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Journal of the Optical Society of America B

Calculation of the disentropy of the Wigner function using the Lambert–Tsallis Wq function with non-integer q values

José leonardo Esteves da Silva, Rubens Viana Ramos

Published May 19, 2020• 9 citations
Publisher

Abstract

The Wigner function is a mathematical tool that provides important information about a quantum light state, like entanglement and quantumness. For example, in a recent work it was shown the disentropy of the Wigner function using the Lambert–Tsallis W q function with q = 2 can be used as a measure of quantumness. When the value of q is non-integer, the disentropy and W q function have fractional powers and, hence, a negative value of the Wigner function can result in a complex value for the disentropy. This prohibits the use of those functions in the calculation of the disentropy of the Wigner function of highly interesting states, such as Schrödinger cats. In order to overcome this problem, we propose a new disentropy equation inspired by the Rényi entropy. The advantages and disadvantages of this new disentropy are discussed and numerical examples are shown.

DOI
10.1364/josab.390323

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The Wigner function is a mathematical tool that provides important information about a quantum light state, like entanglement and quantumness. For example, in a recent work it was shown the disentropy of the Wigner function using the Lambert–Tsallis W q function with q = 2 can be used as a measure of quantumness. When the value of q is non-integer, the disentropy and W q function have fractional powers and, hence, a negative value of the Wigner function can result in a complex value for the disentropy. This prohibits the use of those functions in the calculation of the disentropy of the Wigner function of highly interesting states, such as Schrödinger cats. In order to overcome this problem, we propose a new disentropy equation inspired by the Rényi entropy. The advantages and disadvantages of this new disentropy are discussed and numerical examples are shown.
In this work, the usage of the disentropy of the Wigner function as order and quantumness measure is investigated. Furthermore, the relative disentropy of the Wigner function as a distance measure between two Wigner functions is also discussed. The numerical calculation of the dynamic of the disentropy and relative disentropy of the Wigner function are realized considering an initially pure quantum state that evolves to a classical mixture and when an initially coherent state evolves to a Schrödinger cat state. It is shown that in all cases the disentropy is minimal for the quantum states with maximal quantumness, and it is maximal for maximally ordered states (pure states with positive Wigner function).
Entropy is a famous and well established concept in physics and engineering that can be used for explanation of basic fundamentals as well it finds applications in several areas, from quantum physics to astronomy, from network communication to medical image processing, for example. Now, entropy meets its dual, the disentropy. As such, the disentropy can be used everywhere entropy is used, offering a different point of view: since entropy is a measure of disorder or uncertainty, disentropy is a measure of order or certainty. Thus, important concepts of physics can be rewritten using disentropy instead of entropy. Although there is a large range of problems that can be solved using entropy or disentropy, there are situations where only the disentropy can be used. This happens because the disentropy can provide a real output value when its argument is negative, while the entropy cannot. Thus, it is possible to calculate, for example, the disentropy of quasi-probability distributions like the Wigner function of highly quantum states. In this direction, the present work shows applications of the disentropy in a small list of problems: quantum and classical information theory, black hole thermodynamics, image processing and number theory.
We propose a new method of calculating entanglement entropy of a many-body interacting Bosonic system (open or closed) in a field theoretic approach without replica methods. The Wigner function and Renyi entropy of a Bosonic system undergoing arbitrary non-equilibrium dynamics can be obtained from its Wigner characteristic function, which we identify with the Schwinger Keldysh partition function in presence of quantum sources turned on at the time of measurement. For non-interacting many body systems, starting from arbitrary density matrices, we provide exact analytic formulae for Wigner function and entanglement entropy in terms of the single particle Green's functions. For interacting systems, we relate the Wigner characteristic to the connected multi-particle correlators of the system. We use this formalism to study the evolution of an open quantum system from a Fock state with negative Wigner function and zero entropy, to a thermal state with positive Wigner function and finite entropy. The evolution of the Renyi entropy is non-monotonic in time for both Markovian and non-Markovian dynamics. The entropy is also found to be anti-correlated with negativity of the Wigner function of a 222 -mode open quantum system.
Abstract It is common knowledge that the Wigner function of a quantum state may admit negative values, so that it cannot be viewed as a genuine probability density. Here, we examine the difficulty in finding an entropy-like functional in phase space that extends to negative Wigner functions and then advocate the merits of defining a complex-valued entropy associated with any Wigner function. This quantity, which we call thecomplex Wigner entropy, is defined via the analytic continuation of Shannon’s differential entropy of the Wigner function in the complex plane. We show that the complex Wigner entropy enjoys interesting properties, especially its real and imaginary parts are both invariant under Gaussian unitaries (displacements, rotations, and squeezing in phase space). Its real part is physically relevant when considering the evolution of the Wigner function under a Gaussian convolution, while its imaginary part is simply proportional to the negative volume of the Wigner function. Finally, we define the complex-valued Fisher information of any Wigner function, which is linked (via an extended de Bruijn’s identity) to the time derivative of the complex Wigner entropy when the state undergoes Gaussian additive noise. Overall, it is anticipated that the complex plane yields a proper framework for analyzing the entropic properties of quasiprobability distributions in phase space.
In this paper we define the quantum version of the Tsallis entropy in terms of quantum phase space distribution functions. The quantum Tsallis entropy is compared with Kenfack's nonclassicality indicator, for different systems, such as the Schr\"odinger cat state, the thermal state, a superposition of the ground and the first excited number states, and the harmonic oscillator state. These comparisons indicate that the Wigner representation gives us complete information about the state with the nonextensivity parameter q=1q=1q=1, while the Husimi representation hides some information with the nonextensivity parameter q<1q<1q<1.
In a recent paper, Tilma, Everitt et al. derived a generalized Wigner function that can characterize both the discrete and continuous variable states, i.e., hybrid states. As such, one can expect that the negativity of the generalized Wigner function applied to the hybrid states can reveal their nonclassicality, in analogy with the well-known Wigner function defined for the continuous variable states. In this work, we demonstrate that, indeed, the negativity volume of the generalized Wigner function of the hybrid bipartite states can be used as an entanglement witness for such states, provided that it exceeds a certain critical value. In particular, we study hybrid bipartite qubit-bosonic states and provide a qubit-Schrödinger cat state as an example. Since the detection of the generalized Wigner function of hybrid bipartite states in phase space can be experimentally simpler than the tomographic reconstruction of the corresponding density matrix, our results, therefore, present a convenient tool in the entanglement identification of such states.