1 Introduction

When representing a quantum state with a quasiprobability distribution in phase space, the most common choices are the Glauber–Sudarshan P-function \(P(\alpha )\), the Wigner function \(W(\alpha )\), or the Husimi Q-function \(Q(\alpha )\), where \(\alpha \) is a complex variable [1]. These correspond to the normal, symmetric, and antinormal ordering of annihilation and creation operators, respectively, and belong to a continuous family of s-ordered quasiprobability distributions \(W_s(\alpha )\) parametrized by the real parameter \(s\in [-1,1]\), see [2, 3]. Conventionally, \(W_1(\alpha )=P(\alpha )\), \(W_0(\alpha )=W(\alpha )\), and \(W_{-1}(\alpha )=Q(\alpha )\). The P-function (i.e., \(s=1\)) of a state is often not even a regular function (for instance, it is expressed in terms of derivatives of the Dirac \(\delta \)-function for a squeezed vacuum state). However, by decreasing the value of s (which corresponds to convolving \(P(\alpha )\) with a Gaussian distribution of increasing variance), we reach a point where \(W_s(\alpha )\) becomes regular, see example in Fig. 1. The largest value of s for which \(W_s(\alpha )\) is a regular function for all states is \(s=0\), corresponding to the Wigner function \(W(\alpha )\). The Wigner function is thus always regular but, as is well known, it is often not a positive function, so it cannot be treated as a genuine probability distribution.

Fig. 1
figure 1

Phase-space representations of the Fock state \({|{1}\rangle }\). 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

Full size image

The negativities of \(W(\alpha )\) can be suppressed by further decreasing s below 0, which smears out the quasiprobability distribution up to the point where it becomes positive everywhere in phase space. The largest value of s for which \(W_s(\alpha )\) is a non-negative function for all states is \(s=-1\), corresponding to the Q-function \(Q(\alpha )\), see example in Fig. 1. This is the only quasiprobability distribution among \(W_s(\alpha )\) with \(s\in [-1,1]\) that behaves as a genuine probability distribution (this is because it is the probability distribution of the outcome of a POVM measurement in the overcomplete basis of coherent states \({|{\alpha }\rangle }\), often called heterodyne or eight-port homodyne detection in quantum optics).

Defining a sensible measure of uncertainty based on the representation of a state in quantum phase space is an objective that has long been pursued, and a natural candidate for this is the differential entropy as defined in Shannon’s information theory [4, 5]. Notably, the Shannon differential entropy of the Q-function associated with a state is called the Wehrl entropy [6] and it has extensively been analyzed in the literature. For example, the Wehrl conjecture [7] (later proven by Lieb [8]) gives a lower bound to the Wehrl entropy of any state. Let us stress here that the Q-function is the only one quasiprobability distribution \(W_s(\alpha )\) with \(s\in [-1,1]\) whose Shannon differential entropy is always well defined for all states as the entropy definition would otherwise require taking the logarithm of negative numbers for some states.

Here, we revisit the definition of the entropy of a quantum phase-space distribution \(W_s(\alpha )\) by relaxing the strong requirement of having a genuine probability distribution, yet requiring the function \(W_s(\alpha )\) to remain regular for all states. The most natural choice is then the Wigner function \(W(\alpha )\), which also enjoys numerous interesting properties and is by far the most commonly used phase-space representation in quantum optics [9]. Among its main properties, let us mention the phase-space area preservation under Gaussian unitaries, which does not hold for any other \(W_s(\alpha )\) with \(s\ne 0\). Furthermore, its marginals yield the probability distributions of the x and p canonical coordinates, so it can be thought of as a sort of joint (x, p) probability distribution albeit violating Kolmogorov’s non-negativity axiom.

It thus makes sense to define the Wigner entropy [10] of a state as the differential Shannon entropy of \(W(\alpha )\), which is well behaved as long as one restricts to Wigner-positive states, i.e., states such that \(W(\alpha )\ge 0\), \(\forall \alpha \in {\mathbb {C}}\). A great merit of this definition is that it inherits all nice properties of Wigner functions. For example, the Wigner entropy is invariant under Gaussian unitaries as a consequence of the area-preservation property (notice that such an invariance does not hold for the Wehrl entropy, nor for any other \(W_s(\alpha )\) with \(s<0\)). In fact, as long as the considered states are Wigner-positive, all results of Shannon’s information theory remain applicable to \(W(\alpha )\) since the latter is a true probability distribution (see, e.g., the Wigner entropy-power inequality [10] for a beam splitter). The Wigner entropy is thus also a lower bound to the sum of marginal entropies of x and p as a consequence of the subadditivity of Shannon entropy, which results in a stronger version of the entropic uncertainty relation of Białynicki-Birula and Mycielski [11], see [12] or [13] for a review. However, extending this entropic uncertainty measure to states with negative Wigner functions appears precluded because it inherently comes with logarithms of negative numbers.

In the present work, we start by exploring a few possible ways of adapting the definition of Wigner entropy to Wigner-negative states, i.e., states such that \(\exists \alpha \in {\mathbb {C}}: W(\alpha )<0\). Unfortunately, no good alternative real-valued functional emerges that obeys all desirable properties of an entropy measure. Then, we advocate that the same functional should be kept when considering Wigner-negative states by resorting to the analytic continuation of the Shannon entropy in the complex plane. As we shall see, this leads us to define a complex-valued Wigner entropy, whose real and imaginary parts both have a physical meaning (the real part has an interpretation close to that of the ordinary real-valued entropy, while the imaginary part is simply proportional to the negative volume of the Wigner function). Interestingly, both the real and imaginary parts of the complex Wigner entropy remain invariant under Gaussian unitaries. Finally, as an additional argument supporting this continuation in the complex plane, we consider the complex extension of de Bruijn’s identity, which leads us to define a complex-valued Fisher information associated with any Wigner function.

This paper is organized as follows. In Sect. 2 we recall the basics of Wigner functions and summarize the main facts about the (real-valued) Wigner entropy of Wigner-positive states [10]. In Sect. 3 we consider some possible ways of extending this definition to Wigner-negative states while keeping a real-valued entropy. We list the desirable properties of such an extended entropy and discuss the results of our exploration, suggesting that a real-valued entropy functional cannot be found. In Sect. 4 we then make the step to extend the Wigner entropy to the complex plane. We prove some main properties of the complex Wigner entropy and discuss a few examples. In Sect. 5 we move to the complex-valued Fisher information and prove that its real (resp. imaginary) part governs the time-evolution of the real (resp. imaginary) part of the Wigner entropy under Gaussian additive noise. Finally, we discuss the perspectives of our work in Sect. 6.

2 Wigner entropy

2.1 Wigner function

The Wigner function associated with a density operator \({\hat{\rho }}\) is defined as

$$\begin{aligned} W(x,p) = {1\over 2\pi } \int \langle x- {y\over 2} | {\hat{\rho }} | x + {y\over 2} \rangle \, e^{ipy} \, \textrm{d}y, \end{aligned}$$
(1)

where we have set \(\hbar =1\). Note that we use from now on the (x, p) canonical coordinates instead of the complex variable \(\alpha =(x+ip)/\sqrt{2}\). More generally, the Weyl transform A(x, p) of any linear operator \({{\hat{A}}}\) is defined as

$$\begin{aligned} A(x,p) = {1\over 2\pi } \int \langle x- {y\over 2} | {{\hat{A}}} | x + {y\over 2} \rangle \, e^{ipy} \, \textrm{d}y, \end{aligned}$$
(2)

implying that the Wigner function is simply the Weyl transform of the density operator \({\hat{\rho }}\). It is straightforward to check that

$$\begin{aligned} {{\hat{A}}} = {{\hat{A}}}^\dagger \quad \Leftrightarrow \quad A(x,p) = A^*(x,p) ,\quad \textrm{Tr} {{\hat{A}}} \; =\; \int A(x,p) \, \mathrm dx \, \mathrm dp , \end{aligned}$$
(3)

which immediately implies that the Wigner function W of any physical state \({\hat{\rho }}\) (with \({\hat{\rho }} = {\hat{\rho }}^\dagger \) and \(\textrm{Tr} {\hat{\rho }} = 1\)) is real and normalized to one. Furthermore, note that the Weyl transforms of operators \(f({{\hat{x}}})\) and \(g({{\hat{p}}})\) are \(f(x)/2\pi \) and \(g(p)/2\pi \), respectively, where f and g are arbitrary functions.

For any two linear operators \({{\hat{A}}}_1\) and \({{\hat{A}}}_2\), the so-called overlap formula reads

$$\begin{aligned} \textrm{Tr} ( {{\hat{A}}}_1 \,{{\hat{A}}}_2 ) = 2 \pi \int A_1(x,p) \, A_2(x,p) \, \textrm{d}x \, \textrm{d}p, \end{aligned}$$
(4)

which implies, for example, that

$$\begin{aligned} \textrm{Tr} ( {\hat{\rho }} )= \int W(x,p) \, \textrm{d}x \, \textrm{d}p,\quad \textrm{Tr} ( {\hat{\rho }}^2 ) = 2 \pi \int W(x,p)^2 \, \textrm{d}x \, \textrm{d}p, \end{aligned}$$
(5)

since the Weyl transform of the identity operator \({\hat{{\mathbb {1}}}}\) is simply the constant function \(1/2\pi \). Note that

$$\begin{aligned}{} & {} \textrm{Tr} ( {\hat{\rho }}^s ) \ne c\, \int W(x,p)^s \, \textrm{d}x \, \textrm{d}p, \qquad \forall s\ne 1 \text {~or~} 2, \end{aligned}$$
(6)

for any constant c. Furthermore, using the overlap formula and the fact that the Weyl transform of a projector \({|{x_0}\rangle }{\langle {x_0}|}\) in position space is \(\delta (x-x_0)/2\pi \), and, similarly, a projector \({|{p_0}\rangle }{\langle {p_0}|}\) in momentum space is \(\delta (p-p_0)/2\pi \), we get

$$\begin{aligned} \rho _x(x_0)&{:}{=}&\langle x_0 | {\hat{\rho }} | x_0 \rangle = \int W(x_0,p) \, \textrm{d}p,\nonumber \\ \rho _p(p_0)&{:}{=}&\langle p_0 | {\hat{\rho }} | p_0 \rangle = \int W(x,p_0) \, \textrm{d}x, \end{aligned}$$
(7)

confirming that the marginals of W(x, p) yield the probability densities of measuring the coordinates x or p, denoted respectively as \(\rho _x\) and \(\rho _p\). To be complete, note also that the Weyl transform of the particle-number parity operator \((-1)^{{\hat{n}}}\) is \(\delta (x)\delta (p)/2\), so that the overlap formula yields the well-known formula

$$\begin{aligned} W(0,0) = \frac{1}{\pi }\, \textrm{Tr} ( {\hat{\rho }} \,(-1)^{{\hat{n}}} ), \end{aligned}$$
(8)

connecting the expectation value of the particle-number parity with the Wigner function at the origin. Denoting the displacement operator as \({{\hat{D}}}_{x',p'}\), the Weyl transform of \({{\hat{D}}}_{x',p'}(-1)^{{\hat{n}}}{{\hat{D}}}_{x',p'}^\dagger \) becomes \(\delta (x-x')\delta (p-p')/2\), so we may extend Eq. (8) into

$$\begin{aligned} W(x',p') = \frac{1}{\pi }\, \textrm{Tr} ( {{\hat{D}}}^\dagger _{x',p'}\, {\hat{\rho }} \, {{\hat{D}}}_{x',p'}\,(-1)^{{\hat{n}}} ), \end{aligned}$$
(9)

which implies that the Wigner function at \((x',p')\) is proportional to the average particle-number parity of the displaced state by \((-x',-p')\). Hence, we also note that \(|W(x,p)|\le 1/\pi \), \(\forall x,p\). Finally, by taking the Weyl transform of the expression

$$\begin{aligned} W(x,p)=\int W(x',p') \,\delta (x-x')\delta (p-p') \, \textrm{d}x'\, \textrm{d}p' , \end{aligned}$$
(10)

we obtain

$$\begin{aligned} {\hat{\rho }}= 2 \int W(x',p') \, {{\hat{D}}}_{x',p'}\, (-1)^{{\hat{n}}}\, {{\hat{D}}}_{x',p'}^\dagger \, \textrm{d}x'\, \textrm{d}p' , \end{aligned}$$
(11)

which is the inverse Weyl transform of \(W(x',p')\).

2.2 Wigner entropy of Wigner-positive states

In Shannon’s information theory, the differential entropy of a pair of random variables (x, p) is defined as [14]

$$\begin{aligned} h(x,p)=-\int f(x,p)\ln f(x,p) \, \textrm{d}x \, \textrm{d}p \end{aligned}$$
(12)

where f(x, p) is the joint probability density of x and p. Roughly speaking, it measures how spread out is the distribution f(x, p). Note that h(x, p) can be positive but also negative. It satisfies the following scaling property: if the two random coordinates transform as

$$\begin{aligned} \begin{pmatrix}x'\\ p'\end{pmatrix}={\mathcal {S}}\begin{pmatrix}x\\ p\end{pmatrix}, \end{aligned}$$
(13)

where \({\mathcal {S}}\) is a \(2\times 2\) matrix, then the differential entropy is shifted by an additive constant, namely [14]

$$\begin{aligned} h(x',p')=h(x,p)+\ln |\!\det ({\mathcal {S}})|. \end{aligned}$$
(14)

As long as a state \({\hat{\rho }}\) admits a non-negative Wigner function, that is \(W(x,p)\ge 0, \forall x,p\), the latter behaves as a genuine probability distribution (it is non-negative and normalized to one) and we may compute its Shannon differential entropy. Accordingly, the Wigner entropy of a Wigner-positive state can be defined as [10]Footnote 1

$$\begin{aligned} h(W)=-\int W(x,p)\ln W(x,p) \, \textrm{d}x \, \textrm{d}p. \end{aligned}$$
(15)

Note that we write it as a functional of W here instead of h(x, p), but it measures the joint uncertainty of the (x,p) pair—or actually any pair of canonically conjugate coordinates—in phase space as long as the system admits a non-negative Wigner function W. We immediately see from Eq. (14) that the Wigner entropy is invariant under symplectic transformations (i.e., any phase shift or squeezing) since \(\det ({\mathcal {S}})=1\) in such a case [15]. It is also trivially invariant under a displacement in phase space since \(h(x+c)=h(x)\) for any constant \(c\in {\mathbb {R}}\) [14]. Hence, h(W) is invariant under all Gaussian unitaries. Note also that it takes the value \(\ln \pi +1\) for the vacuum state as well as any pure Gaussian state in view of this invariance.

The Wigner entropy appears as a natural functional to measure the uncertainty of Wigner-positive states in phase space as it enjoys reasonable properties, most notably:

  • Symmetric functional—The Wigner entropy can be rewritten as

    $$\begin{aligned} h(W)=\int \varphi \big (W(x,p)\big ) \, \textrm{d}x \, \textrm{d}p, \end{aligned}$$
    (16)

    where \(\varphi (z)=-z \ln z\). Therefore, it is indeed a symmetric functional, which means that it is invariant under any area-preserving transformation \({\mathcal {M}}\), that is, \(h({\mathcal {M}}\left[ W\right] )=h(W)\), where \({\mathcal {M}}[W]\) is any transformation in phase space that keeps the level-function of W unchanged [16]. This property is the continuous counterpart to the fact that the discrete Shannon entropy is invariant under permutations of the probability vector. The set of area-preserving transformations \({\mathcal {M}}\) includes the class of Gaussian unitaries (symplectic transformations and displacements in phase space).

  • Concave functional—The Wigner entropy is a concave functional of the Wigner function, that is,

    $$\begin{aligned} h(\lambda _1 W_1+\lambda _2 W_2)\ge \lambda _1 \, h(W_1)+\lambda _2 \, h(W_2), \end{aligned}$$
    (17)

    where \(\lambda _1,\lambda _2\ge 0\) and \(\lambda _1+\lambda _2=1\). This property is guaranteed by the fact that the function \(\varphi (z)\) appearing in Eq. (16) is itself concave.

  • Lower bound on marginal entropies—The Wigner entropy is a lower bound on the sum of the entropies of the marginal distributions, namely,

    $$\begin{aligned} h(W)\le h(\rho _x)+h(\rho _p). \end{aligned}$$
    (18)

    This is a consequence of the subadditivity of Shannon’s differential entropy [14]: the joint entropy can be written as \(h(x,p)=h(x)+h(p)-I(x\mathrm{:}p)\), where the mutual information \(I(x\mathrm{:}p) \ge 0\).

  • Pure Gaussian extremal states—Although only conjectured as of today [10], it is expected that the Wigner entropy of Wigner-positive states reaches its minimum over the set of pure Gaussian states, viewed as minimum-uncertainty states, that is

    $$\begin{aligned} \ln \pi +1 {\mathop {\le }\limits ^{?}} h(W). \end{aligned}$$
    (19)

    It must be recalled that the entropic uncertainty relation of Białynicki-Birula and Mycielski [11],

    $$\begin{aligned} \ln \pi +1 \le h(\rho _x)+h(\rho _p), \end{aligned}$$
    (20)

    gives a lower bound on the sum of marginal entropies which actually coincides with the Wigner entropy of pure Gaussian states. Provided it holds,Footnote 2 Eq. (19) can thus be viewed as a tighter entropic uncertainty relation involving the Wigner entropy (limited to Wigner-positive states) instead of the sum of marginal entropies as in Eq. (20).

3 Tentative definitions for the entropy of Wigner-negative states

3.1 Desirable properties

We wish to construct a valid extension of h(W) to Wigner-negative states that remains consistent with all (or most) of its properties. Let us denote the set of Wigner functions by \({\mathcal {W}}\), and the subset of non-negative Wigner functions by \({\mathcal {W}}_+\). We seek a real-valued functional \(\Phi :{\mathcal {W}}\mapsto {\mathbb {R}}\) with the following properties:

  1. 1.

    \(\forall W\in {\mathcal {W}}_+:\quad \Phi (W)=h(W)\)

  2. 2.

    \(\forall W\in {\mathcal {W}}:\quad \ \ \Phi (W)\) is a symmetric functional

  3. 3.

    \(\forall W\in {\mathcal {W}}: \quad \ \ \Phi (W)\) is a concave functional

  4. 4.

    \(\forall W\in {\mathcal {W}}:\quad \ \ \Phi (W) \le h(\rho _x)+h(\rho _p) \)

  5. 5.

    \(\forall W\in {\mathcal {W}}:\quad \ \ \ln \pi +1 \le \Phi (W)\).

3.2 Candidate real-valued extensions of h(W)

We consider three possible real-valued extensions of h(W) to Wigner-negative states which do satisfy properties 1 and 2 (i.e., symmetric functionals that reduce to the usual Wigner entropy for Wigner-positive states). We first define the real Wigner entropy as

$$\begin{aligned} h_r(W) = -\int W(x,p) \ln |W(x,p)| \, \textrm{d}x \, \textrm{d}p. \end{aligned}$$
(21)

It is not an entropy stricto sensu as it departs from the usual “\(-z\ln z\)” form, but we use this name because it will appear in Sect. 4 that it coincides with the real part of the complex-valued Wigner entropy. We then define the absolute Wigner entropy as the entropy of the absolute value of the Wigner function, namely

$$\begin{aligned} h_a(W) = -\int |W(x,p)| \ln |W(x,p)|\, \textrm{d}x \, \textrm{d}p. \end{aligned}$$
(22)

Note that \(|W(x,p)|\) is not normalized for Wigner-negative states, so that \(h_{\textit{a}}(W)\) is also not a true entropy. Finally, we define the positive Wigner entropy as the entropy of the positive part of the Wigner function, namely

$$\begin{aligned} h_{\mathrm {+}}(W) = -\int \limits _{W(x,p)\ge 0} W(x,p) \ln W(x,p) \, \textrm{d}x \,\textrm{d}p. \end{aligned}$$
(23)

Note again that W(x, p) is not normalized over the domain \(W(x,p)\ge 0\) in the case of Wigner-negative states, so that \(h_{\mathrm {+}}(W)\) is not a genuine entropy

Fig. 2
figure 2

The three plotted functions \(\varphi _r(W)=-W\ln |W|\), \(\varphi _a(W)=-|W|\ln |W|\), 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 \(W\ge 0\). These functions are used to build the symmetric functionals \(h_r(W)=\int \varphi _r(W(x,p))\, \textrm{d}x\, \textrm{d}p\), \(h_a(W)=\int \varphi _a(W(x,p))\,\textrm{d}x\, \textrm{d}p\) and \(h_+(W)=\int \varphi _+(W(x,p))\,\textrm{d}x\,\textrm{d}p\), 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 \(|W|\le 1/\pi \) but this does not change the problem

Full size image

All three functionals \(h_r\), \(h_a\), and \(h_+\) are symmetric as they are constructed from the integration of some function \(\varphi (W)\) over the entire phase space (see Fig. 2), so they are invariant under Gaussian unitaries. Moreover, it is obvious that the three functionals coincide with the Wigner entropy when W is non-negative, that is, \(h_r(W)=h_a(W)=h_+(W)=h(W)\), for all \(W\in {\mathcal {W}}_+\). However, it appears that none of these three functionals is concave over the full set \({\mathcal {W}}\). This can be understood by looking at the corresponding functions \(\varphi _r\), \(\varphi _a\), and \(\varphi _+\) in Fig. 2. These three functions coincide with \(\varphi (z)=-z \ln z\) when \(z\ge 0\), so they are concave over the restricted domain \(z\in {\mathbb {R}}^+\), but they are not concave anymore in the whole domain \(z\in {\mathbb {R}}\). Hence, the functionals \(h_r\), \(h_a\), and \(h_+\) are concave for \(W\in {\mathcal {W}}_+\), but not concave for \(W\in {\mathcal {W}}\). These functionals are, of course, not unique—other functions \(\varphi (z)\) could be built—but it appears that the construction of a symmetric concave functional over \({\mathcal {W}}\) is bound to fail because it is impossible to build a concave function that extends the function \(\varphi (z)=-z \ln z\) to negative z (this is because the derivative of \(\varphi (z)\) is infinite at \(z=0^+\)).

The construction of a real-valued extension of h(W) satisfying properties 1, 2, and 3 thus seems hopeless, which motives the introduction of a complex-valued extension of h(W) in Sect. 4. Before doing so, we examine whether some of the above three real-valued extensions of h(W) may nevertheless satisfy properties 4 or 5. Note first that these three functionals can be ordered as

$$\begin{aligned} h_r(W)\le h_+(W)\le h_a(W). \end{aligned}$$
(24)

This can easily be proven by defining the negative Wigner entropy as

$$\begin{aligned} h_{\mathrm {-}}(W) = \int \limits _{W(x,p)\le 0} W(x,p) \ln |W(x,p)| \, \textrm{d}x \, \textrm{d}p, \end{aligned}$$
(25)

and noting that \(h_{\mathrm {-}}(W)\ge 0\) since \(|W(x,p)|\le 1/\pi \), so that \(\ln |W(x,p)|< 0\). It is easily seen that the following relations hold:

$$\begin{aligned} h_r(W) = h_+(W)-h_-(W), \end{aligned}$$
(26)
$$\begin{aligned} h_a(W) = h_+(W)+h_-(W), \end{aligned}$$
(27)

which implies Eq. (24) since \(h_{\mathrm {-}}(W)\ge 0\). We have investigated whether these three functionals may be a lower bound to the sum of marginal entropies (property 4) or whether they may themselves be lower bounded by \(\ln \pi +1\) (property 5). Since the results of this exploration are not fully satisfactory, we report them in Appendix A. From numerical evidence, it seems that \(h_+(W)\) is lower bounded by \(\ln \pi +1\), hence the same is true for \(h_a(W)\), but we lack a proof. We can also lower bound the sum of marginal entropies \(h(\rho _x)+h(\rho _p)\) by a quantity that depends on \(h_{\mathrm {+}}(W)\), but the result is not tight.

4 Complex-valued entropy of a Wigner function

4.1 Definition

The definition (15) based on the differential Shannon entropy is only valid provided \(W(x,p)\in {\mathcal {W}}_+\) because it requires taking the logarithm of W(x, p) and the logarithm function in the real domain is not defined for negative values. In order to extend the Wigner entropy to all states, including those with negative Wigner functions, we turn to a complex extension of the Wigner entropy by using the complex logarithm function

$$\begin{aligned} \text {Ln} \,(z)=\ln |z| +i \, \arg (z), \qquad z\in {\mathbb {C}}. \end{aligned}$$
(28)

We must recall here that Ln(z) is a multivalued function since the argument is defined modulo \(2\pi \). Here, we conventionally use the principal value of the logarithm, meaning that its imaginary part lies in \(]-\pi ,\pi ]\). We will also use a lazy notation and write \(\ln (z)\) instead of \(\text {Ln} \,(z)\) for the complex logarithm of z.

Thus, we define the complex Wigner entropy as

$$\begin{aligned} h_c(W)= & {} -\int \, W(x,p) \, \ln W(x,p) \, \mathrm dx \, \mathrm dp \nonumber \\= & {} h_r(W) + i \, h_i(W) \end{aligned}$$
(29)

where the real and imaginary parts of \(h_c(W)\) are respectively given by

$$\begin{aligned} h_r(W)= & {} -\int W(x,p) \, \ln |W(x,p)| \, \mathrm dx \, \mathrm dp \end{aligned}$$
(30)
$$\begin{aligned} h_i(W)= & {} -\int W(x,p) \, \arg W(x,p) \, \mathrm dx \, \mathrm dp \nonumber \\= & {} \pi \int \frac{|W(x,p)|-W(x,p)}{2}\, \mathrm dx \, \mathrm dp. \end{aligned}$$
(31)

Note that \(\arg W(x,p)\) is equal to 0 if the Wigner function is positive and is equal to \(\pi \) if it is negative, hence the second definition of \(h_i(W)\) [Eq. (31)]. Thus, by convention, we define \(h_i(W)\) as non-negative.

The imaginary part of the complex Wigner entropy happens to be simply proportional to the volume of the negative part of the Wigner function, namely

$$\begin{aligned} h_i(W) = \pi \, \text {Vol}_{-}(W) \end{aligned}$$
(32)

with

$$\begin{aligned} \textrm{Vol}_{-}(W) =-\int \limits _{W(x,p) < 0} W(x,p) \, \mathrm dx \, \mathrm dp. \end{aligned}$$
(33)

The negativity of the Wigner function has long been known to witness the non-classicality of a state and, along this line, the negative volume \(\textrm{Vol}_{-}(W)\) has even been used to build a measure of non-classicality [19] (see also [20, 21] for a resource-theoretical approach to Wigner negativity). It is remarkable that the negative volume appears naturally here simply from the expression of the complex Wigner entropy. Note that \(h_i(W)\) is unbounded as it is known that one can build Wigner functions with an arbitrarily large negative volume [22].

4.2 Properties of the complex Wigner entropy

Let us examine some properties of the real and imaginary parts of the complex Wigner entropy \(h_c(W)\). Note first that \(h_c(W)\) is defined only if the integral in Eq. (29) exists. Since any Wigner function tends to zero at the limit of infinite distance from the origin in phase space, we have the complex logarithm of \(W(x,p)\rightarrow 0\). However, using the standard continuity argument, we see that this does not cause a problem since \(\lim _{W\rightarrow 0}W \ln |W| =0\) and \(\lim _{W\rightarrow 0}W \arg W = 0\).

Property 1

Both the real and imaginary parts of the complex Wigner entropy are invariant under symplectic transformations (Gaussian unitaries).

Proof

Consider the symplectic transformation such that the vector of canonically conjugate variables \(\textbf{X}=(x,p)^T\) transforms into \(\mathbf {X'}=(x',p')^T\) according to \(\mathbf {X'}={\mathcal {S}}\textbf{X}+\textbf{d}\), with \({\mathcal {S}}\) being a symplecticFootnote 3 matrix and \(\textbf{d}\) being a (real) displacement vector. Following this transformation, the Wigner function can thus be expressed as \(W'(x',p')= |\det ({\mathcal {S}})|^{-1}W(x,p)\), where (x, p) is expressed in terms of \((x',p')\) by inverting the transformation. Therefore, using the change of variable \(\textrm{d}x'\, \textrm{d}p' = |\det ({\mathcal {S}})|\, \textrm{d}x\, \textrm{d}p\),

$$\begin{aligned} h_r(W')= & {} -\int W'(x',p')\ln |W'(x',p')|\, \textrm{d}x'\, \textrm{d}p'\nonumber \\= & {} -\int W(x,p)\ln \left| \frac{W(x,p)}{\det ( {\mathcal {S}})}\right| \textrm{d}x \, \textrm{d}p\nonumber \\= & {} h_r(W)+ \ln |\det ({\mathcal {S}})| \nonumber \\= & {} h_r(W) \end{aligned}$$
(34)

and

$$\begin{aligned} h_i(W')= & {} \pi \int \frac{|W'(x',p')|-W'(x',p')}{2} \, \textrm{d}x'\, \textrm{d}p'\nonumber \\= & {} \pi \int \frac{|W(x,p)|-W(x,p)}{2 |\det ( {\mathcal {S}})|}\, \textrm{d}x'\, \textrm{d}p'\nonumber \\= & {} \pi \int \frac{|W(x,p)|-W(x,p)}{2 }\, \textrm{d}x\, \textrm{d}p\nonumber \\= & {} h_i(W). \end{aligned}$$
(35)

\(\square \)

Note that the proof of Eq. (34) looks similar to the corresponding proof for Wigner-positive states in Sect. 2.2 as we have used \(\det ({\mathcal {S}})=1\) since \({\mathcal {S}}\) is symplectic. Yet, the proof of Eq. (34) exploits the normalization of W(x, p) in \(\mathcal W\) (encompassing Wigner-negative states).

Together, Eqs. (34) and (35) thus imply that the complex Wigner entropy \(h_c\) is indeed invariant under all Gaussian unitaries. Note that this property remains valid when extending the Wigner function to n modes.

Property 2

Both the real and imaginary parts of the complex Wigner entropy are symmetric functionals.

Proof

The proof simply goes by noting that the real and imaginary parts of \(h_c\) can be written as

$$\begin{aligned} h_r(W)= & {} \int \varphi _r\left( W(x,p)\right) \, \textrm{d}x \, \textrm{d}p, \end{aligned}$$
(36)
$$\begin{aligned} h_i(W)= & {} \int \varphi _i\left( W(x,p)\right) \, \textrm{d}x \, \textrm{d}p, \end{aligned}$$
(37)

with \(\varphi _r(z)=-|z|\ln z\) and \(\varphi _i(z)=\pi (|z|-z)/2\). The fact that W can be negative does not make any difference. \(\square \)

It can be shown that integrals such as those appearing in Eqs. (36) and (37) only depend on W(x, p) through the level function, hence they are equal for two Wigner functions that have the same level function [16]. This property extends the invariance of \(h_c\) under symplectic transformations (which are area-preserving) to the invariance of \(h_c\) under all (even nonphysical) area-preserving transformations (those that keep the level function of the Wigner function unchanged, so that the initial and final Wigner functions are equivalent in the sense of continuous majorization theory, see [16]).

Property 3

The real part of the complex Wigner entropy is additive when considering a product state.

Proof

Let us consider a two-mode Wigner function defined as \(W(x_1,p_1,x_2,p_2)=W_1(x_1,p_1)W_2(x_2,p_2)\). Then,

$$\begin{aligned} h_r(W)= & {} -\int W\ln |W|\,\textrm{d}x_1 \,\textrm{d}p_1\, \textrm{d}x_2\, \textrm{d}p_2 \nonumber \\= & {} -\int W_1\ln |W_1| \,\textrm{d}x_1 \,\textrm{d}p_1\int W_2\, \textrm{d}x_2\, \textrm{d}p_2 -\int W_1\,\textrm{d}x_1 \,\textrm{d}p_1 \int W_2\ln |W_2|\, \textrm{d}x_2\, \textrm{d}p_2\nonumber \\= & {} h_r(W_1) + h_r(W_2) \end{aligned}$$
(38)

where we have omitted the arguments of Wigner functions for notational simplicity. \(\square \)

Again, the proof is similar to that for Wigner-positive states but here we have used the normalization of Wigner functions in \(\mathcal W\), including negative Wigner functions. Interestingly, numerics seems to indicate that the real part of the complex Wigner entropy is subadditive for a two-mode Wigner function \(W(x_1,p_1,x_2,p_2)\) that is not in a product form, namely \(h_r(W)\le h_r(W_1)+h(W_2)\), but we have not found a proof.

Property 4

The imaginary part of the complex Wigner entropy is superadditive when considering a product state.

Proof

Using the identity

$$\begin{aligned} |XY|-XY = (|X|-X)Y+X(|Y|-Y) +(|X|-X)(|Y|-Y) \end{aligned}$$
(39)

and setting \(W(x_1,p_1,x_2,p_2)=W_1(x_1,p_1)W_2(x_2,p_2)\), we obtain

$$\begin{aligned} h_i(W)= & {} \pi \int \frac{|W|-W}{2} \,\textrm{d}x_1 \,\textrm{d}p_1\, \textrm{d}x_2\, \textrm{d}p_2 \nonumber \\= & {} \pi \int \frac{|W_1 W_2|-W_1 W_2}{2}\,\textrm{d}x_1 \,\textrm{d}p_1\, \textrm{d}x_2\, \textrm{d}p_2 \nonumber \\= & {} \pi \int \frac{(|W_1|-W_1)}{2}\,\textrm{d}x_1 \,\textrm{d}p_1 \int W_2\, \textrm{d}x_2\, \textrm{d}p_2 +\pi \int W_1\, \textrm{d}x_1\, \textrm{d}p_1 \int \frac{(|W_2|-W_2)}{2}\,\textrm{d}x_2 \,\textrm{d}p_2 + \Delta \nonumber \\= & {} h_i(W_1)+h_i(W_2) + \Delta \end{aligned}$$
(40)

where we have defined

$$\begin{aligned} \Delta= & {} \pi \int \frac{(|W_1|-W_1)(|W_2|-W_2)}{2}\,\textrm{d}x_1 \,\textrm{d}p_1\, \textrm{d}x_2\, \textrm{d}p_2 \nonumber \\= & {} \frac{2}{\pi }\, h_i(W_1) \,h_i(W_2)\nonumber \\\ge & {} 0 \end{aligned}$$
(41)

\(\square \)

Thus, \(h_i(W)\ge h_i(W_1)+h_i(W_2)\). We see that \(\Im (h_c)\) becomes additive in the special case where at least one of the two states is Wigner-positive, but otherwise it is superadditive because the product of the negative volumes is nonzero.Footnote 4 Note that the superadditivity of \(\Im (h_c)\) is linked to our convention \(\arg W(x,p)=\pi \) for \(W(x,p)<0\), but it should be realized that \(\Im (h_c)\) becomes subadditive if we choose \(\arg W(x,p)=-\pi \). From numerics, we also observe that the imaginary part of the complex entropy \(h_i\) apparently remains superadditive for arbitrary two-mode Wigner functions \(W(x_1,p_1,x_2,p_2)\), but we have not proven this.

Property 5

The imaginary part of the complex Wigner entropy is a convex function of the Wigner function.Footnote 5

Proof

Let us consider the following mixture \(W(x,p)=\lambda W_1(x,p)+(1-\lambda )W_2(x,p)\), where \(\lambda \in [0,1]\). Then,

$$\begin{aligned} h_i(W)= & {} \pi \int \left( \frac{|\lambda W_1+(1-\lambda )W_2|}{2} -\frac{\lambda W_1+(1-\lambda )W_2}{2}\right) \, \textrm{d}x \, \textrm{d}p\nonumber \\\le & {} \pi \int \left( \frac{\lambda | W_1|+(1-\lambda )|W_2|}{2} -\frac{\lambda W_1+(1-\lambda )W_2}{2}\right) \, \textrm{d}x \, \textrm{d}p\nonumber \\= & {} \lambda \, h_i(W_1)+(1-\lambda ) \, h_i(W_2) \end{aligned}$$
(42)

where we have used the triangle inequality, namely \(|X+Y|\le |X|+|Y|\). \(\square \)

Again, convexity originates here from the choice that \(\arg W(x,p)=\pi \) for \(W(x,p)<0\). The opposite choice \(\arg W(x,p)=-\pi \) would instead imply the concavity of \(\Im (h_c)\). Unfortunately, we cannot say anything about the concavity of the real part of the Wigner entropy. Numerical tests show that it is neither concave nor convex.

4.3 Expressing the Wigner entropy in state space

By using the overlap formula (4), the definition (29) of the complex Wigner entropy in phase space may be reexpressed in state space as

$$\begin{aligned} h_c(W) = \textrm{Tr} ( {\hat{\rho }} \, {\hat{\Theta }} ), \end{aligned}$$
(43)

where \({\hat{\Theta }}\) is defined as the operator whose Weyl transform is given by

$$\begin{aligned} {\mathcal {W}}[{\hat{\Theta ]}} = - \frac{1}{2\pi }\ln W(x,p) \end{aligned}$$
(44)

Thus, we may also rewrite \({\hat{\Theta }}={\mathcal {S}}[{{\hat{L}}}] \), where \({\mathcal {S}}[{{\hat{L}}} ]\) denotes the Weyl (or symmetric) ordered form of the operator \({{\hat{L}}} = - \ln W({{\hat{x}}},{{\hat{p}}})\), hence

$$\begin{aligned} h_c(W) = \textrm{Tr} ( {\hat{\rho }} \, {\mathcal {S}}[{{\hat{L}}}] ). \end{aligned}$$
(45)

Unfortunately, Eq. (45) does not seem to provide a very convenient method to compute \(h_c\), except in some special cases. As an example, consider a thermal state with a mean photon number \(\nu \). Its Wigner function is

$$\begin{aligned} W_\nu (x,p) = \frac{1}{\pi (1+2\nu )} \text {e}^{-\frac{x^2 + p2}{1+2v}}. \end{aligned}$$
(46)

Here, the operator

$$\begin{aligned} {{\hat{L}}} = -\ln W_\nu ({{\hat{x}}}, {{\hat{p}}}) = \ln \pi (1+2\nu ) + \frac{{{\hat{x}}}^2+{{\hat{p}}}^2}{1+2\nu } \end{aligned}$$
(47)

is already written in symmetric order, so we have

$$\begin{aligned} {\mathcal {S}}[{{\hat{L}}}] = \ln \pi (1+2\nu ) + \frac{{\hat{{\mathbb {1}}}} + 2 {{\hat{n}}}}{1+2\nu }. \end{aligned}$$
(48)

Since \(\textrm{Tr} ( {\hat{\rho }} \, {{\hat{n}}} )=\nu \), Eq. (45) gives

$$\begin{aligned} h_c(W_\nu )= & {} \ln \pi +1 + \ln (1+2\nu ), \end{aligned}$$
(49)

which is indeed the Wigner entropy of a Gaussian thermal state.

4.4 Examples and numerical exploration

We have numerically computed the complex Wigner entropy of a variety of quantum states in order to get a grasp on its meaning. First of all, we know that \(h_c\) lies on the real axis as soon as the state is Wigner positive. Furthermore, provided that conjecture (19) is true, we know that \(\Re h_c \ge \ln \pi +1\) for all Wigner-positive states. This is visible in Fig. 3, where we have plotted the complex Wigner entropy \(h_c\) of randomly generated statesFootnote 6, in particular pure states (orange) and mixed states (blue). The red dotted line indicates \(h_r=\ln \pi +1\). We note that mixed states typically have a lower value of \(\Im h_c \), which is of course associated with a lower (or even zero) negative volume. The first Fock states \({|{n}\rangle }\) with n ranging from 0 to 8 are represented by red stars in the complex entropy plane; we observe that both \(\Re h_c\) and \(\Im h_c\) increase with n. The vacuum state (as well as any pure Gaussian state) corresponds to the point \((\ln \pi +1,0)\).

Fig. 3
figure 3

Complex Wigner entropy \(h_c = h_r + i \, h_i\) of randomly generated states. Each blue (orange) point is associated with a random pure (mixed) state. Red stars represent Fock states \({|{n}\rangle }\) with \(n=0,1,\ldots , 8\). Note that \(h_r\) and \(h_i\) increase monotonically with n for Fock states. The dotted red line corresponds to \(h_r=\ln \pi +1\simeq 2.145\), which is a (conjectured) lower bound on the real part of the entropy of Wigner-positive states [10]

Full size image

In Fig. 4, we look more precisely at the complex entropy near this point and compare the superpositions and mixtures of the lowest Fock states. The situation is more subtle than anticipated as we see that for some pure state near the vacuum, \(\Im h_c\) becomes positive (which logically reflects the non-Gaussianity of the state as a consequence of Hudson’s theorem) but \(\Re h_c\) goes below \(\ln \pi +1\). This conjectured lower bound only applies along the real axis (i.e., for Wigner positive states), so it is not contradicted. Sadly, however, it seems difficult to determine the lowest possible value of \(\Re h_c\) for all states. Furthermore, we observe that the path connecting two neighboring Fock states is rather complicated. We see that the superpositions (solid blue curves) generally have much larger values of \(\Im (h_c)\) than the corresponding mixtures (dashed orange curves), which follows from the comparison of their negative volumes. The value of \(\Im (h_c)\) reaches precisely zero if the mixed state becomes Wigner positive (which happens if the weight of \({|{0}\rangle }\) is large enough). For example, we have \(\Im (h_c)=0\) for balanced mixtures of \({|{0}\rangle }\) and \({|{1}\rangle }\), or balanced mixtures of \({|{0}\rangle }\) and \({|{2}\rangle }\), but never for any mixture of \({|{1}\rangle }\) and \({|{2}\rangle }\). When increasing the weight of \({|{0}\rangle }\) beyond this point, \(h_c\) follows the real axis down to the lower bound attained for \({|{0}\rangle }\).

Fig. 4
figure 4

Plots of the complex Wigner entropy \(h_c = h_r + i \, h_i\) 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., \({|{\psi }\rangle }=\sqrt{p}{|{m}\rangle }+e^{i\varphi }\sqrt{1-p}{|{n}\rangle }\) with \(p\in [0,1]\) (note that the complex Wigner entropy of \({|{\psi }\rangle }\) is independent of \(\varphi \)). The dotted orange lines are the corresponding binary mixtures of Fock states, i.e., \(\hat{\rho }=p{|{m}\rangle }{\langle {m}|}+(1-p){|{n}\rangle }{\langle {n}|}\). The following binary superpositions and mixtures are plotted: \((m,n)=(0,1)\), (0, 2), (0, 3), and (1, 2)

Full size image

Let us now perform the analytical calculation of \(h_c\) for another example, namely cat states. With the appropriate displacement and rotation (hence, some Gaussian unitary, which do not affect \(h_c\)), any cat state can be brought to the following canonical expression:

$$\begin{aligned} {|{\psi }\rangle }_\alpha = \frac{1}{\sqrt{C}} \left( \sqrt{m}{|{\alpha }\rangle }+e^{i\varphi }\sqrt{1-m}{|{-\alpha }\rangle } \right) \end{aligned}$$
(50)

where \(\alpha \in {\mathbb {R}}_+\), \(\varphi \in [0,2\pi )\), \(m\in (0,1)\), and the normalization constant is \(C=1+2\sqrt{m(1-m)}\exp (-2\alpha ^2)\cos \varphi \). The density operator is thus

$$\begin{aligned} {|{\psi }\rangle }_\alpha \!{\langle {\psi }|}&= \frac{1}{C}\Big ( m{|{\alpha }\rangle }{\langle {\alpha }|}+(1-m){|{-\alpha }\rangle }{\langle {-\alpha }|} +2\sqrt{m(1-m)} (e^{i\varphi }{|{-\alpha }\rangle }{\langle {\alpha }|} + e^{-i\varphi }{|{\alpha }\rangle }{\langle {-\alpha }|}) \Big ) \end{aligned}$$
(51)

so the corresponding Wigner function is given by

$$\begin{aligned} W_\alpha (x,p)&= \frac{m}{C}\, W_0(x-\sqrt{2}\alpha ,p) + \frac{1-m}{C}\, W_0(x+\sqrt{2}\alpha ,p) + \frac{2\sqrt{m(1-m)}}{C} \, W_0(x,p) \cos (2\sqrt{2}\alpha p+\varphi ) \end{aligned}$$
(52)

where \(W_0(x,p)=\exp (-x^2-p^2)/\pi \) is the Wigner function of the vacuum state. In Fig. 5, we plot the real and imaginary parts of the complex Wigner entropy of this cat state (taking \(m=1/2\) and \(\varphi =0\)) and compare with their values for the corresponding incoherent mixture of the coherent states \({|{\alpha }\rangle }\) and \({|{-\alpha }\rangle }\). Of course, \(\Im (h_c)\) vanishes in the incoherent case, while it is non-zero for cat states (it is shown in Appendix B that \(\Im h_c\) tends to 1 in the limit \(\alpha \rightarrow \infty \)). In contrast, the curves of \(\Re (h_c)\) are more comparable for the cat state and its corresponding mixture (as shown in Appendix B, the two curves tend to each other in the limit \(\alpha \rightarrow \infty \), which is understandable as the two Wigner functions almost coincide up to a series of infinitely narrow fringes whose contributions cancel each other out). Of course, the cat state and mixture both coincide with the vacuum state for \(\alpha =0\), hence their complex entropies coincide. It can also be verified that the difference between \(h_r\) for \(\alpha =0\) and \(h_r\) for \(\alpha \rightarrow \infty \) tends to \(\ln 2\).

Fig. 5
figure 5

Plots of the real and imaginary parts of the complex Wigner entropy \(h_r+i\, h_i\) for \(W_{\textrm{cat}}\) and \(W_{\textrm{mix}}\), which are respectively the Wigner functions of an even cat state and of a balanced mixture of coherent states \({|{\alpha }\rangle }\) and \({|{-\alpha }\rangle }\). 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 \(|\alpha |\gg 1\), \(h_r\left( W_{\textrm{cat}}\right) \simeq h_r(W_{\textrm{mix}})\) and \(h_i\left( W_{\textrm{cat}}\right) \simeq 1\)

Full size image

5 Complex Fisher information and extended de Bruijn’s identity

5.1 Classical overview

Let us consider a vector \(\textbf{X}_0\) of classical random variables and a normally-distributed random vector \(\textbf{N}\) with covariance matrix \(\textbf{I}\). From these two random vectors, we define the random vector , which can be seen as a noisy version of \(\textbf{X}_0\) with the amount of noise growing in time t (the variance increases linearly with t). The probability density of \(\textbf{X}_t\) is the probability density of \(\textbf{X}_0\) convolved with a Gaussian distribution of covariance \(t\textbf{I}\), which describes a Gaussian diffusion process in time t characterized by the diffusion equation

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \, p_t(\textbf{x}) = \frac{1}{2} \, \Delta p_t \end{aligned}$$
(53)

where \(\Delta =\varvec{\nabla }^2\) is the Laplacian and \(p_t(\textbf{x})\) denotes the probability density of \(\textbf{X}_t\).

De Bruijn’s identity expresses the rate of increase of the Shannon entropy of the probability density \(p_t(\textbf{x})\) when it undergoes such a Gaussian diffusion. It is written as [14]

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \, h(\textbf{X}_0+\sqrt{t}\textbf{N}) = \frac{1}{2} \, J(\textbf{X}_0+\sqrt{t}\textbf{N}) \end{aligned}$$
(54)

where \(J(\textbf{X})\) is the Fisher information of \(\textbf{X}\). The latter is defined based on the logarithmic gradient \(\varvec{\nabla }\ln p\), namely

$$\begin{aligned} J(\textbf{X})&=\int p(\textbf{x}) \, \Vert \varvec{\nabla }\ln p\Vert ^2\textrm{d}\textbf{x}, \end{aligned}$$
(55)
$$\begin{aligned}&= \int \varvec{\nabla } p \cdot \varvec{\nabla } \ln p \,\, \textrm{d}\textbf{x}, \end{aligned}$$
(56)
$$\begin{aligned}&= \int p^{-1}(\textbf{x}) \, \Vert \varvec{\nabla } p \Vert ^2\textrm{d}\textbf{x}, \end{aligned}$$
(57)

where we have used \(\varvec{\nabla } \ln p =(\varvec{\nabla } p ) / p\). Note that the above three definitions of the Fisher information are equivalent as long as \(p(\textbf{x})\ge 0\), \(\forall \textbf{x}\), but we shall see that only the second one is suitable when moving to the complex plane. De Bruijn’s identity (54) can be verified by explicitly calculating the time derivative of \(h(\textbf{X}_t)\) and using Eq. (53) as well as the definition of the Fisher information \(J(\textbf{X})\), see [14]. Note also that \(J(\textbf{X})\ge 0\), expressing the fact that \(\textbf{X}_t\) can only spread over time.

Note that de Bruijn’s identity makes use of a special case of the definition of the Fisher information. In general, the Fisher information is introduced for a family of distributions \(p _{\varvec{\theta }}\) which depend on a set of parameters \({\varvec{\theta }} = (\theta _1, \theta _2, ...)\). The Fisher information is then defined as \(J _{\varvec{\theta }} (p _{\varvec{\theta }}) = \int p_{\varvec{\theta }} ({\textbf {x}}) \Vert \nabla _{\varvec{\theta }} \ln p _{\varvec{\theta }} ({\textbf {x}}) \Vert ^2 d\textrm{x}\). When the parameters \({\varvec{\theta }}\) act as a translation over the distributions, so that \(p_{\varvec{\theta }}({\textbf {x}}) = p({\textbf {x}}-\varvec{\theta })\), the Fisher information simplifies to Eq. (55) and is independent of \(\varvec{\theta }\) [14]. It is this particular non-parametric definition of the Fisher information that we will extend to the complex plane.

5.2 Complex Fisher information

Here, we will show that it is possible to derive a similar relation for an arbitrary Wigner function, even when it becomes negative. In this case, both terms of de Bruijn’s identity (54) become complex-valued, so we need to introduce a complex-valued Fisher information too. Crucially, the diffusion equation (53) holds regardless of the sign of \(p(\textbf{x})\), so it describes the evolution of any Wigner function under additive Gaussian noise (it is valid for positive as well as negative Wigner functions). Since we consider the Wigner function of a single mode, we deal with the 2-dimensional vector \(\textbf{x}=(x,p)^T\) of canonically conjugate coordinates. We define the complex Fisher information associated with any Wigner function W(x, p) as the complex continuation of definition (56), that is,

$$\begin{aligned} J_c(W)&= \int \varvec{\nabla } W \cdot \varvec{\nabla } \ln W \,\, \textrm{d}x \, \textrm{d}p \nonumber \\&= J_r(W) + i \, J_i(W), \end{aligned}$$
(58)

where \(\ln W\) stands for the complex logarithm of W. The real part of the complex-valued Fisher information is thus given by

$$\begin{aligned} J_r(W)&= \int \varvec{\nabla } W \cdot \varvec{\nabla } \ln |W| \,\, \textrm{d}x \, \textrm{d}p . \end{aligned}$$
(59)

One can write (notice the absence of absolute value on the right hand side)

$$\begin{aligned} \varvec{\nabla } \ln |W| = \frac{1}{W} \, \varvec{\nabla } W, \end{aligned}$$
(60)

so we also recover the other two definitions

$$\begin{aligned} J_r(W)&= \int W(x,p) \, \Vert \varvec{\nabla } \ln |W| \, \Vert ^2\,\, \textrm{d}x \, \textrm{d}p ,\nonumber \\&= \int W^{-1}(x,p) \, \Vert \varvec{\nabla } W \Vert ^2\,\, \textrm{d}x \, \textrm{d}p , \end{aligned}$$
(61)

which are the counterparts of Eqs. (55) and (57). These alternative definitions seem to suggest that \(J_r(W) \ge 0\) similarly as in the classical case, but we do not have a proof of this inequality (neither do we know if it holds).

From Eq. (58), the imaginary part of the complex Fisher information is expressed as

$$\begin{aligned} J_i(W)&= \int \varvec{\nabla } W \cdot \varvec{\nabla } \arg W \,\, \textrm{d}x \, \textrm{d}p, \nonumber \\ {}&= \pi \int \varvec{\nabla } W \cdot \varvec{\nabla } {\chi } \,\, \textrm{d}x \, \textrm{d}p , \end{aligned}$$
(62)

where we have defined the Wigner-negativity indicator function

$$\begin{aligned} {\chi }(x,p) {:}{=}\varvec{1}_{(x,p)\in \mathcal D} = {\left\{ \begin{array}{ll} 0\qquad \text {if}\;W(x,p)\ge 0 \\ 1\qquad \text {if}\;W(x,p)< 0 \end{array}\right. } \end{aligned}$$
(63)

with \(\mathcal D\) being the negative domain of W in phase space (it may consist of several non-contiguous areas). Applying the nabla \(\varvec{\nabla }\) operator onto an indicator function gives the so-called surface delta-function, which vanishes everywhere except on the boundary \(\partial D\) of \(\mathcal D\), where it points in the (inward) normal direction. It is well defined when integrated over phase space, so using the identity

$$\begin{aligned} \varvec{\nabla }(f \varvec{G})=\varvec{\nabla }f \cdot \varvec{G} + f \, \varvec{\nabla } \varvec{G} \end{aligned}$$
(64)

and substituting f with \(\chi \) and \(\varvec{G}\) with \(\varvec{\nabla }W\), we may rewrite Eq. (62) as

$$\begin{aligned} J_i(W)&= \pi \underbrace{\int \varvec{\nabla }({\chi } \, \varvec{\nabla }W) \,\, \textrm{d}x \, \textrm{d}p \,}_{\displaystyle \oint _{C_\infty } \!\!\! {\chi } \, \varvec{\nabla }W \cdot \varvec{n} \, \textrm{d}s =0} - \, \pi \int {\chi } \, \Delta W \,\, \textrm{d}x \, \textrm{d}p , \end{aligned}$$
(65)

where the volume integral \(\int \varvec{\nabla }(\cdots )\, \textrm{d}x \, \textrm{d}p\) can be replaced by a contour integral \(\oint (\cdots )\, \varvec{n} \, \textrm{d}s\) on the contour \(C_\infty \) at infinity, which vanishes since since \({\chi }=0\) (and \(\varvec{\nabla }W\) is finite) on the contour \(C_\infty \). Thus, the imaginary part of the Fisher information can be expressed as

$$\begin{aligned} J_i(W)&= - \pi \int {\chi } \, \Delta W \,\, \textrm{d}x \, \textrm{d}p , \nonumber \\&= - \pi \int _\mathcal{D} \Delta W \,\, \textrm{d}x \, \textrm{d}p , \nonumber \\&= - \pi \oint _{\partial D} \varvec{\nabla } W \cdot \varvec{n} \, \textrm{d}s . \end{aligned}$$
(66)

which is a contour integral over the boundary \(\partial D\) of the (possibly multiple) negative domain \(\mathcal D\) in phase space. Here, \(\varvec{n}\) is a normal vector to the boundary (conventionally pointing outwards, hence the minus sign) and \(\textrm{d}s\) denotes the infinitesimal element along the boundary. Note that \(J_i(W)\le 0\) since any Wigner function W can only have (possibly multiple) negative domains in an overall positive infinite domain, thus \(\varvec{\nabla } W\) always points outwards along the boundary \(\partial D\).

As expected, the complex Fisher information enjoys invariance under displacements and rotations in phase space (i.e., passive Gaussian unitaries), but not squeezing operations.

Property 6

The real and imaginary parts of \(J_c(W)\) are invariant under translations and orthogonal symplectic transformations in phase space.

Proof

As in the proof of Property 1, we consider the symplectic transformation \(\mathbf {X'}={\mathcal {S}}\textbf{X}+\textbf{d}\) but we add the restriction that \({\mathcal {S}}\) is an orhogonal symplectic matrix, that is, \({\mathcal {S}}^{-1}={\mathcal {S}}^T\). Then, we use the change of basis formula for the nabla operator, namely \(\varvec{\nabla }= {\mathcal {S}}^T \varvec{\nabla '}\), which implies \(\varvec{\nabla '}= {\mathcal {S}}\, \varvec{\nabla }\) since \({\mathcal {S}}\) is orthogonal. Therefore,

$$\begin{aligned} J_r(W')= & {} \int (W'(x',p'))^{-1} \Vert \varvec{\nabla '} W'(x',p') \Vert ^2\,\, \textrm{d}x' \, \textrm{d}p'\nonumber \\= & {} \int \frac{|\textrm{det}\,{\mathcal {S}}|}{W(x,p)}\frac{\Vert {\mathcal {S}}\varvec{\nabla }W(x,p)\Vert ^2}{|\textrm{det}\,{\mathcal {S}}|^2} |\textrm{det}\,{\mathcal {S}}|\,\textrm{d}x\,\textrm{d}p\nonumber \\= & {} J_r(W). \end{aligned}$$
(67)

For the imaginary part, we use the second definition in Eq. (66) involving the Laplacian of W and exploit the invariance of the latter operator under rigid motions, \(\Delta '=\varvec{\nabla '}^T\varvec{\nabla '}=\varvec{\nabla }^T {\mathcal {S}}^T {\mathcal {S}}\,\varvec{\nabla }=\varvec{\nabla }^T \varvec{\nabla }=\Delta \), hence

$$\begin{aligned} J_i(W')= & {} - \pi \int _\mathcal{D'} \Delta ' W'(x',p') \,\, \textrm{d}x' \, \textrm{d}p'\nonumber \\= & {} - \pi \int _\mathcal{D} \frac{\Delta W(x,p)}{|\textrm{det}\,{\mathcal {S}}|} \, |\textrm{det}\,{\mathcal {S}}|\,\, \textrm{d}x \, \textrm{d}p \nonumber \\= & {} J_i(W). \end{aligned}$$
(68)

\(\square \)

For illustration, the value of the complex Fisher information has been plotted in Fig. 6 for randomly generated (pure and mixed) states as well as for the first few Fock states.

Fig. 6
figure 6

Complex Fisher information \(J_c = J_r + i \, J_i\) of randomly generated states. Each blue (orange) point is associated with a random pure (mixed) state. Red stars represent Fock states \({|{n}\rangle }\) with \(n=0,1,\ldots , 9\). It appears that \(J_i\) decreases monotonically with n for Fock states

Full size image

5.3 Complex de Bruijn’s identity

Let us now prove the complex version of de Bruijn’s identity, which is expressed as

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} h_c(W) = \frac{1}{2} J_c(W). \end{aligned}$$
(69)

Proof

We must calculate the time derivative of the complex Wigner entropy, namely

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} h_c(W)&= -\frac{\textrm{d}}{\textrm{d}t} \int W \ln W \, \textrm{d}x \, \textrm{d}p \nonumber \\ {}&= - \int \left( \frac{\textrm{d}}{\textrm{d}t} W \right) \ln W \, \textrm{d}x \, \textrm{d}p -\underbrace{ \int W \, \frac{\textrm{d}}{\textrm{d}t} \ln W \, \textrm{d}x \, \textrm{d}p. }_{=0} \end{aligned}$$
(70)

It is easy to show that both the real and imaginary parts of the second term of Eq. (70) vanish. For the real part,

$$\begin{aligned} \int W \, \frac{\textrm{d}}{\textrm{d}t} \ln |W| \, \textrm{d}x \, \textrm{d}p = \int \frac{\textrm{d}W}{\textrm{d}t} \, \textrm{d}x \, \textrm{d}p = \frac{\textrm{d}}{\textrm{d}t} \underbrace{\int W \,\textrm{d}x \, \textrm{d}p}_{=1} =0 , \end{aligned}$$
(71)

and for the imaginary part

$$\begin{aligned} \int W \, \frac{\textrm{d}}{\textrm{d}t} \arg W \, \textrm{d}x \, \textrm{d}p&\quad =\pi \int W \, \frac{\textrm{d}}{\textrm{d}t} \left( \frac{1}{2} - \frac{|W|}{2W} \right) \, \textrm{d}x \, \textrm{d}p \nonumber \\&\quad =-\frac{\pi }{2} \int \left( \frac{\textrm{d}|W|}{\textrm{d}t} - \frac{|W|}{W} \, \frac{\textrm{d}W}{\textrm{d}t} \right) \, \textrm{d}x \, \textrm{d}p \nonumber \\&\quad =-\frac{\pi }{2} \int \underbrace{\left( \frac{\textrm{d}|W|}{\textrm{d}W} - \frac{|W|}{W} \right) }_{=0} \, \frac{\textrm{d}W}{\textrm{d}t} \, \textrm{d}x \, \textrm{d}p \nonumber \\&\quad =0. \end{aligned}$$
(72)

This integral is zero simply because W(x, p) vanishes along the boundary \(\partial D\) of the negative domain \(\mathcal D\).

Using the diffusion equation for the Wigner function (which is valid for all Wigner functions in \(\mathcal W\)), namely

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} W = \frac{1}{2}\Delta W, \end{aligned}$$
(73)

we may rexpress Eq. (70) as

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} h_c(W)&\quad = -\frac{1}{2} \int \Delta W \ln W \, \textrm{d}x \, \textrm{d}p \nonumber \\&\quad = -\frac{1}{2} \int \varvec{\nabla }(\varvec{\nabla }W \ln W) \, \textrm{d}x \, \textrm{d}p +\frac{1}{2} \int \varvec{\nabla }W \!\cdot \! \varvec{\nabla }\ln W \, \textrm{d}x \, \textrm{d}p \nonumber \\&\quad = -\frac{1}{2} \underbrace{ \oint _{C_\infty } \ln W \, \varvec{\nabla }W \cdot \varvec{n} \, \textrm{d}s}_{=0} +\frac{1}{2} \underbrace{ \int \varvec{\nabla }W \!\cdot \! \varvec{\nabla }\ln W \, \textrm{d}x \, \textrm{d}p}_{J_c(W)} , \end{aligned}$$
(74)

where we have used the identity (64) substituting f with \(\ln W\) and \(\varvec{G}\) with \(\varvec{\nabla }W\). The first term vanishes as it is a contour integral whose integrand \(\ln W\, \varvec{\nabla }W\) tends to zero at infinite distance from the origin in phase space. \(\square \)

It should be emphasized that this proof, although it looks partly similar to the corresponding one for probability distributions, applies to the complex plane (all logarithms should be understood as complex logarithms). Let us also mention that a proof for the real part of the complex de Bruijn’s identity, Eq. (69), has been presented in Appendix A of Ref. [23].

Finally, we remark that the diffusion equation obeyed by the Wigner function, Eq. (73), can be used in order to rewrite the second definition in Eq. (66) as

$$\begin{aligned} J_i(W)&= - 2\pi \int _\mathcal{D} \frac{\textrm{d}W}{\textrm{d}t} \,\, \textrm{d}x \, \textrm{d}p , \nonumber \\&= - 2\pi \, \frac{\textrm{d}}{\textrm{d}t} \int _\mathcal{D} W \,\, \textrm{d}x \, \textrm{d}p , \nonumber \\&= 2\pi \, \frac{\textrm{d}}{\textrm{d}t}\text {Vol}_{-}(W) , \nonumber \\&=2 \, \frac{\textrm{d}}{\textrm{d}t} h_i(W), \end{aligned}$$
(75)

confirming the imaginary part of the complex de Bruijn’s identity, Eq. (69).

6 Discussion and perspectives

We have defined the complex Wigner entropy of an arbitrary state as the complex continuation of the Shannon differential entropy of its Wigner-function W(x, p), namely

$$\begin{aligned} h_c(W) = -\int _{{\mathbb {R}}^2} W(x,p) \ln W(x,p) \, \textrm{d}x \, \textrm{d}p, \end{aligned}$$
(76)

where \(\ln (\cdot )\) stands for the complex logarithm function. Following the same logic, the complex Fisher information has been defined as

$$\begin{aligned} J_c(W) = \int _{{\mathbb {R}}^2} \left( \frac{\textrm{d}W}{\textrm{d}x} \, \frac{\textrm{d}(\ln W)}{\textrm{d}x} + \frac{\textrm{d}W}{\textrm{d}p} \,\frac{\textrm{d}(\ln W)}{\textrm{d}p} \right) \textrm{d}x \, \textrm{d}p \end{aligned}$$
(77)

These two definitions purposely coincide with the usual ones as soon as the state is Wigner positive, that is, \(W(x,p)>0\), \(\forall x,p\), since the Wigner function is then a genuine joint probability distribution. Thus, for such states, \(\Im (h_c)=\Im (J_c)=0\), whereas \(\Re (h_c)\) and \(\Re (J_c)\) inherit all properties of their counterparts in Shannon’s information theory. For example, the Wigner entropy is then a concave and subadditive functional of the state, and it obeys a Wigner entropy-power inequality [10].

In contrast, when the state is Wigner negative, that is, \(\exists (x,p):W(x,p)<0\), its Wigner entropy admits a strictly positive imaginary part \(\Im (h_c)\), which is proportional to the Wigner negative volume. Although the complex-valued entropy \(h_c\) loses some of its properties, such as concavity, it remains invariant when the state undergoes any Gaussian unitary (i.e., displacement, rotation, or squeezing in phase space). Both the real and imaginary parts of \(h_c\) can also be related to the real and imaginary parts of \(J_c\), respectively, when the state undergoes a Gaussian diffusion process (i.e., a convolution with a Gaussian noise whose variance grows linearly with time). As a consequence of a complex extension of de Bruijn’s identity, \(\Re (h_c)\) evolves with a rate \(\propto \Re (J_c)\) and, similarly, \(\Im (h_c)\) evolves with a rate \(\propto \Im (J_c)\). For a Wigner-positive state, \(J_c=\Re (J_c)\ge 0\) becomes the usual Fisher information, hence \(h_c=\Re (h_c)\) can only increase with time. This translates the fact that the Wigner function W(x, p) expands with time. For a Wigner-negative state, \(\Im (J_c)<0\), so that \(\Im (h_c)\) can only decrease with time. This is consistent with the fact that the Wigner negative volume can only decrease along this Gaussian diffusion process, until it reaches zero. Once the state becomes Wigner-positive, \(\Im (h_c)\) remains equal to zero along further time evolution.

Note that both \(\Im (h_c)\) and \(\Im (J_c)\) have a straightforward interpretation involving the Wigner negative domain \(\mathcal D\). As already mentioned, \(\Im (h_c)\) is proportional to (minus) the volume integral of W over \(\mathcal D\), i.e., the negative volume, while \(\Im (J_c)\) is proportional to (minus) the contour integral of the gradient of W over the boundary \(\partial D\) of the negative domain \(\mathcal D\). Just as in Shannon’s information theory, the imaginary Fisher information can thus be linked to the derivative of the imaginary Wigner entropy. Note also that our analysis in the present paper is restricted to Wigner functions associated with a single pair of canonically conjugate coordinates (x, p), that is, a single bosonic mode, but our results should be easily generalizable to a multidimensional (multimode) case.

A possible application of the complex Wigner entropy may arise in the context of entropic uncertainty relations [13]. For Wigner-positive states, we have

$$\begin{aligned} h(\rho _x)+h(\rho _p){\mathop {\ge }\limits ^{\mathrm {(a)}}} \Re (h_c(W)){\mathop {\ge }\limits ^{\mathrm {(b)}}}\ln \pi +1, \end{aligned}$$
(78)

where \(\Re (h_c(W))=h_c(W)\) since it is real-valued. More precisely, inequality (a) directly follows from the subadditivity of Shannon’s differential entropy, while inequality (b) as conjectured in [10] yields a tight entropic uncertainty relation [12]. Of course, chaining inequalities (a) and (b) yields the entropic uncertainty relation due to Białynicki-Birula and Mycielski [11]. Clearly, Eq. (78) implies that there is a forbidden zone in the complex entropy plane along the real axis below \(\ln \pi +1\). Moving to arbitrary (negative) Wigner functions, this strongly suggests that the complex Wigner entropy of a physical state is constrained to lie in some allowed area in the complex plane. The complex entropic uncertainty relation would then be expressed as \(h_c \in {\mathcal {A}}\), with \({\mathcal {A}}\) being the allowed area. This topic is worth further investigation.

Arguably, a weakness of our approach is that we miss a fully satisfactory understanding of the physical meaning behind the complex-valued entropy \(h_c\) (even its real part is not associated with a well-understood property when the state is Wigner negative). We believe—although we have not been able to do it—that a satisfactory operational interpretation of the complex Wigner entropy could be obtained by extending the notion of typical volume to the complex plane. For a Wigner-positive state, Shannon information theory tells us that a sequence of n independent instances of W(x, p) populates with high probability a typical volume \(\sim \exp (n\, h(W))\) in Euclidean space \({\mathbb {R}}^{2n}\). For a Wigner-negative state, it is tempting to keep the same expression and conclude that the typical volume becomes in general complex since the entropy is itself complex, translating the fact that for each instance of the sequence, x and p cannot be simultaneously defined (unlike two classical variables). Just as the entropy is related to the typical volume, the Fisher information is related to the surface area of the typical set, hence the complex Fisher information is another hint at the notion of complex typical volume. If meaningful, this notion should of course be put on firmer grounds.Footnote 7

On a more hypothetical note, the analytic continuation of the Shannon entropy functional in the complex plane suggests that one could go even one step further and consider the entropy of complex-valued functions. Here, the Wigner function is real-valued (albeit it can be both positive or negative) since the density operator is Hermitian, but we may possibly define the complex entropy of the complex-valued Weyl transform of non-Hermitian operators. All tools of complex analysis, such as the residue theorem, could possibly be used here. Yet, there remains to find a good interpretation of such a complex entropy.

Note added: The early steps of this work have been reported in the Ph.D. thesis of one of us [24], but the present paper presents a more complete analysis of complex entropies in phase space. After completion of this work, we have become aware of a few papers where the Shannon differential entropy associated with a Wigner function had been mentioned, although mostly via numerical investigations, for Wigner-positive [25] and Wigner-negative states [26, 27]. Recently, building on the present work, the relative entropy version of the Wigner entropy (including its complex extension) has also been put forward as a means to measure the non-Gaussianity of a quantum state [23].