Quantum Wigner entropy Zacharie Van Herstraeten1, ∗ and Nicolas J. Cerf1, † 1Centre for Quantum Information and Communication, École polytechnique de Bruxelles, CP 165/59, Université libre de Bruxelles, 1050 Brussels, Belgium We define the Wigner entropy of a quantum state as the differential Shannon entropy of the Wigner function of the state. This quantity is properly defined only for states that possess a positive Wigner function, which we name Wigner-positive states, but we argue that it is a proper measure of quan- tum uncertainty in phase space. It is invariant under symplectic transformations (displacements, rotations, and squeezing) and we conjecture that it is lower-bounded by ln π+1 within the convex set of Wigner-positive states. It reaches this lower bound for Gaussian pure states, which are natural minimum-uncertainty states. Our conjecture implies the entropic uncertainty relation and bears a resemblance with the Wehrl-Lieb conjecture, and we prove it over the subset of passive states of the harmonic oscillator which are of particular relevance in quantum thermodynamics. Along the way, we present a simple technique to build a broad class of phase-invariant Wigner-positive states exploiting an optical beam splitter and reveal an unexpectedly simple convex decomposition of extremal passive states. The Wigner entropy is anticipated to be a significant physical quantity, for example in quantum optics where it allows us to establish a Wigner entropy-power inequality. I. INTRODUCTION However, it is well known that the Wigner function is not a true probability distribution since it lacks positiveness. The phase-space formulation of quantum mechanics For example, all pure non-Gaussian states have a Wigner provides a complete framework that echoes classical sta- function that admits negative regions as a consequence tistical mechanics. Quantum states and quantum opera- of Hudson theorem [3]. This is the price to pay to the tors are described within this formulation by continuous uncertainty principle, which forbids the joint definition functions of the pair of canonical variables x and p. These of non-commuting variables x and p. Hence, some func- variables traditionally refer to the position and momen- tionals of probability distributions, such as the Shannon tum observables, but are also isomorphic to the conjugate differential entropy, become in general ill-defined if ap- quadrature components of a mode of the electromagnetic plied to Wigner functions. field (we use this quantum optics nomenclature in the In contrast, there exists a well-known distribution in present paper). The conversion from quantum operators quantum phase space that behaves as a genuine proba- to quantum phase-space distributions is carried out via bility distribution, namely the Husimi Q-function [2], de- the Wigner-Weyl transform [1], which maps any linear fined as Q α α ρ^ α π. It corresponds to the prob- ρ^ α operator A^ into a distribution A x; p as ability to measure state in a coherent state . Re- member that( ) a= coherent⟨ S S ⟩ ~ state α is an eigenstate of the 1 a^ x^ ip^ 2 S ⟩ α A x; p exp 2ipy h (x y) A^ x y dy; (1) annihilation operator with eigenvalue . πh Splitting the complex parameterS ⟩α√into two real param- ̵ eters x and p such that =α( x+ ip) ~gives where( h )denotes= ̵ S the Planck( ~ constant)⟨ − S (weS + set⟩ h 1 in the remainder of this paper). Accordingly, the Wigner 1 ̵ ̵ = Q x; p =x +ip ρ^ x ip : (3) function of a quantum state is the Wigner-Weyl trans- π form of its density operator ρ^, written as W x; p . The Wigner function comes as close to a probability distri- Despite lacking( the) nice= properties⟨ + S S of+ Wigner⟩ functions bution in phase space as allowed by quantum( mechanics.) such as the overlap formula (2), the Husimi function has It indeed shares most properties of a classical probabil- the advantage of being positive, hence it has a properly ity distribution. Notably, the marginal distributions of defined entropy. The Shannon differential entropy of the W x; p coincide with the probability distributions for x Husimi function is indeed known as the Wehrl entropy and p, respectively ρx x x ρ^ x and ρp p p ρ^ p , and is defined as h Q Q x; p ln Q x; p dx dp. as( it can) easily be shown that W x; p dp ρx x and This entropy is at the core of the Wehrl conjecture [4], W x; p dx ρp p .( Also,) = ⟨ theS expectationS ⟩ ( value) = ⟨ ofS anyS ⟩ ( ) = − ( ) ( ) arXiv:2105.12843v1 [quant-ph] 26 May 2021 later proven by Lieb [5,6], which∬ states that the Wehrl operator A^ in state ρ^ is straightforwardly∫ ( ) computed= ( ) from entropy is lower-bounded by ln π 1 and that the only its∫ Wigner( ) function= ( ) through the overlap formula [2]: minimizers of h Q are the coherent states. Interestingly, there is a link between+ the Husimi func- A^ Tr A^ ρ^ 2π A x; p W x; p dx dp: (2) tion and Wigner( function) of a state, as can simply be un- derstood using the quantum optics language. To this pur- ⟨ ⟩ = = U ( ) ( ) pose, recall that the vacuum state 0 [or ground state of the harmonic oscillator H^ p^2 x^2 2 in natural units] ∗ [email protected] S ⟩ 2 2 † [email protected] admits the Wigner function W0 x; p exp x p π. = + ~ ( ) = − − ~ 2 AppendixA). In general, we will denote the quantum states admitting a positive Wigner function [i.e., states such that W x; p 0, x; p] as Wigner-positive states. For such states, it is possible to compute the Shannon differential entropy( ) ≥ of their∀ Wigner function. We make the leap and define the Wigner entropy of any Wigner- positive state ρ^ as FIG. 1. Reduced output state σ^ of a balanced beam splitter (of transmittance η = 1~2) when the input state is ρ^, as de- h W W x; p ln W x; p dx dp (8) scribed in Eq. (7). The Wigner function of σ^ coincides with the Husimi Q-function of ρ^, hence it is positive. Consequently, where ( ) = − U ( ) ( ) the Wigner entropy of σ^ is equal to the Wehrl entropy of ρ^. 1 W x; p exp 2ipy x y ρ^ x y dy (9) π α Since a coherent state is a displaced vacuum state, is the Wigner( ) function= S of (ρ^. We)⟨ argue− S that,S + although⟩ it W x; p W x′; p′ its Wigner function is then α 0 , where is limited to Wigner-positive states, the Wigner entropy ′ S ⟩′ x x 2 Re α and p p 2 Im α . Using this, is the natural measure in order to characterize quantum ( ) = ( ) the Husimi√ function can be expressed√ from the overlap uncertainty in phase space : it bears information about = − ( ) = − ( ) formula (2) as: the joint uncertainty of the marginal distributions of the 1 x and p variables as well as their correlations in phase Q x; p Tr x ip x ip ρ^ π space. In contrast with the Wehrl entropy, it is not the classical entropy of the outcome of a specific measure- ( ) = 2 [SW+0 x~⟩⟨ +2x;Sp~ ] 2p W x;~ p~ d~x d~p: (4) ment, namely a joint x; p measurement (called het- √ √ erodyne detection or eight-port homodyne detection in Thus, it= appearsU that Q− is a convolution− between( ) W and quantum optics). Of course,( ) in the special case where the W0, with a rescaling factor of 2. In the language of Wigner-positive state can be prepared using the setup of random variables (and provided√W is non-negative), we Figure1, its Wigner entropy can be viewed simply as the could say that if x;~ p~ is distributed according to W Wehrl entropy of the corresponding input state, but the and x0; p0 is distributed according to W0, then x; p definition goes further and the Wigner entropy remains is distributed according( ) to Q, with relevant for Wigner-positive states that cannot be built ( ) ( ) in this way. x x~ x0 2 and p p~ p0 2: (5) The Wigner entropy h W enjoys interesting proper- √ √ ties. Firstly, unlike the Wehrl entropy h Q , it is in- This is a familiar= ( − relation) ~ in quantum= ( − optics,) ~ describing variant under symplectic( transformations) (displacement, η 1 2 the action of a beam-splitter of transmittance rotation, and squeezing) in phase space. Such( ) transfor- ρ^ σ^ onto the state and the vacuum state. Defining as mations, which are ubiquitous in quantum optics, corre- = ~ the reduced state of the corresponding output of the spond to the set of all Gaussian unitaries in state space. beam-splitter, as shown in Figure1, we conclude that We stress that a sensible measure of phase-space uncer- σ^ the Wigner function of is precisely the Husimi function tainty must remain invariant under symplectic transfor- ρ^ of , namely mations, since these are also area-preserving transforma- tions in phase space. In contrast, h Q is greater for Wσ^ x; p Qρ^ x; p (6) squeezed states than for coherent states. As it can be where ( ) = ( ) understood from Fig.1, this preference( simply) originates from the fact that one input of the balanced beam-splitter ^ ^ † is itself a coherent state. Secondly, the Wigner entropy σ^ Tr2 U 1 ρ^ 0 0 U 1 : (7) 2 2 h W can be related to the entropy of the marginal dis- tributions h ρx and h ρp , but also encompasses the ^ = ( ⊗ S ⟩⟨ S) Here U 1 denotes the beam-splitter unitary of transmit- 2 x-(p correlations.) Shannon information theory establishes tance η 1 2, while Tr2 denotes a reduced trace over one a relation between( ) the entropy( ) of a joint distribution and of the modes, say the second mode.