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).
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).
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.
We introduce an operational criterion to identify Wigner function (WF) negativity for an arbitrary quantum state within the framework of quantum non-demolition measurements. This criterion corresponds to experimentally accessible schemes that enable a direct measurement of the WF, and establishes the coherent-state basis as a privileged basis for determining when the WF exhibits negative regions. We show that the absence (presence) of coherent superpositions in the coherent-state basis provides direct information about the positivity (negativity) of the WF. In particular, the absence of such superpositions constitutes a sufficient condition for WF positivity. Although a general proof of necessity remains elusive, we demonstrate that this condition is also necessary in two relevant cases: Schrödinger-cat states and higher-order cat states on a circle. More precisely, for Schrödinger-cat states we establish a necessary and sufficient condition for the positivity of the WF in full generality, whereas for high-order cat states on a circle we derive an analogous condition in the limit of a large number of densely packed coherent states.
This study lights on the phenomenon of quantum decoherence when Biphoton and Schrödinger cat states are evolve. The primary focus of the work is to explore and analyze the phenomenon of decoherence by virtue of the insightful representations on the way of deriving the Wigner function for the considered states. A crucial aspect of this work involves shedding light on the behaviour of Biphoton and Schrödinger cat states with the same spatial modes and similar theoretical tomography. By considering the influence of an external environment, we further extend our exploration of decoherence, aiming to understand how environment as a factor, impact the stability and coherence of these quantum states.
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.
Much of the discussion of decoherence has been in terms of a particle moving in one dimension that is placed in an initial superposition state (a Schrödinger "cat" state) corresponding to two widely separated wave packets. Decoherence refers to the destruction of the interference term in the quantum probability function. Here, we stress that a quantitative measure of decoherence depends not only on the specific system being studied but also on whether one is considering coordinate, momentum or phase space. We show that this is best illustrated by considering Wigner phase space where the measure is again different. Analytic results for the time development of the Wigner distribution function for a two-Gaussian Schrodinger "cat" state have been obtained in the high-temperature limit (where decoherence can occur even for negligible dissipation) which facilitates a simple demonstration of our remarks.
Quantum coherence is an essential ingredient in quantum information processing and plays an important role in quantum computation. Therefore, it is a hot issue about how to quantify the coherence of quantum states in theoretical framework. The coherence effect of a state is usually described by the off-diagonal elements of its density matrix with respect to a particular reference basis. Recently, based on the established notions from quantitative theory of entanglement, a resource theory of coherence quantification has been proposed[1,2]. In the theory framework, a proper measure of coherence should satisfy three criteria: the coherence should be zero for all incoherent state; the coherence should not increase under mixing quantum states; the coherence should not increase under incoherent operations. Then, a number of coherence measures have been suggested, such as l1 norm of coherence and the relative entropy of coherence[2]. Wigner function is known as an important tool to study the non-classical property of quantum states for continuous-variable quantum systems. It has been generalized to finite-dimensional Hilbert spaces, and named as discrete Wigner function[9-16]. The magic property of quantum states, which promotes stabilizer computation to universal quantum computation, can be generally measured by the absolute sum of the negative items (negativity sum) in the discrete Wigner function of the observed quantum states. In this paper we investigate quantum coherence from the view of discrete Wigner function. From the definition of the discrete Wigner function of the quantum systems with odd prime dimensions, for a given density matrix we analyze in phase space the performance of its diagonal and off-diagonal items. We find that, the discrete Wigner function of a quantum state contains two aspects: the true quantum coherence and the classical mixture, where the part of classical mixture can be excluded by only considering the discrete Wigner function of the diagonal items of the density matrix. Thus, we propose a possible measure method for quantum coherence from the discrete Wigner function of the off-diagonal items of the density matrix. We show that the proposed measure method satisfies the criteria (C1) and (C2) of coherence measure perfectly. For the criteria (C3), we give a numerical proof in three-dimensional quantum system. Meanwhile, we compare the proposed coherence measure with l1 norm coherence, and get an inequality relationship between them. Finally, an inequality is obtained to discuss the relation between quantum coherence and the negativity sum of discrete Wigner function, which shows that the quantum coherence is only necessary but not sufficient for quantum computation speed-up.