Complex-valued Wigner entropy of a quantum state

TL;DR

The paper addresses the challenge of defining an entropy for quantum phase-space distributions that can be negative by introducing a complex-valued Wigner entropy via the analytic continuation of Shannon entropy. It establishes that both the real and imaginary parts of the complex Wigner entropy are invariant under Gaussian unitaries, with the real part reflecting phase-space spreading under Gaussian processing and the imaginary part encoding the negative volume of the Wigner function. It further develops a complex Fisher information and proves a complex de Bruijn identity, linking entropy dynamics under Gaussian diffusion to this complex information. These results provide a coherent framework for entropic analysis of phase-space quasiprobabilities and point toward complex entropic uncertainty relations and extended operational interpretations. Overall, the work extends entropy concepts to the complex plane as a natural tool for studying nonclassical phase-space distributions and their evolution.

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 the complex 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.

Figures (10)

• Figure 1: Phase-space representations of the Fock state $\ket{1}$. Its Glauber-Sudarshan P-function $P(\alpha)$ is expressed in terms of the second derivative of a Dirac $\delta$-function (represented by a red star) and is thus not regular. Smoothing $P(\alpha)$ with a Gaussian distribution of one unit of shot-noise (noted as the convolution $*G$) yields the Wigner function $W(\alpha)$, which is regular but takes negative values (red zone) near the origin. Smoothing again the Wigner function $W(\alpha)$ with the same Gaussian distribution yields the Husimi function $Q(\alpha)$, which is both regular and non-negative. Here, we focus on defining the complex-valued entropy of $W(\alpha)$ instead of the real-valued entropy of $Q(\alpha)$, which is the well-known Wehrl entropy.
• Figure 2: The three plotted functions $\varphi_r(W)=-W\ln\abs{W}$, $\varphi_a(W)=-\abs{W}\ln\abs{W}$, and $\varphi_+(W)=-\Theta(W)\, W\ln W$ are concave in $\mathbb{R}^+$ but not concave in $\mathbb{R}$. Of course, we have $\varphi_r(W)=\varphi_+(W)=\varphi_a(W)=-W\ln W$ provided $W\geq 0$. These functions are used to build the symmetric functionals $h_{\mathrm{r}}(W)=\int\varphi_r(W(x,p))\, \mathrm{d} x\, \mathrm{d} p$, $h_{\mathrm{a}}(W)=\int\varphi_a(W(x,p))\,\mathrm{d} x\, \mathrm{d} p$ and $h_+(W)=\int\varphi_+(W(x,p))\,\mathrm{d} x\,\mathrm{d} p$, which are concave over the set $\mathcal{W}_+$ but not concave over the set $\mathcal{W}$. Note that the functions $\varphi(W)$ must only be defined on a narrow domain since $|W|\leq1/\pi$ but this does not change the problem.
• Figure 3: Complex Wigner entropy $h_{\mathrm{c}} = h_{\mathrm{r}} + i \, h_{\mathrm{i}}$ of randomly generated states. Each blue (orange) point is associated with a random pure (mixed) state. Red stars represent Fock states $\ket{n}$ with $n=0,1,..., 8$. Note that $h_{\mathrm{r}}$ and $h_{\mathrm{i}}$ increase monotonically with $n$ for Fock states. The dotted red line corresponds to $h_{\mathrm{r}}=\ln\pi+1\simeq 2.145$, which is a (conjectured) lower bound on the real part of the entropy of Wigner-positive states PhysRevA.104.042211.
• Figure 4: Plots of the complex Wigner entropy $h_{\mathrm{c}} = h_{\mathrm{r}} + i \, h_{\mathrm{i}}$ for superpositions vs. mixtures of Fock states (red stars stand for Fock states). The blue lines correspond to binary superpositions of Fock states, i.e., $\ket{\psi}=\sqrt{p}\ket{m}+e^{i\varphi}\sqrt{1-p}\ket{n}$ with $p\in[0,1]$ (note that the complex Wigner entropy of $\ket{\psi}$ is independent of $\varphi$). The dotted orange lines are the corresponding binary mixtures of Fock states, i.e., $\hat{\rho}=p\ket{m}\bra{m}+(1-p)\ket{n}\bra{n}$. The following binary superpositions and mixtures are plotted: $(m,n)=(0,1)$, $(0,2)$, $(0,3)$, and $(1,2)$.
• Figure 5: Plots of the real and imaginary parts of the complex Wigner entropy $h_{\mathrm{r}}+i\, h_{\mathrm{i}}$ for $W_{\mathrm{cat}}$ and $W_{\mathrm{mix}}$, which are respectively the Wigner functions of an even cat state and of a balanced mixture of coherent states $\ket{\alpha}$ and $\ket{-\alpha}$. When $\alpha=0$, the two states coincide with the vacuum state. As $\alpha$ increases, the negative volume of the cat state increases. In the regime of $|\alpha|\gg 1$, $h_{\mathrm{r}}\left(W_{\mathrm{cat}}\right) \simeq h_{\mathrm{r}}(W_{\mathrm{mix}})$ and $h_{\mathrm{i}}\left(W_{\mathrm{cat}}\right)\simeq 1$.

Theorems & Definitions (7)

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