Non-negative Wigner-like distributions and Renyi-Wigner entropies of arbitrary non-Gaussian quantum states: The thermal state of the one-dimensional box problem

TL;DR

This paper introduces non-negative Wigner-like distributions and Renyi-Wigner entropies to analyze non-Gaussian quantum states in phase space, providing new tools for understanding quantum thermodynamics and state features.

Contribution

The work develops a novel framework for evaluating Renyi-Wigner entropies of non-Gaussian states using phase-space distributions, extending the analysis to thermal states in a 1D box.

Findings

Explicit evaluation of distributions for thermal states in a 1D box.

Insights into non-Gaussian features compared to Gaussian states.

Analysis of entropy behavior in semiclassical and high-temperature limits.

Abstract

In this work, we consider the phase-space picture of quantum mechanics. We then introduce non-negative Wigner-like (operational) distributions \widetilde{\mathcal W}_{rho;alpha}(x,p) corresponding to the density operator \hat{rho} and being proportional to {W_{rho^(alpha/2)}(x,p)}^2, where W_{rho}(x,p) denotes the usual Wigner function. In doing so, we utilize the formal symmetry between the purity measure Tr(rho^2) and its Wigner representation (2 pi hbar) \int dx dp {W_{rho}(x,p)}^2 and then consider, as a generalization, such symmetry between the fractional moment Tr(\hat{rho}^{alpha}) and its Wigner representation (2 pi hbar) \int dx dp {W_{rho^{alpha/2}}(x,p)}^2. Next, we create a framework that enables explicit evaluation of the Renyi-Wigner entropies for the classical-like distributions \widetilde{\mathcal W}_{rho;alpha}(x,p). Consequently, a better understanding of some non-Gaussian features of a given state rho will be given, by comparison with the Gaussian state rho_G defined in terms of its Wigner function W_{rho_G}(x,p) and essentially determined by its purity measure T(rho_G)^2 alone. To illustrate the validity of our framework, we evaluate the distributions \widetilde{\mathcal W}_{beta;alpha}(x,p) corresponding to the (non-Gaussian) thermal state rho_{\beta} of a single particle confined by a one-dimensional infinite potential well with either the Dirichlet or Neumann boundary condition and then analyze the resulting Renyi entropies. Our phase-space approach will also contribute to a deeper understanding of non-Gaussian states and their properties either in the semiclassical limit (hbar \to 0) or in the high-temperature limit (beta \to 0), as well as enabling us to systematically discuss the quantal-classical Second Law of Thermodynamics on the single footing.

Equations

$$W_{\rho }(x,p)$$

$$\hat{\rho }$$

$$⟨\hat{A}⟩=\text{mbox}Tr(\hat{\rho }\hat{A})=\int dxdp{W}_{\rho }(x,p)W_A(x,p)$$

$$W_A(x,p)={\int }_{-\infty }^{\infty }d\xi ⟨x+\frac{\xi }{2}|\hat{A}|x-\frac{\xi }{2}⟩\exp \left(-\frac{i}{\hslash }p\xi \right).$$

$$\text{mbox}Tr({\hat{\rho }}^2)=(2\pi \hslash ){⟨W_{\rho }⟩}_{W_{\rho }}=\int dxdp{\mathcal{W}}_{{\rho }^2}(x,p),$$

$$⟨p^{\alpha -1}⟩=\text{mbox}Tr({\hat{\rho }}^{\alpha })=\int dxdp{\mathcal{W}}_{{\rho }^{\alpha }}(x,p),$$

$$S_{\alpha }({\tilde{\mathcal{W}}}_{\rho })=(1-\alpha {)}^{-1}\ln \left({\mathcal{N}}_{{\rho }^{\alpha }}\right),$$

$$(2\pi \hslash {)}^{l-1}\int dxdp{W}_{\rho }(x,p){\left\{W_{\rho }\left(x-\frac{\hslash }{2i}\frac{\partial }{\partial p},p+\frac{\hslash }{2i}\frac{\partial }{\partial x}\right)\right\}}^{l-2}{W}_{\rho }(x,p)=\text{mbox}Tr({\hat{\rho }}^l).$$

$${\tilde{\mathcal{W}}}_{\rho }\to {\tilde{\mathcal{W}}}_{\rho ;1}(x,p)={\mathcal{W}}_{\rho }(x,p)=(2\pi \hslash )\{W_{{\rho }^{1/2}}(x,p){\}}^2\ge 0,$$

$$S_1({\tilde{\mathcal{W}}}_{\rho })=-{\partial }_{\alpha }{\ln \left({\mathcal{N}}_{{\rho }^{\alpha }}\right)|}_{\alpha =1},$$

$$(2\pi \hslash {)}^3\int dxdp{W}_r(x,p)W_s\left(x-\frac{\hslash }{2i}\frac{\partial }{\partial p},p+\frac{\hslash }{2i}\frac{\partial }{\partial x}\right)W_u\left(x-\frac{\hslash }{2i}\frac{\partial }{\partial p},p+\frac{\hslash }{2i}\frac{\partial }{\partial x}\right)W_v(x,p)$$

$$={\delta }_{nm}$$

$$\text{mbox}Tr(\hat{\rho }\hat{A})=\text{mbox}Tr({\hat{\rho }}^{1/2}\hat{A}{\hat{\rho }}^{1/2})=(2\pi \hslash )\int dxdp{W}_{{\rho }^{1/2}}(x,p)W_A\left(x-\frac{\hslash }{2i}\frac{\partial }{\partial p},p+\frac{\hslash }{2i}\frac{\partial }{\partial x}\right)W_{{\rho }^{1/2}}(x,p).$$

$$\text{mbox}Tr(\hat{\rho }\hat{A})=(4\pi \hslash {)}^{-1}{\int }_{-\infty }^{\infty }dxdpd{x}_{1}d{p}_{1}d{x}_{2}d{p}_2{W}_{{\rho }^{1/2}}(x/2,p)\exp \left\{-\frac{i}{\hslash }(x_1-x_2)p\right\}\times$$

$$W_A(x_1/2,p_1)\exp \left\{-\frac{i}{\hslash }(x_2-x)p_1\right\}{W}_{{\rho }^{1/2}}(x_2/2,p_2)\exp \left\{-\frac{i}{\hslash }(x-x_1)p_2\right\}.$$

$$\text{mbox}Tr({\hat{\rho }}^{\alpha })$$

$$\text{mbox}Tr({\hat{\rho }}^{\alpha })=({\mathcal{N}}_{{\rho }^{\alpha }}{)}^{-2}{\int }_{-\infty }^{\infty }dxdpd{x}_{1}d{p}_{1}d{x}_{2}d{p}_{2}d{x}_{3}d{p}_3\times$$

$$W_{{\rho }^{\alpha /2}}(x,p)\exp \left\{-\frac{i}{\hslash }(x_1+x_2+x_3)p\right\}{W}_{{\rho }^{\alpha /2}}(x+(x_2+x_3)/2,p_1)\exp \left(\frac{i}{\hslash }{x}_1{p}_1\right)\times$$

$$W_{{\rho }^{\alpha /2}}(x-(x_1-x_3)/2,p_2)\exp \left(\frac{i}{\hslash }{x}_2{p}_2\right)W_{{\rho }^{\alpha /2}}(x-(x_1+x_2)/2,p_3)\exp \left(\frac{i}{\hslash }{x}_3{p}_3\right).$$

$$\text{mbox}Num(\beta ):=(2\pi \hslash )\sum _n\exp \left(-\beta E_n\right)W_n(x,p).$$

$$W_{\beta }(x,p)$$

$${\tilde{\mathcal{W}}}_{\beta ;1}(x,p)$$

$${\psi }_n(x)={\left(\frac{1}{a}\right)}^{1/2}\sin \left\{\frac{n\pi }{2}\left(1+\frac{x}{a}\right)\right\},$$

$${\varphi }_n(x)={\left(\frac{1}{a}\right)}^{1/2}\cos \left\{\frac{n\pi }{2}\left(1+\frac{x}{a}\right)\right\},$$

$$W_{n;D}(x,p)$$

$$W_{n;N}(x,p)$$

$$W_{\beta ;D}(x,p)$$

$$W_{\beta ;N}(x,p)$$

$$W_{\beta }(x,p)$$

$${\vartheta }_3(z,q)=1+2\sum _{n=1}^{\infty }{q}^{n^2}\cos (2nz);{\vartheta }_4(z,q)=1+2\sum _{n=1}^{\infty }(-1{)}^n{q}^{n^2}\cos (2nz);$$

$$W_{\beta }(x,p)$$

$$W_{\beta }(x,p)=\frac{1}{(2\pi \hslash )Z_{\beta }}[\exp \left(-\frac{\beta p^2}{2m}\right)-\cos \left(\frac{2{\xi }_{x}p}{\hslash }\right)\sum _{k=0}^{\infty }{\left(-\frac{2\beta p^2}{m}\right)}^k\times$$

$$\sum _{\nu =1}^{\infty }\left\{i^{2k}\text{mbox}erfc\left\{{\left(\frac{2m}{\beta {\hslash }^2}\right)}^{1/2}\left({\xi }_x+(\nu -1)L\right)\right\}-i^{2k}\text{mbox}erfc\left\{{\left(\frac{2m}{\beta {\hslash }^2}\right)}^{1/2}\left(\nu L-{\xi }_x\right)\right\}\right\}-$$

$$\frac{Z_{\beta ,\text{mbox}cl}}{4{\pi }^{1/2}ap}\sin \left(\frac{2{\xi }_{x}p}{\hslash }\right)\sum _{k=1}^{\infty }{\left(-\frac{2\beta p^2}{m}\right)}^k\times$$

$$\sum _{\nu =1}^{\infty }\left\{i^{2k-1}\text{mbox}erfc\left\{{\left(\frac{2m}{\beta {\hslash }^2}\right)}^{1/2}\left({\xi }_x+(\nu -1)L\right)\right\}+i^{2k-1}\text{mbox}erfc\left\{{\left(\frac{2m}{\beta {\hslash }^2}\right)}^{1/2}\left(\nu L-{\xi }_x\right)\right\}\right\}+$$

$$\frac{B{Z}_{\beta ,\text{mbox}cl}}{2\pi ap}\sin \left(\frac{2{\xi }_{x}p}{\hslash }\right)\sum _{\mu =1}^{\infty }\left\{\exp \left\{-\frac{2m}{\beta {\hslash }^2}{\left({\xi }_x+(\mu -1)L\right)}^2\right\}+\exp \left\{-\frac{2m}{\beta {\hslash }^2}{\left(\mu L-{\xi }_x\right)}^2\right\}\right\}].$$

$$W_{\beta }(x,p)$$

Full text

I Introduction

The Rényi-$\alpha$ entropy of a probability distribution $P=\{p_{j}\}$ was originally introduced in classical information theory, explicitly given by $S_{\alpha}(P)=(1-\alpha)^{-1}\ln\{\sum_{j}(p_{j})^{\alpha}\}$ with order $\alpha>0$ REN61 ; REN65 , as a generalization of the Shannon measure of information (SMI) given by $S_{1}(P)=-\sum_{j}p_{j}\ln p_{j}$ (for a deeper conceptual discussion of SMI, see, e.g., BEN17 ). Then, its quantum analog, given by $S_{\alpha}(\hat{\rho})=(1-\alpha)^{-1}\ln\{\mbox{Tr}(\hat{\rho}^{\alpha})\}$ of a density operator $\hat{\rho}$, has been studied in various contexts of quantum information theory and quantum thermodynamics WEH78 ; LIN13 ; MIS15 ; HAM16 ; ABE16 ; KIM18 ; DON19 , which is accordingly a generalization of the von Neumann entropy $S_{1}(\hat{\rho})=-\mbox{Tr}(\hat{\rho}\,\ln\hat{\rho})$; for instance, for a generalized formulation of quantum thermodynamics, which is built upon the maximum entropy principle applied to $S_{\alpha}(\hat{\rho})$ MIS15 , as well as in the discussion of its time derivative under the Lindblad dynamics, the result of which may be useful for exploring the dynamics of quantum entanglement in the Markovian regime ABE16 . Obviously, the case of $\alpha=2$ gives the well-known quantity $S_{2}(\hat{\rho})=-\ln\{\mbox{Tr}(\hat{\rho}^{2})\}$.

The quantum-mechanical expectation values of observables $\hat{A}$ can be calculated independently of the pictures in consideration (i.e., either the operator picture in the Hilbert space or the $c$-number picture in the classical phase space), and are required to obey, e.g., the Ehrenfest theorem (as a quantum-classical channel) stating that the classical laws of motion also hold true formally for the quantal expectation values GRI05 . On the other hand, the density operator $\hat{\rho}$ itself of the Hilbert-space picture can possess genuine quantum features (or non-classicalities) such as coherence and entanglement, which have been attracting considerable interest, as the need for a better theoretical understanding of them increases in response to the experimental manipulation of them in small quantum systems CHU00 ; MAH04 . However, when it comes to a systematic study of the canonical quantum-classical transition in the limit of $\hbar\to 0$, it would be more favorable in some contexts to take into consideration the quasi-probability distribution of the $c$-number picture, such as the Wigner function WIG32 ; HIL84 ; LEE95 ; SCH01 ; CUR05

[TABLE]

As such, the Wigner function $W_{\rho}(x,p)$ may be regarded as the direct counterpart to the classical probability distribution $P(x,p)$, and therefore the quantum-classical channels between the two $c$-number distributions can also be explored. However, the Wigner function of a generic quantum state may possess negative values, as is well-known.

Therefore, it will be interesting information-theoretically and thermodynamically to discuss the quantal-classical Second Law on the single footing, i.e., in terms of the Rényi-$\alpha$ entropy $S_{\alpha}(W_{\rho})$ for a given distribution $W_{\rho}(x,p)$, albeit with its negative values, in addition to the First Law, expressed in terms of the internal energy $\langle\hat{A}\rangle$ with $\hat{A}\to\hat{H}$, where the expectation value is given by

[TABLE]

with the Weyl-Wigner representation

[TABLE]

Thus far, such a unified approach of the Rényi-$\alpha$ entropies has not extensively been discussed, except for either the case of $\alpha=2$, for which the purity measure $\mbox{Tr}(\hat{\rho}^{2})=(2\pi\hbar)\int dxdp\,\{W_{\rho}(x,p)\}^{2}$ given in compact form, or the case of a Gaussian state $\hat{\rho}_{\mbox{\tiny G}}$ (due to its mathematically simple structure) defined as a quantum state, the Wigner function $W_{\rho_{\mbox{\tiny G}}}(x,p)$ of which is Gaussian, such as the canonical thermal equilibrium state $\hat{\rho}_{\beta}$ (with $\beta=1/k_{{\mbox{\scriptsize B}}}T$) of an $N$-oscillator system (corresponding to its ground state at $T=0$), the coherent state, and the squeezed state, etc. BRA05 ; FER05 ; WOL06 ; WED12 ; OLI12 ; ADE14 . Further, the entropy $S_{2}(W_{\rho_{\mbox{\tiny G}}})$ of a Gaussian state has been shown to coincide with the so-called Wigner entropy $S_{\mbox{\tiny W}}(W_{\rho_{\mbox{\tiny G}}})$ up to a constant, where $S_{\mbox{\tiny W}}(W_{\rho_{\mbox{\tiny G}}}):=-\int dxdp\,W_{{\rho}_{\mbox{\tiny G}}}(x,p)\,\ln\{(2\pi\hbar)\,W_{{\rho}_{\mbox{\tiny G}}}(x,p)\}$ is well-defined due to the non-negativity of $W_{{\rho}_{\mbox{\tiny G}}}(x,p)$ over the entire phase space ADE12 ; PAT17 .

It has also been known GEN10 ; ADE14 that while Gaussian states are crucial resources for quantum information processing (QIP) with continuous variables, non-Gaussian states also are either required or desirable (in terms of efficiency) for some tasks relevant to QIP such as entanglement distillation EIS02 ; GIE02 ; FIU02 , cluster quantum computation RAL03 ; LUN08 , and teleportation DEL07 ; WAN15 , etc. In doing so, the so-called non-Gaussianity (nG) was used as a critical resource of the information processing, and some measures of nG have then been proposed for quantification of the non-Gaussian character of a given state $\hat{\rho}_{\mbox{\tiny nG}}$, such as the measure defined in terms of the Hilbert-Schmidt distance between $\hat{\rho}_{\mbox{\tiny nG}}$ and its reference Gaussian state $\hat{\sigma}_{\mbox{\tiny G}}$ with the same first and second moments of the canonical quadrature operators as $\hat{\rho}_{\mbox{\tiny nG}}$ GEN07 , the quantum relative entropy (QRE) between $\hat{\rho}_{\mbox{\tiny nG}}$ and $\hat{\sigma}_{\mbox{\tiny G}}$ GEN08 ; MAR13 , and the one defined in terms of the difference between the Wehrl entropies of $\hat{\rho}_{\mbox{\tiny nG}}$ and $\hat{\sigma}_{\mbox{\tiny G}}$ IVA12 , as well as the one defined in terms of the Bures distance between $\hat{\rho}_{\mbox{\tiny nG}}$ and $\hat{\sigma}_{\mbox{\tiny G}}$ GHI13 .

However, each measure of nG has its drawback; e.g., a drawback of the QRE, used most widely for a general (non-Gaussian) state $\hat{\rho}$, is that its actual evaluation requires the full information about $\hat{\rho}$ such that it is often not feasible to calculate if only partial information is available GEN10 . Therefore, it will also be desirable to discuss the Rényi-$\alpha$ entropies $S_{\alpha}(\hat{\rho})$ in this context, which will provide the higher-moment information of $\hat{\rho}$ beyond $\mbox{Tr}(\hat{\rho}^{2})$, thus enabling us to approximately determine the state as accurately as possible, while Gaussian states (“tame” in their behaviors) are essentially determined by $S_{2}(\hat{\rho})$ alone OLI12 ; ADE14 ; KIM18 . As such, the entropies $S_{\alpha}(\hat{\rho})$ of a non-Gaussian state may contain the nG information in the operational sense. Further, the non-Gaussian features of $\hat{\rho}$, as the deviation from those of the Gaussian state $\hat{\rho}_{\mbox{\tiny G}}$ being defined in terms of its Wigner function $W_{{\rho}_{\mbox{\tiny G}}}(x,p)$, give impetus to a discussion of the entropies $S_{\alpha}(\hat{\rho})$ (with $\alpha\in{\mathbb{R}}^{+}$) in the phase-space picture, which will, in turn, enable us to explore more rigorously their quantum-classical transitions in the context of information theory and thermodynamics.

Exact evaluations of Rényi-$\alpha$ entropies $S_{\alpha}(W_{\rho})$ for given distributions $W_{\rho}(x,p)$ of non-Gaussian states have been studied in KIM18 . The resulting framework has successfully provided a general expression for calculating the entropies $S_{l}(W_{\rho})$ for integer orders $l$. Within this framework, the entropies $S_{\alpha}(W_{\rho_{\mbox{\tiny G}}})$ of Gaussian states $W_{\rho_{\mbox{\tiny G}}}(x,p)$ for real values of $\alpha>0$ also have been rediscovered in closed form, but with the help of an additional recurrence relation between two consecutive entropies $S_{l}$ and $S_{l+1}$ followed by the analytic continuation of $l\to\alpha$. However, it still remains an open question to directly evaluate the entropies $S_{\alpha}(W_{\rho_{\mbox{\tiny nG}}})$ of non-Gaussian states $W_{\rho_{\mbox{\tiny nG}}}(x,p)$ for real values $\alpha$ for which finding such a recurrence relation would be a formidable task.

In this work, we intend to create another framework, as a generalization of the preceding one, in which a group of Wigner-like (operational) distributions denoted by $\widetilde{\mathcal{W}}_{\rho;\alpha}(x,p)$ (with order $\alpha$ including the case of $\alpha=2$, obviously) will be introduced, corresponding to the same density operator $\hat{\rho}$, and then the Rényi entropies $S_{\alpha}(\widetilde{\mathcal{W}}_{\rho;\alpha})$, being tantamount to $S_{\alpha}(\hat{\rho})$, can be evaluated in the phase-space picture for arbitrary non-Gaussian states, actually with no need for the aforesaid recurrence relation and analytic continuation [cf. Eqs. (6) and (9)]. Remarkably, the distributions $\widetilde{\mathcal{W}}_{\rho;\alpha}(x,p)$ will be shown to be non-negative over the entire phase space (like the classical probability distribution) and well-defined in the genuine quantum regime all the way to the semiclassical limit. Besides, because of the equivalence between $S_{\mbox{\tiny W}}(W_{\rho_{\mbox{\tiny G}}})$ and $S_{2}(W_{\rho_{\mbox{\tiny G}}})$ (up to a constant) for (non-negative) Gaussian states and also the non-negative feature of $\widetilde{\mathcal{W}}_{\rho;\alpha}(x,p)$ for non-Gaussian states, the resulting entropies $S_{\alpha}(\widetilde{\mathcal{W}}_{\rho;\alpha})$, called the Rényi-Wigner entropies, may also be regarded as a generalization of the Wigner entropy $S_{\mbox{\tiny W}}(W_{\rho_{\mbox{\tiny G}}})$. Subsequently, we will consider a specific non-Gaussian state that will be applied for our framework of its Rényi-Wigner entropies; this is the thermal state $\hat{\rho}_{\beta}$ of a single particle confined by a one-dimensional infinite potential well with either Dirichlet or Neumann boundary condition. Its Wigner function (with its negative values) will be shown to tend asymptotically to a Gaussian shape in the limit of $\hbar\to 0$ only.

The general layout of this paper is as follows: In Sec. II we introduce a group of Wigner-like distributions as variants of the Wigner function and then provide a generic framework for the Rényi-Wigner entropies of arbitrary quantum states in the classical phase space. In Sec. III we explicitly evaluate the Wigner function, and its variants, of the thermal state of the one-dimensional box problem and then discuss the relevant issues of quantum-classical transition. In Sec. IV we apply our framework for this thermal state, and discuss some subjects relevant to the resulting Rényi-Wigner entropies. Finally, we provide concluding remarks in Sec. V.

II Non-negative Wigner-like distributions and entropies

We first observe that the purity measure, being the first moment of probability $\langle p\rangle=\sum_{n}\,(p_{n})^{2}$ with the eigenvalues $p_{n}$’s of $\hat{\rho}$, may be rewritten as

[TABLE]

in which the expectation value $\left\langle W_{\rho}\right\rangle_{\scriptscriptstyle W_{\scriptscriptstyle\rho}}:=\int dxdp\,W_{\rho}(x,p)\,W_{\rho}(x,p)$, and the symbol ${\mathcal{W}}_{\rho^{\scriptscriptstyle 2}}(x,p):=(2\pi\hbar)\{W_{\rho}(x,p)\}^{2}\geq 0$. This quantity ${\mathcal{W}}_{\rho^{\scriptscriptstyle 2}}(x,p)$ should be distinguished from its counterpart $W_{\rho^{\scriptscriptstyle 2}}(x,p)$, which is directly obtained from Eq. (1a) with $\hat{\rho}\to\hat{\rho}^{2}$ and thus may be negative valued like $W_{\rho}(x,p)$ itself. However, such formal symmetry between $\hat{\rho}^{2}$ and ${\mathcal{W}}_{\rho^{\scriptscriptstyle 2}}(x,p)$ is not available for higher moments, for which $\mbox{Tr}(\hat{\rho}^{l})\not\propto\int dxdp\,\{W_{\rho}(x,p)\}^{l}$ with $l=3,4,5,\cdots$ KIM18 . To directly discuss higher moments $\mbox{Tr}(\hat{\rho}^{\alpha})$ with $\alpha\in{\mathbb{R}}^{+}$ in the phase space, we therefore generalize Eq. (4) in such a way that the $(\alpha-1)$th fractional moment of probability is given by

[TABLE]

in which the quantity ${\mathcal{W}}_{\rho^{\scriptscriptstyle\alpha}}(x,p)=(2\pi\hbar)\{W_{\rho^{{\scriptscriptstyle\alpha/2}}}(x,p)\}^{2}\geq 0$ correspondingly results from Eq. (1a) with $\hat{\rho}\to\hat{\rho}^{\alpha/2}$. Here, a fractional operator $\hat{\rho}^{\alpha}$ is obtained from the spectral expansion of $\hat{\rho}$ by substituting its eigenvalues $p_{n}$’s with their positive $\alpha$th powers. We stress that its phase-space counterpart $W_{\rho^{{\scriptscriptstyle\alpha}}}(x,p)$, on the other hand, cannot directly be obtained from $W_{\rho}(x,p)$.

Now we introduce non-negative distributions $\widetilde{\mathcal{W}}_{\rho;\alpha}(x,p):={\mathcal{W}}_{\rho^{\scriptscriptstyle\alpha}}(x,p)/{\mathcal{N}}_{\rho^{\scriptscriptstyle\alpha}}$ with the normalizing ${\mathcal{N}}_{\rho^{\scriptscriptstyle\alpha}}=\int dxdp\,{\mathcal{W}}_{\rho^{\scriptscriptstyle\alpha}}(x,p)=\mbox{Tr}(\hat{\rho}^{\alpha})$ such that $\int dxdp\,\widetilde{\mathcal{W}}_{\rho;\alpha}(x,p)=1$. Then, this fractional moment may also be interpreted as the expectation value ${\mathcal{N}}_{\rho^{\scriptscriptstyle\alpha}}=(\langle\widetilde{\mathcal{W}}_{\scriptscriptstyle\rho;\alpha})^{-1}\,{\mathcal{W}}_{\rho^{\scriptscriptstyle\alpha}}\rangle_{\scriptscriptstyle\widetilde{\mathcal{W}}_{\scriptscriptstyle\rho;\alpha}}$, expressed in terms of $\langle A\rangle_{\scriptscriptstyle\widetilde{\mathcal{W}}_{\scriptscriptstyle\rho;\alpha}}:=\int dxdp\,A(x,p)\,\widetilde{\mathcal{W}}_{\scriptscriptstyle\rho;\alpha}(x,p)$. For comparison only, we introduce other distributions $\utilde{W}_{\rho;\alpha}(x,p):=W_{\rho^{\scriptscriptstyle\alpha}}(x,p)/N_{\rho^{\scriptscriptstyle\alpha}}$, as well, with $N_{\rho^{\scriptscriptstyle\alpha}}=\int dxdp\,W_{\rho^{\scriptscriptstyle\alpha}}(x,p)$ such that $\int dxdp\,\utilde{W}_{\rho;\alpha}(x,p)=1$. Clearly, these distributions $\utilde{W}_{\rho;\alpha}(x,p)$ can be negative-valued, though. Now, let the set $\widetilde{\mathcal{W}}_{\rho}=\{\widetilde{\mathcal{W}}_{\rho;\alpha^{\prime}}(x,p)|\alpha^{\prime}>0\}$, all elements of which correspond to the same density operator $\hat{\rho}$. Then, the Rényi entropy of $\hat{\rho}$ with order $\alpha$ ($\in{\mathbb{R}}^{+}$) can be expressed as the Rényi-Wigner entropy of $\widetilde{\mathcal{W}}_{\rho}$ such that

[TABLE]

which is well-defined. Here, the particular selection of $\widetilde{\mathcal{W}}_{\rho;\alpha}(x,p)$ with $\alpha^{\prime}\to\alpha$ from the set $\widetilde{\mathcal{W}}_{\rho}$ is required for actual evaluation of ${\mathcal{N}}_{\rho^{\scriptscriptstyle\alpha}}$ for a given order $\alpha$. This necessarily means that $S_{\alpha}(\widetilde{\mathcal{W}}_{\rho})\to S_{\alpha}(\widetilde{\mathcal{W}}_{\rho;\alpha})$ indeed. Obviously, this entropy $S_{\alpha}(\widetilde{\mathcal{W}}_{\rho})$ differs from the expression $\overline{S}_{\alpha}(W_{\rho}):=(1-\alpha)^{-1}\ln\left[\int dxdp\,(2\pi\hbar)\{W_{\rho}(x,p)\}^{\alpha}\right]$ for $\alpha\neq 2$; also note Eqs. (29a)-(29d). Eq. (6) is the first central result of our paper. This result enables us to evaluate the Rényi-$\alpha$ entropy in the phase-space picture in a more compact way than its counterpart provided in Ref. KIM18 (cf. Eq. (13) thereof) which has been derived, on the other hand, for positive integers $\alpha\to l=2,3,4,\cdots$ only and expressed in terms of the product of plain Wigner functions $W_{\rho}(x,p)$ with the Bopp shift; as a result, the normalizing ${\mathcal{N}}_{\rho^{\scriptscriptstyle\alpha}}$ in Eq. (6) now replaces the lengthy expression

[TABLE]

Then, we can compute the entropy with no need for an analytic continuation of $l\to\alpha$ for arbitrary non-Gaussian states. In fact, it is easy to expect that this analytic continuation will be a formidable task for generic non-Gaussian states.

We will be interested especially in the case of $\alpha=1$, for which

[TABLE]

obviously with ${\mathcal{W}}_{\rho}(x,p)\neq W_{\rho}(x,p)$. Then, the von-Neumann entropy simply reduces to

[TABLE]

directly obtained without considering any analytic continuation at all. We remark that the Shannon measure of information (or Shannon’s entropy), as the classical counterpart of the von-Neumann entropy, will directly appear from Eq. (9) with $\hbar\to 0$; in fact, it is known that all Rényi entropies $S_{\alpha}(\hat{\rho})$, and so $S_{\alpha}(\widetilde{\mathcal{W}}_{\rho})$, tend asymptotically to the von-Neumann entropy in the classical limit (e.g., LIN13 ; BRA15 ; PAT17 ). It is also worthwhile to point out that the “Wigner entropy” in the form of $S_{\mbox{\tiny W}}(\widetilde{\mathcal{W}}_{\rho;\alpha}):=-\int dxdp\,\widetilde{\mathcal{W}}_{\rho;\alpha}(x,p)\,\ln\{(2\pi\hbar)\,\widetilde{\mathcal{W}}_{\rho;\alpha}(x,p)\}$ is ill-defined, though $\widetilde{\mathcal{W}}_{\rho;\alpha}(x,p)\geq 0$, because the Bopp shift has not been employed at all and thus this expression, e.g., cannot appropriately distinguish pure states from mixed states [cf. Eqs. (29a)-(29d) and the discussion thereafter ].

Now we examine some properties of a distribution $\widetilde{\mathcal{W}}_{\rho;\alpha}(x,p)$. First, we point out that its non-negative nature (like that of its classical counterpart) cannot suitably reflect the orthogonality relation between any pair $(n,m)$ of two different eigenstates, because of the trace $\int dxdp\,\widetilde{\mathcal{W}}_{n}(x,p)\,\widetilde{\mathcal{W}}_{m}(x,p)>0$. Therefore, this distribution cannot be interpreted as a genuine probability distribution satisfying the required quantum feature. For comparison, we simply point out that this problem can be remedied finally with the help of the Bopp shift, as in Eq. (7), such that, with $\hat{\rho}^{\alpha}\to\hat{\rho}$ for pure states, $\mbox{Tr}(\hat{\rho}_{n}\hat{\rho}_{m})$ is expressed, instead, as

[TABLE]

indeed, in which $(r,s,u,v)=(n,n,m,m);(n,m,m,n);(m,m,n,n);(m,n,n,m)$, thus equivalent to $\mbox{Tr}\{(\hat{\rho}_{n})^{2}(\hat{\rho}_{m})^{2}\}=\delta_{nm}$. In the limit of $\hbar\to 0$, the left-hand side of Eq. (II) would simply reduce to $\int dxdp\,\widetilde{\mathcal{W}}_{n}(x,p)\,\widetilde{\mathcal{W}}_{m}(x,p)>0$ and therefore the orthogonality relation would be gone completely, which is exactly the case for any classical probability distribution $P(x,p)$.

Second, let us consider the expectation value of an observable $\hat{A}$ within our formulation. For the sake of simplicity, we now restrict our discussion into the case of $\alpha=1$. It is then straightforward to show that $\mbox{Tr}(\hat{\rho}\hat{A})\neq\int dxdp\,\widetilde{W}_{\rho;1}(x,p)\,W_{\scriptscriptstyle A}(x.p)$, but the expression

[TABLE]

This result is reminiscent of the expression $\langle\hat{A}\rangle_{\psi}=\int dx\,\psi^{\ast}(x)\,A(x,-i\hbar\,\partial_{x})\,\psi(x)$ in formal similarity. Subsequently, with the help of the Fourier transform ${\mathfrak{W}}_{\scriptscriptstyle A}({\mathfrak{x}},{\mathfrak{p}})=(2\pi\hbar)^{-1}\int dxdp\,W_{\scriptscriptstyle A}(x,p)\,\exp\{i\,(x{\mathfrak{p}}+p{\mathfrak{x}})/\hbar\}$, Eq. (11) will finally be transformed, after some algebraic manipulations, into

[TABLE]

Considering in Eq. (II) the (diagonal) terms with $x=x_{1}=x_{2}$ and $p=p_{1}=p_{2}$ only, then Eq. (II) would reduce to the compact form given by $\int dxdp\,\widetilde{W}_{\rho;1}(x,p)\,W_{\scriptscriptstyle A}(x.p)$, which is obviously not tantamount to $\mbox{Tr}(\hat{\rho}\hat{A})$ for a generic observable $\hat{A}\neq\openone$. This also shows that the distribution $\widetilde{W}_{\rho;1}(x,p)$ has the conceptual drawback that this quantity cannot be interpreted as a quasi-probability distribution over phase space, either. However, it is easy to show that for $\hat{A}\to\hat{\rho}^{\alpha-1}$ leading to $[\hat{A},\hat{\rho}^{1/2}]=0$, Eqs. (11) and (II) exactly reduce to Eq. (5). Therefore, the distribution $\widetilde{\mathcal{W}}_{\rho;\alpha}(x,p)$, despite its conceptual drawback discussed above, is still useful (in the operational sense) for evaluations of the Rényi-$\alpha$ entropies in the phase-space picture.

From the discussions provided in the preceding paragraphs, we arrive at the following conclusion concerning our central result: The usual Wigner function $W_{\rho}(x,p)$ of a non-Gaussian state $\hat{\rho}$ (as well as any product of the Wigner functions) cannot produce its Rényi-$\alpha$ entropies (with $\alpha\in{\mathbb{R}}^{+}$). On the other hand, the non-negative distribution $\widetilde{\mathcal{W}}_{\rho;\alpha}(x,p)$ cannot suitably be interpreted as a (quasi)-probability distribution, but it is actually this distribution that can produce the Rényi entropies $S_{\alpha}$ in the phase-space picture. Therefore, it is legitimate to say that this distribution is an operational one for exact evaluation of the Rényi entropies and thus for systematic access to the higher-moment information (including $S_{1}$) of a non-Gaussian state $\hat{\rho}$; again, without ${\mathcal{W}}_{\rho^{\scriptscriptstyle\alpha}}(x,p)$, the analytic continuation for order $\alpha$ would necessarily be required for evaluation of, e.g., $S_{1}$, as discussed already, which will however be a formidable task for generic non-Gaussian states. As a result, we may also claim that while the usual Wigner function $W_{\rho}(x,p)$, as a well-defined quasi-probability distribution, enables us to compute the expectation value of an observable via Eqs. (2) and (3), it cannot cover the full information available in a given density operator $\hat{\rho}$, without, e.g., the supplemental quantities ${\mathcal{W}}_{\rho^{\scriptscriptstyle\alpha}}(x,p)$ (or $W_{\rho^{\scriptscriptstyle\alpha}}(x,p)$).

It is also tempting to ask about difference between the two fractional quantities, ${\mathcal{W}}_{\rho^{\scriptscriptstyle\alpha}}(x,p)$ and $W_{\rho^{\scriptscriptstyle\alpha}}(x,p)$, in addition to whether they are non-negative or not. To do so, let us rewrite the fractional moment given in Eq. (5) as

[TABLE]

Then, we apply the same technique as for Eq. (II) with the help of the Fourier transform ${\mathfrak{W}}_{\rho^{\scriptscriptstyle\alpha}}({\mathfrak{x}},{\mathfrak{p}})=(2\pi\hbar)^{-1}\int dxdp\,W_{\rho^{\scriptscriptstyle\alpha}}(x,p)\,\exp\{i\,(x{\mathfrak{p}}+p{\mathfrak{x}})/\hbar\}$, which will enable Eq. (13) to have the form

[TABLE]

As observed, this expression, being not in terms of our non-negative quantity ${\mathcal{W}}_{\rho^{\scriptscriptstyle\alpha}}(x,p)$, has higher computational complexity (as its drawback) than its counterpart, Eq. (5), even for integer orders $\alpha\to l$ (cf. Eq. (IV) for an actual evaluation of Eq. (14) with respect to a particular state).

Now, we consider the canonical thermal equilibrium state $\hat{\rho}_{\scriptscriptstyle\beta}$ for explicit evaluation of $\widetilde{W}_{\rho;\alpha}(x,p)$. First, its Wigner function is given by $W_{\beta}(x,p)=\{(2\pi\hbar)\,Z_{\beta}\}^{-1}\,\mbox{Num}(\beta)$, where the partition function $Z_{\beta}$ and the numerator

[TABLE]

By noting that $\mbox{Tr}\{(\hat{\rho}_{\scriptscriptstyle\beta})^{\alpha}\}=Z_{\alpha\beta}/(Z_{\beta})^{\alpha}$, it is straightforward to show that $\widetilde{\mathcal{W}}_{\rho;\alpha}(x,p)=:\widetilde{\mathcal{W}}_{\beta;\alpha}(x,p)=(2\pi\hbar)\,\{W_{\rho^{\scriptscriptstyle\alpha/2}}(x,p)\}^{2}\,\{\mbox{Tr}(\rho^{\alpha})\}^{-1}=\{(2\pi\hbar)\,Z_{\alpha\beta}\}^{-1}\,\{\mbox{Num}(\alpha\beta/2)\}^{2}\geq 0$, where $W_{\rho^{\scriptscriptstyle\alpha}}(x,p)=(Z_{\beta})^{-\alpha}\,(Z_{\alpha\beta})\,W_{\alpha\beta}(x,p)$. Therefore, $\widetilde{\mathcal{W}}_{\beta;\alpha}(x,p)$ can be obtained from $W_{\beta}(x,p)$ simply by substitution of both $Z_{\beta}\to Z_{\alpha\beta}$ and $\mbox{Num}(\beta)\to\{\mbox{Num}(\alpha\beta/2)\}^{2}$, which is actually valid for the canonical thermal state of an arbitrary quantum system. As a simple example, the thermal state of a single linear oscillator, being Gaussian, is considered such that $Z_{\beta}=2^{-1}\,\mbox{csch}(\beta\hbar\omega/2)$ ING02 , and

[TABLE]

where $\kappa=(m\omega/\hbar)^{1/2}$, as well as the Wigner entropy $S_{\mbox{\tiny W}}(W_{\beta})=S_{2}(W_{\beta})+1-\ln(2)\neq S_{1}(W_{\beta})$. It is then straightforward to verify Eqs. (6) and (9) for this system. We also note that in the limit of $\hbar\to 0$ leading to $(2\pi\hbar)\,Z_{\beta}\to(Z_{\beta,\mbox{\scriptsize cl}})$, Eqs. (16a) and (16b) will reduce to their classical counterpart $P_{\beta}(x,p)=(Z_{\beta,\mbox{\scriptsize cl}})^{-1}\,e^{-\beta\,(p^{2}/2m\,+\,m\omega^{2}x^{2}/2)}$.

Finally, we stress that our formulation for the study of non-Gaussian states, consisting of the non-negative phase-space distributions and Rényi-Wigner entropies, essentially differs from the approach based on the (non-negative) Husimi functions $Q_{\rho}(x,p)=\langle\gamma|\hat{\rho}|\gamma\rangle/\pi$, defined in terms of the coherent state $|\gamma\rangle$ with $\gamma=2^{-1/2}\,(\kappa x+ip/\hbar\kappa)$, and the resulting Wehrl entropies defined as $(-1)\int dqdp\;Q_{\rho}(x,p)\,\ln\{Q_{\rho}(x,p)\}$; this entropy has been known to have a conceptual weakness that results from the non-orthogonality $|\langle\gamma_{1}|\gamma_{2}\rangle|^{2}=e^{-|\gamma_{1}-\gamma_{2}|^{2}}$, where $|\gamma_{1}\rangle$ and $\gamma_{2}\rangle$ denote different coherent states (e.g., GNU01 ; KIM18 ).

III Wigner function of thermal state for one-dimensional box problem

The system under consideration is a single particle confined in the region of $-a\leq x\leq a$ (with $a>0$) by a one-dimensional infinite potential well with either the Dirichlet boundary condition $\Psi(a)=\Psi(-a)=0$ (Dbc) or Neumann boundary condition $\Phi^{\prime}(a)=\Phi^{\prime}(-a)=0$ (Nbc). As is well-known, its $n$th eigenstate for Dbc is given by LEE83 ; ALM90 ; GRO94 ; LEE95 ; DIA02

[TABLE]

where $n=1,2,3,\cdots$ and $|x|\leq a$, and $\phi_{\scriptscriptstyle 0}(x)=(2a)^{-1/2}$; therefore, $\phi_{n}(x)$ is discontinuous at $x=\pm a$ if the analytic continuation is under consideration that $\phi_{n}(x)\equiv 0$ for $|x|>a$. The corresponding energy eigenvalue is $E_{n}(L)=(p_{n})^{2}/(2m)$ for both Dbc and Nbc, where $m$ is the mass of the particle, and $p_{n}=\hbar k_{n}$ with $k_{n}=\pm n\pi/L$; here $L=2a$ denotes the width of the potential well (note that $E_{0}=0$ for Nbc FAC15 ). Then, it is straightforward to compute the Wigner function corresponding to the eigenstate $|n\rangle$ such that for Dbc,

[TABLE]

where $n=1,2,3,\cdots$ and $W_{\scriptscriptstyle 0;N}(x,p)=(2\pi ap)^{-1}\sin(2\xi_{x}p/\hbar)$; here, $\xi_{x}=a-|x|$ is required by the boundary condition of $|x+\xi|\leq a$ and $|x-\xi|\leq a$. Therefore, $W_{n}(\pm a,p)=0$ for both Dbc and Nbc, as required (cf. Ref. DIA02 ). We observe that Eqs. (18a) and (18b), as well as $W_{\scriptscriptstyle 0;N}(x,p)$, are even functions of both $x$ and $p$, non-Gaussian, and can be negative valued indeed (cf. Figs. 1 and 2).

It is also instructive, for later purposes, to consider Eqs. (18a) and (18b) in the limit of $\hbar\to 0$, in particular the respective ground states. First, with the help of the identity $\delta(p)=\lim_{\epsilon\to 0}\,(\pi p)^{-1}\,\sin(p/\epsilon)$ GAS99 , it is easy to show that $W_{0;N}(x,p)\to(2a)^{-1}\,\delta(p)$ in this limit. It is also straightforward to observe that $W_{1;D}(x,p)\to 0$ for $p\neq 0$ and $W_{1;D}(x,0)\to\infty$, thus yielding $W_{0;D}(x,p)\to(2a)^{-1}\,\delta(p)$ as well.

Now, we are ready to discuss the thermal Wigner function of this system, which is given for Dbc and Nbc by [cf. Eq. (15)]

[TABLE]

with $\lambda=\beta\hbar^{2}\pi^{2}\,(8ma^{2})^{-1}$ and $(Z_{\beta})_{\scriptscriptstyle N}=(Z_{\beta})_{\scriptscriptstyle D}+1$, respectively. These can also be expressed as the integral form

[TABLE]

in terms of the Jacobi theta functions GRA07

[TABLE]

here, $Z_{\beta}\to(Z_{\beta})_{\scriptscriptstyle D}=2^{-1}\,\{\vartheta_{3}(0,e^{-\lambda})-1\}$ and $B=-1$ for $W_{\beta;D}(x,p)$ while $Z_{\beta}\to(Z_{\beta})_{\scriptscriptstyle N}=2^{-1}\,\{\vartheta_{3}(0,e^{-\lambda})+1\}$ and $B=1$ for $W_{\beta;N}(x,p)$.

To study the quantum-classical transition, we intend to rewrite Eqs. (19a) and (19b); after some algebraic manipulations, every single step of which is provided in detail in the Appendix, we can finally arrive at the expression

[TABLE]

in terms of the error function $\mbox{erf}(z)$, where the width $L=2a$ and the classical partition function $Z_{\beta,\mbox{\scriptsize cl}}=(8\pi ma^{2}/\beta)^{1/2}$ for both Dbc and Nbc. Then, in the classical limit, Eq. (22) reduces to its classical counterpart $P_{\beta}(x,p)=(Z_{\beta,\mbox{\scriptsize cl}})^{-1}\,e^{-\beta p^{2}/2m}>0$, being Gaussian, which results from the term of $\nu=0$ (with $\hbar\to 0$). Eq. (22) is the second central result of our paper.

Comments are deserved here. First, we observe that $W_{\beta}(\pm a,p)=0$ and thus $W_{\beta}(x,p)$ is continuous in the entire phase space. On the other hand, $P_{\beta}(\pm a,p)\neq 0$ and thus $P_{\beta}(x,p)$ is discontinuous at both boundary points. This discontinuity also implies disappearance of the wave properties. Second, the classical probability distribution $P_{\beta}(x,p)$ further reduces to $(2a)^{-1}\,\delta(p)$ at $T=0$, in accordance with both $W_{1;D}(x,p)$ and $W_{0;N}(x,p)$ within $\hbar\to 0$, as discussed after Eq. (18b). Third, all other terms of $\nu\neq 0$ of Eq. (22) will then represent the purely quantum correction; the value $2L$ denotes the length of an arbitrary primitive periodic orbit (i.e., a closed path traversed only once from an arbitrary phase-space position $(x_{0},p_{0})$ to the same one after two reflections on the potential walls at $x=\pm a$) KEA87 ; BRA97 ; STE98 . In fact, if an index $\nu=\nu_{\mbox{\scriptsize e}}$ (or $\mu=\mu_{\mbox{\scriptsize e}}$) is even, then it represents a periodic orbit with its length $\nu_{\mbox{\scriptsize e}}L$ (or $\mu_{\mbox{\scriptsize e}}L$), corresponding to $\nu_{\mbox{\scriptsize e}}/2$ (or $\mu_{\mbox{\scriptsize e}}/2$) repetitions of its primitive periodic orbit. On the other hand, if an index $\nu=\nu_{\mbox{\scriptsize o}}$ (or $\mu=\mu_{\mbox{\scriptsize o}}$) is odd, then it represents an orbit moving from $(x_{0},p_{0})$ to $(-x_{0},-p_{0})$ with its length $\nu_{\mbox{\scriptsize o}}L$ (or $\mu_{\mbox{\scriptsize o}}L$), which is also needed due to the even parity of this system; note that the cases of $\nu,\mu<0$ simply denote periodic orbits initially moving in the negative direction.

To explicitly discuss Eq. (22) in the high-temperature regime, we employ both identities $\mbox{erf}=1-\mbox{erfc}$ and $e^{2z_{1}z_{2}+z_{2}^{2}}\,\mbox{erfc}(z_{1}+z_{2})=\sum_{k=0}^{\infty}(-2z_{2})^{k}\,i^{k}\mbox{erfc}(z_{1})$ ABR65 , which will yield the exact expression

[TABLE]

With the help of $i^{k}\mbox{erfc}(0)=\{2^{k}\,\Gamma(k/2+1)\}^{-1}$ with $i^{-1}\mbox{erfc}(z)=(2/\sqrt{\pi})\exp(-z^{2})$ ABR65 , the boundary condition $W_{\beta}(\pm a,p)=0$ can be confirmed. We note here that the classical Gaussian part and the quantal non-Gaussian part compete with each other, which is not the case for a single linear oscillator, Eq. (16a). This non-Gaussian part is actually expressed as two different kinds of contributions; the sums over the periodic orbits $(\nu,\mu)$ are responsible for the purely quantum effect (i.e., temperature-independent) while the sums of $k$ for the thermal effect (also note, for comparison, that $\hbar$ and $\beta$ are always non-separable in form of $\beta\hbar\omega$ for Eq. (16a)). As is well-known, the sums over periodic orbits with non-zero lengths are responsible for the stepwise nature of the spectral staircase $N(E)=\sum_{n}\Theta(E-E_{n})$ while the trivial orbits with zero lengths solely contribute to the smooth increase of $N(E)$ with $E$ KEA87 ; BRA97 ; STE98 .

Therefore, it is interesting to consider two different limits of Eq. (III) separately; first, the purely semiclassical limit, by neglecting all periodic orbits with non-zero lengths (i.e., weakening the oscillatory quantum correction), and second, the high-temperature limit ($\beta\to 0$). First, in the semiclassical limit, Eq. (III) easily reduces to

[TABLE]

with the corresponding normalizing $\overline{N}_{\beta}$. Eq. (24) actually meets the boundary condition $W_{\beta}(\pm a,p)=0$, as long as $\hbar$ is finite, albeit sufficiently small. Then, Figs. 3 and 4 show that even this semiclassical result with the weakened oscillatory quantum correction can possess negative values indeed. On the other hand, in the high-temperature limit, Eq. (III) turns out to be

[TABLE]

where the quantum fluctuation

[TABLE]

with $\int dxdp\;\mbox{QF}_{\beta}(x,p)=0$. Eq. (25) also can be negative valued, as long as $\beta$ is finite, albeit sufficiently small. This cannot meet the boundary condition (cf. Fig. 5).

Now, the non-negative distribution $\widetilde{\mathcal{W}}_{\beta;\alpha}(x,p)$ for the canonical thermal state can directly be obtained from the usual Wigner function $W_{\beta}(x,p)$ by utilizing the scenario for the canonical thermal state, discussed after Eq. (15). Therefore, it is straightforward to have

[TABLE]

for an arbitrary order $\alpha$, which can easily be evaluated explicitly with the help of Eqs. (22)-(25). The resulting expressions of $\widetilde{\mathcal{W}}_{\beta;\alpha}(x,p)$ and those of the Rényi-Wigner entropy $S_{\alpha}(\widetilde{\mathcal{W}}_{\beta})$ given in Eqs. (6) and (9) are clearly straightforward to obtain but simply too large in size, and so we do not provide them here.

IV Evaluations of Rényi-Wigner Entropies in the Phase Space

We begin with numerical evaluations of the entropies $S_{\alpha}(\widetilde{\mathcal{W}}_{\beta})$ given in Eq. (27b) for the one-dimensional box problem. Then, we observe good agreement between $S_{\alpha}(\widetilde{\mathcal{W}}_{\beta})$ and its counterpart $S_{\alpha}(\hat{\rho}_{\scriptscriptstyle\beta})=(1-\alpha)^{-1}\{(\ln Z_{\alpha\beta})-\alpha(\ln Z_{\beta})\}$ (in the high-temperature regime) (cf. Fig. 6). It is also interesting to compare Eq. (4) (or Eq. (6) with $\alpha=2$) and Eq. (14) by performing their actual evaluations for the pure state $\widetilde{\mathcal{W}}_{\scriptscriptstyle 0;N}(x,p)=(2\pi\hbar)\,\{W_{\scriptscriptstyle 0;N}(x,p)\}^{2}$ for Nbc as a simple illustration of our formulation, where $W_{\scriptscriptstyle 0;N}(x,p)=(2\pi ap)^{-1}\sin\{2\,(a-|x|)\,p/\hbar\}$; for Eq. (4), $\mbox{Tr}(\hat{\rho}^{2})=(2\pi\hbar)\int_{-a}^{a}dx\int_{-\infty}^{\infty}dp\,\{W_{\scriptscriptstyle 0;N}(x,p)\}^{2}=1$ and thus $S_{2}(\widetilde{\mathcal{W}}_{\scriptscriptstyle 0;N})=0$, as is easily verified. On the other hand, Eq. (14) will take the form, after some steps of algebraic manipulations, of

[TABLE]

as long as $|a-(\cdots)|\leq a$ in the argument of $\sin$, where $y_{1}:=x+(x_{1}+x_{2}+x_{3})/2;y_{2}:=x-(x_{1}-x_{2}-x_{3})/2;y_{3}:=x-(x_{1}+x_{2}-x_{3})/2;y_{4}:=x-(x_{1}+x_{2}+x_{3})/2$. Then we will have $\int_{-\infty}^{\infty}dx\int_{-\infty}^{\infty}dx_{1}\int_{-\infty}^{\infty}dx_{2}\int_{-\infty}^{\infty}dx_{3}\to\int_{-a}^{a}dy\int_{-a}^{a}dy_{1}\int_{-a}^{a}dy_{2}\int_{-a}^{a}dy_{3}$, which will finally lead to $\mbox{Tr}(\hat{\rho}^{2})=1$. As is explicitly shown here, an evaluation of $S_{2}(\hat{\rho})$ via Eq. (6), considered one of our central results, is much simpler than employing Eq. (14). Besides, we already notice that it would be a formidable task to find the recurrence relation between entropies $S_{l}$ and $S_{l+1}$, with $l=2,3,4,\cdots$ for a given distribution $\widetilde{\mathcal{W}}_{\rho;\alpha}(x,p)$ (or $W_{\rho}(x,p)$) [cf. Eqs. (22) and (III)], if needed for the analytic continuation of $l\to\alpha$.

For comparison, we briefly discuss other “entropies” as well, without considering the Bopp shift. First, some moments of the Wigner function $W_{n;D}(x,p)$ for Dbc can explicitly be evaluated such that

[TABLE]

Because of $n$-dependence, Eqs. (29b)-(29d) cannot appropriately reflect the higher moments $\mbox{Tr}(\hat{\rho}^{3})=\mbox{Tr}(\hat{\rho}^{4})=\mbox{Tr}(\hat{\rho}^{5})=1$, as expected. For $W_{n;N}(x,p)$ of the Nbc case, similar results will appear. Therefore, it is obvious that the “Wigner entropy” given by $S_{\mbox{\tiny W}}(\widetilde{\mathcal{W}}_{\scriptscriptstyle n;D})=-\int dxdp\,\widetilde{\mathcal{W}}_{\scriptscriptstyle n;D}(x,p)\,\ln\{(2\pi\hbar)\,\widetilde{\mathcal{W}}_{\scriptscriptstyle 0;N}(x,p)\}$ will be $n$-dependent and so cannot at all be used as an appropriate entropy for our purpose. This confirms that the same will also apply for the resulting “Wigner entropy” $S_{\mbox{\tiny W}}(\widetilde{\mathcal{W}}_{\beta})=-\int dxdp\,\widetilde{\mathcal{W}}_{\beta}(x,p)\,\ln\{(2\pi\hbar)\,\widetilde{\mathcal{W}}_{\beta}(x,p)\}$ of the thermal state. On the other hand, the Wigner entropy of the classical thermal distribution $W_{\beta}(x,p)\to P_{\beta}(x,p)=(Z_{\beta,\mbox{\scriptsize cl}})^{-1}\,e^{-\beta p^{2}/2m}$ with $|x|<a$ is given by the closed form

[TABLE]

Likewise, by applying Eq. (9) with substitution of $(2\pi\hbar)\,Z_{\beta}\to Z_{\beta,\mbox{\scriptsize cl}}$ and $W_{\rho^{\scriptscriptstyle\alpha}}(x,p)\to(2\pi\hbar)^{\scriptscriptstyle\alpha-1}(Z_{\beta,\mbox{\scriptsize cl}})^{\scriptscriptstyle\alpha}\,e^{-\alpha\beta p^{2}/2m}$, it is straightforward to obtain the entropy $S_{1}(P_{\beta})\to\ln(Z_{\beta,\mbox{\scriptsize cl}})+1/2-\ln(2\pi\hbar)$. By setting $(2\pi\hbar)\to 1$, we see that $S_{\mbox{\tiny W}}(P_{\beta})$ and $S_{1}(P_{\beta})$ become identical.

V Conclusions

We have introduced the Wigner-like operational distributions $\widetilde{\mathcal{W}}_{\rho;\alpha}(x,p)$ in the classical phase space, all of which are non-negative and well-defined over the entire phase space, by utilizing the properties of fractional moments $\mbox{Tr}(\hat{\rho}^{\alpha})$ with $\alpha>0$ of the density operator $\hat{\rho}$. Then we have provided a framework for exact evaluations of Rényi-Wigner entropies for the classical-like distributions $\widetilde{\mathcal{W}}_{\rho;\alpha}(x,p)$, in particular for arbitrary non-Gaussian states, which enables us to go beyond the study of Rényi entropies restricted to the Gaussian states with non-negative valued Wigner functions $W_{\rho}(x,p)$. This result can be regarded as a generalization of the preceding one developed in Ref. KIM18 , which has enabled to evaluate the entropies but essentially restricted to integer values of $\alpha$ only.

Subsequently, we have rigorously evaluated the Wigner function $W_{\rho}(x,p)$, directly leading to $\widetilde{\mathcal{W}}_{\rho;\alpha}(x,p)$, of the thermal state of a single particle confined by a one-dimensional infinite potential well with either the Dirichlet or Neumann boundary condition (as a simple non-Gaussian state), in order to illustrate our concept. We have successfully applied our framework for this non-Gaussian state. Our analysis has also been useful for a study of the quantal-classical transition by expressing the density operator $\hat{\rho}$ (mixed state) itself in terms of its phase-space counterpart $W_{\rho}(x,p)$ (or $\widetilde{\mathcal{W}}_{\rho;\alpha}$) which can be decomposed into its classical part and the quantum correction [cf. Eqs. (III)-(25)]. This aspect will provide further insights for deeper semiclassical analysis.

Our study will overall contribute to a better understanding of non-Gaussian states and their transitions either in the semiclassical limit ($\hbar\to 0$) or in the high-temperature limit ($\beta\to 0$). This phase-space approach will also be useful information-theoretically and thermodynamically for deeper discussions of the quantal-classical Second Law on the single footing. We may expect that our analysis of the Rényi-Wigner entropies for non-Gaussian states will contribute to making some additional classification among non-Gaussian states and its quantification (beyond the non-Gaussianity as their deviations from the respective reference Gaussian states) to be pursued and also that our approach will apply for other billiard systems (i.e., confined systems with different boundary shapes in two dimensions), which are well-known to possess quantum signatures of classically regular and chaotic motions.

Acknowledgments

The author gratefully acknowledges the financial support provided by the US Army Research Office (Grant No. W911NF-15-1-0145).

Appendix A Derivation of Eq. (22)

We begin by rewriting Eqs. (19a) and (19b) as

[TABLE]

where $y=\xi/a$. Employing the Poisson summation rule GAS99

[TABLE]

and then with the help of the identity GRA07

[TABLE]

Eq. (31) will be transformed into

[TABLE]

We perform the integration over $y$ by applying the identity (33) with $n\to y$, which will result in Eq. (22). In doing so, we also used, with the help of Poisson’s sum rule, the relation

[TABLE]

which reduces to $Z_{\beta,\mbox{\scriptsize cl}}$ in the limit of $\hbar\to 0$.