore, the obtained continuity equations can be recast in the form of [8, 14]
for von Neumann entropy and purity, respectively, through which one can quantify the role of quantum fluctuations driven by the contributions from \@footnotemark\@footnotetextFor quantum systems which account for the all order corrections from Eq. (2), quantumness and classicality can be quantified through the above obtained continuity equation framework through the results from Eqs. (2), (29), and (33) [8, 12, 14]..
3 Rényi entropy continuity equation
The Rényi entropy, , generalizes the Hartley entropy, the Shannon entropy, the collision entropy and the minimal entropy . Once contextualized in the phase-space formulation, it can be written as
where the index is introduced to denote an estimative of the fractal dimension. By following the same derivative structure reported in the previous section, one could notice that
which is consistent with the previously obtained continuity equations.
In this work, it has been shown that the Weyl transform quantum states lead to phase-space continuity equations which allow for evaluating departures from Liouvillian properties of Wigner currents and fluxes. The results provide the tools for discussing quantumness and classicality in terms of phase-space quantum decoherence, purity and von Neumann entropy fluxes, in the context of (an)harmonic periodic quantum systems. The informational content analysis was extended in order to provide an equivalent formulation for the phase-space Rényi entropy, in terms of an additional continuity equation. Our results suggest that the Wigner flow framework can be universally applied to distinguish how quantum and classical regimes are quantitatively distinguished from each other [8, 12, 14].
The work of A. E. B. was supported by the Brazilian agencies FAPESP (grant 2018/03960-9) and CNPq (grants 451081/2018-8 and 300831/2016-1). The work of O. B. was partially supported by the COST action MP1405 Quantum Structure of Spacetime (QSPACE).
-  Wigner E 1932 Phys. Rev. 40 749
-  Case W B 2008 Am. J. Phys. 76 937
-  Bernardini A E and Chinaglia M 2015 Mod. Phys. Lett. A30 1550118
-  Steuernagel O, Kakofengitis D and Ritter G 2013 Phys. Rev. Lett. 110 030401
-  Meekhof D M, Monroe C, King B E, Itano W M and Wineland D J 1996 Phys. Rev. Lett. 76 1796
-  Moyal J 1949 Proc. Camb. Phil. Soc. 45 99
-  Ballentine L E 1998 Quantum Mechanics: a Modern Development pp. 633 (World Scientific).
-  Bernardini A E and Bertolami O 2017 EPL 120 20002
-  Ferraro A and Paris M G A 2012 Phys. Rev. Lett. 108 260403
-  Donoso A and Martens C C 2001 Phys. Rev. Lett. 87 223202
-  Domcke W, Hänggi P and Tannor D 1997 Chem. Phys. 217 117
-  Bernardini A E, Leal P and Bertolami O 2018 JCAP 02 025
-  Jizba P and Arimitsu T 2004 Annals of Physics 312 17
-  Bernardini A E 2018 Phys. Rev. A98 052128