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 ρˆ α π. It corresponds to the prob- ρˆ α operator Aˆ 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 aˆ xˆ ipˆ 2 S ⟩ α A x, p exp 2ipy h (x y) Aˆ 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 ρˆ x ip . (3) function of a quantum state is the Wigner-Weyl trans- π form of its density operator ρˆ, 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 ρˆ x and ρp p p ρˆ 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 Aˆ in state ρˆ 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- Aˆ Tr Aˆ ρˆ 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 Hˆ pˆ2 xˆ2 2 in natural units] ∗ [email protected] S ⟩ 2 2 † [email protected] admits the Wigner function W0 x, p exp x p π. = ‰ + Ž ~ ( ) = ‰− − Ž ~ 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 ρˆ as FIG. 1. Reduced output state σˆ of a balanced beam splitter (of transmittance η = 1~2) when the input state is ρˆ, as de- h W W x, p ln W x, p dx dp (8) scribed in Eq. (7). The Wigner function of σˆ coincides with the Husimi Q-function of ρˆ, hence it is positive. Consequently, where ( ) = − U ( ) ( ) the Wigner entropy of σˆ is equal to the Wehrl entropy of ρˆ. 1 W x, p exp 2ipy x y ρˆ x y dy (9) π α Since a coherent state is a displaced vacuum state, is the Wigner( ) function= S of (ρˆ. 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 ρˆ 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 thatŠ 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- ρˆ σˆ 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- σˆ the Wigner function of is precisely the Husimi function tainty must remain invariant under symplectic transfor- ρˆ of , namely mations, since these are also area-preserving transforma- tions in phase space. In contrast, h Q is greater for Wσˆ x, p Qρˆ 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 ˆ ˆ † is itself a coherent state. Secondly, the Wigner entropy σˆ Tr2 U 1 ρˆ 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 ˆ =  ( ⊗ 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. From Eq. (6), it its marginal entropies, namely h x, p h x h p I, appears= that~ the entropy of the Wigner function of σˆ is where I 0 is the mutual information [7]. Applied to nothing else but the Wehrl entropy of ρˆ in this particu- the Wigner entropy, this gives( the) inequality= ( ) + h( W) − lar setup. A natural question then arises : can we give h ρx h≥ ρp . This means that a lower bound on the an intrinsic meaning to the entropy of a Wigner function Wigner entropy implies in turn a lower bound on the( sum) ≤ independently of this particular setup? of( the) marginal+ ( ) entropies. In this paper, we will answer by the affirmative. First, In the light of these considerations, we introduce a con- let us notice that the setup of Fig.1 ensures that the jecture on the Wigner entropy, which resembles the Wehrl output state σˆ always has a positive Wigner function (see conjecture. As anticipated in [8], we conjecture that the 3

Wigner entropy of any Wigner-positive state ρˆ satisfies where , stands for the anticommutator. The set of Gaussian states are those for which the Wigner function h W ln π 1. (10) W x, p{⋅ is⋅} Gaussian, hence these states are completely As we will show, this( bound) ≥ is reached+ by all Gaussian characterized by their first- and second-order moments pure states, which appears consistent with Hudson the- c and( Γ) . The set of Gaussian unitaries in state space orem [3]. It implies (but is stronger than) the entropic is isomorphic to the set of symplectic transformations in uncertainty relation of Białynicki-Birula and Mycielski phase space. Formally, a symplectic transformation is an [9], namely h ρx ρp ln π 1. Importantly, our con- affine map on the space of quadrature operators which jecture also implies the Wehrl conjecture since we have is defined by a symplectic matrix S and a displacement shown that the( ) Husimi+ ( ) ≥ function+ of any state ρˆ is the vector d, namely Wigner function of some Wigner-positive state σˆ in a particular setup, see Fig.1. However, the converse is not ˆx Sˆx d. true as there exist Wigner-positive states whose Wigner (14) function cannot be written as the Husimi function of a → + physical state (an example will be shown in SectionIV). The symplectic matrix S is a real matrix that must pre- The paper is organized as follows. In SectionII, we serve the symplectic form, that is, SΩS⊺ Ω, which start by recalling some basics of the symplectic formalism implies in particular that det S 1. The displacement and then define the Wigner entropy of a Wigner-positive vector d is an arbitrary real vector. The first-= and second- state as a distinctive information-theoretical measure of order moments of a state ρ evolve= under such a symplectic its uncertainty in phase space. In SectionIII, we discuss transformation as the characterization of the set of Wigner-positive states and focus on the particular subset of phase-invariant Wigner-positive states. Then, in SectionIV, we turn to c Sc d , Γ SΓS⊺ . (15) our main conjecture and provide a proof for some special case of phase-invariant Wigner-positive states, namely → + = the passive states. Finally, we conclude in SectionV and In the special case of Gaussian states, this completely provide an example of the application of Wigner entropy, characterizes the evolution of the state under the Gaus- namely the Wigner entropy-power inequality. Further, sian unitary. in AppendixA, we present a quantum-optics inspired The core of this paper is the definition of a new method for generating a large variety of Wigner-positive information-theoretical measure of uncertainty in phase states with a balanced beam-splitter, extending on Fig.1. space, which we call the Wigner entropy h W , where AppendixB is devoted to the detailed analysis of the h denotes Shannon differential entropy functional and set of Wigner-positive states when considering the Fock W x, p is the Wigner function of ρˆ, see Eqs.( (8)) and (9). spaces restricted to two photons as this provides a helpful As(⋅) already mentioned, it only applies to Wigner-positive illustration of our results. states( since,) otherwise, the definition of the entropy en- tails the logarithm of a negative number. We write it as a functional of W but, of course, it is eventually a II. WIGNER ENTROPY OF A STATE functional of the state ρˆ since W itself depends on ρˆ. In contrast with the Shannon entropy of a discrete vari- In this paper, we restrict our considerations to a sin- able, the Shannon differential entropy of a continuous gle bosonic mode (one harmonic oscillator) for simplic- variable does not have have an absolute meaning (it de- ity, although the definition of the Wigner entropy and pends on the scale of the variable) and it becomes neg- our corresponding conjecture should extend to the mul- ative if the probability distribution is highly peaked [7]. tidimensional case. Let us briefly review the symplectic However, when applied to a Wigner function, the natural ˆx x,ˆ pˆ ⊺ formalism for one bosonic mode. Let be the scale is provided here by the area h of a unit cell in phase vector of quadrature operators (or position and momen- space. Hence, the Wigner entropy has a meaning per se xˆ , xˆ= ( i Ω) tum canonical operators) satisfying j k jk, with and it is legitimate to conjecture̵ a lower bound, namely the matrix Eq. (10) when setting h 1. 0 1 [ ] = Ω The Wigner entropy h W has the nice property to 1 0 (11) ̵ = be invariant under symplectic transformations. Consider = ‹  ( ) ˆx ˆx′ Sˆx d being the symplectic form.− The coherence vector (also the symplectic transformation and let us called displacement vector) of a state ρˆ is defined as denote as W and W ′ the Wigner function of the input and output states, respectively.→ The change= + of variables c ˆx Tr ˆx ρˆ (12) corresponding to this transformation gives ρˆ where stands for the= expectation⟨ ⟩ ∶= ( value) in state , while Γ ρˆ the covariance matrix of state is defined as W x, p ⟨⋅⟩ W ′ x′, p′ (16) Γjk xˆj xˆj , xˆk xˆk (13) det S ( ) ( ) = = ⟨{ − ⟨ ⟩ − ⟨ ⟩}⟩ S S 4 which indeed implies that h W ′ W ′ x′, p′ ln W ′ x′, p′ dx′ dp′ W x, p ( ) = − U W (x, p ln) ( )dx dp det S ( ) = −h UW ln( det) SŒ ‘ S S h W (17) = ( ) + S S FIG. 2. Schematic of a convex set. The black points form its W boundary, while the red points are the boundary points that where we have= used( ) the fact that is normalized and the fact that S is a symplectic matrix (det S 1). are extremal points (note the continuum of extremal points). Note that this invariance can also be understood as a sole consequence of the fact that symplectic= transfor- mations conserve areas in phase space since det S 1. positive Wigner functions. For this reason, we devote Indeed, for any functional F , we have this section to the quantum states with positive Wigner = functions, which we call Wigner-positive states. Note ′ ′ ′ ′ ′ F W x , p dx dp that Wigner-positivity is a particular case of η-positivity η 0 W x, p for [10, 11]. Quantum Wigner-positive states of a U ‰ ( F )Ž det S dx dp single mode are described by a Wigner function W x, p det S = ( ) that respects the condition ( ) = U F ŒW x, p ‘dx S dp S (18) S S W x, p 0 x, p. (20) The special case= U of‰ Gaussian( )Ž states is very easy to deal with. A straightforward calculation shows that the Restricting to pure( states,) ≥ the set∀ of Wigner-positive Wigner entropy of a Gaussian state ρˆ is given by states is well known: Hudson theorem establishes that Gaussian pure states are the only pure quantum states h W ln 2π det Γ 1 ln π µ 1 (19) with a positive Wigner function [3]. When it comes to √ mixed states, however, the situation becomes more diffi- 2 where µ (Trˆρ) = 1Š 2 det Ï + 1 =stands( ~ for) + the purity cult since the mixing of states enables one to build non- of the state. All Gaussian√ states that are connected with Gaussian Wigner-positive states. The characterization a symplectic= transformation= ~( obviously) ≤ conserve their pu- of the set of Wigner-positive mixed states has been at- rity since det Γ′ det SΓS⊺ det Γ, which confirms tempted [11, 12], but the resulting picture is somehow that their Wigner entropy is invariant. The lowest value complex. Just like writing a necessary and sufficient con- of h W among Gaussian= ( states) = is then reached for pure dition for a Wigner function to correspond to a positive- states (µ 1) and is given by ln π 1, as expected. This definite density operator is a hard task, it appears cum- is the( value) of the Wigner entropy of all coherent states bersome to express a necessary and sufficient condition and squeezed= states (regardless the+ squeezing parameter, for a density operator to be associated with a positive squeezing orientation, and coherence vector). Accord- Wigner function. ingly, the Gaussian pure states would be the minimum- On a more positive note, the set of Wigner-positive Wigner-uncertainty states. The difficult task remains, states is convex since a mixture of Wigner-positive states however, to prove that non-Gaussian Wigner-positive is itself Wigner-positive. Taking advantage of this prop- states cannot violate this lower bound, see SectionIV. erty, we may focus on the extremal states of the con- Provided our conjecture is valid, then the Wigner func- vex set, as pictured in Figure2. These are states that tion of any Wigner-positive state can be simulated from cannot be obtained as a mixture of other states of the the Wigner function of the vacuum state (or any Gaus- set. Conversely, any state of the set can be generated as sian pure state). More precisely, information theory tells a mixture of these extremal states. This brings a sim- us that the difference h W ln π 1 can be viewed as plification in the proof of our main conjecture, namely the number of equiprobable random bits that are needed, expressing a lower bound on the Wigner entropy of an on average, to generate one( )x,− p instance− of the Wigner arbitrary Wigner-positive state (see SectionIV). Indeed, function of state ρ from a random x, p instance that is the Shannon entropy being concave, the entropy of a mix- drawn from the Wigner function( ) of the vacuum state (or ture is lower-bounded by the entropy of its components, any Gaussian pure state). ( ) that is

h p1W1 p2W2 p1 h W1 p2 h W2 , (21) III. WIGNER-POSITIVE STATES

where p1 (and p2+are positive) ≥ reals( such) + that( p)1 p2 1. As explained in SectionII, the Wigner entropy natu- Hence, it is sufficient to prove the lower bound on h W rally appears as an information-theoretic measure of un- for the extremal Wigner-positive states in order+ to have= certainty in phase space, but is only properly defined for a proof over the full set. ( ) 5

Several classes of Wigner-positive states

To be more specific, we define several sets of Wigner- positive quantum states, which will be useful in the rest of this paper. As we shall see, our conjecture is trivially verified for some of them, while it remains hard to prove for others.

• : Physical quantum states FIG. 3. Beam-splitter state σˆ obtained at the output of a bal- anced beam-splitter (of transmittance η = 1~2). If the input It is the convex set of all single-mode quantum is an arbitrary product state ρˆ ⊗ ρˆ , then the reduced state Q ρˆ A B states. Their density operator satisfies the of the output σˆ is guaranteed to be Wigner-positive. This three physicality conditions: hermicity, positive- generalizes Fig.1, where ρˆB = S0⟩⟨0S. The set of beam-splitter definiteness and unit-trace. Of course, they can states is denoted as B. The convex hull of these states, de- have partly negative Wigner functions. noted as Bc, is obtained by sending any separable state (i.e., a mixture of product states) into a balanced beam splitter and Wigner-positive quantum states • + : tracing over one of the output modes. The whole set Bc is It is the subset of states in that have positive strictly included in the set of Wigner-positive states Q+. QWigner functions. It is a convex set. Hence, all states in this set have a well-definedQ Wigner entropy and are the subject of our conjecture. It can be shown with a simple argument that the set • : Gaussian states of Gaussian states is a subset of . Indeed, it is well It is the subset of states in that have a Gaussian known that the product of two identical copies of a Gaus- + γˆ G B GWigner functions. It does not form a convex set sian state is invariant under the action of a beam- since the mixture of GaussianQ states does not need splitter (see symplectic formalism [13]). We have the Uˆ γˆ γˆ Uˆ † γˆ γˆ γˆ to be Gaussian, so we refer to its convex hull as c. identity η η , where is any single-mode ˆ Gaussian state and Uη is the unitary of a beam-splitter • : Classical states G with transmittance( ⊗ ) η.= One⊗ can then easily reconstruct According to Glauber’s definition, classical states the set of Gaussian states with the above setup, so it areC mixtures of coherent states. They are char- follows that . Note that it is easy to build beam- acterized by a positive Glauber-Sudarshan P- splitter states as in Fig.3 that are not Gaussian states, function. By definition, is a convex set and c hence this isG a⊂ strictB inclusion relation. The analog re- since coherent states are Gaussian states. lation also applies to the respective convex hull of these C C ⊂ G sets, namely c c. Unfortunately, the set c does not The extremal states of and c are respectively coher- coincide with + as we will see that there exist Wigner- ent states and Gaussian pure states. For these two sets, positive statesG that⊂ B do not belong to c (see,B e.g., the C G our conjecture is immediately verified since the Wigner dark blue regionQ in Fig.6). In summary, we have the ln π 1 entropy of Gaussian pure states is precisely and following chain of strict inclusion relationsB since the entropy is concave. Unfortunately, the convex closure of Gaussian states c is yet but a small fraction+ of + c c (23) the set of Wigner-positive states +. As an evidence of G this, we construct a wider set of Wigner-positive states Q ⊃ Q ⊃ B ⊃ G ⊃ C by exploiting a technique relyingQ on a balanced beam- as pictured in Fig.4. splitter (hence, we name this set as ).

• : Beam-splitter states B Phase-invariant states in Q+ These are the states σˆ resulting from the setup de- Bpicted in Figure3. More precisely, a beam-splitter As it appears, the set of Wigner-positive states is state σˆ denotes the reduced output state of a beam- + hard to fully encompass. Therefore, in order to make a splitter with transmittance η 1 2 fed by a tensor concrete step towards theQ proof of our conjecture, we re- product of two arbitrary states ρˆA and ρˆB, = ~ strict our attention in this paper to a class of quantum states known as phase-invariant states. Phase-invariant ˆ ˆ † σˆ Tr2 U 1 ρˆA ρˆB U 1 . (22) 2 2 states have a Wigner function that is invariant under rotation, so they are fully characterized by their radial We show in= Appendix ( A ⊗that) state σˆ always pos- Wigner function. Such states have the advantage of being sesses a positive Wigner function, regardless of ρˆA characterized in simpler mathematical terms as they can and ρˆB. The sole condition is that the input state is be written as mixtures of Fock states, which are eigen- a tensor product and the beam splitter is balanced states of the harmonic oscillator. The wave function and (η 1 2). the Wigner function of the nth Fock state (starting at

= ~ 6

FIG. 5. Beam-splitter state σˆ(m, n) obtained at the output = 1 of a balanced beam-splitter (of transmittance η 2 ) that is fed with Fock states of m and n photons. The states σˆ(m, n) are Wigner-positive phase-invariant states, hence they belong FIG. 4. Schematic view of the various sets considered here. to the set . The full set of quantum states is denoted as Q, while the set S+ of Wigner-positive states is denoted as Q+. Then, Bc stands for convex hull of the set B of beam-splitter states, while G c corresponds to a specific value of r through the relation stands for the convex hull of the set G of Gaussian states. 2 Further, C stands for the set of classical states. In all these t 2r . When condition (28) is matched for some t, the sets, we distinguish the states that are phase invariant, which Wigner function is non-negative at r t 2. We call = are characterized by a probability vector p ∈ S. For states in S+ the restriction of S satisfying the Wigner-positivity» Q+, the vector p ∈ S+, while for states in Bc, the vector p ∈ Sb. conditions (28), so that any vector p in= S+ is~ associated with a unique phase-invariant Wigner-positive state in +. n 0 for vacuum) are the following: The characterization of S+ can be operated as follows. 2 QEach value of t in Eq. (28) gives the equation of a hy- 1 n − 1 x = − 4 − 2 2 p ψn x π 2 n! Hn x exp , (24) perplane dividing S in two halves [ must be located on 2 one side of the hyperplane to guarantee that W r 0 ( ) = 1 n ( ) 2 ( 2) Œ− 2 ‘ 2 for the corresponding r]. Two hyperplanes associated re- Wn x, p 1 Ln 2x 2p exp x p , (25) π spectively to t and t dt intersect in a (lower-dimensional)( ) ≥ hyperplane which is at the boundary of the convex set ( ) = (− ) ‰ + Ž ‰−th − Ž where Hn and Ln are respectively the n Hermite and S+. When t goes from+ 0 to , the set of all these in- Laguerre polynomials. A phase-invariant state is thus tersections form a locus of points which determines the expressed as the mixture curved boundary of S+. Mathematically,∞ the condition +∞ that a point p S belongs to the curved boundary of S+ ρˆ pk k k (26) is equivalent to the following condition: k=0 ∈ +∞ k = thQ S ⟩⟨ S pk 1 Lk t 0 with k denoting the k Fock state, so that it is fully k=0 N t s.t. ⎧ (29) described by the probability vector p R , with com- ⎪+∞ ⎪ ∑ (− )k d ( ) = ponentsS ⟩ pk. In order to be an acceptable probability ⎪ pk 1 Lk t 0 p ∈ ∃ ⎨k=0 dt distribution, must satisfy the physicality conditions ⎪ ⎪ ∑ (− ) ( ) = +∞ Note that since S+ is⎩⎪ convex, all the points in its curved pk 0 k, pk 1. (27) boundary are extremal points. However, other isolated k=0 extremal points may exist, as illustrated in Figure2. ≥ ∀ Q = We call S the restriction of RN satisfying the physicality conditions (27). Any vector p that belongs to S corre- Phase-invariant beam-splitter states in B sponds to a unique phase-invariant state in . Now, we turn to the phase-invariant states in . In + The above considerations reflect the fact that charac- order to check that the phase-invariant stateQ that is char- terizing the set of phase-invariant Wigner-positive states acterized by a vector p is Wigner-positive, we needQ to S (associated with p ) remains complex. For this rea- verify that the corresponding mixture of Fock states has a S+ son, we consider a subset of states that are built by using positive Wigner function∈ everywhere in phase space. This a balanced beam-splitter,∈ following the same idea as for is done by using Eq. (25), so that the Wigner-positivity the construction of set but injecting phase-invariant condition on p read as Fock states at the input. As pictured in Fig.5, we define +∞ the beam-splitter state Bσˆ m, n as the reduced output k pk 1 Lk t 0 t 0, (28) state of a balanced beam-splitter fed by m and n pho- k=0 tons at its two inputs, that( is, ) Q (− ) ( ) ≥ ∀ ≥ where we define t 2x2 2p2. Let us also define the usual ˆ ˆ † 2 2 σˆ m, n Tr2 U 1 m m n n U 1 . (30) radial parameter r x p , so that each value of t 2 2 = »+ = + ( ) =  (S ⟩⟨ S ⊗ S ⟩⟨ S)  7

to mixtures of Fock states up to n 2, that is

ρˆ 1 p1 p2 0 0 p1 1 =1 p2 2 2 (31)

with p1=, p(2 − 0 and− p)S1 ⟩⟨p2S + 1.S We⟩⟨ areS + interestedS ⟩⟨ S in the 2 2 Wigner-positive restriction of S , namely S+. Restrict- 2 ing ourselves≥ to n 2 +makes≤ it possible to represent S+ in a two-dimensional plane with coordinates p1 and p2, 2 see Fig.6. The mathematical= description of S+ was also given in [11], but we analyze it here from a physical per- spective, through the prism of quantum optics. Since the beam splitter conserves the total photon number, we know that only the states σˆ m, n such that m n 2 2 belong to S+. These states are expressed as ( ) + ≤ σˆa σˆ 0, 0 0 0 1 1 σˆ σˆ 1, 0 0 0 1 1 b ≡ ( ) = 2 S ⟩⟨ S 2 2 FIG. 6. Two-dimensional representation of S (large white 1 1 (32) 2 σˆc ≡ σˆ(1, 1) = S0⟩⟨0S + S2⟩⟨2S triangle) and S+ (blue zone, including dark and light blue). 2 2 The points a, b, c, and d correspond to the beam-splitter 2 1 1 1 states whose convex closure yields Sb, represented as the light σˆd ≡ σˆ(2, 0) = S0⟩⟨0S + S1⟩⟨1S 2 2 blue zone. The dark blue zone stands for the subset of phase- 4 2 4 invariant Wigner-positive states that cannot be expressed as ≡ ( ) = S ⟩⟨ S + S ⟩⟨ S + S ⟩⟨ S mixtures of beam-splitter states. As discussed in Sec.IV, the and their corresponding Wigner functions are displayed triangle a-b-e encompasses the set of passive states while the in Figures7 and8. We observe that the minimum value of triangle a-b-d encompasses the states whose Wigner function the Wigner functions always reaches zero (except for the coincides with the Husimi Q-function of a state. vacuum state σˆa), which reflects that these are extremal states of the set of Wigner-positive phase-invariant states 2 (associated with S+). Thus, any state σˆ m, n is Wigner-positive and phase- This is confirmed in Figure6, where the four beam- splitter states are represented by points a, b, c, and d: invariant. It is a mixture of Fock states with mixture 2 coefficients given in( Appendix) A. We denote as Sb the they are indeed extremal points of the convex set S+, set of probability vectors p corresponding to all mixtures which appears as the blue zone (including light and dark blue). The complete characterization of 2 can be done of states σˆ m, n . It is clear that Sb S+ S, as depicted S+ in Fig.4. by using the Wigner-positivity conditions (28) and (29). Interestingly,( ) as will be illustrated⊂ by the⊂ following ex- The derivation is done in AppendixB and leads to the p p ample, the Wigner function associated with any state following conditions on 1 and 2: σˆ m, n happens to have a minimum value that reaches 1 precisely zero [except for σˆ 0, 0 , which is simply the vac- p1 2 uum( state].) In fact, it is shown in AppendixA that (33) ⎪⎧ whenever m n, the Wigner( function) of σˆ m, n always ⎪ ≤ 1 1 2 ⎪p2 1 4p1 cancels at the origin in phase space. This suggests that ⎨ 4 4 » the states σˆ ≠m, n are the extremal states( of the) set of ⎪ ⎪ ≤ + − phase-invariant Wigner-positive states (those associated Any state in the⎩ form (31) is Wigner-positive if and only p p with S+). However,( ) as we will show in the following ex- if its components 1 and 2 satisfy conditions (33). ample, the situation is more tricky as this set also admits Several observations can be made from Figure6. First, other extremal states that are not of the form σˆ m, n . state σˆa, which coincides with the vacuum state, is a 2 Hence, Sb S+ is a strict inclusion and there exist phase- trivial extremal state of S+ even if its Wigner function invariant Wigner-positive states that cannot be written( ) does not reach zero. As already mentioned, σˆb, σˆc, and 2 2 as mixtures⊂ of beam-splitter states σˆ m, n . σˆd are other extremal states of S+. The convex set S+ has three facets. Two of them correspond to the physicality ( ) conditions (27), i.e. p1 0 and p2 0. The third facet correspond to (28) where we have set t 0, which gives Example : restriction to two photons us p0 p2 1 2 or p1 1≥2. Note that,≥ in general, the set S+ always has a facet corresponding to = n n Let us denote by S and S+ the restriction of respec- + ≥ ~ ≤ ~ 1 tively S and S+ that have components pk 0 for k n. As 2 pk . (34) an example, let us consider the set S , which corresponds k even 2 = > Q = 8

FIG. 9. Expressing the positivity of the radial Wigner func- tion W (r) for increasing values of r corresponds to a contin- uum of straight lines, which are all tangents of ellipse (35). As an illustration, we plot as dashed√ lines the tangents√ associated with W (r) = 0 for r = 0, r = 1~ 2, r = 1, r = 2, and r → ∞. For instance, expressing W (0) ≥ 0 implies p1 ≤ 1~2, while ex- pressing W (1) ≥ 0 implies p2 ≤ 1~2. For r > 1, the positivity condition becomes redundant, and, at the limit r → ∞, it FIG. 7. Wigner functions of the four beam-splitter states σˆa, gives p2 ≥ 0, which is equivalent to the physicality condition. σˆb, σˆc, and σˆd, denoted respectively as Wa(x, p), Wb(x, p), Wc(x, p), and Wd(x, p). These four states are Wigner-positive phase-invariant states, but, in addition, their Wigner func- in 0, 1 4 , namely tions touch precisely zero (except for the vacuum state σˆa) as is more evident from Fig.8. This fact reflects that these are 2 2 ( ~ ) p1 p2 1 4 extremal states of the set of Wigner-positive phase-invariant 1 . (35) states (associated with 2 ). 1 2 1 4 S+ − ~ Œ ‘ + Œ ‘ = The resulting constraints~ on p~1 and p2 for all r’s is ex- pressed by Eq. (33). 2 Overall, Figure6 shows that the subspace Sb, which is spanned by the extremal states σˆa, σˆb, σˆc, and σˆd, covers 2 indeed a large region of S+ (indicated in light blue) so any point in this region can thus be generated by a convex 2 mixture of them. However, S+ also includes a small region (indicated in dark blue) that is located under the ellipse defined by Eq. (35) and above the straight line c-d. This 2 region is thus outside the polytope Sb generated by the σˆ 2 states, which confirms that S+ also admits a continuum of extremal points along this ellipse.

IV. CONJECTURED LOWER BOUND FIG. 8. Radial Wigner functions of the four phase-invariant beam-splitter states σˆa, σˆb, σˆc, and σˆd, denoted respectively Our conjectured lower bound on the Wigner entropy as Wa(r), Wb(r), Wc(r), and Wd(r). As advertised, the min- reads imum value of these Wigner functions touches zero (except for the vacuum σˆa, for which Wa(r) → 0 as r → ∞). h Wρˆ ln π 1 ρ + (36)

We wish to prove( it) for≥ all+ Wigner-positive∀ ∈ Q states in + but it appeared in SectionIII that this set is hard to characterize. In this Section, we will expose the centralQ which expresses the positivity of the Wigner function at result of our paper, namely a proof of this conjecture r 0 (recall that t 2r2). As pictured in Fig.9, ex- over a subset of phase-invariant Wigner-positive states pressing the positivity of the radial Wigner function for with thermodynamical relevance that are called passive increasing= values of r=yields a continuum of straight lines, states. As a side-result, we will exhibit an unexpect- whose locus of intersecting points form an ellipse centered edly simple relation between the extremal passive states 9 and the beam-splitter states σˆ m, n , which guides us to extremal passive state as εˆn and to its Wigner function test our conjecture over the much larger set Sb of phase- as En. They are defined as follows: invariant Wigner-positive states.( ) n Before doing so, let us discuss the implication of our 1 εˆn k k conjecture in the restricted subspace of phase-invariant n 1 k=0 2 Wigner-positive states associated with S+. First, as a = Q S ⟩⟨ S (39) consequence of Eq. (6), we know that the Wigner func- +1 n σˆ σˆ σˆ En x, p Wk x, p tions of a, b, and d coincide respectively with the n 1 Husimi Q-functions of 0 , 1 , and 2 . Hence, the k=0 (proven) Wehrl conjecture applied to 0 , 1 , and 2 im- ( ) = Q ( ) The states εˆn are called extremal+ [16] in the sense that plies that the Wigner entropyS ⟩ S of⟩ σˆa, σˆb,S and⟩ σˆd is indeed any passive state ρˆp can be expressed as a unique convex ln π 1 S ⟩ S ⟩ S ⟩ lower bounded by . Further, this naturally extends mixture of extremal passive states, namely to the subspace spanned by σˆa, σˆb, and σˆd, correspond- ing to the triangle a-b+-d in Fig.6. Thus, the states that +∞ are located in the blue region but do not belong to this ρˆp ek εˆk , (40) triangle are Wigner-positive states whose Wigner func- k=0 tion cannot be expressed as a physical Q-function. This = Q where pk and ek are probabilities that are linked through underlies the fact that our conjecture is stronger than the relation ek k 1 pk pk+1 . the Wehrl conjecture. In particular, let us prove that the In the special case of phase-invariant states within the σˆ Wigner function of state c cannot be written as the Q- restricted space= with( + up)( to− two photons,) the set of pas- function of a physical state. Reasoning by contradiction, sive states corresponds to the triangle a-b-e in Fig.6, ρˆ 2 assume there exists a state in the setup of Fig.1 such which belongs to as expected. Of course, a, b, and σˆ ρˆ S+ that the resulting output state is c. Since must be e correspond respectively to the extremal passive states phase-invariant in order to get a phase-invariant output εˆ0, εˆ1, and εˆ2. state such as σˆc and since σˆc does not contain more than 2 photons, it is clear that ρˆ can only be a mixture of 0 , 1 , and 2 . However, the output state corresponding to Proof of the conjecture for passive states any such mixture precisely belongs to the triangle a-bS-d⟩, whichS ⟩ doesS ⟩ not contain c. Hence, there is no state ρˆ. Let us prove the lower bound (36) for the subset of pas- sive states ρˆp. First, note that passive states are known Passive states to be Wigner-positive [17], as will soon become clear [see Eq. (41)]. Thus, their Wigner entropy is well defined. Second, notice that, as a consequence of the concavity Passive states are defined in quantum thermodynam- of entropy, it is sufficient to prove the Wigner conjecture ics as the states from which no work can be extracted for extremal passive states εˆn. through unitary operations [14]. If ρˆp is the density op- The main tool that we will use to carry out our proof erator of a passive state, then the following relation holds is a formula that we have derived from an expression in true for any unitary operator Uˆ: Ref. [18], making a nontrivial link between the Wigner ˆ ˆ ˆ † ˆ n Tr ρˆpH Tr UρˆpU H , (37) functions and wave functions of the first Fock states. It reads as follows (to the best of our knowledge, it has where Hˆ is the Hamiltonian
 ≤ of
the system. Passive states never appeared as such in the literature): are useless in the sense that it is not possible to decrease n n 2 2 their energy by applying a unitary (since a unitary con- Wk x, p ψk x ψn−k p (41) serves the entropy, any work extraction should come with k=0 k=0 a decrease of internal energy). It can be shown that pas- Q ( ) = Q ( ) ( ) where Wk and ψk are respectively the Wigner function sive states are decreasing mixtures of energy eigenstates, th in the sense that if the eigenstates are labeled with in- and the wave function of the k Fock state as defined creasing energy, the associate probabilities must be de- in Eqs. (24) and (25). As a by-product, note that Eq. creasing [15]. In the present paper, we are considering (41) immediately implies that all extremal passive states eigenstates of the harmonic oscillator, which are the Fock εˆn admit a positive Wigner function, hence the Wigner states. A passive state is then written as function of an arbitrary passive state is necessarily posi- tive. +∞ Let us denote the x and p probability densities of the ρˆp pk k k with pk pk+1. (38) th 2 2 n Fock state as ρn x ψn x and ρn p ψn p . k=0 Their corresponding Shannon differential entropy is de- = Q S ⟩⟨ S ≥ Among the set of passive states, extremal passive states fined as h ρk x ( )ρ=k Sx (ln )Sρk x dx and( )h= Sρk (p )S are defined as equiprobable mixtures of the low energy ρk p ln ρk p dp. In the following, we refer to these th eigenstates up to some threshold. We refer to the n quantities( as (h ))ρk= − ∫h ρk( x) h( ρ)k p . We( are( now)) = − ∫ ( ) ( ) ( ) ≡ ( ( )) = ( ( )) 10 ready to lower bound the Wigner entropy of the nth ex- tropy of any passive state which reads as tremal passive state εˆn by using Eq. (41): ∞ ∞ h pk Wk 2 pk h ρk . (44) n 1 k=0 k=0 h En h Wk x, p n 1 k=0 ŒQ ‘ ≥ Q ( ) and is valid as soon as the probabilities pk are decreasing, n ( ) = Œ 1 Q ( 2 )‘ 2 that is, pk pk+1. h + ψk x ψn−k p n 1 k=0 It is tempting to extrapolate that the bound (44) re- ≥ = Œ1 n Q ( ) ( ) ‘ mains valid beyond the set of passive states. We know + h ρk x ρn k p indeed that there exist phase-invariant Wigner-positive n 1 − k=0 (42) states that are not passive states (in Fig.6, these are the ≥ 1 Qn ‰ ( ) ( )Ž states within the light blue region, which do not belong + h ρk h ρn−k a b e pk n 1 k=0 to the triangle - - ). As long as the coefficients are such that the corresponding state is Wigner-positive, it = 2 Qn ‰ ( ) + ( ) Ž + h ρk has a well-defined Wigner entropy and we may infer that n 1 k=0 the lower bound (44) applies. Unfortunately, our numer- = ln π 1Q ( ) ical simulations have shown that relation (44) does not + hold in general for non-passive (Wigner-positive) states. ≥ + The first inequality in (42) results from the concavity Of course, we conjecture that relation (42) does hold for of the entropy. Then, we use the fact that the entropy such states and we have not found any counterexample. of a product distribution is the sum of the marginal en- tropies. Finally, we apply the entropic uncertainty rela- Relation between the extremal passive states and tion of Białynicki-Birula and Mycielski [9] on Fock states, the beam-splitter states namely 2 h ρk ln π 1, k. Our conjecture is thus proven for all extremal passive states and the proof nat- urally expands( ) to≥ the whole+ ∀ set of passive states. Let us now highlight an interesting relation between εˆ Let us now prove that a slightly tighter lower bound extremal passive states n and the beam-splitter states σˆ m, n can be derived for the Wigner entropy of passive states that we defined in SectionIII. To this purpose, by exploiting Eq. (40), namely the fact that these states we consider a mixed quantum state of two modes (or ( ) τˆ can be expressed as convex mixtures of extremal passive harmonic oscillators) which we denote as n. It is defined as an equal mixture of all two-mode states with a total states εˆn (in place of decreasing mixtures of Fock states). photon number (or energy) equal to n, namely We denote the Wigner function of the passive state ρˆp as W x, p P and bound its Wigner entropy as follows: 1 n τˆn k k n k n k . (45) ( ) +∞ n 1 k=0 h WP h ek Ek x, p = Q S ⟩⟨ S ⊗ S − ⟩⟨ − S k=0 This state is maximally+ mixed over the set of states with ( ) = ∞ŒQ ( )‘ total energy n, so that it is invariant under any unitary ek h Ek x, p transformation that preserves the total energy. In partic- k=0 ular, it is invariant under the action of a balanced beam- ≥ ∞ ‰ ( )Ž † Q Uˆ τˆn Uˆ τˆn k 1 pk pk+1 h Ek x, p splitter, which implies the identity 1~2 1~2 . k=0 After partial tracing over the second mode, we obtain = ∞ ( + )( − ) ‰ ( k )Ž = Q 2 n k 1 pk pk+1 h ρj (43) 1 k=0 k 1 j=0 Tr2 τˆn k k , (46) n 1 k=0 ≥ Q∞( k+ )( − ) Q ( ) [ ] = S ⟩⟨ S 2 pk pk 1 h ρj+ Q + which is simply the extremal+ state εˆn. Alternatively, ex- k=0 j=0 ploiting the invariance under Uˆ and recalling the defi- = ∞ ∞ ( − ) ( ) 1~2 Q Q nition of the beam-splitter states σˆ m, n , we have 2 pk pk+1 h ρj j=0 k=j 1 n ( ) = Q∞ Q ( − ) ( ) Tr τˆ σˆ k, n k . 2 p h ρ 2 n (47) j j n 1 k=0 j=0 [ ] = Q ( − ) = Q ( ) This establishes an interesting+ link between the extremal The first inequality in (43) comes from the concavity of passive states and the beam-splitter states, namely entropy over the convex set of extremal states, while the n second inequality is obtained from Eq. (42). The final 1 εˆn σˆ k, n k . (48) expression is a stronger lower bound on the Wigner en- n 1 k=0 = Q ( − ) + 11

Expressed in terms of Wigner function, this translates as n n Wk x, p S(k,n−k) x, p , (49) k=0 k=0 Q ( ) = Q ( ) where S(m,n) denotes the Wigner function of σˆ m, n . It is instructive to compare Eq. (49) with Eq. (41). Extremal passive states εˆn are defined as mixtures( ) of Fock states [see Eq. (38)], which possess each a non- positive Wigner function (except for the vacuum). This is at the heart of the difficulty of proving our conjecture: we cannot give a meaning to the Wigner entropy of a Fock state (except for the vacuum), so the convex de- composition of a state into Fock states cannot be used to bound its Wigner entropy. In this context, both Eqs. (41) and (49) have the crucial interest to provide the de- composition of the Wigner function of an extremal pas- sive state into a sum of positive functions. However, with Eq. (41), these positive functions do not correspond to physical Wigner functions. Numerical simulations indeed FIG. 10. Wigner entropy of the beam-splitter states σˆ(m, n) 2 2 show that in general ψk x ψn−k p is not a physically as computed numerically for m, n = 0, 1, ⋯ 30. It appears that acceptable Wigner function (it is positive but does not the Wigner entropy increases monotonically for increasing val- correspond to a positive-semidefinite( ) ( ) density operator). ues of m and n. On the contrary, Eq. (49) exhibits the decomposition of an extremal passive state into states σˆ m, n , which are Wigner-positive quantum states as we have shown. conjecture also implies a lower bound on the sum of The set spanned by the states σˆ m, n( associated) with the marginal entropies of x and p, hence it results in Sb is obviously bigger than the set of passive states and a tightening of the entropic uncertainty relation due to offers a nice playground for testing( our) conjecture. In- Bialynicki-Birula and Mycielski that is very natural from deed, each state σˆ m, n is Wigner-positive so it has the point of view of Shannon information theory (the a well-defined Wigner entropy. Figure 10 displays the Wigner entropy accounts for x-p correlations since it is Wigner entropy of the( states) σˆ m, n as computed nu- the joint entropy of x and p). Of course, it also im- merically up to m, n 30. The minimum Wigner entropy plies the Heisenberg uncertainty relation formulated in ln π 1 is reached for the vacuum( state) σ 0, 0 0 0 , terms of variances of x and p. Furthermore, our con- so it follows that the= conjecture holds for whole set Sb jecture implies (but is stronger than) Wehrl conjecture, due+ the concavity of the entropy. Of course,( this) = isS based⟩⟨ S notoriously proven by Lieb. It is supported by several on numerical evidence since we do not have an analyt- elements. First, we have provided in Sec.IV an analyti- ical proof that h S(m,n) ln π 1. Further, although cal proof for a subset of phase-invariant Wigner-positive the set Sb is much bigger than the set of passive states, states, namely the passive states. Second, this was com- it does not encompass( the) ≥ whole+ set of phase-invariant plemented by a semi-analytical semi-numerical proof for Wigner-positive states S+, as evidenced by Figure6. the larger set of phase-invariant states associated with Sb. Third, we also carried out an extensive numerical search for counter-examples in + but could not find any. V. CONCLUSION Given that the Wigner entropy is only properly de- fined for Wigner-positiveQ states, we have also been led We have promoted the Wigner entropy of a quantum to investigate the structure of such states in Sec.III. state as a distinct information-theoretical measure of its We have put forward an extensive technique to produce uncertainty in phase space. Although it is, by defini- Wigner-positive states using a balanced beam-splitter. In tion, restricted to Wigner-positive states, the fact that particular, we have focused on the beam-splitter states such states form a convex set makes it a useful physical σˆ m, n and have highlighted their connection with the quantity. Since it is a concave functional of the state, (smaller) set of passive states and (larger) set of phase- we naturally turn to its lower bound over the convex set invariant( ) Wigner-positive states. We have also found an of Wigner-positive states. We conjecture that this lower unexpectedly simple relation between the states σˆ m, n bound is ln π 1, which is the value taken on by the and the extremal passive states. Wigner entropy of all Gaussian pure states. The latter The Wigner entropy enjoys various reasonable proper-( ) then play the role+ of minimum Wigner-uncertainty states. ties, in particular it is invariant over all symplectic trans- Our conjecture is consistent with Hudson theorem, formations in phase space or equivalently all Gaussian whereby all Wigner-positive pure states must be Gaus- unitaries in state space. Its excess with respect to ln π 1 sian states, thus states reaching the value ln π 1. Our is an asymptotic measure of the number of random bits + + 12 that are needed to generate a sample of the Wigner func- Note added tion from the vacuum state. More generally, since the Wigner entropy is the Shannon differential entropy of the After completion of this work, we have learned that Wigner function, viewed as a genuine probability distri- our method for generating positive Wigner functions with bution, it inherits all its key features. For example, we a 50:50 beam splitter as explained in AppendixA has may easily extend to Wigner entropies the celebrated en- recently also been described in [22]. tropy power inequality [7], which relates to the entropy of the convolution of probability distributions. Consider the setup of Fig.3 where the input state is again a product Acknowledgments state ρˆA ρˆB but the beamsplitter now has an arbitrary transmittance η, so that the output state reads The authors warmly thank Anaelle Hertz and Michael ⊗ ˆ ˆ † σˆ Tr2 Uη ρˆA ρˆB Uη . (50) G. Jabbour for many helpful discussions on this subject. Z.V.H. acknowledges a fellowship from the FRIA foun- Let us restrict to the special case where both ρˆA and =
( ⊗ )  dation (F.R.S.-FNRS). N.J.C. acknowledges support by ρˆB are Wigner-positive states, which of course implies the F.R.S.-FNRS under Project No. T.0224.18 and by that ρˆA ρˆB is Wigner-positive as well as σˆ (even if the EC under project ShoQC within ERA-NET Cofund η 1 2). Thus, ρˆA, and ρˆB, and σˆ all have a well- defined Wigner⊗ entropy, which we denote respectively in Quantum Technologies (QuantERA) program. as ≠hA~, hB, and hout. Since the beam splitter effects the affine transformation xout η xA 1 η xB and Appendix A: A balanced beam-splitter produces pout η pA 1 η pB in phase√ space, we√ may simply = + − Wigner-positive states treat this√ as a convolution√ formula for probability distri- butions.= Hence,+ the− entropy power inequality directly applies to the Wigner entropy. Defining the Wigner In this appendix, we show that when a balanced beam- entropy-power [19] of the two input states as splitter is fed by a two-mode separable input, then its reduced 1-mode output is Wigner positive. We consider −1 hA −1 hB NA 2πe e ,NB 2πe e (51) the following set-up: and the Wigner entropy-power of the output state as = ( ) = ( ) † ˆ 1 ˆ −1 hout σˆ TrB U ρˆA ρˆB U 1 , (A1) Nout 2πe e (52) 2 2 we obtain the Wigner entropy-power inequality ˆ =  ( ⊗ )  = ( ) where Uη is the unitary operator of the beam-splitter: Nout η NA 1 η NB. (53) ˆ †ˆ ˆ† Uη exp θ aˆ b aˆb (A2) This is equivalent to≥ a nontrivial+ ( − ) lower bound on the Wigner entropy of the output state σˆ, namely h Wσˆ such that η cos2 θ=and ‰0 ‰ θ −π 2.ŽŽ We are going to h Wσˆ , where σˆG denotes the the Gaussian output state G σˆ η 1 2 obtained if each of the two input states is replaced( ) by≥ show that is Wigner-positive when : = ≤ ≤ ~ a( Gaussian) state with the same Wigner entropy. This Wσˆ x, p 0 x, p.= ~ (A3) illustrates the physical significance of the Wigner entropy. Finally, a logical extension of the present work is Our proof has two parts.( ) In≥ the first∀ part we show that to consider more than a single harmonic oscillator (or the action of a beam-splitter in phase space corresponds bosonic mode) as we expect that all properties of the to a convolution between the Wigner functions of the two Wigner entropy and especially our conjecture will gener- inputs. In the second part, we show that the convolution alize. Let us mention that this work is part of a broader of two Wigner functions corresponds in state space to the project. The key observation is that the Wigner function overlap between two density operators, which is always of Wigner-positive states can be interpreted as a true a non-negative quantity. Finally, we give some further probability distribution. Hence, we can take advantage observations regarding the σˆ m, n states which are build of this observation and adapt the standard tools of proba- using that set-up. bility theory (here, we have applied Shannon information ( ) theory to define the Wigner entropy). In this context, the theory of majorization [20] has proved to be another pow- 1. Convolution in phase space erful too, and it notably allows to formulate a generaliza- tion of Wehrl conjecture [6]. In a forthcoming paper [21], we use the theory of majorization to state a stronger con- The action of a beam-splitter on the modes operators jecture on the uncertainty content of Wigner functions. is described by the following transformation: This enables us, for instance, to demonstrate analytically h W aˆ′ η 1 η aˆ the lower bound on for all phase-invariant Wigner- . 2 ˆ′ η η ˆ (A4) positive states in S+, including the dark blue region. b √ √ b ( ) − Œ ‘ = ‹ √ √  Œ ‘ − 13

From this, we derive the expression of the old quadra- placement operators acting on mode ˆb: tures operators xˆA, xˆB, pˆA, pˆB as a function of the new ˆ ˆ†ˆ quadratures operators xˆ′ , xˆ′ , pˆ′ , pˆ′ : R ϕ exp iϕb b , A B A B (A11) Dˆ α exp αˆb† α∗ˆb . ′ ′ ( ) = (− ) xˆA η xˆA 1 η xˆB, ′ √ ′ √ ′ Let us define ρˆB as( the) = result‰ of a− particularŽ combination ⎧pˆA = η pˆA − 1 −η pˆB, ˆ ˆ ⎪ of R and D acting on ρˆB: ⎪ (A5) ⎪xˆ √1 η xˆ′√ η xˆ′ , ⎪ B = − A − B ′ ˆ ˆ ˆ† ˆ † ⎪ ρˆB D xα ipα R π ρˆBR π D xα ipα . (A12) ⎨ √ ′ ′ ⎪pˆB 1 η pˆ √η pˆ . ⎪ = − A + B ρˆ′ ⎪ The Wigner= ( function+ ) ( of) B can( ) be expressed( + ) from the ⎪ √ √ ′ ρˆ The 2-mode Wigner⎩⎪ = function− of+ the output W is then Wigner function of B as follows: obtained through the following relation: ρˆB WB x, p ′ ′ ′ ′ ′ ′ (A13) W xA, pA, xB, pB W xA, pA, xB, pB ρˆ W 2x x, 2p p . (A6) B ←→ B( )α α WA xA, pA WB xB, pB , √ √ ( ) = ( ) The last ingredient←→ of( our proof− is the expression− ) of the where WA and WB are= the( Wigner) functions( of) respec- overlap between two quantum states, which is always tively ρˆA and ρˆB. Computing the Wigner function of σˆ non-negative. That quantity can be expressed equiva- ′ ′ ′ ′ ′ is done by integrating W xA, pA, xB, pB over the vari- lently in the formalism of state space or phase space: ′ ′ ables xB, pB: ( ) Tr ρˆ1ρˆ2 2π dxdpW1 x, p W2 x, p 0. (A14) ′ ′ ′ ′ Wσˆ xA, pA dxBdpB Combining[ ] Eqs.= U (A10), (A13() and) ((A14)) gives≥ us the ′ ′ ′ ′ expression of Wσˆ as an overlap between two quantum WA η xA 1 η xB(, η pA) = U1 η pB (A7) states: √ »′ ′ √ »′ ′ WBŠ 1 η− xA − η xB, 1 η− pA − η pB ×. ′ ′ 1 ′ Wσˆ x , p Tr ρˆAρˆ 0, (A15) » √ » √ A A π B That expressionŠ − finds+ a natural− writing+ by introducing ′′ ′′ ′ ′ ′ the new variables x , p : where the xA, pA(dependence) = is[ hidden] ≥ in ρˆB. This con- cludes our proof, and we have shown that σˆ is Wigner- ′′ ′ ′ x η xA η 1 η xB, positive. The proof naturally expands from product in- » (A8) puts to separable inputs since the mixing of Wigner- ⎪⎧ ′′ = ′ − ( − ) ′ ⎪p η pA η 1 η pB, positive states remains Wigner-positive. ⎨ » ⎪ so that it reduces⎩⎪ = to − ( − ) 3. σˆ(m, n) states 1 x′′ p′′ W x′ , p′ dx′′dp′′ W , σˆ A A η A η η σˆ m, n states are defined in SectionIII and play a ′ ′′ ′ ′′ (A9) ( ) = U1 xA x Œp√A x√ ‘ particular role in this paper. We give here the expression WB , . 1 η 1 η 1 η of their( density) operator decomposed onto the basis of − − × Œ √ √ ‘ Fock states: Eq. (A9) shows that− the action of a beam-splitter on a − − m+n min(z,m) min(z,m) product input corresponds to a convolution between WA σˆ m, n m!n!2m2n −1 W η 1 η and B rescaled respectively by a factor and . z=0 i=max(0,z−n) j=max(0,z−n) In the case of a balanced beam-splitter, the√ value√ of the ( m ) =n( m n) Q Q Q parameter η is 1 2, and the rescaling values are equal.− 1 i+jz! m n z ! z z , i z i j z j Introducing the new variables x˜ 2x′′ and p˜ 2p′′, (A16) we can then write~ the previous expression as: ‹ ‹ ‹ ‹ (− ) ( + − ) S ⟩⟨ S √ √ − − = = z z zth ′ ′ where is a projector onto the Fock state. Wσˆ xA, pA 2 d˜xd˜pWA x,˜ p˜ Taking advantage of Eqs. (A12) and (A15), we can (A10) ′ ′ easily showS ⟩⟨ S that the Wigner function of σˆ m, n (that we ( WB) = 2xA x,˜ 2pA( p˜ ). U write S(m,n)) cancels at the origin when m n. Indeed, √ √ since Fock states are invariant under rotation,( ) it follows × Š − −  that: ≠ 2. State space picture 1 1 S 0, 0 Tr m m n n δmn, (A17) (m,n) π π Eq. (A10) can be expressed in state space formalism. To that purpose, we recall the usual rotation and dis- where δmn(is the) = Kronecker[S ⟩⟨ delta.SS ⟩⟨ S] = 14

Appendix B: Wigner-positivity conditions for mixtures up to 2 photons

In this appendix, we derive the conditions that apply to a mixture of the three first Fock states (0, 1 and 2) such that it has a non-negative Wigner function. We are considering any state that can be written as:

ρˆ 1 p1 p2 0 0 p1 1 1 p2 2 2 , (B1) where =p(1, p−2 −0 and)S ⟩⟨p1 S +p2 S ⟩⟨1.S + WeS first⟩⟨ S identify a condition on p1 and p2 that is equivalent to the Wigner- positivity of ρˆ≥. Then we compute+ ≤ the geometrical locus of p1, p2 that ensures Wigner-positivity, and extremal Wigner-positivity. ( ) FIG. 11. Geomtrical locus of Wigner-positive values of (p1, p2). The ellipse corresponds to the region where (p1, p2) 1. Equivalent condition for Wigner-positivity is such that the parabola (B5) never becomes negative. The other region corresponds to (p1, p2) such that the parabola becomes negative only for negative values of t. The dashed The Wigner function of a Fock state is radial and reads lines defines the limits such that p1, p2 ≥ 0 and p1 + p2 ≤ 1. as:

1 n 2 2 Wn r 1 exp r Ln 2r , (B2) product is non-negative. Applied to Eq. (B5), this gives π the following conditions: ( ) = (− ) (− ) ( ) where we use the non-negative radial parameter r p1 2p2 x2 p2. We recall the three first Laguerre polynomials: (B6) = ⎪⎧p ≥ 1 » ⎪ 1 2 + L x 1, ⎨ 0 ⎪ Condition (B6) describes⎪ a≤ locus which is the intersec- L1 x x 1, ⎩ ( ) = (B3) tion of two half-planes. We now check the discriminant 1 2 2 2 condition. The discriminant is equal to ∆ 4p2 2p2 p1, L2(x) = − x+ 2x 1. 2 so that the condition ∆ 0 can be written as: 2 2 = − + ( ) = − + ρˆ p1 p2 1 4 Using this, we can express the Wigner function of as: ≤ 1. (B7) 1 2 1 4 1 − ~ W r exp r2 2p r4 2p 4p r2 1 2p . Œ ‘ + Œ ‘ ≤ π 2 1 2 1 Condition (B7) describes~ an ellipse.~ Note that the union (B4) of the sets determined by conditions (B6) and (B7) along- ( ) = (− ) ‰ + ( − ) + − Ž We want to identify the set of possible values of p1, p2 side with the physicality conditions can be summarized such that ρˆ is Wigner-positive. Introducing the param- as: eter t 2r2, we can write an equivalent condition to 1 p1 2 , W r 0 r 0 as: (B8) = ⎪⎧p ≤ 1 1 1 4p2, ⎪ 2 4 4 1 ( )1≥ 2∀ ≥ ⎨ p2t p1 2p2 t 1 2p1 0 t 0. (B5) ⎪ » p , p 0 2 with the additional⎪ constraint≤ + that− 1 2 . Figure 11 illustrates the geometrical⎩ locus associated to the differ- + ( − ) + − ≥ ∀ ≥ ent conditions. ≥

2. Locus of (p1, p2) for Wigner-positivity

3. Locus of (p1, p2) for extremal Wigner-positivity Eq. (B5) is the equation of a parabola with a non- negative coefficient associated to t2. We want that P t parabola to have non-negative values for t 0. This Let us define as the polynomial of the parabola is possible either if its discriminant ∆ is non-positive described in Eq. (B5). We refer to the first order deriva- P t ( ) t P ′ t (∆ 0), or if both its roots corresponds to t ≥0. tive of with respect to as . Let us examine the latter possibility first. If we have 1 2 ( )P t p t p 2p (t) 1 2p a parabola≤ defined by at2 bt c 0, the≤ sum of its 2 2 1 2 1 roots is b a and their product is c a. The two roots (B9) ′( ) = + ( − ) + − are non-positive if their sum+ is+ non-positive= and their P t p2t p1 2p2 − ~ ~ ( ) = + − 15

The locus of extremal Wigner-positive states is the set of as follows: p1, p2 such that 1 2 p1 1 a ( ) 2 P t 0 ⎧ √ (B12) ⎪ = 1 − t 0 s.t. ′ (B10) ⎪p a 1 P t 0 ⎪ 2 4 ( ) = ⎨ ∃ ≥ œ ⎪ ⎪ a = ( + ) 0 1 ′ ( ) = where the parameter⎩ goes from to . Injecting that The condition P t 0 is satisfied at t 2 p1 p2. In- parametrization in (B4) yields the following expression: jecting that value of t in P t 0 gives us the following equation: ( ) = = − ~ 2 1 1 1 a ( ) = W r exp r2 a 1 r2 1 . a ¾ 2 2 π 2 1 a 4p 2p2 p 0. (B11) ⎛ − ⎞ 2 1 ( ) = ‰− Ž ( + ) − + (B13) W r ⎝ + ⎠ − + = a is the radial Wigner function of the extremal with the additional constraint that 2p2 p1, since t 0. Wigner-positive states located on the arc of ellipse. This describes an arc of ellipse, that we can parameterize ( ) ≥ ≥

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