The local structure present in Wigner and Husimi phase-space distributions and their marginals are studied and quantified via information-theoretic quantities. Shannon, R\'enyi, and cumulative residual entropies of the Wigner and Husimi distributions are examined in the ground and excited states of a harmonic oscillator. The entropies of the Wigner function marginals are lower than the corresponding entropies of the Husimi function marginals, which illustrates how the nodal structure present in the Wigner function is lost upon consideration of the Husimi function. Shannon and cumulative residual entropies of the Wigner function yield entropies which are complex valued. Absolute values and real components of these quantities increase with quantum number, displaying a behavior which is consistent with their real-valued Husimi function counterparts. This comportment is also similar to the real-valued R\'enyi entropies of the Wigner function. The entropies of the Wigner function are observed to be lower than the corresponding Husimi function ones, in agreement with the results for the marginals. The real components of the Wigner function entropies are seen to be closer to the uncertainty relation bound compared to the corresponding Husimi function entropies. These real components are also closer to the bound when contrasted to the entropic sum of the marginal densities. The R\'enyi or collision entropy of the Wigner function sits exactly on the bound. Related statistical correlation measures show that the position-momentum correlation is larger in the Wigner function compared to the Husimi function and increases with quantum number.
The local structure present in Wigner and Husimi phase-space distributions and their marginals are studied and quantified via information-theoretic quantities. Shannon, R'enyi, and cumulative residual entropies of the Wigner and Husimi distributions are examined in the ground and excited states of a harmonic oscillator. The entropies of the Wigner function marginals are lower than the corresponding entropies of the Husimi function marginals, which illustrates how the nodal structure present in the Wigner function is lost upon consideration of the Husimi function. Shannon and cumulative residual entropies of the Wigner function yield entropies which are complex valued. Absolute values and real components of these quantities increase with quantum number, displaying a behavior which is consistent with their real-valued Husimi function counterparts. This comportment is also similar to the real-valued R'enyi entropies of the Wigner function. The entropies of the Wigner function are observed to be lower than the corresponding Husimi function ones, in agreement with the results for the marginals. The real components of the Wigner function entropies are seen to be closer to the uncertainty relation bound compared to the corresponding Husimi function entropies. These real components are also closer to the bound when contrasted to the entropic sum of the marginal densities. The R'enyi or collision entropy of the Wigner function sits exactly on the bound. Related statistical correlation measures show that the position-momentum correlation is larger in the Wigner function compared to the Husimi function and increases with quantum number.
Abstract The Shannon information entropy sum is examined as systems comprised of two-level harmonic oscillator states are passed through dephasing and depolarization quantum channels. We show how the entropy sum increases as the dephasing parameter is increased. The interpretation here is a loss of information in the system as the quantum coherences are removed. This result and interpretation also holds in the depolarization channel as the system transits to the maximally mixed state. The increase in the entropy sum is associated with a delocalization of both position and momentum densities. The behaviour of phase-space Wigner functions and their Shannon entropies is also addressed. Wigner negativity is seen to decrease and disappear with an increase of the dephasing parameter. Here, Wigner negativity is associated with the presence of quantum coherences. The imaginary component of the Wigner function entropy is shown to capture the changes in Wigner negativity. On the other hand, Wigner functions with diagonal input density matrices are non-negative in the depolarization channel. Their real-valued Shannon entropies increase with the depolarization parameter as the system approaches the maximally mixed state. The mutual information between the Wigner function and the product of position and momentum space densities is used to probe the statistical correlation between position and momentum variables. This measure decreases in both channels as the parameter increases. Thus, there is a loss of correlation as quantum coherences are eliminated in the dephasing case, and as the system transits to the maximally mixed case in the depolarizing channel.
We use phase space distributions, specifically the Wigner and Husimi quasi probability distributions, to study harmonically trapped Bose--Einstein condensate described by the Gross Pitaevskii equation. From the mean field ground state wavefunction we construct both distributions and their position and momentum space marginals and we use these to compute a comprehensive set of information theoretic measures: Shannon, Wehrl, and Rényi entropies; Fisher information; cumulative and cross cumulative residual entropies; mutual information; and Kullback--Leibler, Jeffreys, Cauchy Schwarz, and Rényi divergences. Studying these quantities as a function of the s-wave scattering length for a representative Rb-85 condensate, we find that stronger repulsive interactions drive increased phase space delocalization, seen by a monotonic growth of Shannon and Wehrl entropies, while the Fisher information shows the complementary trend -- increasing in position space and decreasing in momentum space in a manner consistent with the global Fisher uncertainty bound. Rényi entropies and divergence measures further reveal a systematic suppression of non classical interference and a shift toward more classical phase space structure in moving from the Wigner to the Husimi representation, with Wigner and Husimi based mutual informations converging at larger interaction strength. We note that, because the Gross Pitaevskii framework treats the many body state as a mean field product, the mutual information computed here quantifies statistical dependence between the conjugate phase space variables of the effective one body distribution rather than genuine particle particle entanglement.
We define the Wigner entropy of a quantum state as the differential Shannon entropy of the Wigner function of the state. This quantity is properly defined only for states that possess a positive Wigner function, which we name Wigner-positive states, but we argue that it is a proper measure of quantum uncertainty in phase space. It is invariant under symplectic transformations (displacements, rotations, and squeezing) and we conjecture that it is lower bounded by $ln\ensuremath{\pi}+1$ within the convex set of Wigner-positive states. It reaches this lower bound for Gaussian pure states, which are natural minimum-uncertainty states. This conjecture bears a resemblance with the Wehrl-Lieb conjecture, and we prove it over the subset of passive states of the harmonic oscillator which are of particular relevance in quantum thermodynamics. Along the way, we present a simple technique to build a broad class of Wigner-positive states exploiting an optical beam splitter and reveal an unexpectedly simple convex decomposition of extremal passive states. The Wigner entropy is anticipated to be a significant physical quantity, for example, in quantum optics where it allows us to establish a Wigner entropy-power inequality. It also opens a way towards stronger entropic uncertainty relations. Finally, we define the Wigner-R\'enyi entropy of Wigner-positive states and conjecture an extended lower bound that is reached for Gaussian pure states.
We explore the role of majorization theory in quantum phase space. To this purpose, we restrict ourselves to quantum states with positive Wigner functions and show that the continuous version of majorization theory provides an elegant and very natural approach to exploring the information-theoretic properties of Wigner functions in phase space. After identifying all Gaussian pure states as equivalent in the precise sense of continuous majorization, which can be understood in light of Hudson's theorem, we conjecture a fundamental majorization relation: any positive Wigner function is majorized by the Wigner function of a Gaussian pure state (especially, the bosonic vacuum state or ground state of the harmonic oscillator). As a consequence, any Schur-concave function of the Wigner function is lower bounded by the value it takes for the vacuum state. This implies in turn that the Wigner entropy is lower bounded by its value for the vacuum state, while the converse is notably not true. Our main result is then to prove this fundamental majorization relation for a relevant subset of Wigner-positive quantum states which are mixtures of the three lowest eigenstates of the harmonic oscillator. Beyond that, the conjecture is also supported by numerical evidence. We conclude by discussing some implications of this conjecture in the context of entropic uncertainty relations in phase space.
The time-dependent behaviour of harmonic oscillator squeezed states is examined from the viewpoint of position and momentum space Shannon entropies and their sum. Analytical expressions for these entropies are obtained from a symplectic group approach. The entropy sum returns to the entropic uncertainty bound at different time intervals, and is characterized by the presence of either a maximum (minimum) in the position space Shannon entropy and a minimum (maximum) in the momentum space one. The maximal value of the entropy sum is characterized by equality of position and momentum Shannon entropies. The transition of the entropy sum from the bound to its maximal value is studied, and relations between position and momentum space Shannon entropies are probed. The relation between the position and momentum space entropies is deformed as the value of the squeezing parameter increases. An analysis of the dependence of the Shannon entropies on the squeezing parameter shows different behaviours at different times. Both position and momentum space Shannon entropies exhibit a linear dependence on the parameter when their sum lies on the uncertainty bound. Here, the entropy sum is independent of the parameter. At other times, the relation between the entropies and squeezing parameter is non-linear, and extremal points are present. The entropy sum here is seen to be an increasing function of the parameter. Kullback–Leibler distances between squeezed and unsqueezed densities illustrate that the position or momentum space densities of the squeezed system can revert back to the unsqueezed ones during the time evolution. The phase-space Wigner function, entropy and mutual information are also examined and discussed. The statistical correlation as measured by mutual information is the entropy sum minus its value at the uncertainty bound.
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.