Phasespace continuity equations for quantum decoherence, purity, von Neumann and Rényi entropies
Abstract
Phasespace features of the Wigner flow are examined so to provide a set of continuity equations that describe the flux of quantum information in the phasespace. The reported results suggest that the nonclassicality 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 phasespace quantifiers are also investigated in the context of the socalled Rényi entropy.

Departamento de Física, Universidade Federal de São Carlos, PO Box 676, 13565905, São Carlos, SP, Brasil.

Email: alexeb@ufscar.br

Departamento de Física e Astronomia, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre, 687, 4169007, Porto, Portugal

Email: orfeu.bertolami@fc.up.pt
1 Introduction
The formulation of the phasespace dynamics of quantum systems according to the WeylWigner 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 WeylWigner phasespace 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, ,
(1) 
such that
(2) 
the Weyl transform of a density matrix, , is interpreted as the Fourier transform of the offdiagonal elements of .
In spite of being consistent with the Moyal’s picture of quantum mechanics [6], where the noncommutative nature of canonical operators supports the Moyal starproduct definition, the above definition does not yield a realvalued probability distribution, given that it admits local negative values, which is inconsistent with the positivevalued probability interpretation. This provides a quasiprobability interpretation which admits an enhanced phasespace framework and eventually fixes this issue [7].
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
(3) 
for any product of two operators, and .
(4) 
which allows for the probability distribution interpretation, with the normalization condition established by , . The WeylWigner definitions also admit extensions from pure states to statistical mixtures. In particular, the purity,
(5) 
is constrained by the conditions of for pure states. In the investigation of classical and quantum regimes of onedimensional 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 fluidanalog of the phasespace quantum entropy (information) flux in the phasespace can be recovered from the WeylWigner formalism of quantum mechanics [8]. In this contribution, phasespace 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 phasespace quantifiers are also investigated in the context of the Rényi entropy definition. The conclusion is that, the set of phasespace continuity equations discussed below emphasizes the relevancy of the Wigner formalism in discussing the flux of quantum information in the phasespace.
2 Phasespace 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 phasespace [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],
(6) 
where
(7) 
(8) 
and is the potential for a nonrelativistic quantum system described by the Hamiltonian
(9) 
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 phasespace trajectories.
In particular, for the generalized discussion of nonrelativistic 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
(10)  
(11)  
(12) 
which can be recast in the form [14]
(13)  
(14)  
(15) 
where is a dimensionless quantity\@footnotemark\@footnotetextNotice that is consistent with the normalization condition given by
(16) 
and the Eq. (2) can be multiplied by so to yield the dimensionless continuity equation,
(17) 
where the phasespace coordinate vector, , is defined.. From this framework, the phasespace information flux associated to extensions of the above continuity equation can be obtained [12, 8].
To implement a selfcontained procedure for obtaining the continuity equations derived from the fluidanalog properties of the Wigner functions, one notices that locally and globally conservative properties associated to a point in the phasespace, , and to a phasespace volume integral bounded by a comoving closed surface, , are connected by a convective (or substantial) derivative [4, 8] operator defined by
(18) 
with , and where corresponds to the classical phasespace vector velocity, . Assuming that a twodimensional classical path, , can be identified, from the definition of , one sets
(19) 
which implies into a conservation law, . In this case, the divergenceless behavior of the classical fluidanalog 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 phasevelocity, , satisfies the constraint equation given by
(20) 
that can be used to quantify the departure from the Liouvillian behavior [4, 8].
For Hamiltonians describing periodic motions (in particular, for those driven by symmetric quantum wells), the parameterization to the twodimensional volume enclosed by the classical path, , results into
(21) 
where corresponds to the phasespace volume integrated probability flux.
Using the properties of , after some manipulation involving Eqs. (2)(2), one obtains
(22) 
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
(23) 
where the unitary vector is defined by and has been used in the last step. One thus has a parametric integral given by
(24) 
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 [8]
(25) 
and
(26) 
In these cases, some suitable mathematical manipulations involving Eq. (2)\@footnotemark\@footnotetextIn particular, if one notices that . result into
(27)  
and
(28)  
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 phasespace 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]
(29)  
and
(30)  
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 [13]. Once contextualized in the phasespace formulation, it can be written as
(31) 
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
(32)  
and that
(33)  
which is consistent with the previously obtained continuity equations.
4 Conclusions
In this work, it has been shown that the Weyl transform quantum states lead to phasespace 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 phasespace 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 phasespace 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].
Acknowledgments
The work of A. E. B. was supported by the Brazilian agencies FAPESP (grant 2018/039609) and CNPq (grants 451081/20188 and 300831/20161). The work of O. B. was partially supported by the COST action MP1405 Quantum Structure of Spacetime (QSPACE).
References
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