Phase Spaces, Parity Operators, and the Born-Jordan Distribution

Phase spaces as given by the Wigner distribution function provide a natural description of infinite-dimensional quantum systems. They are an important tool in quantum optics and have been widely applied in the context of time-frequency analysis and pseudo-differential operators. Phase-space distribution functions are usually specified via integral transformations or convolutions which can be averted and subsumed by (displaced) parity operators proposed in this work. Building on earlier work for Wigner distribution functions [A. Grossmann, Comm. Math. Phys. 48(3), 191 (1976)], parity operators give rise to a general class of distribution functions in the form of quantum-mechanical expectation values. This enables us to precisely characterize the mathematical existence of general phase-space distribution functions. We then relate these distribution functions to the so-called Cohen class [L. Cohen, J. Math. Phys. 7(5), 781 (1966)] and recover various quantization schemes and distribution functions from the literature. The parity-operator approach is also applied to the Born-Jordan distribution which originates from the Born-Jordan quantization [M. Born, P. Jordan, Z. Phys. 34(1), 858 (1925)]. The corresponding parity operator is written as a weighted average of both displacements and squeezing operators and we determine its generalized spectral decomposition. This leads to an efficient computation of the Born-Jordan parity operator in the number-state basis and example quantum states reveal unique features of the Born-Jordan distribution.


Introduction
There are at least three logically independent descriptions of quantum mechanics: the Hilbert-space formalism [30], the path-integral method [45], and the phase-space approach such as given by the Wigner function [23,31,47,66,70,79,104,106,117]. The phase-space formulation of quantum mechanics was initiated by Wigner in his ground-breaking work [115] from 1932, in which the Wigner function of a spinless non-relativistic quantum particle was introduced as a quasi-probability distribution. The Wigner function can be used to express quantum-mechanical expectation values as classical phase-space averages. More than a decade later, Groenewold [60] and Moyal [94] formulated quantum mechanics as a statistical theory on a classical phase by mapping a quantum state to its Wigner function and they interpreted this correspondence as the inverse of the Weyl quantization [111][112][113].
Coherent states have become a natural way to extend phase spaces to more general physical systems [5, 8-12, 20, 49, 95]. In this regard, a new focus on phasespace representations for coupled, finite-dimensional quantum systems (as spin systems) [48, 71-77, 82, 101, 103, 108] and their tomographic reconstructions [76,80,81,102] has emerged recently. A spherical phase-space representation of a single, finite-dimensional quantum system has been used to naturally recover the infinitedimensional phase space in the large-spin limit [74,76]. These spherical phase spaces have been defined in terms of quantum-mechanical expectation values of rotated parity operators [72,74,76,82,103,108] (as discussed below) in analogy with displaced reflection operators in flat phase spaces. But in the current work, we exclusively focus on the (usual) infinite-dimensional case which has Heisenberg-Weyl symmetries [20,49,85,95]. This case has been playing a crucial role in characterizing the quantum theory of light [52] via coherent states and displacement operators [3,4,21,22] and has also been widely used in the context of time-frequency analysis and pseudo-differential operators [15-17, 27, 28, 54, 56, 59]. Many particular phase spaces have been unified under the concept of the so-called Cohen class [27,28,54] (see Definition 2 below), i.e. all functions which are related to the Wigner function via a convolution with a distribution (which is also known as the Cohen kernel).
However, parity operators similar to the one by Grossmann and Royer [54,61,97] have still been lacking for general phase-space distribution functions. (Note that such a form appeared implicitly for -parametrized distribution functions in [21,93].) In the current work, we generalize the previously discussed parity operator Π [54,61,97] for the Wigner function by introducing a family of parity operators Π (refer to Definition 3) which is parametrized by a function or distribution . This enables us to specify general phase-space distribution functions in the form of quantum-mechanical expectation values (refer to Definition 4) as We will refer to the above operator Π as a parity operator following the lead of Grossmann and Royer [61,97] and given its resemblance and close analogy to the reflection operator Π discussed in prior work [14,72,74,76,82,103,108]. Here, D (Ω) denotes the displacement operator and Ω describes suitable phase-space coordinates (see Sec. 3.1). (Recall that ℏ = ℎ/(2 ) is defined as the Planck constant ℎ divided by 2 .) The quantum-mechanical expectation values in the preceding equation give rise to a rich family of phase-space distribution functions (Ω, ) which represent arbitrary (mixed) quantum states as given by their density operator . In particular, this family of phase-space representations contains all elements from the (above mentioned) Cohen class and naturally includes the pivotal Husimi Q and Born-Jordan distribution functions.
We would like to emphasize that our approach to phase-space representations averts the use of integral transformations, Fourier transforms, or convolutions as these are subsumed in the parity operator Π which is independent of the phase-space coordinate Ω. Although our definition also relies on an integral transformation given by a Fourier transform, it is only applied once and is completely absorbed into the definition of a parity operator thereby avoiding redundant applications of Fourier transforms. This leads to significant advantages as compared to earlier approaches: conceptual advantages (see also [71,76,93,103,108]): -The phase-space distribution function is given as a quantum-mechanical expectation value. And this form nicely fits with the experimental reconstruction of quantum states [7,13,41,65,76,88,102]. -All the complexity from integral transformations (etc.) is condensed into the parity operator Π . -The dependence on the distribution and the particular phase space is separated from the displacement D (Ω). computational advantages: -The repeated and expensive computation of integral transformations (etc.) in earlier approaches is avoided as Π has to be determined only once. Also, the effect of the displacement D (Ω) is relatively easy to calculate.
In this regard, the current work can also be seen as a continuation of [76] where the parity-operator approach has been emphasized, but mostly for finite-dimensional quantum systems. Moreover, we connect results from quantum optics [21,22,52,83], quantum-harmonic analysis [29, 36-39, 54, 56, 69, 110], and group-theoretical approaches [20,49,85,95]. It is also our aim to narrow the gap between different communities where phase-space methods have been successfully applied. On the other hand, a major contribution of our work is the analysis of existence properties of generalized phase-space distributions and their parity operators. While the Wigner function has been known to exist for the general class of tempered distributions (a class of generalized functions that includes the pivotal 2 space), we further illuminate which classes of Cohen kernels yield well-defined generalized phase-space distribution functions. Such existence questions are fully absorbed into the parity operators and precise conditions are used to guarantee their mathematical existence.
Similarly as the parity operator Π (which is the Weyl quantization of the delta distribution), we show that its generalizations Π are Weyl quantizations of the corresponding Cohen kernel (refer to Sec. 4.3 for the precise definition of the Weyl quantization used in this work). We discuss how these general results reduce to wellknown special cases and discuss properties of phase-space distributions in relation to their parity operators Π . In particular, we consider the class of -parametrized distribution functions [21,22,52,93], which include the Wigner, Glauber P, and Husimi Q functions, as well as the -parametrized family, which has been proposed in the context of time-frequency analysis and pseudo-differential operators [15][16][17]56]. We derive spectral decompositions of parity operators for all of these phase-space families, including the Born-Jordan distribution. Relations of the form Π = • Π motivate the name "parity operator" as they are in fact compositions of the usual parity operator Π followed by some operator that usually corresponds to a geometric or physical operation (which commutes with Π). In particular, is a squeezing operator for the -parametrized family and corresponds to photon loss for the -parametrized family (assuming < 0). This structure of the parity operators Π connects phase spaces to elementary geometric and physical operations (such as reflection, squeezing operators, photon loss) and these concepts are central to applications: the squeezing operator models a non-linear optical process which generates non-classical states of light in quantum optics [53,83,89]. These squeezed states of light have been widely used in precision interferometry [58,91,105,116] or for enhancing the performance of imaging [87,109], and the gravitational-wave detector GEO600 has been operating with squeezed light since 2010 [1,62].
The Born-Jordan distribution and its parity operator constitute a most peculiar instance among the phase-space approaches. This distribution function has convenient properties, e.g., it satisfies the marginal conditions and therefore allows for a probabilistic interpretation [56]. The Born-Jordan distribution is however difficult to compute. But most importantly, the Born-Jordan distribution and its corresponding quantization scheme have a fundamental importance in quantum mechanics. In particular, there have been several attempts in the literature to find the "right" quantization rule for observables using either algebraic or analytical techniques. In a recent paper [55], one of us has analyzed the Heisenberg and Schrödinger pictures of quantum mechanics, and it is shown that the equivalence of both theories requires that one must use the Born-Jordan quantization rule (as proposed by Born and Jordan [19]) while its Born-Jordan quantization is the different expression (Recall that the operatorsˆandˆsatisfy the canonical commutation relations [ˆ,ˆ] = ℏ using the spatial coordinates , ∈ { , , } and the Kronecker delta .) One of us has shown in [57] that the use of (2) instead of (1) solves the so-called "angular momentum dilemma" [33,34]. To a general observable ( , ), the Weyl rule associates the operator where F is the symplectic Fourier transform of and D ( , ) the displacement operator (see Sec. 3.1); in the Born-Jordan case this expression is replaced with where the filter function BJ ( , ) is given by .
We obtain significant, new results for the case of Born-Jordan distributions and therefore substantially advance on previous characterizations. In particular, we derive its parity operator Π BJ in the form of a weighted average of geometric transformations where D ( , ) is the displacement operator and ( ) is the squeezing operator (see Eq. (45) below) with a real squeezing parameter . We have used the sinus cardinalis sinc( ) := sin( )/ and the hyperbolic secant sech( ) := 1/cosh( ) functions. The parity operator Π BJ in Eq. (3) decomposes into a product Π = • Π containing the usual reflection operator Π. This is another example of the above-discussed motivation for our terminology of parity operators. We prove in Proposition 2 that Π BJ is a bounded operator on the Hilbert space of square-integrable functions and therefore gives rise to well-defined phase-space distribution functions of arbitrary quantum states. We derive a generalized spectral decomposition of this parity operator based on a continuous family of generalized eigenvectors that satisfy the following generalized eigenvalue equation for every real (see Theorem 5): Facilitating a more efficient computation of the Born-Jordan distribution, we finally derive explicit matrix representations in the so-called Fock or number-state basis, which constitutes a natural representation for bosonic quantum systems such as in quantum optics [53,83,89]. Curiously, the parity operator Π BJ of the Born-Jordan distribution is not diagonal in the Fock basis as compared to the diagonal parity operators of -parametrized phase spaces (cf. [76]) that enable the experimental reconstruction of distribution functions from photon-count statistics [7,13,41,88]  along with their operator norms (Theorem 2) and generalized spectral decompositions in Sec 5.2; the Born-Jordan parity operator is a weighted average of displacements (Theorem 3) or equivalently a weighted average of squeezing operators (Theorem 4), and it is bounded (Proposition 2); the Born-Jordan parity operator admits a generalized spectral decomposition (Theorem 5); its matrix representation is calculated in the number-state basis in Theorem 6; and an efficient, recursion-based computation scheme is proposed in Conjecture 1. Our work has significant implications: General (infinite-dimensional) phase-space functions can now be conveniently and effectively described as natural expectation values. We provide a much more comprehensive understanding of Born-Jordan phase spaces and means for effectively computing the corresponding phase-space functions. Working in a rigorous mathematical framework, we also facilitate future discussions of phase spaces by connecting different communities in physics and mathematics.
We start by recalling precise definitions of distribution functions and quantum states for infinite-dimensional Hilbert spaces in Sec. 2. In Sec. 3, we discuss phasespace translations of quantum states using coherent states, state one known formulation of translated parity operators, and relate a general class of phase spaces to Wigner distribution functions and their properties. We note that an experienced reader can skip most of the introductory Sections 2 and 3 and jump directly to our results. These preparations will however guide our study of phase-space representations of quantum states as expectation values of displaced parity operators in Section 4. We present and discuss our results for the case of the Born-Jordan distribution and its parity operator in Section 5. Formulas for the matrix elements of the Born-Jordan parity operator are derived in Section 6. Explicit examples for simple quantum systems are discussed and visualized in Section 7, before we conclude. A larger part of the proofs have been relegated to Appendices.

Distributions and Quantum States
All of our discussion and results in this work will strongly rely on precise notions of distributions and related descriptions of quantum states in infinite-dimensional Hilbert spaces. Although most (or all) of this material is quite standard and wellknown [54,63,68,96], we find it prudent to shortly summarize this background material in order to fix our notation and keep our presentation self-contained. This will also help to clarify differences and connections between divergent concepts and notations used in the literature. We hope this will also contribute to narrowing the gap between different physics communities that are interested in this topic.

Schwartz Space and Fourier Transforms
We will now summarize function spaces that are central to this work, refer also to [54,Ch. 1.1.3]. The set of all smooth, complex-valued functions on R that decrease faster (together with all of their partial derivatives) than the reciprocal of any polynomial is called the Schwartz space and is usually denoted by S(R ), refer to [ The topological dual space S (R ) of S(R ) is often referred to as the space of tempered distributions, and we will denote the distributional pairing for ∈ S (R ) and ∈ S(R ) as , := ( ) ∈ C. In Sec. 2, we will consistently use the symbol to denote distributions and , to denote Schwartz or square-integrable functions. Also note that S(R ) is dense in 2 (R ) and tempered distributions naturally include the usual function spaces S(R ) ⊂ 2 (R ) ⊂ S (R ) via distributional pairings in the form of an integral , . This inclusion is usually referred to as a rigged Hilbert space [25,50] or the Gelfand triple. Recall that the Lebesgue spaces (R ) with 0 < < ∞ are subspaces of equivalence classes of measurable functions : R → C that differ only on a set of measure zero such that the -th power of their absolute value is Lebesgue integrable, i.e. ∫ R | ( )| d < ∞ [96]. Remarkably, every tempered distribution is the derivative of some polynomially bounded continuous function, that is, given ∈ S (R ) there exists : R → C continuous such that | ( )| ≤ (1+ 2 ) for some , ≥ 0 and all ∈ R , as well as a multi-index such that , For the rest of our work, we will restrict the general space of R with ≥ 1 to the case of R which is most relevant for the applications we highlight. This simplifies our notation, even though many statements could be generalized.
Recall that for all ∈ S(R 2 ) the symplectic Fourier transform [F ] ( , ) (see App. B in [54]) is related to the usual Fourier transform ∈ (R 2 ) and ∈ S(R 2 ). Thus this is the extension of F with respect to the distributional pairing in our sense, cf. also Appendix A. In particular the symplectic Fourier transform generalizes to phase-space distribution functions ( , ) without further adjustment and all the properties of F on S(R 2 ) transfer to S (R 2 ).

Quantum States and Expectation Values
Let us denote the abstract state vector of a quantum system by | which is an element of an abstract, infinite-dimensional, separable complex Hilbert space (here and henceforth denoted by) H . The Hilbert space H is known as the state space and it is equipped with a scalar product · | · [63]. Considering projectors P := | | defined via the open scalar products P = | · | , an orthonormal basis of H is given by {| , ∈ N} if | = for all , ∈ N and ∞ =0 P = 1 in the strong operator topology. For a broader introduction to this topic we refer to [63].
Depending on the given quantum system, explicit representations of the state space can be obtained by specifying its Hilbert space [51]. In the case of bosonic systems, the Fock (or number-state) representation is widely used. A quantum state | is an element of the Hilbert space ℓ 2 of square-summable sequences of complex numbers [63]. It is characterized by its expansion | = ∞

=0
| into the orthonormal Fock basis {| , = 0, 1, . . . } of number states using the expansion coefficients = | ∈ C, refer to, e.g., [22] and [63,Ch. 11]. The scalar product | then corresponds to the usual scalar product of vectors, i.e. to the absolutely convergent sum ∞ =0 ( ) * =: | ℓ 2 . The corresponding norm of vectors is then given by For a quantum state | , the coordinate representation ( ) ∈ S(R) and its Fourier transform (or momentum representation) ( ) ∈ S(R) are given by complex, square-integrable, and smooth functions that are also fast decreasing. The quantum state | = ∫ R ( )| d of ( ) = | is then defined via coordinate eigenstates | . The coordinate representation of a coordinate eigenstate is given by the distribution ( − ) ∈ S (R), refer to [51,63]. The scalar product | is then fixed by the usual 2 scalar product, i.e. by the convergent integral ∫ R * ( ) ( ) d =: | 2 . This integral induces the norm of square-integrable functions via ( ) 2 = [ | 2 ] 1/2 . The above two examples are particular representations of the state space, which are convenient for particular physical systems, however these representations are equivalent via For the position operatorˆ: S (R) → S (R), ( ) ↦ → ( ) one can consider the dualˆ : S (R) → S (R), ↦ → •ˆ. This map satisfies the generalized eigenvalue equationˆ | 0 = 0 | 0 for all 0 ∈ R where its generalized eigenvector | 0 ∈ S (R) is the delta distribution, which allows for the resolution of the position operatorˆ= ∫ R | | d . For more details, we refer to [50] or [51, p.1906].
refer to Theorem 2 in [51]. In particular, any coordinate representation ( ) ∈ S(R) of a quantum state | can be expanded in the number-state basis into ( ) = ∞ =1 are eigenfunctions of the quantum-harmonic oscillator. For any ( ), ( ) ∈ S(R), the 2 scalar product is equivalent to the ℓ 2 scalar product and it is invariant with respect to the choice of orthonormal basis, i.e. any two orthonormal bases are related via a unitary transformation. The Plancherel formula ∫ In the following, we will consistently use the notation · | · for scalar products in Hilbert space, without specifying the type of representation. This is motivated by the invariance of the scalar product under the choice of representation. However, in order to avoid confusion with different types of operator or Euclidean norms, we will use in the following the explicit norms ( ) 2 and | ℓ 2 , despite their equivalence. We will now shortly define the trace of operators on infinite-dimensional Hilbert spaces, refer to [96, Ch.VI.6] for a comprehensive introduction. Recall that the trace of a positive semi-definite operator , where the sum of non-negative numbers on the right-hand side is independent of the chosen orthonormal basis {| , ∈ N} of H , but it does not necessarily converge. Moreover recall that the set of trace-class operators is given by where K (H ) denotes the set of compact operators on H and † is the adjoint of (which is in finite dimensions given by the complex conjugated and transposed matrix). The expression Tr( . For ∈ B 1 (H ), the mapping ↦ → Tr( ) is linear, continuous with respect to the trace norm, and independent of the chosen orthonormal basis of H . Trace-class operators ∈ B 1 (H ) have the important property that their products with bounded operators ∈ B (H ) are also in the trace class, i.e. , ∈ B 1 (H ). Using this definition, one can calculate the trace independently from the choice of the orthonormal basis or representation that is used for evaluating scalar products.
A density operator or state ∈ B 1 (H ) is defined to be positive semi-definite with Tr( ) = 1. It therefore admits a spectral decomposition [92,Prop. 16.2], i.e. there exists an orthonormal system {| , ∈ N} in H such that Here, B ( H) denotes the set of bounded linear operators on H, and one has | | | < ∞ for every ∈ B ( H) and , ∈ H . An operator ∈ B ( H) is said to be positive semi-definite if is self-adjoint and | ≥ 0 for all ∈ H.

Coherent States, Phase Spaces, and Parity Operators
We continue to fix our notation by discussing an abstract definition of phase spaces that relies on displaced parity operators. This usually appears concretely in terms of coherent states [20,49,85,95], for which we consider two equivalent but equally important parametrizations of the phase space using the coordinates or ( , ) (see below). This definition of phase spaces can be also related to convolutions of Wigner functions which is usually known as the Cohen class [27,28,54]. We also recall important postulates for Wigner functions as given by Stratonovich [20,107] and these will be later considered in the context of general phase spaces.

Phase-Space Translations of Quantum States
We will now recall a definition of the phase space for quantum-mechanical systems via coherent states, refer to [20,49,85,95]. We consider a quantum system which has a specific dynamical symmetry group given by a Lie group . The Lie group acts on the Hilbert space H using an irreducible unitary representation D of . By choosing a fixed reference state as an element |0 ∈ H of the Hilbert space, one can define a set of coherent states as | := D ( )|0 where ∈ . Considering the subgroup ⊆ of elements ℎ ∈ that act on the reference state only by multiplication D (ℎ)|0 := |0 with a phase factor , any element ∈ can be decomposed into = Ωℎ with Ω ∈ / . The phase space is then identified with the set of coherent states |Ω := D (Ω)|0 . In the following, we will consider the Heisenberg-Weyl group 3 , for which the phase space Ω ∈ 3 / (1) is a plane. We introduce the corresponding displacement operators that generate translations of the plane. Displacement operators are also known as Heisenberg-Weyl operators [54] or, in the physics literature, simply as Weyl operators [6,40,67].
In particular, for harmonic oscillator systems, the phase space Ω ≡ ∈ C is usually parametrized by the complex eigenvalues of the annihilation operatorˆand Glauber coherent states can be represented explicitly [22] in the so-called Fock (or number-state) basis as Here, the second equality specifies the displacement operator D ( ) as a power series of the usual bosonic annihilationˆand creationˆ † operators, which satisfy the commutation relation [ˆ,ˆ †] = 1, refer to Eq. (2.11) in [22]. In particular, the number state representation of displacements is given by [22] [ where ( − ) ( ) are generalized Laguerre polynomials. This is the usual formulation for bosonic systems (e.g., in quantum optics) [83], where the optical phase space is the complex plane and the phase-space integration measure is given by dΩ = 2ℏ d 2 = 2ℏ d ( ) d ( ) (where one often sets ℎ = 2 ℏ = 1, cf., [20][21][22]). The annihilation operator admits a simple decomposition with respect to its eigenvectors, see, e.g., [22, Eqs.
Let us now consider the coordinate representation ( ) ∈ S(R) of a quantum state. The phase space is parametrized by Ω ≡ ( , ) ≡ ∈ R 2 and the integration measure is dΩ = d = d d . The displacement operator acts via (see also [111][112][113]) where , 0 , 0 ∈ R. The right hand side of Eq. (10) specifies the displacement operator as a power series of the usual operatorsˆandˆ, which satisfy the commutation relation The most common representations of these two unbounded operators areˆ as integrals from Section 2.1 (cf. Example 3(2), Appendix A) for all : R → C such that , · ∈ S (R), and all ∈ S(R), Ω ∈ R 2 . In particular it does not matter whether D (Ω) acts on a function : R → C or on the induced functional ↦ → , .
The two (above mentioned) physically motivated examples are particular representations of the displacement operator for the Heisenberg-Weyl group in different Hilbert spaces while relying on different parametrizations of the phase space. Let us now highlight the equivalence of these two representations. In particular, we obtain the formulasˆ= 2ℏ for any non-zero real , refer to Eqs. (2.1-2.2) in [22]. In the context of quantum optics, the operatorsˆandâ re the so-called optical quadratures [83]. The operatorsˆandˆ † are now defined on the Hilbert space 2 (R), whereasˆandˆ † act on elements of the Hilbert space ℓ 2 . For any ≠ 0 they reproduce the commutator [ˆ, This differs from other approaches where one considers the embedding : for all ( ) ∈ 2 (R), and they correspond to raising and lowering operators of the quantum harmonic-oscillator eigenfunctions Fock ( ), refer to [63]. Substituting noŵ = √︁ ℏ/2 −1 (ˆ+ˆ † ) andˆ= − √︁ ℏ/2 (ˆ−ˆ † ) into the exponent on the right-hand side of (10) yields This then confirms the equivalence where the phase-space coordinate is defined by is independent of the choice of . Let us also recall two properties of the displacement operator [111][112][113] (see, e.g., [54, p. 7]): In the following, we will use both of the phase-space coordinates and ( , ) interchangeably. The displacement operator is obtained in both parametrizations, and they are equivalent via (11). Motivated by the group definition, we will also use the parametrization Ω for the phase space via D (Ω), where Ω corresponds to any representation of the group, including the ones given by the coordinates and ( , ).

Phase-Space Reflections and the Grossmann-Royer Operator
Recall that the parity operator Π reflects wave functions via Π ( ) := (− ) and Π ( ) := (− ) for coordinate-momentum representations [14,54,61,86,97], and Π|Ω := |−Ω for phase-space coordinates of coherent states [14,21,86,97]. This parity operator is obtained as a phase-space average of the displacement operator from (10). One finds for all ∈ S(R), ∈ R that For example, the choice = √ corresponds to the quantum-harmonic oscillator of mass and angular frequency . And = √ is related to a normal mode of the electromagnetic field in a dielectric.
for short. Thus the parity operator equals evaluating the symplectic Fourier transform of the displacement operator at the phase-space point Ω = 0. This is related to the Grossmann-Royer operator which is the parity operator transformed by the displacement operator [14,54,61,86,97]. Here, we use in both (14) and (15) an abbreviated notation for formal integral transformations of the displacement operator.
Remark 1 This abbreviation in Eq. (15) is justified as the existence of the correspond- dΩ is guaranteed by, e.g., [54, Sec. 1.3., Prop. 8] for all ∈ S (R). In the following, we will use this abbreviated notation for formal integral transformations of the displacement operator, i.e. by dropping . However, we might need to restrict the domain of more general parity operators to ensure the existence of the respective integrals.

Wigner Function and the Cohen class
The Wigner function ( , ) of a pure quantum state | was originally defined by Wigner in 1932 [115] and it is (in modern terms) the integral transformation of a pure state ∈ 2 (R), i.e.
The second and third equalities specify the Wigner function using the Grossmann-Royer operator [54,61] from (15), refer to [54, Sec. 2.1.1., Def. 12]. We use this latter form to extend the definition of the Wigner function to mixed quantum states as in [4,14,21,97].
The Wigner representation is in general a bĳective, linear mapping between the set of density operators (or, more generally, the trace-class operators) and the phase-space distribution functions that satisfy the so-called Stratonovich postulates [20,107]: The not necessarily bounded operator is the Weyl quantization of the phase-space function (or distribution) (Ω) ∈ S (R 2 ), refer to Sec. 4.3. Based on these postulates, the Wigner function was defined for phase-spaces of quantum systems with different dynamical symmetry groups via coherent states [20,49,75,76,95,108]. Before finally presenting the definition of the Cohen class for density operators following [54, Sec. 8.1., Def. 93] or [28], let us first recall the concept of convolutions. Given Schwartz functions , ∈ S(R 2 ) one defines their convolution via * : which is again in S(R 2 ). In principle this formula extends to general functions, although convergence may become an issue. These extensions are used in Theorem 1 as well as Section 4.3. Now Eq. (17) as well as the fact that for all Ω ∈ R 2 . This definition extends in a natural way to general linear functionals : → C on some subspace ⊆ (R 2 → C), and general functions : for all Ω ∈ R 2 . Defining the convolution via Eq. (18) is consistent with the distributional pairing in the sense that * | * ≡ * , if * |( ) := * , on S(R 2 ). Moreover one readily verifies the identity ( * ) * , = ( ∨ * ) for all ∈ S (R 2 ), , ∈ S(R 2 ). This shows that Eq. (18) is equivalent to other extensions of convolutions commonly found in the literature, e.g., [96, p. 324]. Be aware that * is always a function of slow growth, that is, For unbounded operators , this postulate still makes sense if is has a finite representation in the number state basis, that is, = | | * = ∫ * dΩ, see also Appendix C.

Definition 2
The Cohen class is the set of all linear mappings from density operators to phase-space distributions that are related to the Wigner function via a convolution. More precisely a linear map : , · ∈ S (R 2 ) for all ∈ B 1 ( 2 (R)). Then belongs to the Cohen class if there exists ∈ S (R 2 ) (called "Cohen kernel") such that This is a generalization of the definition commonly found in the literature [54,Def. 93]: there one restricts the domain of from the full trace class to only rankone operators = | | for some , ∈ 2 (R) or even ∈ S(R). As a simple example [54, p. 90] the Wigner function is in the Cohen class: To see this choose = in the above definition: Remark 2 Given some ∈ S (R 2 ) associated to an element of the Cohen class, one formally obtains the symplectic Fourier transform of is generated by a function : R 2 → C via the usual distributional pairing (we will call this "admissible" later, cf. Section 4.1). The reason we make this observation is that this object always exists: it is a product of two classical functions where F [ ] is a bounded and square-integrable function, i.e. |F [ ] (Ω)| = |Tr[D (Ω) ] | ≤ D (Ω) sup 1 = 1 due to unitarity of D (Ω), and ∈ 2 (R 2 ) [54, Proposition 68] so the same holds true for its Fourier transform. Thus-while the expression * may be ill-defined for certain ∈ S (R 2 ), ∈ B 1 ( 2 (R))-going to the Fourier domain yields a well-defined object which can be studied rather easily.

Phase-Space Distribution Functions via Parity Operators
We propose a definition for phase-space distributions and the Cohen class based on parity operators, the explicit form of which will be calculated in Section 4.4. A similar form has already appeared in quantum optics for the so-called -parametrized distribution functions, see, e.g., [21,93]. In particular, an explicit form of a parity operator that requires no integral-transformation appeared in (6.22) of [22], including its eigenvalue decomposition which was later re-derived in the context of measurement probabilities in [93], refer also to [86,97]. Apart from those results, mappings between density operators and their phase-space distribution functions have been established only in terms of integral transformations of expectation values, as in [3,4,21].
For a convolution kernel ∈ (R 2 ), we introduce the corresponding filter kernel where F denotes the symplectic Fourier transform (see Section 2.1). Henceforth we say ∈ S (R 2 ) is admissible if its filter kernel is generated by a function via the More precisely has to be a linear functional on a subspace of , Ω ∈ R 2 . However we will keep things informal by assuming henceforth that all convolutions we encounter are well-defined in the sense of Eq. (18). usual integral form of the distributional pairing , = ( ) ∈ C for ∈ S (R) and ∈ S(R) (see Section 2.1): More precisely is admissible if there exists a function from R 2 to C such that 2 ℏ F ( ) ( ) = * , for ∈ S(R). In this case we call the filter function associated with . Most importantly if the convolution kernel is admissible and itself is generated by a function, i.e. if we consider * , · ∈ (R 2 ) admissible, then Eq. (19) simplifies to for all Ω ∈ R 2 . As before ∨ (Ω) = (−Ω). The technical condition of being admissible is always satisfied in practice (cf . Tables 2 and 3). The advantage of only considering admissible kernels is that the definition of the (generalized) parity operator makes for an obvious generalization of the parity operator from Section 3.2.
For an even more general definition we refer to Remark 12 in Appendix A.

Definition 3
Given any admissible convolution kernel ∈ (R 2 ) with associated filter function we define a parity operator Π on S(R) via This extends to a parity operator on the tempered distributions Π | : The derivation of the extension (22) of Π to tempered distributions is detailed in Appendix A. Displacements of tempered distributions ∈ S (R) are understood via the distributional pairing D (Ω) , = , D † (Ω) and (22) gives rise to a well-defined linear operator Π | from to S (R) acting on ∈ S(R) via The definition of Π is independent of the object it acts on (see Appendix A): Π | , · = Π , · for all ∈ S(R) where , · denotes the functional ↦ → , ∈ S (R). All filter functions used in practice (refer to Tables 2 and 3) obey In this case, Π | is not only compatible with the inner product on 2 (R), but also with the embedding S(R) ↩→ S (R) usually employed in mathematical physics (see Lemma 3 in Appendix A). This motivates us to henceforth write Π both in the case of (21) and instead of Π | in (22). While our definition above is pleasantly intuitive, we have to explicitly consider the domain of the parity operator. For a general (admissible) kernel , one needs to restrict the domain ⊆ S (R) of Π to tempered distributions for which the integral in Eq. (21) exists, as done in Eq. (23) and already hinted at in Remark 1.
Example 2 Domain considerations are illustrated using the standard ordering with Table 2). Given any , ∈ S(R), we have This reproduces known properties as in Eq. (5.39) of [29] (cf. Remark 3); however we emphasize that, although Eq. (25)  , Π for all suitable . In particular, contains all Schwartz functions showing that Π is densely defined. However the functional , · ∈ S (R) fails to be in for most functions : R → C of slow growth including non-zero constant ones such as := 1 ∈ S (R). In particular, Π does not extend to a well-defined operator on 2 (R) as not all square-integrable functions will be contained in .
Following this line of thought, we investigate the well-definedness and boundedness of Π on the Hilbert space 2 (R). As in Example 2, we observe that S(R) ⊆ for all filter functions which is particularly relevant for applications. This follows by interpreting Π as a Weyl quantization (cf. Section 4.3) whereby ↦ → Π is specified as a map from S (R 2 ) to the linear maps between S(R) and S (R) (cf. Chapter 6.3 in [56] or Lemma 14.3.1 in [59]). Consequently, every parity operator has a welldefined matrix representation in the number-state basis (which is a subset of S(R), cf. Section 2.2). The following stronger statement is shown in Appendix B.1: Lemma 2 Given any convolution kernel ∈ S (R 2 ) the following are equivalent: Also the following statements are equivalent: Recalling from Section 3.3, , is the usual cross-Wigner transform given by Let us highlight that condition (ii,b) in Lemma 2 is a known sufficient condition from time-frequency analysis to ensure that a tempered distribution is an element of the Cohen class, cf. Theorem 4.5.1 in [59]. Now the almost magical result of Lemma 2 is that Π being well defined on 2 (R) automatically implies boundedness as long as is admissible. This can also be attributed to the folklore that unbounded operators "cannot be written down explicitly": As the operator Π for admissible kernels is defined via an explicit integral, one gets the boundedness of Π "for free." Indeed the proof that all five statements from the above lemma are equivalent breaks down if one considers not only admissible but arbitrary kernels.
We define a general class of phase-space distribution functions (Ω, ) via the For general , however, this only makes sense if all displaced quantum states D † (Ω) D (Ω) are supported on . We avoid these technicalities by restricting the definition to those filter functions which give rise to operators Π that are bounded on 2 (R) and thereby allow for general .

Definition 4
Given any ∈ S (R 2 ) such that Π ∈ B ( 2 (R)) we define a linear mapping (·, ) on the density operators ∈ B 1 ( 2 (R)) in the form of the quantummechanical expectation value While our definition considers the practically most important case of bounded parity operators, we give a detailed account in Appendix C of the extension of (Ω, ) to arbitrary ∈ S (R 2 ) whereby the associated parity operators may be unbounded. This is of importance for, e.g., the standard and antistandard orderings as shown in Example 2. The prototypical case where these extensions may not apply due to ∉ S (R 2 ) is the case of the Glauber P function which is well known to be singular except for classical thermal states. However, most other convolution kernels appearing in practice are induced by a tempered distribution and thus fit into the framework of either Definition 4 or its extension in Appendix C.
Either way Definition 4 has many conceptual and computational advantages as we have detailed in the introduction. To further clarify the scope of said definition we now-similarly to the proof of If the convolution kernel ∈ S (R 2 ) is additionally admissible-meaning it is the reflected symplectic Fourier transform = (2 ℏ) −1 F * | of its filter functionthen in analogy to (15) one finds The proof of Theorem 1 is given in Appendix B.2. The construction of a particular class of phase-space distribution functions was detailed in [4], where the term "filter function" also appeared in the context of mapping operators. However, these filter functions were restricted to non-zero, analytic functions. Definition 4 extends these cases to the Cohen class via Theorem 1 which allows for more general phase spaces. For example, the filter function of the Born-Jordan distribution has zeros (see Theorem 3 below), and is therefore not covered by [4]. Most of the well-known distribution functions are elements of the Cohen class. We calculate important special cases in Sec. 4.4. The Born-Jordan distribution and its parity operator are detailed in Sec. 5.
Our approach to define phase-space distribution functions using displaced parity operators also nicely fits with the characteristic [21,28,83] as given in Eq. (5.2) of [29] translates into the definition (30) with the parity operator. Both Eqs. (29) and (30) need to respect domain restrictions as discussed in Example 2 and neither equation is well defined for tempered distributions in (R) or squareintegrable functions in 2 (R) that are not contained in the domain .

Common Properties of Phase-Space Distribution Functions
We detail now important properties of (Ω, ) and their relation to properties of (Ω) and Π . These properties will guide our discussion of parity operators and this allows us to compare the Born-Jordan distribution to other phase spaces. Table 1 provides a summary of these properties and the proofs have been deferred to Appendix D. Recall that we are dealing exclusively with convolution kernels ∈ S (R 2 ) which give rise to bounded operators Π so the induced phase-space distribution (Ω, ) is well defined everywhere. The filter function in [29] agrees with our (− , ) up to substituting − with and switching arguments, which is usually immaterial as (− , ) = ( , ) for all filter functions seen in practice.
Proposition 1 Let , ∈ S (R 2 ) be given such that is admissible and the parity operator Π from Definition 3 is in B ( 2 (R)). If is generated by a phase-space function : R 2 → C (i.e. ≡ * , · ) and if ∨ * , · ∈ S (R 2 ), then in analogy to (33) as quadratic forms on S(R).
Proof The Plancherel formula The second equality follows from (21) and it specifies the parity operator as a phasespace average of quantizations of single Fourier components. But it is even more instructive to consider the case of the delta distribution (2) , the Weyl quantization of which yields the Grossmann-Royer parity operator Op Weyl ( (2) ) = ℏ Π (as obtained in [61]). Applying (34), the Cohen quantization of the delta distribution yields the parity operator from (21). In particular, the operator Π from Definition 3 is a -type quantization of the delta distribution as or equivalently, the Weyl quantization of the Cohen kernel, up to coordinate reflection. Since Π is the Weyl quantization of the tempered distribution ∨ ∈ S (R 2 ), one can adapt results contained in [37] to precisely state conditions on , for which bounded operators Π are obtained via their Weyl quantizations, refer also to Property 1.

Explicit Form of Parity Operators
Expectation values of displaced parity operators are obtained via the kernel function in (37) and recover well-known phase-space distribution functions for particular cases of or , which are motivated by the ordering This family of phase-space representations is related to the one considered in [3,4] by setting = /2 and = − = 2 − 1/4. schemes Op ( Ω 0 ) from Table 2. Important special cases of these distribution functions and their corresponding filter functions and Cohen kernels are summarized in Table 3.
In particular, the parameters = 1/2 and = 0 identify the Wigner function with 1/2,0 (Ω) ≡ 1 and (38) reduces to (14). Note that the corresponding Cohen kernel from Theorem 1 is the 2-dimensional delta distribution (2) (Ω) and that convolving with (2) (Ω) is the identity operation, i.e. (2) * = [see (27)]. The filter function from (37) results for a fixed parameter of = 1/2 in the Gaussian (Ω) : . The corresponding parity operators are diagonal in the number-state representation (refer to Property 4), and they can be specified for −1 ≤ < 1 in terms of number-state projectors [21,93,97] as where the second equality specifies Π in the form of a spectral decomposition. This form has implicitly appeared in, e.g., [21,93,97]. We provide a more compact proof in Appendix E. Equation (39) readily implies Π sup = (1− ) −1 for ≤ 0, and for < 0 one even finds that Π are trace-class operators due to Note that for > 0 the corresponding filter functions lie outside of our framework as then ∉ S (R 2 ) due to its superexponential growth. While one can still formally write down their distribution functions, one runs into convergence problems resulting in singularities. However, their symplectic Fourier transform always exists and it is related to the Wigner function via (Ω)F [ ] (Ω) by multiplying with the filter function (Ω) (cf. Remark 2). This class of -parametrized phase-space representations has gained widespread applications in quantum optics and beyond [31,53,89,106,117], and they correspond to Gaussian convolved Wigner functions for < 0 such as the Husimi Q function for = −1. Note that the Cohen kernel via Theorem 1 corresponds to the vacuum state |0 (Ω, +1) of a quantum harmonic oscillator [21]. Gaussian deconvolutions of the Wigner function are formally obtained for > 0, which includes the Glauber P function for = 1 [21]. Due to the rotational symmetry of its filter function (Ω), the -parametrized distribution functions are covariant under phase-space rotations, refer to Property 4. Another important special case is obtained for a fixed parameter of = 0, which results in Shubin's -distribution, refer to [15][16][17]56]. Its filter function from (37) reduces to the chirp function 0 (Ω) =: (Ω) = exp [ (2 −1) /(2ℏ)] while relying on the parametrization with and . The resulting distribution functions (Ω, ) are in the Cohen class due to Theorem 1 and they are square integrable following Property 2 as the absolute value of (Ω) is bounded. We calculate the explicit action of the corresponding parity operator Π .

Theorem 2
The action of the -parametrized parity operator Π := Π 0 on some coordinate representation ( ) ∈ 2 (R) is explicitly given for any ≠ 1 by which for the special case = 1/2 reduces (as expected) to the usual parity operator Π. It follows that Π is bounded for every 0 < < 1 (or in general for every real that is not equal to 0 or 1) and its operator norm is given by Π sup = 1/ √︁ 4( − 2 ).
Proof By (38), the parity operator Π acts on the coordinate representation ( ) via This integral can be evaluated using the explicit form of 0 (Ω) form (37) and the action of D on coordinate representations ( ) from (10) yields where the change of variables = + ( −1) 0 with 0 = ( − )/( −1) and d = | −1| d 0 was used. Therefore, the right-hand side is
In the following, we will rely on this -parametrized family to construct and analyze the parity operator of the Born-Jordan distribution.

Parity-Operator Description of the Born-Jordan Distribution
The Born-Jordan distribution (Ω, BJ) is an element of the Cohen class [16,27,56] and is obtained by averaging over the -distributions (Ω, ) ∈ 2 (R 2 ): (Ω, BJ) := As in Definition 4, this distribution function is also obtained via the expectation value of a parity operator.

Theorem 3 The Born-Jordan distribution (Ω, BJ) of a density operator ∈ B 1 (H ) is an element of the Cohen class, and it is obtained as the expectation value
of the (displaced) parity operator Π BJ that is defined by the relation and the corresponding parity operator can be expanded as Using the explicit form of 0 (Ω) from (37), the evaluation of the integral over concludes the proof.

Proposition 2 The Born-Jordan parity operator Π BJ is bounded as an upper bound of its operator norm is given by Π BJ sup ≤ /2.
Proof Using the Π -representation of Π BJ we compute for arbitrary ( ) ∈ 2 (R) where in the second-to-last step we used Theorem 2.
It is well-known that the Born-Jordan distribution is related to the Wigner function via a convolution with the Cohen kernel BJ , refer to [54,56]. However, calculating this kernel, or the corresponding parity operator directly might prove difficult. In the following, we establish a more convenient representation of the Born-Jordan parity operator which is an "average" of Π from Theorem 2 via the formal integral transformation which-as in Sect. 3.2-is interpreted as Π BJ ( ) = ∫ 1 0 Π ( ) d for all ( ) ∈ 2 (R). Recall that the parity operator Π is well defined and bounded for every 0 < < 1.
Theorem 4 Let us consider the squeezing operator ( ) which depends on the real squeezing parameter ∈ R. The Born-Jordan parity operator is a composition of the reflection operator Π followed by a squeezing operator (and the two operations commute), and this expression is integrated with respect to a well-behaved weight function sech( /2) = 2/( /2 + − /2 ). Note that the function sech( /2) ∈ S(R) is fast decreasing and invariant under the Fourier transform (e.g., as Hermite polynomials).
Proof The explicit action of Π BJ on a coordinate representation ( ) ∈ 2 (R) is given by (see Theorem 2) Let us recognize that (− ) = − /2 ( ) Π ( ) is the composition of a coordinate reflection and a squeezing of the pure state ( ); also the two operations commute. This results in the explicit action The expression for the parity operator in Theorem 4 is very instructive when compared to Theorem 3, and this confirms that the parity operator Π BJ decomposes into the usual parity operator Π followed by a geometric transformation, refer also to Section 5.3. In the case of the Born-Jordan parity operator this geometric transformation is an average of squeezing operators.

Spectral Decomposition of the Born-Jordan Parity Operator
We will now adapt results for generalized spectral decompositions, refer to [25,26,50,90]. This will allow us to solve the generalized eigenvalue equation for parity operators and to determine their spectral decompositions.
Recall the distributional pairing for smooth, well-behaved functions ( ) ∈ S(R) in Section 2.1 with respect to tempered distributions ∈ S (R) (such as functions of slow growth ( )). We will use this distributional pairing to construct 2 scalar products of the form , = ∫ R * ( ) ( ) d , which corresponds to a rigged Hilbert space [25,50] or the Gelfand triple S(R) ⊂ 2 (R) ⊂ S (R). This rigged Hilbert space allows us to specify the generalized spectral decomposition of the Born-Jordan parity operator with generalized eigenvectors in S (R) as functions of slow growth. It was shown in the previous section that the Born-Jordan parity operator Π BJ is a composition of a coordinate reflection and a squeezing operator. We now recapitulate results on the spectral decomposition of the squeezing operator from [18,25,26], up to minor modifications. Recall that the squeezing operator forms a unitary, strongly continuous one-parameter group ( ) = − with ∈ R that is generated by the (unbounded) self-adjoint Hamiltonian This Hamiltonian admits a purely continuous spectrum ∈ R, and satisfies generalized eigenvalue equations for every ∈ S(R), where the last equation is equivalent to | ± = | ± . The Gelfand-Maurin spectral theorem [25,50,90] results in a spectral resolution of Here the generalized eigenvectors are specified in terms of their coordinate representations as slowly increasing functions, i.e. ± ( ) := | ± ∈ (R) with refer to [25,26,50,90] and Appendix F for more details. Note that ± ( ) are generalized eigenfunctions: they are not square integrable, but the integral ∫ R [ ± ( )] * ( ) d exists as a distributional pairing for every ∈ S(R). Also note that these generalized eigenvectors can be decomposed into the number-state basis with finite expansion coefficients that decrease to zero for large , refer to Appendix F. The spectral decomposition of the squeezing operator is then given by refer to Eq. 6.12 in [25] and Eq. 2.14 in [26]. Note that these eigenvectors are also invariant under the Fourier transform (e.g., as Hermite polynomials). It immediately follows that the squeezing operator satisfies the generalized eigenvalue equation which can be easily verified using the explicit action ( ) ± ( ) = /2 ± ( ) = − ± ( ). One can now specify the Born-Jordan parity operator using its spectral decomposition.
Proof The generalized eigenvalues can be computed via where Π | ± = ±| ± . Using (48), one obtains Remark 7 Recall that Π BJ is a bounded (by Proposition 2) and self-adjoint operator. Consequently, the usual spectral theorem in multiplication operator form [63,Thm. 7.20] yields a -finite measure space ( , ), a bounded, measurable, real-valued function ℎ on , and unitary : 2 (R) → 2 ( , ) such that for all ∈ 2 ( , ) and ∈ . While this undoubtedly is a nice representation, the spectral decomposition in Theorem 5 is more readily determined with the help of the Gelfand-Maurin spectral theorem [25,50,90]. In particular, said theorem lets us directly work with the generalized eigenfunctions in Eq. (47), even though they are not square integrable.

Geometric Interpretation of Parity Operators
While above we have comprehensively explored analytic properties of the Born-Jordan and other practically important parity operators, here we relate these mathematical objects to geometric transformations. Even the rather complex Born-Jordan parity operator admits a surprisingly simple decomposition into two elementary geometric transformations. Equation (46) decomposes the Born-Jordan parity operator into an ordinary reflection of the wave function's coordinate followed by a weighted average of squeezing operations as As such, the action on any wave function ( ) ∈ 2 (R) can be summarized as the reflected, squeezed function ( )Π ( ) = /2 (− ) averaged over all parameters ∈ (−∞, ∞) with respect to the rapidly decaying weight function sech( /2).
It is not only the Born-Jordan parity operator that admits a simple geometric interpretation, but it rather seems to hint at a universal property, at least in the classes of practically important phase-space representations. In particular, we now state that both Π and the pivotal parity operator Π , which contains the most popular variants of Wigner, Husimi and Glauber P phase-space functions as special cases, can be decomposed into elementary geometric transformations. Consequently, the parity operator Π admits a spectral decomposition where ± |Π = ± ± | has been used.

Explicit Matrix Representation of the Born-Jordan Parity Operator
Recall that the -parametrized parity operators Π are diagonal in the Fock basis and their diagonal entries can be computed using the simple expression in (39). This enables the experimental reconstruction of distribution functions from photon-count statistics [7,13,41,88] in quantum optics.

Remark 10
The Born-Jordan parity operator Π BJ is not diagonal in the numberstate basis, as its filter function BJ (Ω) is not invariant under arbitrary phase-space rotations, refer to Property 4. The filter function BJ (Ω) is, however, invariant under /2 rotations in phase space, and therefore only every fourth off-diagonal is non-zero.
We now discuss the number-state representation of the parity operator Π BJ , which provides a convenient way to calculate (or, more precisely, approximate) Born-Jordan distributions.

Theorem 6 The matrix elements [Π BJ ] := |Π BJ of the Born-Jordan parity operator in the Fock basis can be calculated in the form of a finite sum
Here, Φ denotes the th and th partial derivatives of the function ( , ) =

arcsinh[1/
√ ] with respect to its variables and , respectively, evaluated at = , then differentiated again times and finally its variable is set to = 1.
Refer to Appendix G for a proof. The derivatives in (51) can be calculated in the form of a finite sum where + + ≥ 1 and are recursively defined integers, refer to (67) in Appendix H. Substituting ℓ for 2ℓ in (49), the matrix elements [Π BJ ] then depend only on these integers via the finite sum +4, +8, . . . } and = ( − −2ℓ)/2. Figure 1(a) shows the first 8 × 8 entries of [Π BJ ]. One observes the following structure: only every fourth off-diagonal is non-zero, the matrix is real and symmetric, and the entries along every diagonal and off-diagonal decrease in their absolute value. In particular, the diagonal elements of Π BJ admit the following special property.
In particular, [Π BJ ] → 0 as → ∞. For a proof we refer to Appendix I.
Also note that the sum of these decreasing diagonal entries results in a trace Tr[Π BJ ] = 1/2 (Property 6) in the number-state basis. However, this trace does not necessarily exist in an arbitrary basis, as Π BJ is not a trace-class operator.

Remark 11
Let us emphasize that the boundedness of Π BJ (Proposition 2) guarantees that using a (large enough) finite block of Π BJ for computations yields a good approximation. The reason for this is that such a block does not differ too much (in trace norm) from the full operator, which readily transfers to the phase-space distribution function. Details can be found at the end of Appendix C.
In the following, we specify a more convenient form for the calculation of these matrix elements, i.e. a direct recursion without summation, which is based on the following conjecture (see Appendix J).

Conjecture 1 The non-zero matrix elements
of the Born-Jordan parity operator are determined by a set of rational numbers ℓ where Γ ℓ = 2 −2ℓ+1/2 √︁ !/( +4ℓ)! and , ℓ ∈ {0, 1, 2, . . . }. The indexing is specified relative to the diagonal (where ℓ = 0) and ℓ is the Kronecker delta. The rational numbers ℓ can be calculated recursively using only 8 numbers as initial conditions, refer to Appendix J for details. This form does not require a summation.   Figure 1(a). The direct recursion in Conjecture 1 enables us to conveniently and efficiently calculate the matrix elements [Π BJ ] and we have verified the correctness of this approach for up to 6400 matrix elements, i.e. by calculating a matrix representation of size 80 × 80. This facilitates an efficient calculation and plotting of Born-Jordan distributions for harmonic oscillator systems, such as in quantum optics [53,83,89]. Note that a recursively calculated 80 × 80 matrix representation, which we have verified with exact calculations, is sufficient for most physical applications, i.e. Figures 2 and  3 (below) were calculated using 30 × 30 matrix representations. However, a matrix representation of size 2000 × 2000 can be easily calculated on a current notebook computer using the recursive method. Numerical evidence shows that the matrix representation of Π BJ can be well-approximated by a low-rank matrix, i.e, diagonalizing the matrix Π BJ reveals only very few significant eigenvalues. In particular, the sum of squares of the first 9 eigenvalues corresponds to approximately 99.97% of the sum of squares of all the eigenvalues of a 2000 × 2000 matrix representation.

Example Quantum States
Matrix representations of parity operators are used to conveniently calculate phasespace representations for bosonic quantum states via their associated Laguerre polynomial decompositions. The -parametrized distribution functions of Fock states | The Born-Jordan distribution of coherent states, i.e. the displaced vacuum states, closely matches the Wigner functions, see Fig. 2(a). The first part in Eq. (55) contains the diagonal elements of the parity operator which correspond to the radially symmetric part of | (Ω, BJ), see Fig. 3(b) (left). The second part in Eq. (56) results in a radially non-symmetric function, see Fig. 3(b) (right). The radially symmetric parts are quite similar to the Wigner function and have + 1 wave fronts enclosed by the Bohr-Sommerfeld band [43,83], i.e. the ring with radius √ 2 +1. The radially non-symmetric functions have + 1 local maxima along the outer squares, i.e. along phase-space cuts at the Bohr-Sommerfeld distance , ∝ √ 2 +1. The sum of these two contributions is the Born-Jordan distribution and it is not radially symmetric for number states, see Fig. 3(a).

Conclusion
We have introduced parity operators Π which give rise to a rich family of phasespace distribution functions of quantum states. These phase-space functions have been previously defined in terms of convolutions, integral transformations, or Fourier transformations. Our approach using parity operators is both conceptually and computationally advantageous and now allows for a direct calculation of phase-space functions as quantum-mechanical expectation values. This approach therefore averts the necessity for the repeated and expensive computation of Fourier transformations. We motivate the name "parity operator" by the fact that parity operators Π = • Π are composed of the usual parity operator and some specific geometric or physical transformation. We detailed the explicit form of parity operators for various phase spaces and, in particular, for the Born-Jordan distribution. We have also obtained a generalized spectral decomposition of the Born-Jordan parity operator, proved its boundedness, and explicitly calculated its matrix representation in the number-state basis. We conjecture that these matrix elements are determined by a proposed recursive scheme which allows for an efficient computation of Born-Jordan distribution functions. Moreover, large matrix representations of the Born-Jordan parity operator can be well approximated using rank-9 matrices. All this will be useful to connect our results with applications in (e.g.) quantum optics, where techniques such as squeezing operators and the number-state representation are widely used.

A Extension of Operators from Schwartz Functions to Distributions
Assume we have a linear operator : S (R) → S (R) or, more generally, : S (R) → (R → C) and we want to extend its action to tempered distributions. Usually this is done by introducing some operator : S (R) → (R → C) (which can-but does not have to-be the same as ) such that Having introduced the concept of operator extensions we may apply it to generalized parity operators. But first let us generalize Definition 3 to arbitrary tempered distributions , even though this is beyond what we need in the main sections of this article. (21)  for all ∈ S (R), ∈ R. Here D ( ) ∈ S (R 2 ) defined via Ω ↦ → ( D (Ω) ) ( ) is the displacement of at as a function of Ω. One readily verifies that for admissible kernels this definition reproduces (24) as well as the definition of the parity operator in (14). Similar to Example 3(1), let us extend Π with respect to the embedding : S (R) ↩→ S (R), that is, we have to find an operatorΠ : S (R) → (R → C) such that ( ) (Π ) = (Π ) ( ) for all , ∈ S (R). We claim thatΠ

Remark 12 Formally
Together with the linearity of the integral as well as the linearity and continuity of this implies for all , ∈ S (R). Thus by setting = Π andˆ=Π in (57) withΠ from (58), that is, In general these two extensions will be different so one has to be careful about which framework one uses. However from the explicit form ofΠ one knows that for any ∈ S (R 2 ) these extensions coincide if and only if ∧ ≡ * • • * . This translates to filter functions as follows:

Lemma 3 Consider any admissible
∈ S (R 2 ) with associated filter function : R 2 → C. The extension of Π with respect to coincides with the extension of Π with respect to · | if and only if * ≡ ∧ . In this case (57) becomes , and all ∈ S (R).
We emphasize that all filter functions used in practice satisfy * ≡ ∧ (cf . Tables 2 and 3), so it does not matter for applications whether one extends Π with respect to or · |.

B Proofs of Lemma 2 and Theorem 1
Before we dive into the proofs of the results in question we first need a lemma which relates convolutions of the cross-Wigner transform with matrix elements of the generalized Grossmann-Royer operator.

D.3 Proof of Property 3
As in (7) This is conveniently shown in the coherent-state representation as detailed in Eq. (8).

E Proof of (39)
Due to Property 4, the parity operator is diagonal in the number-state representation |Π | ∝ . Its diagonal elements can be calculated where (9) was used for [ D (Ω) ] . One applies the polar parametrization of the complex plane via Then where the second equality is due to d = d /2 with = 2 and the integral with respect to results in the multiplication by 2 . The Laguerre polynomial decomposition of the exponential function

F Spectral Decomposition of the Squeezing Operator
The eigenvectors from (47) are orthogonal and normalized in terms of the delta function as detailed by The integral can be calculated using a change of variables d = d with = ln( | |). One obtains a complete basis by applying an integral of two different Fourier components indexed by and , refer to [25,26] for more details. The eigenfunctions ± ( ) are not square integrable, but they can be decomposed into the number-state basis with finite coefficients. The coefficients shrink to zero, but are not square summable. The resulting integrals | ± can be specified in terms of a finite sum. In particular, 0 + = | | −1/2 /(2 √ ) has the largest eigenvalue. Its number-state representation is given by where every fourth entry is non-zero and the entries decrease to zero for large .

G Matrix Representation of the Born-Jordan Parity Operator
The matrix elements of the parity operator can be computed via Theorem 3 as It was discussed in Section 3.1 that one can substitute = ( + −1 )/ √ 2ℏ, which results in the integral Let us now apply a change of variables ↦ → −1 √ ℏ and ↦ → √ ℏ which yields d d ↦ → ℏd d and the integral We now substitute the explicit form of (9) and obtain The integral in (64)  Note that now denotes the variable of the function ( , ) and should not be confused with the scaling parameter = ( + −1 )/ √ 2ℏ from Section 3.1, which has also been used in the beginning of this section. This finally results in can be computed recursively. Note that, obviously, is smooth. The inner derivatives of Φ gives rise to the following lemma. Proof Note that the symmetry of the holds due to Schwarz's theorem [100, pp. 235-236] as is smooth. Then this statement is readily verified via induction over = + . First, = 1 corresponds to = 1, = 0 so arcsinh[ ( ) −1/2 ] = 1/(−2 √︁ +1) which reproduces (65). For ↦ → + 1 it is enough to consider ( , ) ↦ → ( +1, ) due to the stated symmetry. The key result here is that which is readily verified. Straightforward calculations conclude the proof.
For + ≥ 1, the above result immediately yields Now the are used to initialize the recursion of the coefficients for + ≥ 1, the sum of which determines the resulting derivatives as we will see now.
where the coefficients have the symmetry = and are defined by Proof The key result here is for any , ∈ N which can be easily seen. We have to distinguish the cases + = 0 and + ≥ 1. which recovers the recursion formula of for = 0 and = 0. Now assume + ≥ 1 such that we can carry out the proof via induction over ∈ N 0 (where = 0 is obvious as it is simply Lemma 5). Using (68) in the inductive step for = + + recovers the recursion formula of the by straightforward computations.

I Proof of Proposition 3
The proof which is given below was informed by a discussion on MathOverflow [44] and its idea was provided GH and M. Alekseyev. We consider the generating function of the entries [Π BJ ] . In total, (71) then converges due to Dirichlet's test [64, p. 328].
With these intermediate results we can finally prove the proposition in question.
Proof Again using the generalized Leibniz rule, Lemma 7 yields that