Overview of the Phase Space Formulation of Quantum Mechanics with Application to Quantum Technologies

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www.advquantumtech.com Overview of the Phase Space Formulation of Quantum Mechanics with Application to Quantum Technologies

Russell P. Rundle* and Mark J. Everitt

taking a quantum wavefunction and repre- The phase-space formulation of quantum mechanics has recently seen senting it in phase space—the Wigner func- increased use in testing quantum technologies, including methods of tion. tomography for state verification and device validation. Here, an overview of Subsequently, in the 1940s, both [3] [4] quantum mechanics in phase space is presented. The formulation to generate Gronewold and Moyal then developed Wigner’s phase-space function to fit a generalized phase-space function for any arbitrary quantum system is around the earlier language Weyl used to Q shown, such as the Wigner and Weyl functions along with the associated represent quantum mechanics in phase and P functions. Examples of how these different formulations are used in space. They did this by developing tools to quantum technologies are provided, with a focus on discrete quantum transform any arbitrary operator into the systems, qubits in particular. Also provided are some results that, to the phase-space representation. This is done by authors’ knowledge, have not been published elsewhere. These results taking the expectation value of the so-called kernel. Further they described the evolu- provide insight into the relation between different representations of phase tion of quantum mechanics in phase space. space and how the phase-space representation is a powerful tool in This created a complete theory of quantum understanding quantum information and quantum technologies. mechanics as a statistical theory, as long as one only wished to consider systems of position and momentum. The phase-space representation was missing an important aspect of quantum me- 1. Introduction chanics, the representation of finite quantum systems. Using the language of Weyl, Groenewold, and Moyal, in 1956 Stratonovich The phase-space formulation is just one of many approaches then introduced the representation of discrete quantum systems to consider quantum mechanics, where the well-known into phase-space methods.[5] Schrödinger wave function and Heisenberg matrix formula- This development went largely unnoticed, while the phase- tions of quantum mechanics were the first to be devised in space formulation of systems of position and momentum started 1925. During the same decade, Hermann Weyl realized that the to pick up popularity in the quantum optics community, where structure of quantum mechanics closely followed the rules of additions to the Wigner function were introduced, the Husimi group theory; from this realization he presented the, now called, Q-function[6] and the Glauber-Sudarshan P-function.[7,8] The de- Weyl transform, that takes a Hamiltonian in phase space and velopment of the phase-space representation by Glauber in par- transforms it into a quantum mechanical operator.[1] It was not ticular led to important contributions. First the development of until 1932 that the inverse transform was presented by Wigner,[2] the displacement operator, and then the relation of this displacement operator to the generation of phase-space functions.[9,10] The above mentioned kernel that maps an arbitrary operator Dr.R.P.Rundle to phase space was realized then discovered to be a displaced School of Mathematics parity operator, the wording came from Royer,[11] where the Q University of Bristol Fry Building, Bristol BS8 1UG, UK and P functions have a similar construction, albeit with a dif- E-mail: [email protected] ferent “parity”. The displaced parity construction is an impor- Dr.M.J.Everitt tant discovery for much of this review and in the development Quantum Systems Engineering Research Group of phase-space methods in experimental settings and quantum Department of Physics technologies in general. However, the displaced parity formal- Loughborough University ism was not used for discrete quantum systems until more Leicestershire LE11 3TU, UK recently.[12,13] The ORCID identification number(s) for the author(s) of this article In the late 1980s, interest in representing qubits, or even qu- can be found under https://doi.org/10.1002/qute.202100016 dits, in phase space started to reemerge. In 1987, Feynman[14] © 2021 The Authors. Advanced Quantum Technologies published by and Wootters[15] both independently constructed a Wigner func- Wiley-VCH GmbH. This is an open access article under the terms of the tion for discrete systems that differs from the earlier construction Creative Commons Attribution License, which permits use, distribution from Stratonovich; however, a relation between these construc- and reproduction in any medium, provided the original work is properly tions will be shown later. Feynman’s work was a writing up of an cited. earlier, back-of-the-envolope musings about negative probabili- DOI: 10.1002/qute.202100016 ties, and only focuses on spin-1/2 systems. Wootters on the other

Adv. Quantum Technol. 2021, 2100016 2100016 (1 of 29) © 2021 The Authors. Advanced Quantum Technologies published by Wiley-VCH GmbH www.advancedsciencenews.com www.advquantumtech.com hand, was more interested in creating a Wigner function for any where we will provide the general formula to create any quasi- discrete system, where he created a Wigner function based on a probability distribution function and any characteristic function. discrete toroidal lattice that holds for dimension d that is a prime This will be followed by looking into the dynamics of such states power, due to the underlying Galois field structure. and how representing quantum mechanics in phase space allows Since this initial construction of a discrete phase space, there one to see to what extent one can treat dynamics classically. This has been effort to extend this formulation to be useful more section will end with a general framework and a summery of the generally (see refs. [16–22] for an overview of this progress). This results in ref. [44]. also resulted in some key discoveries for finite systems in phase space, such as considering coupled qubits,[23,24] qudits,[25] and the link between Wigner function negativity and contextuality.[25–30] 2.1. Systems of Position and Momentum Shortly after Feynman and Wootters presented their find- ings, Várilly and Gracia-Bondía came across Stratonovich’s early Wigner’s original formulation was generated in terms of a wave- 𝜓 work and brought it to public attention in 1989.[31] Filling in function, (q), and is in essence the Fourier transform of a corre- some of the earlier gaps and also presenting the coherent-state lation function, where for a function defined in terms of position construction[32–34] to relate it more closely with Glauber’s earlier and momentum work. This was then followed by further development.[12,13,35–41] ∞ ( ) ( ) = 𝜓 ∗ + z 𝜓 − z ipz These developments focussed on an SU(2) structure of phase W𝜓 (q, p) q q e dz (1) ∫−∞ 2 2 space, restricting larger Hilbert spaces to the symmetric subspace. Alternatively, there has been development in describing where we will assume that ℏ = 1. We note that in the original how larger systems in SU(N), although requiring many more de- construction, the Wigner function was presented to describe the grees of freedom, can fully describe a quantum state. The devel- position-momentum phase space for many particles, here we will opment of SU(N) Wigner functions can be found in refs. [42–46]. restrict the discussion to one concomitant pair of degrees of free- As quantum technologies advanced into the twenty-first cen- dom where we will discuss the composite case later in this sec- tury, direct measurement of phase space became increasingly tion. more possible and useful. Experiments then utilized the dis- This formulation of the Wigner function was then soon gen- placed parity formulation to displace a state and make a direct eralized to apply to any arbitrary operator, A, as an expectation parity measurement experimentally; for example, the measure- value of a kernel, Π(𝛼), such that[3,4] ment of optical states in QED experiments,[47] and the rotation of [ ] 𝛼 = Π 𝛼 qubit before taking a generalized parity measurement in ref. [48]. WA( ) Tr A ( ) (2) More examples of such experiments are given within this review. where we now define the√ Wigner function in terms of the com- In this review, we begin by presenting the formulation of plex variable 𝛼 = (q + ip)∕ 2 to fit inline with the quantum op- quasiprobability distribution functions for systems of position tics literature. and momentum, where we will then generalize this formulation The kernel in Equation (2) is fully constructed as a displaced for use in any arbitrary system in Section 2. This will provide parity operator an understanding of the general framework needed to understand how to map a given operator to its representation in phase Π(𝛼) = D (𝛼) Π D †(𝛼)(3) space, and how such a mapping is bijective and informationally complete. Following this in Section 3, we will use this gener- which consists of the modified parity operator alized formulation to introduce the phase-space formulation of Π= i𝜋a†a various finite-dimensional quantum systems. This will include 2e (4) an in-depth discussion on different representations of qubits in † which is two-times the standard parity operator, where a and a phase space and how different representations are related; we will are the annihilation and creation operators, respectively. The sec- also present multiple ways one can represent multi-qubit states ond component of Equation (3) is the displacement operator in phase space dependent on different group structures. We will ( ) then describe how these methods can be used in practice for use † ∗ − 1 |𝛼|2 𝛼a† −𝛼∗a 1 |𝛼|2 −𝛼∗a 𝛼a† D (𝛼) = exp 𝛼a − 𝛼 a = e 2 e e = e 2 e e (5) in quantum technologies in Section 4; where we will translate some important figures of merit for quantum computation into The displacement operator displaces any arbitrary state around the language of phase space, as well discussing the importance of phase space and is central to the definition of a coherent negative values in the Wigner function. This will be followed by state,[7,33,49] which is generated by displacing the vacuum state, examples of experiments performed on qubits where the phase |0⟩, such that space distributions were directly measured. D (𝛼)|0⟩ = |𝛼⟩ (6) 2. Formulation of Phase Space where |𝛼⟩ is an arbitrary coherent state. The coherent states form In this section we will lay the groundwork for how one can gener- an overcomplete basis, where the resolution of identity requires ate a phase-space distribution for any arbitrary operator, where we integration over the full phase space will provide a general framework for any group structure in Sec- ∞ tion 2.3. We will begin with the original formulation in position- 1 |𝛼⟩⟨𝛼| d2𝛼 = 𝟙 (7) momentum space, within the Heisenberg–Weyl group (HW) 𝜋 ∫−∞

Adv. Quantum Technol. 2021, 2100016 2100016 (2 of 29) © 2021 The Authors. Advanced Quantum Technologies published by Wiley-VCH GmbH www.advancedsciencenews.com www.advquantumtech.com and the overlap between any two coherent states is non-zero, function—this is the usual definition of the P function where ∞ ∞ ( ) † [ ( )] A = P (𝛼) |𝛼⟩⟨𝛼| d𝛼 = P (𝛼) D (𝛼)|0⟩⟨0| D (𝛼) d𝛼 1 ∗ ∫ A ∫ A ⟨𝛼 |𝛼 ⟩ = exp |𝛼 |2 + |𝛼 |2 − 2𝛼 𝛼 (8) −∞ −∞ 1 2 2 1 2 1 1 (12) Note that there is a factor of 2 in Equation (4), and subsequently 𝛼 in Equation (3), this is chosen for two reasons. First, it aligns where PA( ) is the P function for an arbitrary operator, A.TheP with the formulation of the Wigner function that was introduced function seems to sacrifice more, as it does not have the correct by Glauber when writing it in a displaced parity formalism. But marginals and it is not non-negative; in fact coherent states in more importantly, it lines up more nicely with the definition of the P functions are singular, which may be an attractive analogy the Wigner function for discrete systems. Later, we will introduce when considering coherent states as so-called “classical stats”, al- a condition that requires the trace of the kernel to be one, which though we must note that coherent states are still fundamentally requires Equation (4) to have the factor of 2 out the front to hold. quantum. Despite the apparent drawbacks, there have been a set The Wigner function is just one of a whole class of potential of key results by the analysis of the P function (see, e.g., ref. [50]— quasi-probability distribution functions for quantum systems, this will also be discussed further in Section 4. what makes it a preferable choice is that the kernel to transform to Apart from the Wigner, Q,andP functions, there are fur- the Wigner function is the same as the kernel used to transform ther choices of quasi-probability distribution function for quan- back to the operator, which is not always the case, such that tum systems. In order to completely describe the full range of possible phase-space functions, including the appropriate ker- ∞ = 1 𝛼 Π 𝛼 𝛼 nels to transform both to and from the given function, it is first A WA( ) ( )d (9) 𝜋 ∫−∞ necessary to introduce the quantum mechanical characteristic functions. The characteristic function is a useful tool in math- showing that the Wigner function is an informationally complete ematical analysis, and is the Fourier transform of the probability transformation of any arbitrary operator. Another important ad- distribution function. In the same way, the characteristic func- vantage is that the marginals of the Wigner function are the prob- tions in quantum mechanics are Fourier transforms of the quasi- ability distribution of the wavefunction. That is for the Wigner probability distribution functions. The Fourier transform of the function of a state 𝜓(q) Wigner function is also known as the Weyl function and is defined ∞ |𝜓 |2 = [ ] (q) ∫ W(q, p)dp (10) 𝜒 (0) 𝛼̃ = 𝛼̃ −∞ A ( ) Tr AD( ) (13) integrating over all the values of p to produce the probability for such that 𝜓 𝜓 (q), likewise for (p) when integrating over q. ∞ (0) 1 ∗ ∗ To represent quantum mechanics in phase space, some prop- 𝜒 𝛼̃ = 𝛼 (𝛼̃𝛼 − 𝛼 𝛼̃) 𝛼 A ( ) ∫ WA( )exp d , and erties one would usually desire from a probability distribution 𝜋 −∞ must be sacrificed—hence the prefix “quasi” in quasi-probability ∞ 1 (0) ∗ ∗ distribution functions. For the Wigner function, this sacrifice is W (𝛼) = 𝜒 (𝛼̃)exp(𝛼𝛼̃ − 𝛼̃ 𝛼) d𝛼̃ (14) A 𝜋 ∫ A on positive-definiteness, a more in-depth conversation on the −∞ negativity of the Wigner function will be given in Section 4.2. If relate the Wigner and Weyl functions. however one does not want to sacrifice a non-negative probability The superscript in brackets (0) in Equation (13) is a specific distribution function, the alternative is to consider the Husimi Q case of the characteristic function, 𝜒 (s)(𝛼), for −1 ≤ s ≤ 1; the function,[6] where instead the sacrifice comes in the form of los- A Weyl function is just the special case when s = 0. In general, a ing the correct marginals. quantum characteristic function is defined The Q function is generated in terms of coherent states, [ ] where (s) 1 s|𝛼̃|2 𝜒 (𝛼̃) = Tr AD(𝛼̃) , where D (𝛼̃) = D (𝛼̃)e 2 (15) [ ( )] A s s Q (𝛼) = ⟨𝛼|A |𝛼⟩ = Tr [A |𝛼⟩⟨𝛼| ] = Tr A D (𝛼)|0⟩⟨0| D †(𝛼) A = 𝛼̃ Note that when s 0, D 0( ) is the standard displacement opera- (11) tor from Equation (5). When s = 0 we say that the displacement operator is symmetrically, or Weyl, ordered. Alternatively, when where the last form of the Q function can be thought of as an s =−1ors = 1, the corresponding displacement operators, and expectation value of a general coherent state, where instead of therefore characteristic functions, are anti-normal and normal, the kernel being a displaced parity operator it is a displacement respectively; this can be realized by showing the operators explic- of the projection of the vacuum state. itly Another drawback to the Q function for some is that trans- 𝛼̃ = −𝛼∗a 𝛼a† 𝛼̃ = 𝛼a† −𝛼∗a forming back to the operator will require a different kernel, this D −1( ) e e and D 1( ) e e (16) is the kernel that transforms an arbitrary operator to the Glauber– Sudarshan P function.[8,9] Likewise, the kernel to transform back demonstrating the opposite order of the arguments in the expo- from the P function to the operator is the kernel for the Q nents. Note that, like the quasi-probability distribution functions,

Adv. Quantum Technol. 2021, 2100016 2100016 (3 of 29) © 2021 The Authors. Advanced Quantum Technologies published by Wiley-VCH GmbH www.advancedsciencenews.com www.advquantumtech.com the characteristic functions are informationally complete, such we return to classical Hamiltonian physics and understand the that phase space approach in terms of the Wigner function from this ∞ perspective. The classical dynamics of any distribution A for a = 𝜒 (s) 𝛼̃ † 𝛼̃ 𝛼̃ given Hamiltonian ℋ is given by A A ( ) D −s( )d (17) ∫−∞ 𝜕 dA = ℋ + A where the reverse transform kernel is the Hermitian conjugate {A, } 𝜕 (23) of the generalized displacement operator with minus the s value dt t of the function. where the Poisson bracket The Q and P functions are then the Fourier transforms of the anti-normally ordered and normally ordered characteristic func- ∑ 𝜕 𝜕 𝜕 𝜕 = f g − f g tions, respectively, where {f, g} 𝜕 𝜕 𝜕 𝜕 (24) i qi pi pi qi ∞ 𝛼 = 1 𝜒 (−1) 𝛼̃ 𝛼𝛼̃ ∗ − 𝛼̃ ∗𝛼 𝛼̃ Q( ) A ( )exp( ) d , and 𝜋 ∫−∞ defines the way that quantities flow in phase space; as the Poisson ∞ bracket is an example of a Lie bracket, the fact that this represents 1 (1) ∗ ∗ 𝛼 = 𝜒 𝛼̃ (𝛼𝛼̃ − 𝛼̃ 𝛼) 𝛼̃ a specific kind of infinitesimal transform for classical dynamics P( ) ∫ A ( )exp d (18) 𝜋 −∞ is natural. The Poisson bracket thus specifies the way in which the state of the system transforms incrementally in time for a More generally, we can say that they are the normally and anti- given Hamiltonian as a flow in phase space. For example, in a normally quasi-probability distribution, where we can define a time increment dt the coordinates general s-values quasi-probability distribution function ∞ (q ,p) → (q + {q , ℋ}dt, p + {p , ℋ}dt) (25) (s) 𝛼 = 1 𝜒 (s) 𝛼̃ 𝛼𝛼̃ ∗ − 𝛼̃ ∗𝛼 𝛼̃ i i i i i i FA ( ) A ( )exp( ) d (19) 𝜋 ∫−∞ and we can see how the Hamiltonian defines a “flow” in phase which holds for any value of s, generating an array of quasi- space (the above picture illustrates this in one-dimension). If we probability distribution functions, however, only s =−1, 0, 1 will change the definition of the Poisson bracket, this would result in be considered here. different physics and the phase space formulation of quantum Further, by using the definition of the characteristic function mechanics follows exactly this line of reasoning. In this way, we in Equation (15) and substituting it into Equation (19), we can will see that in redefining the Poisson bracket, quantum mechan- see that by bringing the integral inside the trace operation, we ics can be seen as a continuous deformation of this underlying can define the generalized kernel for any quasi-probability dis- “flow” algebra. tribution function as the Fourier transform of the displacement Before proceeding, we first note that if we set A equal to the operator probability density function 𝜌(t) then we observe two important corollaries. The first is that the conservation of probability re- ∞ d𝜌 (s) 1 (s) ∗ ∗ quires = 0 (so it is a constant, but not necessarily an integral, Π (𝛼) = D (𝛼̃)exp(𝛼𝛼̃ − 𝛼̃ 𝛼) d𝛼̃ (20) dt 𝜋 ∫−∞ of the motion) and the second is that {𝜌, ℋ} can be thought of as the divergence of the probability current. This is expressed as Note this can alternatively be expressed as weighted integrals of in the Liouville “conservation of probability” equation (which has the displacement operator, see ref. [41], for example. Where a gen- obvious parallels with the von-Neumann equation for density op- eral function in HW phase space is defined as erators): [ ] s 𝛼 = Π(s) 𝛼 FA( ) Tr A ( ) (21) 𝜕𝜌 = {ℋ, 𝜌} (26) 𝜕t For the reverse transform, to take a phase-space distribution to an operator, one can then perform When Dirac motivated the Heisenberg picture of quantum me- ∞ chanics, he did so by arguing that equations of the above form = (s) 𝛼 Π(−s) 𝛼 𝛼 A ∫ FA ( ) ( )d (22) and the Lie algebraic structure of the Poisson bracket be both re- −∞ tained; if quantum dynamics is to be defined in terms of some providing the transforms between an arbitrary density operator infinitesimal transform, this is a sensible requirement. Dirac’s and an arbitrary s-valued quasi-probability distribution function. argument then was to introduce other mathematical structures To complete the picture, all that is needed is a discussion of the in which to achieve this end—operators in a vector space. dynamics on quantum systems in phase space. The alternative we take here is the deformation quantization approach that leaves all quantities in exactly the same form as in classical Hamiltonian physics but to introduce quantum phe- 2.2. Evolution in Phase Space nomena by deforming the phase space directly. In this view, function multiplication is replaced by a “star” product, such that Dynamics of quantum systems are particularly appealing in the lim f ⋆ g = fg (27) phase space formulation. To understand why this is the case ℏ→0

Adv. Quantum Technol. 2021, 2100016 2100016 (4 of 29) © 2021 The Authors. Advanced Quantum Technologies published by Wiley-VCH GmbH www.advancedsciencenews.com www.advquantumtech.com and that we introduce an alternative to the Poisson bracket— Importantly for our discussion, the star product can also be termed the Moyal bracket written in a convolution-integral form: [ ] 1 1 ′ ′ ′′ ′′ ′ ′ {{ f, g}} = f ⋆ g − g ⋆ f (28) f (q, p) ⋆ g(q, p) = dq dp dq dp f (q + q ,p+ p ) iℏ 𝜋2ℏ2 ∫ ( ( )) that satisfies the property × + ′′ + ′′ 2i ′ ′′ − ′′ ′ . g(q q ,p p )exp ℏ q p q p (36) {f, g} = lim{{ f, g}} (29) ℏ→0 which has the advantage that it both highlights that quantum effects are non-local and, from a computational perspective, nu- This last expression guarantees that any such formulation of merical integration can be less prone to error than differentiation “quantum mechanics” will smoothly reproduce classical dynam- (although more costly). ics in the limit ℏ → 0 and for this reason can be viewed as a de- We now can use a specific example to a result by Brif and formation of classical physics. For this reason, we briefly reintro- Mann[36] to connect this form to a more general expression duce ℏ in this section for comparison with classical mechanics, we will then return to ℏ = 1 in the next section. Understanding ⋆ = ′ ′ ′′ ′′ ′ ′ ′′ ′′ the Wigner function as the exact analog of the probability den- f (q, p) g(q, p) ∫ dq dp dq dp f (q ,p) g(q ,p ) sity function—and as a quantum constant of the motion—means [ ] that the conservation of probability requires dW = 0: we also note × Π Π ′ ′ Π ′′ ′′ dt Tr (q, p) (q ,p) (q ,p ) (37) that {{W, ℋ}}can be thought of as the divergence of this generalized probability current. The Louiville equation is then replaced where Π(q, p) is the usual displaced parity operator. by the more general equation: To summarise the quantum dynamics of any system or composite of systems is given by 𝜕W = {{ ℋ,W}} (30) 𝜕t 𝜕W = {{ ℋ,W}} 𝜕t We see that stationary states are then those for which {{ℋ,W}} = ⋆ 1 0 which lead to left and right -genvalue equations that are the {{ W, ℋ}} = [W ⋆ ℋ − ℋ ⋆ W] phase space analog of the time-independent Schrödinger equa- iℏ tion (whose solutions will be integrals of the motion as the phase- Ω ⋆ ℋ Ω = Ω′ ℋ Ω′′ Ω′ Ω′′ Ω Ω′ Ω′′ space analogue of energy eigenstates). Although this is not a W( ) ( ) W( ) ( ) K( , ; )d d ∫Ω′′ ∫Ω′ foundations paper it is worth noting: i) the quantisation of phase (38) space resolves issues around operator ordering in convectional [3] quantum mechanics (see Groenewalds theorem, e.g. ); ii) that where we see this as an equivalent view to classical mechanics this approach may be of particular value when looking to unify with a deformation of function multiplication that in the limit quantum mechanics and gravity as time is treated on the same ℏ → 0 will, if the limit exits, reduce these dynamics to classical footing as all other coordinates and the geometric view of intro- physics. This can be viewed as moving from a non-local to local ducing curvature through the metric tensor is perfectly natural theory as W(Ω) ⋆ ℋ(Ω) → W(Ω)ℋ(Ω)asℏ → 0. in a phase space formulation. We note that owing to the marginals of the Wigner function The star-product is usually written, for canonical position and producing a positive distribution, one can formulate quantum dy- momentum, in its differential form as namics in terms of the marginal distribution of the shifted and [ ( ← → ← → )] squeezed quadrature components. Thus producing an alterna- ∑N 𝜕 𝜕 𝜕 𝜕 tive formulation of the quantum evolution.[52] f ⋆ g = f exp iℏ − g (31) 𝜕 𝜕 𝜕 𝜕 As a final note the classical dynamics of particles should even i=1 qi pi qi pi be recoverable in this limit as ℏ → 0 the Wigner function of, for where the arrows indicate the direction in which the derivative is example, coherent states tends to Dirac∏ delta functions centered 𝜌 = 𝛿 − − to act. Which can be shown to be: at one point in phase space (qi,pi) i (qi Qi,pi Pi)and ( ) the Liouville equation for point particles will recover that clas- ℏ ℏ ⋆ = + i 𝜕⃖⃖⃗ − i 𝜕⃖⃖⃗ sical point trajectories in an analogous way to solutions of the f (q, p) g(q, p) f q p,p q g(q, p) (32) 2 2 Kontsevich equation used in plasma physics.[53] As Moyal brack- ( ) iℏ iℏ ets tend to Poisson brackets as ℏ → 0, it is natural to see that the = f (q, p)g q − 𝜕⃖⃖ ⃖ ,p+ 𝜕⃖⃖ ⃖ (33) 2 p 2 q Kontsevich equation is recoverable in this limit. ( ) ( ) ℏ ℏ = + i 𝜕⃖⃖⃗ − i 𝜕⃖⃖ ⃖ f q p,p g q p,p (34) 2 2 2.3. Quantum Mechanics as a Statistical Theory ( ) ( ) iℏ iℏ = f q, p − 𝜕⃖⃖⃗ g q, p + 𝜕⃖⃖ ⃖ (35) 2 q 2 q We will now generalize the discussion of phase space for use for any group structure. This will include us generalizing important ℏ ℏ where (q + i 𝜕⃖⃖⃗)and(p − i 𝜕⃖⃖⃗) are known as the Bopp equations from Section 2.1 and providing the generalization for 2 p 2 q operators.[51] the evolution in Equation (38).

∫ Ω Ω Ω= To generalize the formulation of phase-space functions, a suit- S-W. 4 Traciality: The deﬁnite integral Ω WA( )WB( )d able candidate for the given group structure needs to be chosen Tr[AB] exists. to replace the Heisenberg–Weyl displacement operator.[44] Such a S-W. 5 Covariance: Mathematically, any Wigner function gen- generalized displacement operator, 𝖣(Ω) then displaces the cor- erated by “rotated” operators (Ω′) (by some unitary responding choice of vacuum state for the given system, |vac⟩, transformation V) must be equivalent to “rotated” creating the generalized coherent state in that system[34,49] Wigner functions generated from the original operator ((Ω′) ≡ V(Ω)V†)—that is, if A is invariant un- |Ω⟩ = 𝖣 Ω | ⟩ Ω ( ) vac (39) der global unitary operations then so is WA( ).

For examples of such operators and the generation of generalized Note that from S-W. 1, the transform back to the operator coherent states and the generation of generalized displacement operators for different systems, see, for example, refs. [34, 42, = Ω  Ω Ω A ∫ WA( ) ( )d (43) 54–58]. Note that when generating quasi-probability distribution Ω functions, any suitable generalized displacement operator can Ω Ω produce an informationally complete function; however, when is integrated over the full volume ,andd is the volume nor- considering just the Weyl characteristic function much more care malized differential element (see Appendices in ref. [42] or ref. needs to be taken in considering the correct operator,[44,59] since it [44] for more details. Also note that, given the kernel for a sys- is crucial to determine a generalized displacement operator that tem, the generalized parity can be recovered by taking generates an informationally complete function, where the in-  = (0) (44) verse transform to the operator exists, such that [ ] this is a helpful result in future calculations.[12,13,44] We can there- 𝜒 (s) Ω̃ = 𝖣 Ω̃ = 𝜒 (s) Ω̃ 𝖣† Ω̃ Ω̃ ( ) Tr A s( ) , where A ( ) − ( )d fore say in general that any quasi-probability distribution func- A ∫ ̃ A s Ω tion is defined (40) [ ] F(s)(Ω) = Tr A (s)(Ω) (45) ̃ A where dΩ is the volume-normalized differential element or Haar measure for the given system. where Once a generalized displacement operator has been chosen, (s) Ω = 𝖣 Ω (s)𝖣† Ω the generation of the Q and P functions simply requires the gen- ( ) ( ) ( ) (46) eralized coherent state from Equation (39), where is the generalized kernel for any s-valued phase-space function. As is demonstrated from Equation (43), the Wigner function Ω = ⟨Ω| |Ω⟩ = Ω |Ω⟩⟨Ω| Ω QA( ) A and A PA( ) d (41) is an informationally complete representation of the state, that ∫Ω is, it contains all the same information as the density operator as However, the construction of the Wigner function is somewhat one can transform between the two representations without loss more involved. of information. The same can be said of the Q and P functions, To generate the Wigner function, we now need to construct the where the operator can be returned from the Q function by inte- corresponding generalized parity operator, , to create the kernel grating over the kernel for the P function, likewise the operator (Ω), yielding the Wigner function can be returned by integrating the P function over the Q function [ ] kernel, explicitly Ω =  Ω WA( ) Tr A ( ) (42) = (s) Ω (−s) Ω Ω A ∫ FA ( ) ( )d (47) In ref. [5], Stratonovich set out five criteria that such a kernel Ω needs to fulfil to generate a Wigner function, known as the Stratonovich-Weyl correspondence. These are, taken and adapted which is the generalization of Equation (22). from ref. [43]: It is now natural to ask whether there is a simple comparison between the different phase-space functions without first going Ω =  Ω to the density operator. Indeed this can be done by way of a gener- S-W. 1 Linearity: The mappings WA( ) Tr[A ( )] and = ∫ Ω  Ω Ω alized Fourier transform.[31,36,44,60] We can simply bypass the cal- A Ω WA( ) ( )d exist and are informationally complete. This means that A can be fully recon- culation of the density operator by substituting Equation (47) for Ω one value of s into Equation (45) for a different s value, yielding structed from WA( ) and vice versa. Ω [ ] S-W. 2 Reality: WA( ) is always real valued when A is Her- (s ) ′ (s ) − ′ mitian, therefore (Ω) must be Hermitian. From the 2 Ω = 1 Ω ( s1) Ω Ω (s2) Ω FA ( ) Tr ∫ FA ( ) ( )d ( ) structure of the kernel in Equation (42), this also Ω means that the generalized parity operator, ,must [ ] (s ) − ′ = 1 Ω ( s1) Ω (s2) Ω Ω (48) also be Hermitian. ∫ FA ( )Tr ( ) ( ) d Ω Ω S-W. 3 Standardization: WA ( ) is “standardized” so that the definite integral over all space ∫Ω W (Ω)dΩ=Tr[A ] A = (s1) Ω  Ω′ Ω Ω ∫  Ω Ω=𝟙 FA ( ) ( ,s2; ,s1)d exists and Ω ( )d . ∫Ω

Adv. Quantum Technol. 2021, 2100016 2100016 (6 of 29) © 2021 The Authors. Advanced Quantum Technologies published by Wiley-VCH GmbH www.advancedsciencenews.com www.advquantumtech.com where general framework from Section 2.3 can be applied to specific [ ] finite-dimensional quantum systems to represent them in phase ′ − ′  Ω Ω = ( s1) Ω (s2) Ω ( ,s2; ,s1) Tr ( ) ( ) (49) space. Where consideration of such structures is important in considering the use of phase-space methods in quantum tech- is the generalized Fourier operator. This can further be general- nologies. As such, we will show the important case of represent- ized to transform between any phase-space function, including ing qubits in phase space, providing a link between the two most the Weyl function, by substituting in Equation (40) instead. well-known formulations of qubits in phase space. This will be We can further extend the idea behind this Fourier kernel to followed by looking at how more general qudits and collections perform a convolution between any two phase-space functions to of qubits can be represented in phase-space in different situa- create a third function, such that tions. Given the phase-space representation of all these systems, along with the continuous-variable construction for optical quan- (s) Ω = (s1) Ω′ (s2) Ω′′  Ω′ Ω′′ Ω Ω′ Ω′′ tum systems, we will then be able to consider many approaches FAB( ) ∫ ∫ FA ( )FB ( ) ( , ; )d d (50) Ω′ Ω′′ to quantum technologies in phase space. where [ ] ′ ′′ − ′ − ′′ (Ω , Ω ; Ω) = Tr (s)(Ω) ( s1)(Ω )( s2)(Ω ) (51) 3.1. Qubits in Phase Space is the convolution kernel. This then provides a generalization of The most important system to introduce now for use in quan- Equation (31) allowing us to define the evolution of any general tum technologies is the construction of phase-space functions for system by [44] qubits. The qubit Wigner function has two main formulations, there is the original construction by Stratonovich that considers 휕 W휌 [ ] the qubit over the Bloch sphere, with continuous degrees of free- =−i W휌 (Ω) − W 휌(Ω) (52) 휕t H H dom. The second formulation was generalized by Wootters, and considers the qubit states on a discrete toroidal lattice, with dis- For specific examples of the application of this equation on dif- crete degrees of freedom. ferent quantum systems, see refs. [15, 31, 36–38, 52, 60–63]. Both formulations will be considered here, as each has had Stratonovich showed in ref. [5] that the kernels can be com- important results in understanding the properties of quantum bined for the individual systems to create a kernel for a compos- mechanics in a quantum technology setting. These two formula- ite system. The procedure to do so is straight-forward, it simply tions are also specific examples of a general formulation of qu- requires one to take the Kronecker product of the individual sys- dits in phase space. It is also interesting to note that the discrete- tems lattice qubit Wigner function from Wootters can be considered ⨂n a sub-quasi-probability distribution of the continuous-spherical  𝛀 =  Ω ( ) i( i) (53) Wigner function. This relation will be shown later, but this rela- i=1 tion does not hold for general qudit states. We will also consider the generation of the Q and P functions 𝛀 = Ω Ω … Ω Ω for n subsystems, where { 1, 2, , n}and i are the de- for qubits, which are already well-known in the Stratonovich for-  Ω grees of freedom for the individual systems. Similarly i( i)is mulation, however to our knowledge does not exist in the dis- the kernel for each individual system. The full composite Wigner crete case. function is then generated by using Equation (53) in Equa- tion (42). Transforming back to the operator from the composite Wigner 3.1.1. The Stratonovich Kernel function requires a simple modification of Equation (43), where the new volume normalized differential operator is Following the steps from Section 2, a suitable generalized dis- ∏n placement operator and generalized parity operator is necessary 𝛀 = Ω d d i (54) for the construction of a Wigner function for qubits. We will start i=1 by considering an appropriate generalized displacement operator. Since we are now concerned with distributions on the sphere, More details on this construction can be found in ref. [44]. the generalized displacement operator is a rotation operator. An We now have all the ingredients necessary to generate a example of the change in geometry and in displacement can be quasi-distribution function for any system, or any composition seen in Figure 1b. A complicating factor in choosing an operator of individual systems, to represent quantum technologies in is that rotation around a sphere isn’t a unique procedure, there phase space. are many ways in which one can perform such an operation. Out of the options, we highlight two choices that are heavily used in the appropriate literature and give equivalent results upon rota- 3. Phase-Space Formulation of Finite Quantum tion of the generalized parity operator. These are Systems 1 휙 휃 Φ = 휙휎𝗓 ∕ 휃휎𝗒 ∕ Φ휎𝗓 ∕ U2 ( , , ) exp (i 2) exp (i 2) exp (i 2), We will now turn our attention to the generation of phase space ( ) functions for finite quantum systems. It will be shown how the and R (휉) = exp 휉휎+ − 휉∗휎− (55)

Figure 1. Qubits in phase space. (a) shows the action of the displacement operator on the vacuum state in phase space for the Heisenberg–Weyl group. When considering qubits, the geometry of the phase space is diﬀerent, where two geometries can be chosen to represent qubits. (b) shows a spherical phase space, similar to the Bloch sphere. Instead of the displacement operator in (a), a rotation operator is needed to rotate a state around the sphere. Alternatively, (c) and (d) show how qubit states can be represented on discrete toroidal lattices. (c) shows how the entries for the discrete Wigner function and (d) is the discrete phase space for the discrete Weyl function. Examples of the qubit Wigner function for the eigenstates of the Pauli matrices are shown below, where each state is marked with the ket representation of the state, | ↑⟩ and | ↓⟩ are the eigenstates of 휎𝗓, | →⟩ and | ←⟩ are the eigenstates of 휎𝗑 and |+⟩,and|−⟩ are the eigenstates of 휎𝗒 with eigenvalues ±1, respectively. For each of the states, the top row shows the Wigner function in Stratonovich formalism on a sphere and then the Wootters formalism on a 4 × 4 grid. Four points are highlighted on the Wigner function generated with the Stratonovich kernel that correspond to the values of the discrete Wootters formulation. Below we show three plots of the Weyl function, where the we have the real and imaginary values of the continuous Weyl function, respectively. This is followed by the discrete Weyl function. which are the standard Euler angles, where we introduce the We therefore need to construct an informationally complete Weyl M = [44] general notation UN that in this case is set to SU(N), M 2j, function by including all three Euler angles, for the quantum number j. The second operator R (휉), where [ ] 휉 = 휃 − 휙 ∕ 휎± = 휎𝗑 ± 휎𝗒 ∕ 휒 휙̃ 휃̃ Φ̃ = 1 휙̃ 휃̃ Φ̃ exp( i ) 2and ( i ) 2, is a generalization of A( , , ) Tr AU2 ( , , ) (58) the Heisenberg–Weyl displacement for SU(2), and can be found in much of the spin-coherent state literature—for example, see and so we prefer to keep this Φ in the deﬁnition of the rotation refs. [32, 34]—and can be related to the Euler angles by[44] operator. Further, keeping all three Euler angles allows us to directly relate to operations on a quantum computer, where it is usual to include an arbitrary rotation in terms of Euler angles— R (휉) = U1(휙, 휃, −휙) (56) 2 or at least some operation with three degrees of freedom. This is 1 휙 휃 Φ 휉 also the reason we prefer to use U2 ( , , )overR ( ), despite the 훼 Although it is much more tempting to have a preference latter having the satisfying connection to D ( ). toward this rotation operator, due to the similarity to the The reason for the requirement of all three angles can be Heisenberg–Weyl case, this rotation operator generates a Weyl thought of by considering the Weyl function as giving the expec- function that is not informationally complete. For instance tation value of the rotation operator used. When considering the rotation operators R (휉), the rotation is on P1 rather than the full SU(2). This means we have rotations restricted to a 2-sphere. ⟨↑| R (휉)|↑⟩ = ⟨↓| R (휉)|↓⟩ (57) In turn, the rotation operator cannot produce the state 휎𝗓, which

Adv. Quantum Technol. 2021, 2100016 2100016 (8 of 29) © 2021 The Authors. Advanced Quantum Technologies published by Wiley-VCH GmbH www.advancedsciencenews.com www.advquantumtech.com is necessary for discerning its two eigenstates | ↑⟩ and | ↓⟩.This not satisfy the Stratonovich–Weyl correspondence. Further, us- is the reason for the result in Equation (57). ing the 휎𝗓 operator as a parity creates a function similar to the It is interesting to note that when our goal is to use the Weyl Weyl function—that is, it is the expectation value of some arbi- function to consider the expectation value of different operators, trary operator in phase space. The only problem is the identity we can instead consider the Weyl function on a torus, where we operator is not represented in this distribution like the 휎𝗓 opera- set Φ=0. It turns out that this restriction is informationally com- tor not represented in the case of the P1 rotation operator. plete and the last degree of freedom is not strictly necessary in Examples of the single-qubit Wigner function for the eigen- the qubit case, as long as the range is extended to −휋<휙̃, 휃̃ ≤ 휋. states of the Pauli operators can be seen in Figure 1d, where | ↑⟩ However, we will generally choose keep the Φ degree of freedom and | ↓⟩ are the eigenstates of 휎𝗓 with eigenvalues ±1, respec- to allow the rotation operator to also define the coherent state rep- tively. Likewise, | →⟩ and | ←⟩ are the eigenstates of 휎𝗑 with resentation on P1, and more importantly to keep this formula- eigenvalues ±1and|+⟩ and |−⟩ are the eigenstates of 휎𝗒 with tion informationally complete for larger Hilbert spaces. eigenvalues ±1. To the right of each function that is generated by We now need to use our choice of rotation operator to rotate the the Stratonovich kernel is the same state generated by the Woot- corresponding generalized parity operator. For the Wigner func- ters kernel, that will be shown in the following section. More tion, this is explicitly comparison between the two is therefore to follow. It is imme- ( √ ) diately clear that when generated with the Stratonovich kernel 1 𝗓 Π1 = 𝟙 + 3휎 (59) on a spherical geometry, the Wigner function is similar to the 2 2 more familiar Bloch sphere, where the highest value of the prob- 𝗓 ability distribution is where the Bloch vector would point. It is where 휎 is the standard Pauli z operator. The appearance of the √ also worth noting that, unlike the Heisenberg–Weyl functions 3 in Equation (59) may seem somewhat strange, as is gen- and further the discrete Wigner functions, every spin coherent erally assumed that the generalized parity should simply be 휎𝗓, 휎𝗓 state has a negative region. This is a result of the spherical ge- without the identity or the coefficients. Choosing as a parity ometry and the role of quantum correlations within phase space; operator can be found in some works, and has proven useful in this will be discussed in more detail in Section 4.2. verification protocols, such as in ref. [64], this choice, however, Alternatively, one may be interested in constructing the Q or P leads to a function that is not informationally complete, that is, function for the qubit, where the generation of the suitable kernel Equation (43) does not hold. Note also that S-W. 3 implies that  = 휎𝗓 is just as simple as in the qubit case. Like the Wigner function Tr[ ] 1, which does not hold for . To generate a generalized generalized parity operator, these too are diagonal operators. The parity that is informationally complete, it is therefore necessary two can be calculated explicitly as to follow the Stratonovich–Weyl correspondence S-W. 1–5. For ( ) more detail of how this can be done, see ref. [5] or in the Appendix 1(−1) 1 𝗓 10 of ref. [65]. This yields the Stratonovich qubit Wigner function Π = (𝟙 + 휎 ) = , 2 2 00 kernel ( ) 1(1) 1 𝗓 20 Π1(휃, 휙) = U1(휙, 휃, Φ)Π1U1(휙, 휃, Φ)† (60) and Π = (𝟙 + 3휎 ) = (63) 2 2 2 2 2 2 0 −1 Φ where the degree of freedom gets cancelled out due to the gen- for the Q and P function, respectively, where the ±1 in the bracket eralized parity being a diagonal operator, this is the reason behind signifies the value of s for the generalized phase-space function. the equivalence of the rotation operators in Equation (55) when These both result in kernels where the Φ degree of freedom also rotating the generalized parity operator. cancels out when taking a similarity transform with the rotation We can then simply return the operator by applying this case operator. The cancellation of the Φ term can therefore mean that to Equation (47) the last Euler angle is not necessary for the calculation of quasi- 휋 휋 probability distribution functions, and one will occasionally see 1 2 A = W (휃, 휙)Π1(휃, 휙)sin휃 d휃 d휙 (61) the kernel generated ignoring this term, for good reason to ex- 휋 ∫ ∫ A 2 2 0 0 clude redundant information. This also lines up nicely with the toroidal definition of the Weyl function, where we only care about If one, on the other hand, insisted on using just the 휎𝗓 operator, the first two degrees of freedom in the rotation operator. As we creating a phase-space function WZ(Ω) = Tr[휌 U1(Ω)휎𝗓 U1(Ω)†], 휌 2 2 will soon see, a toroidal geometry is a natural alternative to the to transform back to the density operator, the identity needs to be Bloch sphere in considering qubit states. reinserted

2휋 휋 휌 = 1 𝟙 + 1 Z 휃 휙 휎𝗓 휃 휙 휃 휃 휙 휋 ∫ ∫ W휌 ( , ) ( , )sin d d (62) 3.1.2. The Wootters Kernel 2 2 0 0

However, we therefore have to assume that the function is of The alternative formulation to consider qubits in phase space can a trace-1 operator, and this does not hold for an arbitrary oper- be seen in the framework set out by Wootters.[15] In the same year, ator. This result is clearly shown by considering either ref. [5] Feynman also gave an identical construction of a Wigner function or ref. [65] and setting the coeﬃcient of the identity operator to for qubits.[14] Feynman’s construction treated only the qubit case, 0. Although this can be useful for examining certain properties whereas in the work by Wootters, the formalism was extended to of a quantum state, it is important to bear in mind that it does any prime-powered dimensional phase space. More importantly,

Adv. Quantum Technol. 2021, 2100016 2100016 (9 of 29) © 2021 The Authors. Advanced Quantum Technologies published by Wiley-VCH GmbH www.advancedsciencenews.com www.advquantumtech.com in ref. [15] Wootters presented the discrete Wigner function in tionship between the Euler-angle parameterization and toroidal the Moyal formalism—the expectation value of a kernel, which lattice approach. Wootters intuitively names the “phase-point operator”. Despite Note first that by setting Φ=0 in the SU(2) case, the rotation this, we will give the formulation of the discrete Wigner function operator defines translations around a torus. This then allows presented by Feynman, as we feel it provides a different insight us to describe the values of the toroidal lattice as points in the into the problem, and then subsequently relate it to Wootters’s continuous torus generates by the Euler angles. We also need to phase-point operator. Following this, we will show how this for- note that there can be a difference in phase between elements in 1 휙 휃  ̃ ̃ mulation can be considered a sub-quasi-probability distribution U2 ( , , 0) and 2(z, x), therefore, a factor of i may be necessary of the Stratonovich formulation. to equate the two. However, the complex behavior of characteris- The idea behind Feynman’s formulation was to write down tic functions is well-known, and it is instead standard practice to probability distributions define the characteristic as the absolute value or absolute value squared, where such practice is commonplace in the signal pro- 1 𝗑 𝗒 𝗓 1 𝗑 𝗒 𝗓 cessing community, for example when analyzing the analog of p++ = ⟨𝟙 + 휎 + 휎 + 휎 ⟩ ,p+− = ⟨𝟙 − 휎 − 휎 + 휎 ⟩ 2 2 the Weyl function—the ambiguity function.[69,70]

1 𝗑 𝗒 𝗓 1 𝗑 𝗒 𝗓 This allows us to relate the two approaches to a Weyl function p−+ = ⟨𝟙 + 휎 − 휎 − 휎 ⟩ ,p−− = ⟨𝟙 − 휎 + 휎 − 휎 ⟩ 2 2 by taking the absolute value squared of each or by adding an extra phase, where (64) | ̃ ̃ |2 = |휒 ̃휋∕ ̃휋∕ |2 (z, x) A(z 2, x 2, 0) (69) which are the joint probabilities, p±±, of finding the qubit having the spin aligned with the ±z and ±x axes, respectively. The choices of operator and expectation values in Equation (64) were It is also interesting to see the Weyl function as a generalized 휒 휙 휃 Φ = presented by Feynman without any derivation or motivation; al- autocorrelation function, where for a pure state A( , , ) ⟨휓| 1 휙 휃 Φ |휓⟩ though, as we will see later Wootters uses the same basis, this U2 ( , , ) . We can think of the rotation operator as act- choice is not unique, in fact there is no unique choice of a prob- ing on the ket state and then taking the inner product. It then ability distribution of this form, see refs. [66–68] for examples of makes sense to take the absolute value squared to get a real- generalizations of these results. valued autocorrelation function. Following the results of Wootters, we can take the expressions Note that in Equation (69), we relate the continuous Weyl func- Φ= in Equation (64) and turn them into a phase-point operator. We tion to the discrete case by setting 0, this is the simplest first introduce two degrees of freedom to span the discrete phase way to reduce to two degrees of freedom for easy calculation and Φ=−휙 space, these are z, x ∈ {0, 1} which cover all four phase points. equivalent results. If, on the other hand, we chose to set , The phase-point operator is then defined like we do when equating with the Heisenberg–Weyl case, there would be no way to produce the 휎𝗓 operator, hence the lack of ( ) 휎𝗓 1 𝗓 𝗑 + 𝗒 informational completeness in this slice for the eigenstates of  (z, x) = 𝟙 + (−1)z 휎 + (−1)x 휎 + (−1)z x 휎 (65) 2 2 in Equation (57). The informational completeness for Equation (68) can be seen resulting in the discrete Wigner function for any arbitrary opera- by observing tor ∑ [ ] = 1  ̃ ̃  ̃ ̃  =  A (z, x) 2(z, x) (70) A(z, x) Tr A 2(z, x) (66) 2 z,̃ x̃ where the subscript is dimension d = 2for (z, x). We can then d which demonstrates the power of the use of the Weyl function as relate the results in Equation (64) by noting that (0, 0) = p++, a tool for practical purposes in state reconstruction. For example, (0, 1) = p+−, (1, 0) = p−+,and(1, 1) = p−−. the discrete form of the Weyl function has proven to be useful in Like the Stratonovich kernel, we can define a generalized dis- verification protocols,[71] where the Weyl function can be used as placement operator to both generate a Weyl function and further a probability distribution for choice of measurement bases. In the to be used to construct the Wootters kernel in terms of a gener- case of stabilizer states,[72] this results in the Weyl function hav- alized displaced parity operator. The displacement operator for ing values of 0 over the majority of the distribution and 1 when discrete systems such as this is the Weyl kernel is the stabilizer for that state. This is simply seen ( ) by noting that a stabilizer state is an eigenstate of a product of 1 𝗑 𝗓  (z, x) = exp i휋xz (휎 )x(휎 )z (67) 2 2 Pauli operators. For a single qubit, this result is simply seen by noting which, like the SU(2) Wigner function, can be used to generate |휓⟩ =  |휓⟩ ⇒ ⟨휓|휓⟩ = = ⟨휓| |휓⟩ a discrete Weyl characteristic function 2(z, x) 1 2(z, x) [ ] [ ]  ̃ ̃ =  ̃ ̃ = Tr 휌  (z, x) (71) (z, x) Tr A 2(z, x) (68) 2

The four values of the discrete displacement operator are simply The same naturally holds when extending to multiple qubits. the Pauli and identity operators. As such, there is a direct rela- Note, however, that in ref. [71], the Weyl function is normalized

√ ∕ n parity operators from Equation (63), we can deﬁne the general by a factor of 1 2 ,forn qubits. This∑ is done so the factor of ∕  ̃ ̃ 2 = kernel 1 2 is not needed in Equation (70) and z,̃ x̃ (z, x) 1. Having deﬁned an appropriate displacement operator that (s) = U1(휗, 휑) Π1(s)U1(휗, 휑)† (79) generates an informationally complete Weyl characteristic func- 2 2 2 2 tion, to complete the discrete formulation in terms of a gener- that yields alized displaced parity operator, it is necessary to generate the generalized parity operator. The generalized parity is, from Equa- s (s) (s) 1 3 2 𝗓 𝗒 𝗒 tion (44), simply  =  (0, 0) = 𝟙 + (휎 + 휎 + 휎 ) (80) 2 2 2 2  =  = 1 𝟙 + 휎𝗓 + 휎𝗒 + 휎𝗒 2 2(0, 0) ( ) (72) For the Q function, this results in a set of symmetric information- 2 ally complete positive operator-valued measures.[73] Considering [22,75] and this in terms of frame theory in quantum measurement, consider this discrete Q function as a frame, where the dual  =   † frame is then the P function kernel, such that 2(z, x) 2(z, x) 2 2(z, x) (73)

(−1) 1 1 𝗓 𝗒 𝗒 is the kernel written in displaced parity form. M =  = 𝟙 + √ (휎 + 휎 + 휎 ) (81)  2 2 Interestingly, the√ eigenvalues for 2 are the same as those for 2 3 Π1 ± ∕ 2, namely (1 3) 2. This means that the non-diagonal Woot- ters generalized parity can be diagonalized as √ (−1) 1 3 𝗓 𝗒 𝗒 (1) M̃ = 3 − 𝟙 = 𝟙 + (휎 + 휎 + 휎 ) =  (82)  = 1 휗 휑 Π1 1 휗 휑 † 2 2 2 U2 ( , ) 2U2 ( , ) (74) 2 2 where More generally we can transform between any of these kernels by 1 휋 휗 = arccos √ and 휑 =− (75) 1 (s2−s1) 4 1 − 2 3 (s2) (s2−s1) (s1) 1 3  = 3 2  + 𝟙 (83) 2 2 2 In fact, like the Weyl function, the full Wootters kernel can be written in terms of the Stratonovich kernel, where By understanding these phase-space functions in these terms, ( ) they can prove to be powerful tools for verification of quantum 2z − x states, for example ref. [76] which we will discuss further in Sec-  (z, x) =Π1 휗 + x휋, 휑 + 휋 (76) 2 2 2 tion 4. The relation between the Stratonovich and Wootters formu- Therefore, the Wootters kernel can be considered a subset of lation of phase-space functions for qubits is a special case that the Stratonovich kernel, and the Wootters qubit Wigner func- does not generalize as the dimension increases. This is because tion is a sub-quasi-distribution function of the Stratonovich qubit of the difference in geometry between the representations. Fol- Wigner function. lowing this discussion, we will see two variations in how a qudit Note that the four points on the Stratonovich qubit Wigner can be represented on a continuous phase space. One representa- function where the Wootters function lies form a tetrahedron tion keeps the SU(2) group structure and is the natural extension within the Bloch sphere, as can be seen in Figure 1, coincid- of the Stratonovich kernel, that was also introduced in his semi- ing with the formulation of symmetric informationally com- [5] [73,74] nal paper. This method creates a discrete Wigner function on a plete projective measurements. We can transform between sphere, where the radius is proportional to the dimension of the the Wigner and Weyl functions here by performing the discrete Hilbert space. The second continuous method uses the structure Fourier transform of SU(d), where d is the dimension of the Hilbert space. Geomet- ∑ rically, this is an oblate spheroid.  = 1  ̃ ̃ 휋 ̃ − ̃ (z, x) (z, x)exp(i [xz xz]) (77) The Wootters function, on the other hand, is defined on (d) × 2 ̃ ̃ z,x (d) discrete toroidal lattice, where d is prime. There have been and many formulations to extend this form of the Wigner function ∑ to higher dimensions, the original Wootters formulation was 1 (z,̃ x̃) = (z, x)exp(−i휋[xz̃ − xz̃]) (78) defined with respect to fields and so the dimensions size was d z,x restricted to primes. Wootter then extended this formulation with Gibbons et al. to work for dimensions that are powers of The Fourier transform between the Wigner and Weyl function primes.[18] Later work also considered the extension to odd num- can therefore be seen as the transformation between Pauli basis bered dimensions,[16,19,21], and even numbered dimensions.[17] measurements and symmetric informationally complete projec- Given the different choices, we refer the reader to any of these tive measurements. articles for specifics, or to refs. [21, 22, 77] for more detailed From Equation (74) and Equation (75), where we consider the overview and comparison. Since this construction above the qubit Wootters kernel as a subset of the full Stratonovich kernel, we can case is not important to this review, the explicit forms will not define Q and P functions in discrete phase space. By using the be given.

3.2. SU(2), Spin-j such that, when c → 0

+ = † − = 𝗓 = † We will now show the generalization of continuous phase space lim A j a , lim A j a, and lim A a a (89) c→0 c→0 c→0 for larger finite systems. Here we begin by staying within the SU(2) Lie group structure before going on to the SU(N) formula- The rotation operator therefore becomes tion in Section 3.3. Both cases can be used to consider similar sit- ( ) uations, but also have some subtle differences. The SU(2) formu- 휉 = 휉 + ∕ − 휉∗ − ∕ R j( ) exp A c A c (90) lation requires another variable M = 2j, two times the azimuthal quantum number, that allows us to represent a spin-j qudit where by setting the dimension of the Hilbert space is d = M + 1. It can alternatively be used to model an M-qubit state in the 휉 휉∗ lim = 훼, lim = 훼∗ (91) symmetric subspace, and can be used over composite meth- c→0 c c→0 c ods for certain processes, such as single-axis twisting or Tavis- Cummings interactions.[78–81] These methods can further be we yield used on states that are not fully symmetric, by taking the ap- [82–85] 휉 = 훼 proach from Zeier, Glaser, and colleagues a multi-qubit state lim R j( ) D ( ) (92) c→0 can be represented by a collection of SU(2) Wigner functions, n ≥ M ≥ 0forn qubits, to show the Wigner function for every M- A similar treatment can be applied to the Euler angles by not- valued symmetric-subspace. This leads to many different Wigner 𝖩M = + − − ing that 2 (2) i(J j J j). This leads to the result functions as the number of qubits increases. [ ] ( ) ( ) We begin by considering the general SU(2) rotation operator. 휙+Φ ∕ 2 † lim UM(−휙, −휃, −Φ) = lim ei( ) 2c exp −|훼|2 exp 훼a Like with the qubit case, there are many ways of formulating ro- c→0 2 c→0 tations on a sphere. The extension to the two ways given in Equa- ( ) × (−훼∗ ) 휙 +Φ † tion (55) are simply exp a exp ( )a a (93) ( ) ( ) ( ) that, when taking Φ=−휙 the limit on the right goes to 1, yield- UM(휙, 휃, Φ) = exp i𝖩M(3)휙 exp i휃𝖩M(2) exp iΦ𝖩M(3) , 2 2 2 2 ing an equivalence to D (훼). Further, when applying this to the ( ) 휉 = 휉 + − 휉∗ − vacuum state, we yield and R j( ) exp J j J j (84) | lim UM(−휙, −휃, −Φ)|j, j⟩ where c→0 2 [ ] ( ) ( ) 2 ∑j √ = lim ei(휙+Φ)∕2c exp −|훼|2 exp 훼a† exp (−훼∗a ) 𝗑 1 → 𝖩M(1) = J = j(j + 1) − m(m + 1) c 0 2 j ( ) 2 =− m j × exp i(휙 +Φ)a† a |0⟩ ( ) × | ⟩⟨ + | + | + ⟩⟨ | [ ] ( ) ( ) |j, m j, m 1| |j, m 1 j, m| (85) 2 (94) = lim ei(휙+Φ)∕2c exp −|훼|2 exp 훼a† exp (−훼∗a ) c→0 ∑j √ 𝗒 i × ( 휙 + Φ)| ⟩ 𝖩M(2) = J = j(j + 1) − m(m + 1) exp i i 0 2 j 2 m=−j [ ] ∞ ∑ 훼n ( ) i(휙+Φ)∕2c2 i휙+iΦ−|훼|2∕2 | | | | = lim e e √ |n⟩ × j, m⟩⟨j, m + 1 − j, m + 1⟩⟨j, m (86) c→0 | | | | n=0 n!

∑j equivalent to the coherent state generated by the standard dis- 𝖩M = 𝗓 = | ⟩⟨ | 휙 Φ 2 (3) J j m|j, m j, m| (87) placement operator, where the and degrees of freedom act m=−j as a global phase and cancel out when we set Φ=−휙. Note also in this result that the limit of the highest weighted spin state is are the the angular momentum operators, the higher- the vacuum state. dimensional extension of the Pauli operators in SU(2), and By analogy, the definition of a generalized spin coherent state is ± = 𝖩M ± 𝖩M J j 2 (1) i 2 (2) are the angular momentum raising and just the highest weight state rotated by a suitable operator. Either lowering operators. Here we also introduce the generalized rotation operator will work, however, they may differ by a phase 𝖩M SU(N) operator notation N (i). factor. For simplicity, we define the spin coherent state by The intuition from looking at R (휉) is that as j → ∞, the j | ⟩ Heisenberg–Weyl displacement operator from Equation (5) is re- | 휃 휙 = 휃 휙 | ⟩ |( , )j R j( , )|j, j (95) covered. This result was shown by Arecchi et al. in ref. [32]. The procedure to show this was also demonstrated in ref. [12], where This results in the simple definition of the Q and P functions first we need a contraction of the operators, where we take with respect to the spin coherent states

± ± 𝗓 1 ⟨ | | ⟩ A = cJ ,A= 𝟙 − 𝖩M(3) (88) Q (휃, 휙) = (휃, 휙) | A |(휃, 휙) , and j j 2c2 j 2 A j| | j

∗ 2휋 휋 | ⟩⟨ | where Y (휃, 휙) are the conjugated spherical harmonics. By fol- = 휃 휙 | 휃 휙 휃 휙 | 휃 휃 휙 lm A ∫ ∫ PA( , ) |( , )j ( , )j| sin d d (96) lowing the same procedure from Equation (44), the SU(2) gen- 0 0 eralized parity can be derived where it has been shown in refs. [60, 61] that these two functions ∑M ∑j → ∞ 2l + 1 jn | | also reach the Heisenberg–Weyl Q and P functions for j . ΠM =ΠM(0, 0) = C j, n⟩⟨j, n (100) 2 2 + jn,l0| | The general idea for the infinite limit of these functions is to l=0 M 1 n=−j imagine a plane tangent to the north pole of the qubit’s sphere. As the value of j increases, the radius of the sphere correspondingly wherewenoteagainthatj = M∕2. Note that when taking the increases, coming closer and closer to the tangential plane at the limit j → ∞, this generalized parity operator becomes the parity north pole. At the limit, the sphere completely touches the plane. operator defined in Equation (4). The derivation and proof of this This visualization exercise also helps to understand the differ- can be found in ref. [12]. Resulting in the generalized displaced ence in Weyl function between SU(2) and the Heisenberg–Weyl parity operator group, where the general SU(2) Weyl function is ΠM(휃, 휙) = UM(휙, 휃, Φ)ΠMUM(휙, 휃, Φ)† (101) [ ] 2 2 2 2 휒 (휙̃, 휃̃, Φ̃ ) = Tr AUM(휙̃, 휃̃, Φ̃ ) (97) A 2 where since the Φ degrees of freedom cancel out with the diagonal generalized parity, the full kernel is equivalent to the As mentioned in the single qubit case, the general SU(2) Weyl Heisenberg–Weyl kernel from Equation (3) as j → ∞.[12] function requires all three Euler angles. This can be seen in a va- More generally, the Q and P functions from Equation (96), and riety of ways, most importantly it is because when using R (휃, 휙) j any s-parameterized quasi-probability distribution function, can as the kernel, the resulting function for |j, j⟩ and |j, −j⟩ are iden- be generated by a simple extension of Equation (99) and Equa- tical. Furthermore, any mixed state on the line inside the Bloch tion (100), where the general kernel can be calculated[36,60] sphere between these two states are also identical. To resolve this, √ a third degree of freedom is needed, resulting in the preference ∑M ∑l 휋 ∕ for the Euler angles. This is not a problem for the Heisenberg– ΠM(s)(휃, 휙) = (Cjj )−s Y∗ (휃, 휙)T M 2 (102) 2 M + 1 jj,l0 lm lm Weyl case as any state on the equator is mapped to infinity, and l=0 m=−l only the equivalent of the top hemisphere is considered in the jj −s = phase space. It is also worth noting that in the qubit case, the where the extra value (Cjj,l0) is 1 when s 0, returning Equa- Weyl function is still informationally complete when Φ=0, this tion (99). The generalized parity for any s-valued function can is not the case with a general qudit. As we saw when consider- then be calculated ing the Wootters kernel, there’s a special connection between the M ( ) j ∑ −s ∑ qubit on the sphere and the qubit on a torus, this means that for M(s) M(s) 2l + 1 jj jn | | Π =Π (0, 0) = C C |j, n⟩⟨j, n| (103) the Weyl function, setting Φ=0, we are modelling the qubit state 2 2 + jj,l0 jn,l0 = M 1 =− on a continuous torus. However, due to how the 𝔰𝔲(2) algebra l 0 n j scales as as M increases, the state can no longer be considered which for s =−1 yields on a torus and the Φ degree of freedom is necessary to generate M(−1) | | an informationally complete function. Π = |j, j⟩⟨j, j| (104) Next we need to show how the Wigner function can be gener- 2 ated. Since the original formulation of the general SU(2) Wigner where, as mentioned earlier, these functions also become their function, there have been many variations in how it is presented. infinite-dimensional counterparts as j → ∞.[60,61] From a harmonic series approach to presenting it as a kernel with From Equation (48), we can then relate and transform between which a group action can be taken. Since all of these representa- these different quasi-distribution functions by way of a general- tions are equivalent, we will present how the kernel can be gen- ized Fourier transform. However, there is an alternative method erated through a multipole expansion. by way of spherical convolution.[60] By using the highest-weight For this, we need a new set of generators to generalize the Pauli state in the Wigner representation and performing a convolution operators from the qubit case. These are known as the tensor or using Equation (51), we can transform an s-valued function to [86] Fano multipole operators, the corresponding (s − 1)-valued function for the same state. For √ example, we can transform from the P function to the Wigner + ∑j ⟨ function, and the Wigner function to the Q function by using j = 2l 1 jn | ⟩ ′| T lm Cjm′,lm|j, n j, m | (98) this method. Similarly, we can transform from the P function of 2j + 1 ′ m ,n=−j a state to the corresponding Q function by using the convolution with the Q function highest weighted state. jn where Cjm′,lm are Clebsch-Gordan coefficients that couple two representations of spin j and l to a total spin j. The kernel can then N be generated from there operators by 3.3. SU( )

√ We now consider an alternative route to treat a general d-level or ∑M ∑l 휋 ∕ ΠM(휃, 휙) = Y∗ (휃, 휙)T M 2 (99) multi-qubit quantum system. Like the SU(2) case, an SU(N)rep- 2 + lm lm M 1 l=0 m=−l resentation can also be used for a general qudit state, where

Adv. Quantum Technol. 2021, 2100016 2100016 (13 of 29) © 2021 The Authors. Advanced Quantum Technologies published by Wiley-VCH GmbH www.advancedsciencenews.com www.advquantumtech.com in this case d = N and we only consider M = 1. In order to ̂1 𝚽 of freedom, N ( ), such that represent an n-qubit state, the construction of an SU(2d)ker- ∏ ∏ ( ) ( ) nel is required—it is also possible to take the same approach ̂ 1 𝝓 𝜽 ≡ 𝙹𝗓 휙 𝙹𝗒 휃 N ( , ) exp i N (3) a−1+k(b) exp i N (1,a) a−1+k(b) as with SU(2) and consider only the symmetric subspace with N≥a≥2 2≤b≤a an SU(d) representation. Alternatively, the generalized displace- ments and generalized parities can be mixed and matched to (107) some extent as was shown in ref. [43], this case will also be considered here. where Although we provide a formalism to construct these Wigner { 0,b= N functions, the details to actually construct the kernels can prove k(b) = ∑ (108) N−b ≠ to be complicated, there are therefore some open questions for i=1 ,bN generalized SU(N) kernels. the first open problem is generalizing SU(N) for any value of M, effectively creating the analog of and the symmetric subspace for MN-level qudits. However, the pro- ∏ ( ) cedure to create the generalized displacement operators for this ̂1 𝚽 ≡ 𝙹𝗓 + 2 Φ N ( ) exp i N ([c 1] ) N(N−1)∕2+c (109) case can be found in ref. [42]. 1≤c≤N−1 The second open problem is the consideration of the hyperbolic space of SU(Q, P). A construction of an SU(1, 1) Wigner Finally yielding the SU(N) rotation operator function has recently been provided in refs. [87, 88]; however, the 1 𝝓 𝜽 𝚽 = ̂ 1 𝝓 𝜽 ̂1 𝚽 general SU(Q, P) case remains to be solved. Insight into these UN ( , , ) N ( , ) N ( ) (110) cases may lead to some interesting results in ideas relating to AdS/CFT in a phase space formalism. Note that if one is also in- Just from this generalization of the Euler angles, many of the terested in some of the more fantastical routes to the unification results from the SU(2) case can be extended to SU(N). Most of quantum mechanics and gravity, in theory a construction of E8 straight-forward is an informationally complete Weyl character- should be possible, however unwieldy the number of degrees of istic function that takes the form freedom prove to be. Here we will settle ourselves with providing [ ] the results from the SU(1, 1) phase space construction to close 휒 𝝓̃ 𝜽̃ 𝚽̃ = 1 𝝓̃ 𝜽̃ 𝚽̃ A( , , ) Tr AUN ( , , ) (111) this section. The kernel for an SU(N) phase space now requires a specific Note that a discrete alternative can be yielded by constructing all generalized displacement and generalized parity. The first step is N2 − 1 generators for SU(N), resulting in to construct the generators of SU(N), which can be represented [ ] by N2 − 1 matrices of size N × N. The method to construct these 휒 = 𝖩1 A(k) Tr A N (k) (112) 𝖩1 can be found in ref. [89]. This results in a set of operators N (i), where 1 ≤ i ≤ N2 − 1. In fact, only a subset of the generators are ≤ ≤ 2 − 𝖩1 = 𝟙 𝖩1 where 0 k N 1, N (0) N and the other N (k)matrices needed to construct the SU(N) Euler angles, these are general- are the generators of the 𝔰𝔲(N) algebra. 𝗒 𝗓 izations of the 휎 and 휎 operators, which are Also from the rotation operator in Equation (110) one can generate generalized coherent states.[89] Like earlier examples of co- 𝗒 𝙹 (1,a) = 𝖩1 ((a − 1)2 + 1) = i(|a⟩⟨1| − |1⟩⟨a|) (105) herent states, SU(N) coherent states are generated by applying N N the rotation operator to a suitable analog to the vacuum state. In a 𝙹𝗓 2 − = 𝖩1 2 − N (a 1) N (a 1) similar procedure SU(N) coherent states can be generated by the ( ) √ |0⟩ 1 state, where we define this state as being the eigenstate of √ − N ∑a 1 1 2(a − 1) 휆 2 − − − ∕ = 2 | ⟩⟨ | − | ⟩⟨ | N (N 1) with the lowest eigenvalue, [2(N 1) N] 2 , making − i i a a (106) a(a 1) i=1 a it the lowest-weighted state, resulting in ⟩ | |(𝝓, 𝜽)1 = U1 (𝝓, 𝜽, 𝚽)|0⟩ 1 where 2 ≤ a ≤ N. Note that this is a subset of the full set of gen- N N N 𝔰𝔲 2 − 𝔰𝔲 휙 +휙 +⋯+휙 erator of (N); N 1 elements are needed to generate (N); i( 1 2 N−1) 휃 휃 … 휃 휃 ⎛ e cos( 1)cos( 2) cos( N−2) sin( N−1) ⎞ − −휙 +휙 +⋯+휙 however, just 2(N 1) elements of the algebra are needed to gen- ⎜ − i( 1 2 N−1) 휃 휃 … 휃 휃 ⎟ e sin( 1)cos( 2) cos( N−2) sin( N−1) ⎜ 휙 +휙 +⋯+휙 ⎟ erate the generalized Euler angles for SU(N). We also note that − i( 3 3 N−1) 휃 휃 … 휃 휃 ⎜ e sin( 2)cos( 3) cos( N−2) sin( N−1) ⎟ there has been an alternative construction of phase space for = ⎜ ⋮ ⎟ 휙 +휙 SU(3) where they instead used three dipole and five quadrupole ⎜ − i( N−2 N−1) 휃 휃 휃 ⎟ e sin( N−3)cos( N−2) sin( N−1) [90] ⎜ 휙 ⎟ operators as the generators of the Lie algebra. − i( N−1) 휃 휃 ⎜ e sin( N−2) sin( N−1) ⎟ Following this, there is a procedure to generate the gener- ⎝ 휃 ⎠ cos( N−1) alization of the Euler angles for SU(N), that can be found in refs. [55, 56, 89]. This results in the generalized Euler angles (113) 1 𝝓 𝜽 𝚽 𝝓 = 휙 … 휙 𝜽 = 휃 … 휃 UN ( , , ), where { 1, , N(N−1)∕2}, { 1, , N(N−1)∕2} 𝚽 = Φ … Φ and { 1, , N−1}. This rotation operator can be split up ignoring an overall global phase. We note that it is also possible 𝝓 𝜽 𝖩1 into two parts, first one dependent on the and degrees of to set the analog to the vacuum state as the eigenstates of N (3) ̂ 1 𝝓 𝜽 𝚽 freedom, N ( , ), and the second dependent on the degrees with eigenvalue 1, as is the more standard choice for the use of

SU(2) for qubits. However, given the structure of the genera- + 1 (1+s) (s) 𝝓 𝜽 = 1 + (N 1) 2 tors for 𝔰𝔲(N), taking the highest-weight state results in much F 휌( , ) N,1, N 4 tidier calculations. From the construction of a coherent state, the generations of N∑2+1 ⟨ ⟩ [ ] × 𝝓 𝜽 1 |𝖩1 | 𝝓 𝜽 1 휌𝖩1 the Q and P functions are simply ( , )N | N (k)|( , )N Tr N (k) (118) k=1 ⟨ ⟩ | | Q (𝝓, 𝜽) = (𝝓, 𝜽)1 A (𝝓, 𝜽)1 , and A N | | N We can then simply transform between functions by taking ⟩⟨ | 1 1 | ( ) A = P(𝝓, 𝜽) (𝝓, 𝜽) (𝝓, 𝜽) d𝛀 (114) 1 − | N N | (s1) 1 (s1 s2) (s2) 1 ∫𝛀 F (𝝓, 𝜽) = + (N + 1) 2 F (𝝓, 𝜽) − (119) N,1,휌 N N,1,휌 N 𝛀 𝛀 where are the degrees of freedom on SU(N)andd in the providing a general formula to consider qudits in SU(N) sim- volume normalized differential element for SU(N), the construc- ilar to Equation (83) for qubits. Note in the same way as the tion of which can be found in ref. [55]. Similarly, the kernel for qubit case, one can define a set of symmetric informationally the SU(N) Wigner function can be generated with respect to co- complete measurements from the Q function, this can then be [42] herent states, where transformed into its dual by √ 2− − + N∑1 ⟨ ⟩ Π1(1)(𝝓, 𝜽) = (N + 1)Π1( 1)(𝝓, 𝜽) − 𝟙 (120) Π1 𝝓 𝜽 = 1 𝟙 + N 1 𝝓 𝜽 1 | 𝖩1 | 𝝓 𝜽 1 𝖩1 N N N ( , ) N ( , )N | N (k)|( , )N N (k) N 2 k=1 which corresponds to the transform between an N-dimensional (115) frame and its dual frame.[22,75] This all provides a structure for considering qudits in phase By setting the variable to 0, the generalized parity operator for space, alternatively by considering the SU(2) subgroup structure SU(N) is yielded, where of SU(N), we can provide a link between the different representations of multi-qubit states. It also means that we can alternatively ( ) √ represent multi-qubit states with a hybrid approach of SU(2) and (N − 1)N(N + 1) Π1 = 1 𝟙 − 𝙹N 2 − SU(N) that was introduced in ref. [43] and applied in ref. [48]. N N z (N 1) (116) N 2 From applying the procedure from Equation (53) to generate a kernel for composite systems, for multi-qubit states we get producing an operator constructed by the identity operator and = ⨂n the diagonal operator from Equation (106), where a N.Like 1 1 Π ⊗ (𝜽, 𝝓) = Π (휃 , 휙 ) the construction of an SU(2) generalized parity operator, the 2 n 2 i i i=1 SU(N) parity is a diagonal operator and commutes with the ̂1 ( )( )( )†  𝚽 n n n N ( ) part of the rotation operator, resulting in the degrees of ⨂ ⨂ ⨂ freedom cancelling out. The number of degrees of freedom are = 1 휙 휃 Φ Π1 1 휙 휃 Φ U2 ( i, i, i) 2 U2 ( i, i, i) further reduced to allow the SU(N) Wigner function to live on i=1 i=1 i=1 the complex projective space PN−1 with 2(N − 1) degrees of free- ( ) ⨂n dom. By considering the operators, this can be realized by noting = 𝕌 𝝓 𝜽 𝚽 Π1 𝕌† 𝝓 𝜽 𝚽 n( , , ) 2 n( , , ) (121) that each SU(N) generator acts on N SU(2) subspaces of the full i=1 𝙹N − SU(N), where each subspace is acted on by y (1,a). Since N 1 𝕌 𝝓 𝜽 𝚽 elements of the generalized parity operator are the same, with where n( , , ) is the n-qubit rotation operator. just the last element differing, it is in effect a weighted identity We can take this new n-qubit rotation operator and apply it to operator for the rotations that do not act on the last SU(2) sub- the SU(N) generalized parity operator to create an alternative space. And so, the elements of the rotation operator that have as representation of a multi-qubit state in phase space 𝙹N < the generator y (1,a)fora N cancel out, until we reach the 𝙹N Π1 𝜽 𝝓 = 𝕌 𝝓 𝜽 𝚽 Π1 𝕌† 𝝓 𝜽 𝚽 element that rotates around y (1,N). 2[n] ( , ) n( , , ) N n( , , ) (122) Note that if we stay with the fundamental representation of SU(N) there is a simple transformation between the different that also obeys the Stratonovich-Weyl correspondence S-W. 1–5. quasi-probability distribution functions. This can be shown by Each choice of representing qubit states has its own benefits and noting that the generalization of Equation (115) is [42] drawbacks. We can immediately see that the benefit of using the multi-qubit SU(2) rotation operator with the SU(N) generalized Π1(s)(𝝓, 𝜽) = parity operator is the reduced number of degrees of freedom. The N multi-qubit rotation operator results in 2n degrees of freedom, 1 2 n − + (1+s) N∑+1 ⟨ ⟩ whereas the SU(N) operator will result in 2(2 1) degrees of 1 (N 1) 2 | | 𝟙 + (𝝓, 𝜽)1 | 𝖩1 (k)|(𝝓, 𝜽)1 𝖩1 (k) (117) freedom for n qubits. N N 2 N N N N k=1 While on the topic of the SU(N) representation of quantum states, it’s also worth noting a slightly different formulation; resulting in the general function SU(P, Q), that has the same number of degrees of freedom as

SU(P + Q). However, the geometry of SU(P, Q) differs signifi- systems, as well as the hybridization of the two. We will also cantly from the compact SU(N) geometry, resulting from a dif- mention ideas that haven’t yet been put into practice, but are ference in sign between the first P elements and the remaining worth discussion. Q elements in the metric, yielding a non-compact, hyperbolic Before the practical results, we will consider how some impor- geometry. The case of a generalized SU(P, Q) representation tant metrics, already present in the quantum information com- in phase space is still an open question. However the specific munity, can be described within the phase-space formulation. case of SU(1, 1) has been addressed in a recent paper.[87] The One metric that is unique to phase-space functions is the pres- SU(1, 1) representation of certain quantum systems has proven ence of ‘negative probabilities’. Due to the uniqueness and unin- to be useful in simplifying some particular problems in quan- tuitive nature of such a phenomena, this will receive a full discus- tum mechanics. In particular Hamiltonians involving squeezing. sion shortly, where we will discuss what negative values do and do This is because the SU(1, 1) displacement operator is a squeez- not say about a quantum state. Before looking at negative volume ing operator, resulting in the SU(1, 1) coherent states to actu- of the Wigner function as a metric, we will consider other figures ally be squeezed states. Since the squeezing of quantum states of merit that can be calculated through phase-space methods leads to many important results in quantum technologies, such as metrology in particular, the use of these phase-space methods for squeezing can prove to be a useful tool. 4.1. Measures of Quality The generators of algebra of SU(1, 1) are similar to SU(2) with a difference in the commutation relations, where There are measures that are widely used when considering the density matrix or state vector of a quantum state that can be ex- M M =− M M M = M [Kx ,Ky ] iKz , [Ky ,Kz ] iKx , pressed within the phase-space representation of quantum mechanics. In particular, considering the Wigner function and a key [K M,K M] = iK M (123) z x y property of traciality from S-W. 4, where The raising and lowering operators are then constructed = Ω Ω Ω ( ) Tr [AB] ∫ WA( )WB( )d (127) M =± M ± M Ω K± i Kx Ky (124)

M many similarities naturally fall out. The most natural starting that, along with Kz , constitutes an alternative basis, where the generalized displacement operator is point is to consider the expectation value of some observable A 휌 ( ) with respect to some state such that 휁 = M휁 − M휁 ∗ SM( ) exp K+ K− (125) ⟨ ⟩ = 휌 = Ω Ω Ω A 휌 Tr [ A ] ∫ W휌( )WA( )d (128) much like Equation (84), where 휁 is the complex squeezing pa- Ω rameter. The central result from ref. [87] the generalized displaced parity operator for SU(1, 1) is constructed from Equa- In the case of a qudit in SU(2), we note that the expectation value M tion (125) and the SU(1, 1) generalized parity (−1)Kz , yielding of the angular momentum operators can be related to the center of mass for the corresponding Wigner function, where √ M † 휋 휋 ΠM (휁) = S (휁)(−1)Kz S (휁) (126) ⟨ ⟩ + + 2 1,1 M M M (2j 1) j(j 1) 𝖩 (i) = f (휃, 휙)W휌(휃, 휙)sin휃 d휃 d휙 2 휌 8휋 ∫ ∫ i as the kernel for SU(1, 1). 0 0 Some of the authors from ref. [87] also looked at the equiv- (129) alence between SU(2) and SO(3) in phase space. Such analysis could also be used to go between SU(1, 1) and SO(1, 2) in where [95] phase space, or even generalized to SO(1, 3) or higher, to con- 휃 휙 = − 휃 휙 − 휃 휙 휃 sider relativistic quantum mechanics in phase space. There have fi( , ) { sin cos , sin sin , cos }, since been works that have looked at generalizing the Wigner function √ W𝖩M (Ω) = j(j + 1)f (Ω) (130) for relativistic particles, for instance see refs. [91–94]. Although 2 (i) i the results are interesting and potentially useful, it is beyond the scope of this paper to discuss them here. where the minus signs and normalization come from our formulation of the Wigner function kernel, which may diﬀer in sign is following a diﬀerent convention. 4. Quantum Technologies in Phase Space There are further important properties in quantum information that utilize the trace of two operators, as such their con- Given the mathematical framework, we will now look into how struction in the phase-space representation are straightforward. the representation of quantum systems in phase space can be A simple example of this is the purity of a state, which for density used to understand processes in quantum technologies. We will matrices is 𝖯(휌) = Tr[휌2], can be calculated in phase space by give some examples of experiments that have taken place to directly measure the phase-space distribution of quantum states. 2 𝖯[W휌(Ω)] = W휌(Ω) dΩ (131) This will include both discrete-variable and continuous-variable ∫Ω

휌 Next we consider the fidelity of a state 2 with reference to a target Fidelity is a measure of how close a measured state is to some 휌 state 1,where target state. We may conversely be interested in a distance mea- [ ] sure, that measures how far away (in Hilbert space) a given state √√ √ is from some target state. Such a distance measure is known as 𝖥(휌 , 휌 ) = Tr 휌 휌 휌 2 (132) 1 2 1 2 1 the trace distance, where [√ ] We are often interested in comparing an experimentally gener- 휌 휌 = 1 ‖휌 − 휌 ‖ = 1 휌 − 휌 † 휌 − 휌 휌 휌 dT ( 1, 2) 1 2 1 Tr ( 1 2) ( 1 2) (137) ated state 2 with a pure target state 1, in which case the fi- 2 2 𝖥 휌 휌 = 휌 휌 delity reduces to ( 1, 2) Tr[ 1 2], which when applied to Equa- √ √ tion (127) we simply yield − 𝖥 휌 휌 ≤ 휌 휌 ≤ − 𝖥 휌 휌 where 1 ( 1, 2) dT ( 1, 2) 1 ( 1, 2). [ ] For a qubit, Equation (137) can instead be expressed in terms 𝖥 W휌 (Ω),W휌 (Ω) = W휌 (Ω)W휌 (Ω)dΩ (133) of the Wigner function, where 1 2 ∫Ω 1 2 ( ) 1 [ ] 2 2 The phase-space calculation of fidelity leads to an interesting d (휌 , 휌 ) = W휌 (Ω) − W휌 (Ω) dΩ T 1 2 ∫ 1 2 method of fidelity estimation of measured quantum states. We Ω will provide the procedure how one can do that here, following ( ) 1 [ ] the work on Flammia and Liu.[71] 2 2 = W휌 −휌 (Ω) dΩ (138) First, it is important to note that the fidelity can be calculated ∫Ω 1 2 using any valid distribution F(Ω), including the Weyl function and the Q and P functions, so long as we have the corresponding This can be seen by noting the two states can be expressed as ̃ Ω dual function F( ) resulting in ∑ ∑ 휌 = 1 𝟙 + 1 휎i 휌 = 1 𝟙 + 1 휎i [ ] 1 ai , 2 bi , and so ̃ 2 2 i 2 2 i 𝖥 휌 , 휌 = F휌 (Ω)F휌 (Ω)dΩ (134) 1 2 ∫ 1 2 Ω ∑ 휌 − 휌 = 1 − 휎i 1 2 (ai bi) (139) where in this sense the Wigner function is self-dual, and the P 2 i function is the dual to the Q function, and vice versa. One drawback from the structure of Equation (134) is that we need to in- resulting in the trace distance being tegrate over the full phase space, however we know from Sec- ( ) 1 ( ) 1 tion 3 that only four points in the phase space are required to ⎡ ∑ 2 ⎤ ∑ 2 1 ⎢ ⎥ 1 produce an informationally complete function. Therefore for n d (휌 , 휌 ) = Tr (a − b )2𝟙 = (a − b )2 T 1 2 4 ⎢ i i ⎥ 2 i i qubits we require a sum over 2n points in phase space. We can ⎣ i ⎦ i then calculate fidelity through the sum ∑ (140) ̃ 𝖥(k) = F휌 (k)F휌 (k) (135) 2 1 k Similarly the Wigner function can be expressed √ ∑ Note that for this to hold, and for later calculations,∑ we also re- 3 2 = W휌 −휌 (Ω) = (a − b )f (휃, 휙) (141) quire the functions to be normalized such that (F휌 (k)) 1 2 i i i ∑ k 1 2 ̃ 2 i (F휌 (k)) = 1. k 1 Following ref. [71], we then need to create an estimator for from Equation (115) along with the properties in Equation (130). the fidelity. For this, select any measurement k with probability This results in = ̃ 2 Pr(k) (F휌 (k)) . The estimator is then ( ) 1 ∑ Ω 2 Ω= 3 − − 휃 휙 휃 휙 Ω F휌 (k) ∑ F휌 (k) [ ] [W휌 −휌 ( )] d (ai bi)(aj bj)fi( , )fj( , ) d 2 ̃ 2 2 ∫Ω 1 2 4 ∫Ω X = , such that E[X] = (F휌 (k)) = Tr 휌 휌 i,j ̃ 1 ̃ 1 2 F휌 (k) F휌 (k) 1 k 1 ∑ = 1 − 2 (136) (ai bi) (142) 4 i where E[X] is the expected value of X. When measuring in prac- therefore tice, calculating the function F휌 (k) perfectly is not possible, and 2 ( ) 1 휌 ( ) 1 so we need to make many copies of 2 and perform sufficient 2 2 ∑ measurements to assure a given confidence of fidelity. The de- Ω 2 Ω = 1 − 2 = 휌 휌 [W휌 −휌 ( )] d (ai bi) dT ( 1, 2) (143) ∫Ω 1 2 tails of which can be found in ref. [71]. 2 i Here we would like to note that besides being applicable to any number of qubits, such a procedure can also be used for any Such a procedure can then be generalized to multi-qubit states. number of qudits, where we can use the 𝔰𝔲(N)algebratogen- Further, similarly to the calculation of fidelity, we can exchange erate a qudit Weyl function or a set of informationally complete the integral over the continuous degrees of freedom by perform- measurements on the manifold of pure states in SU(N). ing a sum over discrete degrees of freedom. Where the procedure

Adv. Quantum Technol. 2021, 2100016 2100016 (17 of 29) © 2021 The Authors. Advanced Quantum Technologies published by Wiley-VCH GmbH www.advancedsciencenews.com www.advquantumtech.com can be applied to not only the Wigner function, but the all distri- resulting probability is always non-negative. The Wigner func- butions, albeit with a different normalization out the front. tion is unique among different phase-space distributions as it re- We also note that alternatively, one can find the Monge dis- sults in the correct marginals, see Equation (10). tance with respect to the Q function.[96] This effectively calcu- But, what does the negativity actually tell us when it appears in lated the distance between the centers of mass of two distribu- the Wigner function? It is generally accepted that it is a result of tions. Given the non-negative property of the Q function, it can quantum correlations, or quantum interference. By putting two be useful in translating procedures performed on classical proba- coherent states in a superposition, the coherences from the two bility distributions into quantum mechanics. As such, it can also states will interfere creating oscillating positive a negative inter- be used in calculating the Rényi entropy. The Rényi entropy is ference in between, that we interpret as quantum correlations. defined with respect to the Q function as [97,98] This has led to results where the negative volume of a Wigner distribution was calculated, where =− Ω Ω Ω ( ) SR ∫ Q휌( )logQ휌( )d (144) | | Ω Ω = 1 | Ω | Ω− V[W휌( )] |W휌( )|d 1 (145) 2 ∫Ω This result was extended to Wigner functions in ref. [99]; however, this is limited to the Heisenberg–Weyl case. We note here which can be seen as an additional measure of quality derived that in the case of SU(N), one can simply use Equation (119) to from the Wigner function. Such a measure can be used for any write Equation (144) in terms of the Wigner function. continuous distrbution, and even composite systems.[108] The above set of equations provide a framework that show The full story is that this negativity is actually a consequence of how the phase-space representation of quantum mechanics can non-Gaussianity, a result that is known as Hudson’s theorem.[109] prove a viable alternative to the more traditional density oper- Consequently negativity does not capture the important quantum ator approach. All of the above can also be extended for any correlations that arise from squeezing, in particular two-mode choice of phase-space function, such as the Q, P, and Weyl func- squeezing. Which, since the state is a squeezed Gaussian, has no tions. And where the equations were specifically states for qu- negative quasi-probabilities. So some care needs to be taken in dit states, general expressions can be found to calculate these using negative volume as a measurement of quantumness, non- values for SU(N) structures of even continuous-variable sys- classicality, entanglement and so on. That being said, it does not tems. Further to these measures there are other ways in which mean negativity should not be used, and there have been many phase-space methods have been useful to characterize a quan- results showing its use.[110–113] Itjustmaybemoreimportantto tum state, these are: measures of coherence[100,101]; phase-space look at the context in some cases, and look for how correlations inequalities[102–105]; and the negativity in the Wigner function, manifest in particular ways. which plays an important role. The manifestation of these nega- In the same way, the interpretation of negative values in dis- tive values provide nuanced details for each system. A closer look crete quantum systems needs some care. Any coherent, single- into negative values will be considered next. qubit state generated by the Stratonovich kernel will exhibit negative values. The maximum positive value is in the direction that the Bloch vector is pointing, as we deviate from that point over 4.2. Negative Probabilities phase space, the probability slowly decreases to zero and then negative, where the maximal negative value is at the point orthog- Negative probabilities have been a subject of interest since the onal to the Bloch vector. Because of the term “classical states” for early days of quantum mechanics. From the observation of neg- coherent states, this tends to lead to the confusion that coher- ative values in the Wigner function, Dirac later discussed their ent qubits states, such as the standard choice of basis in quan- | ↑⟩ | ↓⟩ presence along with negative energies in 1942.[106] Dirac stated tum information and , are classical. This term “classical” that negative probabilities “should not be considered as non- is just a misnomer, a qubit state is fundamentally non-classical, sense. They are well-defined concepts mathematically, like a neg- likewise with any quantum coherent state. By definition, any co- ative of money.” This growing acceptance to the concept of nega- herent qubit state is quantum; the only non-quantum state that a tive probabilities then lead to a number of other people to take the qubit can exist in is the fully mixed state. Negativity can therefore concept seriously. A more rigorous exploration was undertaken be used as a sign of quantumness, but it is not necessarily a sign two years later by Bartlett.[107] of entanglement. Later, Groenewold argued that to satisfy other desirable prop- One could also make the argument that this negativity can be erties of a quantum-mechanical probability distribution function considered a√ sign of superposition,√ as |0⟩ is a superposition of in phase space, it is necessary for negative probabilities to exist.[3] (|0⟩ + |1⟩ )∕ 2and(|0⟩ − |1⟩ )∕ 2, and in fact any pure state for In the 1980s, Feynman then gave a simple argument for the exis- a qubit is the superposition of two antipodal pure states. In this tence of negative probabilities in constructing a discrete Wigner way, the negativity is a result of this and the overall symmetry function.[14] Put simply, his argument was that negative probabil- of qubit states; this really shows the beauty of the Stratonovich ities are no more nonsense than a negative quantity of anything kernel, how it presents the symmetry in coherent qudit states, (in his example he gave apples), and that the negativity is just one where every N-level coherent state is just the SU(N) rotation of step in a total sum. The negativity is never considered in isolation. the chosen ground state. Similarly, any probability over phase space needs to be aver- If one decides that this symmetry is not so important and aged over a quadrature, due to the inherent uncertainty of quan- require that specific spin coherent states are non-negative, then tum states. When a measurement is made over a quadrature, the they can consider the Wootters kernel instead. The main benefit

Figure 2. Examples of the phase space distribution of the Dicke state |j, j − 1⟩, also known as the W state if considered as the symmetric state of 2j = 휃 휙 qubits, from ref. [60]. Note that the J in the figure corresponds to our j in the text. (a) shows the Dicke state for j 10 where F|W⟩( , ,s) shows that this is the s-valued phase-space function for the W state. As with Equation (102) s = 0 corresponds to the Wigner function and s =±1 refers to the P and Q functions, respectively. Intermediate values of s have also been taken in between. (b) shows the Wigner function where the value of j increases from j = 30 up to the limit j → ∞, showing that the W state tends toward the 1-photon Fock state as j → ∞. Reproduced with permission.[60] Copyright 2020, American Physical Society. to the Wootters kernel is that the eigenstates of the Pauli ma- state goes to the 1-photon Fock state at the limit j → ∞. Similar trices all produce non-negative Wigner functions. In fact, if an results can also be seen from taking the limit j → ∞ for any Dicke octahedron is drawn within the Bloch sphere, where the vertices state, where |j, j − n⟩ → |n⟩, the n-photon Fock state. are the eigenstates of the Pauli matrices, the discrete Wigner function for every state that lies on and within this octahedron is non-negative. These are the so-called magic states,[114] if 4.3. Methods for Optical Systems a pure state rotates between any of these magic states, then negativity arises—a result that is made clear by considering Phase-space methods have gained much attention in the optics the Wootters kernel as a subset of the Stratonovich kernel. community, where there is a large collection of books on quan- This gives rise to the literature on the equivalence between tum optics that include in-depth treatment of phase-space meth- Wigner function negativity and contextuality, and further work ods and the application thereof. This will therefore not be a thor- on magic state distillation, proving to be an advantage to choose ough account of the advancements made in quantum optics us- the Wootters kernel over the Stratonovich kernel.[25–30,115] The ing these methods, we will instead just point out some results negativity that arises from using Wootters kernel can further be we find particularly interesting and will prove useful later in this summarized by considering the extension of Hudson’s theorem section. For anyone who wants to read more deeply into the ap- for finite-dimension systems produced by Gross.[21] plication of phase-space methods in quantum optics, we sug- Alternatively, when using the Stratonovich kernel, one could gest the following heavily truncated list of books on quantum consider an analog of Hudson’s theorem to apply to the decreas- optics.[116–120] There have been further, in-depth reviews of the ing volume of negativity in the highest weighted state, |j, j⟩,as use of the Wigner function in optical systems[121,122] and in the j → ∞. At the infinite limit, the state becomes Gaussian and the use of optical quantum computing.[123,124] Heisenberg–Weyl group is reached—the highest weighted state When generating the Wigner function experimentally, there is simply the non-negative vacuum state. In the same way, by are many approaches one can take. The two we are interested in choosing the Dicke state with one energy level lower |j, j − 1⟩,as here are first the generation of the density operator and then fol- j → ∞ this slowly transforms into the one-photon Fock state, as lowing Equation (2) to produce the Wigner function. The second is shown in Figure 2. In Figure 2a, we see the phase-space repre- is to take direct measurements of points in phase space, by pass- sentation of the Dicke state |j, j − 1⟩ where j = 10; this also shows ing the need to calculate the full density operator. The real power how the state is represented in phase space for different values of measuring quantum systems in phase space comes in realiz- of s, including giving a good demonstration of how negative val- ing the displaced parity formalism of the kernel. By realizing the ues arise for different parameterizations. We then focus on the invariance of cyclic permutations of elements within a trace oper- Wigner function in Figure 2b, where we can see how this Dicke ation, it is clear that the Wigner function generation can be seen

Adv. Quantum Technol. 2021, 2100016 2100016 (19 of 29) © 2021 The Authors. Advanced Quantum Technologies published by Wiley-VCH GmbH www.advancedsciencenews.com www.advquantumtech.com as the parity measurement of a displaced state: 4.4. Measuring Qubits in Phase Space [ ] [ ] † By building on the techniques used to measure phase space for W휌(훼) = Tr 휌 Π(훼) = Tr 휌 D (훼)ΠD (훼) [ ] [ ] continuous variable systems, it is possible to visualize discrete = Tr D †(훼)휌 D (훼) Π = Tr 휌 (−훼) Π (146) variable systems in phase space. Some early work sought to first recreate the density operator for the state, before applying the appropriate kernel. One such example of this method was to show where D (−훼) = D †(훼), and 휌 (훼) = D (훼)휌 D (훼)† is the rotated the creation of an atomic Schrödinger cat state in an experiment density operator. using a Rydberg atom with angular momentum j = 25.[132] This This insight amounts to an experimental procedure of creating work takes the atom as a large angular momentum space, with a the quantum state, then displacing the state by some value 훼 and Hilbert space of dimension 51. then measuring this state. At this point the parity operator can Alternatively, one can measure a multi-qubit state by consid- be applied through classical computation of diagonal elements, ering the collection of symmetric subspaces that the full Hilbert or by following the procedure in ref. [125], one can perform a space is made up of, in a tomography method known as DROPS parity measurement by coupling the optical state to a two-level (or discrete representation of operators for spin systems).[82–85] atom, where the two levels act as the ±1 parity values. Other methods would rely on taking the j = n∕2 symmetric sub- Direct measurement of the Weyl function, on the other hand, space of an n-qubit system, the power of using the DROPS has not gained so much attention. Similarly to the measurement method is that you get much more information about the state— of the Wigner function, this can be executed by coupling to a including information about antisymmetric states such as the qubit, where instead of taking parity measurements, the qubit two-qubit singlet state, which cannot be represented by the j = 1 is measured in the x and y Pauli basis. Following the procedure symmetric subspace. Another benefit of this method is that the laid out in refs. [126, 127], we start by coupling the optical state, √ authors have made vast libraries of results available, including 휌 |+⟩ = | ↑⟩ + | ↓⟩ ∕ f with a qubit in the state ( ) 2, such that code to generate these functions.[133] The methods already mentioned rely on the full construction 휌 = |+⟩⟨+| ⊗휌 tot(0) f (147) of the density matrix for the state, requiring full tomography before a phase-space representation can be constructed. As the We then perform the unitary transformation number of qubits increases, this can get exponentially difficult, we therefore desire methods to represent a quantum state in phase space that don’t require full density matrix reconstruction. 휌 훼̃ = 𝖱 훼̃ 휌 𝖱† 훼̃ tot( ) ( ) tot(0) ( ), where Our main focus from now on is the Wigner function and to ( [ ]) 1 𝗓 † ∗ present results that utilize the displaced parity structure of the 𝖱(훼̃) = exp 휎 ⊗ a 훼̃ − a 훼̃ (148) 2 kernel, where we can imitate the Heisenberg–Weyl group result from Equation (146). By using the framework outlined in Sec- note that this is the Pauli z operator with the displacement operation 3, these results can be adapted into whichever phase-space tor with a factor of a half. Once the state has been prepared in the distribution is preferred, as they are all informationally com- appropriate point in phase space, 훼, the qubit is then measured plete representations of a quantum state. First we will consider in either the x or y basis, resulting in the simplest case of phase-space functions generated by a single SU(2) kernel construction from Equation (101). This will then be followed up with utilizing Equation (121) to show how a compo- 휒 (0) 훼̃ = ⟨휎𝗑 ⟩ + ⟨휎𝗒 ⟩ 휌 ( ) i (149) sition of single-qubit kernels can be used to measure the phase space of quantum states directly. ⟨휎𝗑 ⟩ = 휎𝗑 휌 훼̃ ⟨휎𝗒 ⟩ = 휎𝗒 휌 훼̃ where Tr[ tot( )] and Tr[ tot( )]. We note Adapting Equation (146) for use in general SU(2) systems that a similar procedure can be performed by coupling a qubit to a results in qudit to generate the Weyl characteristic function for qudit states. [ ] [ ] Once we have generated the Weyl function, we can use Equa- 휃 휙 = 휌 ΠM 휃 휙 = 휌 M 휙 휃 Φ ΠM M 휙 휃 Φ † W휌( , ) Tr 2 ( , ) Tr U2 ( , , ) 2 U2 ( , , ) tion (19) to perform a simple Fourier transformation to generate [ ] ⟨ ⟩ = 휌 휙 휃 Φ ΠM = ΠM any quasi-probability distribution function. An important part of Tr ( , , ) 2 2 휌 (휙,휃,Φ) the analysis of optical states is the investigation of different transforms of this Weyl characteristic function, in order to reveal cer- (150) tain quantum effects that are more difficult to see. One such case 휌 휙 휃 Φ = M 휙 휃 Φ †휌 M 휙 휃 Φ where this approach is insightful is when considering two-mode where ( , , ) U2 ( , , ) U2 ( , , ). The procedure is squeezed states. Such states are Gaussian and so have no nega- to rotate the generated state 휌 to a desired point in phase space, tive values, despite being highly entangled. Generating a charac- creating the state 휌 (휙, 휃, Φ), followed by taking the appropriate teristic function and then filtering to a quasi-probability can re- generalized parity measurement for the group structure. sult in negative values.[128,129] Allowing a phase-space analysis of When experimentally measuring phase space the way that such states.[102,130] A similar approach has also been taken for an- Equation (150) is practically implemented depends on many fac- alyzing discrete states.[131] An experimental way to achieve these tors, including the system that is being measured and how the results should be possible in combination with an extension of measurement results are collected. This means that practically, Equation (149). the rotation and generalized parity measurement needs to be put

Figure 3. Examples of phase-space measurement performed on a single qubit. (a,b) are results and figures from ref. [95], (c,d) are similarly from ref. [134]. In both cases, the Wigner function for a single-qubit state has been constructed, where (a) and (b) have created an eigenstate of 휎𝗒. (a) compares the results from experimental and theoretical construction. (b) shows the same state as it decoheres, plotting the fidelity over time, with snapshots of the Wigner function at three points of decoherence. (c) and (d) show an eigenstate of 휎𝗑, where similarly (c) shows the theoretical plot for this state. (d) shows the experimental results of points in phase space. The dots with error bars are the experimental results, where the curves are the theoretical values, showing good comparison between the experimental and theoretical values. (a,b) Reproduced with permission.[95] Copyright 2019, AIP Publishing. (c,d) Reproduced with permission.[134] Copyright 2018, American Physical Society. in the right context. Two examples of this when measuring a sin- of the Wigner function used throughout the text, where as the gle qubit can be found in recent works, ref. [134] and ref. [95], results in ref. [134] and ref. [95] differ by a factor of 2휋. where in ref. [134] the authors measured a two-level caesium Results from the two experiments are shown in Figure 3, atom, and in ref. [95] authors measure a state produced in a nitro- where Figure 3a,b show the results from ref. [95] and Figure 3c,d gen vacancy center in diamond. In both cases, they first produced are the results from ref. [134]. In ref. [95], they measured 1200 a state in a superposition of the excited and the ground state. points in phase space, where the values of 휃 are separated by Using Equation (150) by applying pulses to rotate this state by a step size of 휋∕60 and the values of 휙 have a step size of values of 휃 and 휙 to given point in phase space. Projective mea- 휋∕10. They also demonstrated the consequence of decoherence surements are then taken at each point in phase space, resulting on the Wigner function for this state, Figure 3b shows how the 휃 휙 =± ∕ in the spin projection probabilities pm( , )form 1 2. These phase space flattens as the fidelity decreases. Figure 3c,d show projective measurements can be expressed as the second experiment. Figure 3c is the theoretical calculation of the Wigner function. Figure 3d shows the experimental results, [ ] 1 1 † shown as dots with error bars, compared to the theoretical curves; p± 1 (휃, 휙) = Tr 휌 U (휃, 휙)P± 1 U (휃, 휙) (151) 2 2 2 2 showing good agreement between experiment and theory. Like with the Wigner function for optical systems, phase-space 𝗓 where the P± 1 are the projectors of the eigenstates of 휎 ∕2 with methods prove useful for showing the coherence in qubits. How- 2 eigenvalues ±1∕2, respectively. These can also be thought of as ever, for systems of qubits to be useful for the advancement of the diagonal entries of the density matrix 휌 (휙, 휃, 0) from Equa- quantum technology, it is important to consider multi-qubit sys- tion (150). tems. This is where the power of phase-space methods becomes The generalized parity is then applied to the spin projection more apparent. probabilities by calculating For a single qubit, full tomography is not that difficult a task, and requires fewer measurements than measuring all of phase space. However, when the number of qubits increases, the size ∑1∕2 [ ] of the Hilbert space increases exponentially, and so measuring W(휃, 휙) = p (휃, 휙) Π1 m 2 mm points in phase space becomes more of a viable alternative. m=−1∕2 (152) [ √ ( )] As we have already mentioned, there are two main approaches 1 to represent a multi-qubit state. We start with the symmetric = 1 + 3 p 1 (휃, 휙) − p− 1 (휃, 휙) 2 2 2 subspace, where we are limited to considering symmetric states of composite qubits. In many situations, symmetric states are ΠM th where [ 2 ]mm is the m diagonal entry of the generalized par- all that are desired from the output for use in computational ity operator. Note that here we are using the same normalization settings or metrology. Examples of such symmetric states are

Adv. Quantum Technol. 2021, 2100016 2100016 (21 of 29) © 2021 The Authors. Advanced Quantum Technologies published by Wiley-VCH GmbH www.advancedsciencenews.com www.advquantumtech.com coherent states[32–34]; GHZ states[135] or, more generally, atomic taking a generalized parity measurement of the resulting state. Schrödinger cat states[78]; W states, or Dicke states[80];and Each qubit then has two degrees of freedom from the local spin-squeezed states.[78,136,137] rotations, this then results in a function that has 2n degrees of It is therefore important to understand how we can directly freedom. measure the phase space of these larger Hilbert spaces, present- In order to make sense out of this many degrees of freedom, ing an alternative to traditional tomographical methods. It has it is necessary to choose certain slices of this high-dimensional already been noted that the Wootters-kernel Wigner function can phase space. The most natural choice is to take the equal-angle be seen as being a sub-distribution of the Wigner function gen- slice, this is where the number of degrees of freedom is reduced 휃 휙 휃 휃 = 휃 = ⋯ = erated with the Stratonovich kernel. Since both functions are to two, ( , ), where we set every equal, such that 1 2 휃 = 휃 휙 informationally complete, this means that by taking four care- n . Likewise, we set all the degrees of freedom to equal each fully chosen points in phase space, the full state can be recon- other. The reason this is a natural choice is because the result- structed. In fact, these points don’t need to be the four that map ing phase-space function shares similarities with the symmetric the Stratonovich kernel to the Wootters kernel, they just need to subspace variant. For example a j = n∕2 two-component atomic be appropriately distributed around the phase space. Schrödinger cat state is similar to the equal angle slice of an n- Similarly, it is possible to reconstruct the full quantum state, qubit GHZ state, where in both cases the Wigner function has and therefore the full phase-space distribution, by taking a subset two coherent states orthogonal to one-another, say at the north of phase-point measurements for any value of j.Itwasshownin and south pole, with n oscillations around the equator between ref. [60] that one can similarly reconstruct the full state by mea- the two coherent states. These oscillations are the signature of suring a finite number of points. This can be realized by general- quantum correlations arising from superposition in Schrödinger izing Equation (152) to an SU(2) Hilbert space of any arbitrary cat state and entanglement in GHZ states. In fact, there are sim- dimension, where ilarities between any symmetric n-qubit state in the equal-angle slice and the corresponding j = n∕2 function. It’s also worth not- ∑j [ ] W(휃, 휙) = p (휃, 휙) ΠM (153) ing that for an n-qubit Q function for a symmetric state, the equal- m 2 mm = ∕ m=−j angle slice is indistinguishable to the j n 2 Q function for the comparable state. note here we have restricted discussion to the Wigner function, The equal-angle slice for the Wigner function was also shown but this can easily be generalized to any phase-space function, to be useful in considering the graph-isomorphism problem.[142] see ref. [60] for more details. It was shown in ref. [142] that if you produce two graph By using the result from Equation (99), that the kernel can states[143,144] from isomorphic graphs, then their equal-angle be generated from tensor operators and spherical harmonics, slices are equivalent, since taking the equal-angle slice can be points in the phase space can be redefined in terms of a spher- thought of as an analogy of disregarding qubit, and in this ical harmonic expansion, with coefficients clm that correspond case node, order. It was further shown in ref. [142] that non- 휃 휙 to the spherical harmonic Ylm( , ). The full phase-space func- isomorphic graphs have distinguishable equal-angle slices. How- tion can then be reconstructed by calculating clm from the Wigner ever the results get more difficult to analyze as the number of + 2 휃 휙 휃 = function at (4j 2) points in phase space, W( a, b), where a qubits increase, as the detail in the Wigner function gets more 휋∕ + 휙 = 휋∕ + a (4j 2) and b 2b (4j 2). and more intricate as the size of the Hilbert space increases. Note that the 4j + 2 points are put as a lower bound, and more Apart from the equal-angle slice, there are many choices one points may be necessary given the intricacies of the state and may prefer to use, depending on whatever characteristic correla- phase space being measured. However, it is also noted in ref. tions may arise within the system of interest. And some of these [60] that this is just the case by equally distributing the angles different choices, along with the equal-angle slice, will be consid- over phase space resulting in bunching at the poles; using other ered in more detail here. methods to distribute points over the sphere, such as taking a Starting with the simplest case of a composite system, we will Lebedev grid[138–140] may reduce the number of points in phase consider bipartite entanglement between two qubits. Two qubits space needed for reconstruction. For example, taking the case famously exhibit entanglement in the well-known maximally en- where j = 1∕2 would result in 16 points in phase space, where tangled Bell states, explicitly we know from Equation (76) the full state can be reconstructed ⟩ ⟩ from 4 points in phase space, distributed as a tetrahedron over | 1 | 1 |Φ± = √ (|↑↑⟩ ± |↓↓⟩ ) |Ψ± = √ (|↑↓⟩ ± |↓↑⟩ ) (154) the surface of a sphere. It stands to reason that this reduction 2 2 in points will scale as we increase j. A method to reduce these points for specific states was demonstrated in ref. [141]. We add where | ↑↑⟩ is shorthand for | ↑⟩ ⊗ | ↑⟩. Apart from the equal- that since these (4j + 2)2 points in phase space constitute an in- angle slice, there are other slices one can take to gain an un- formationally complete set of measurements, one could then derstanding of the quantum correlations present within the sys- adapt Equation (135) to calculate the fidelity from these points tem. One such example was shown in ref. [48], where the slice 휙 = 휙 = 휃 휃 in phase space. 1 2 0 was taken and the resulting 1 and 2 degrees of Alternatively, to measure systems of multiple qubits, we can freedom were plotted against each other. An example of this slice consider a kernel that is the composition of multiple SU(2) ker- for |Φ−⟩ and |Ψ+⟩ from ref. [48] is shown in Figure 4a. Two- nels from Equation (121). As was shown in Equation (121), qubit Bell-type entanglement in the Wigner function manifests measurement of the phase space for such a composite state in this slice as oscillations between negative and positive quasi- can be performed by local rotations on each qubit and then probabilities in phase space. Figure 4a shows theoretical results

Figure 4. Examples of phase-space measurement performed on multi-qubit systems with a composite kernel calculation. (a–c) are results and figures from ref. [48], and (d–f) are from ref. [64]. In all cases, a composite generalized displaced parity form has been taken to measure the phase space of these states. (a) shows two two-qubit maximally entangled Bell states, where theoretical plots are shown, followed by simulated and experimental results that are both from IBM’s qx. (b) and (c) then show results of measuring a five-qubit GHZ state. (b) shows the equatorial slice – equally spaced points around the equator of the equal-angle slice. (c) shows the full theoretical equal-angle slice with chosen points in phase space measured. (d) and (e) are Q functions of points in a single-axis twisting evolution, where (d) are theoretical simulations and (e) are the experimental measurement results. The resultant states are a spin coherent state, a squeezed spin coherent state, followed by five-, four-, three-, and two-component atomic Schrödinger cat states in order from left to right. The last panel shows the corresponding Wigner function for the five-component atomic Schrödinger cat state. (f) shows the equivalent to the equatorial slice from (b) on N-qubit GHZ states where the 휎𝗓 operator is used as the generalized parity operator. (a)–(c) Reproduced with permission.[48] Copyright 2016, AIP Publishing. (d)–(f) Reproduced with permission.[64] Copyright 2019, The American Association for the Advancement of Science.

for what should be measured at 81 points in phase space for As the number of qudits increases, the choices of slice be- these two states, along with what was actually measured using comes more difficult, due to the increasing number of degrees IBM’s qx. of freedom. In this case, the n-qudits can be represented by n There are further ways to view phase space for bipartite entan- marginal distributions, however if there is any entanglement glement, for instance the approach in works such as ref. [145] present these become less useful. As a default, the equal-angle consider bipartite entanglement between two states that have slice is the simplest starting point to getting an idea of the corre- larger Hilbert space, demonstrating the entanglement between lations that manifest in phase space. two Bose–Einstein condensates. In ref. [145], the authors consid- Examples of the equal-angle slice have been shown to be use- ered the Wigner function of these states by taking a marginal ful in characterizing multi-qubit states in many experimental set- Wigner function and a conditional Wigner function. The tings, where there have been examples ranging from three-qubit marginal Wigner function is simply the Wigner function for one states up to 20-qubit multi-component Schrödinger cat states. In of the states, which can be calculated for either the reduced den- ref. [146], the authors created entangled states of three qubits, sity matrix or by integrating out one set of degrees of freedom generated by two photonic qubits and a further path-encoded qubit. Through this method they generated a cluster GHZ state [ ] Ω = 휌 ΠM Ω = Ω Ω Ω and a W state. W1( 1) Tr 1 2 ( 1) W( 1, 2)d 2 (155) ∫Ω As we move up to a five-qubit state in Figure 4b and Figure 4c, 2 we can see there are now five oscillations. When plotting on a 휌 = 휌 휙 휃 = 휋∕ where 1 Tr 2[ ] is the reduced density matrix for the first sub- sphere, the variation in the degrees of freedom for 2 lies system. Note that for a maximally entangled Bell states, taking the on the equator. This gives rise to the name “equatorial slice”. The marginal Wigner function results in a completely mixed state that equatorial slice on its own has been plotted in Figure 4b with has a value of 1∕2 everywhere, see the video in the supplemen- both theoretical and experimental results plotted, where there tary material of ref. [48] for an example of this. The conditional has been a least-squares best-fit curve plotted for the experimen- Wigner function is calculated by projecting a state onto the ba- tal data. By looking for these oscillations around the equator, this sis states of one of the subsystems, more details can be found in test serves as a verification of the generation of GHZ-type entan- ref. [145]. glement. Although there is clearly a lot of noise in the machine,

Adv. Quantum Technol. 2021, 2100016 2100016 (23 of 29) © 2021 The Authors. Advanced Quantum Technologies published by Wiley-VCH GmbH www.advancedsciencenews.com www.advquantumtech.com the oscillating behavior shows that a considerable about of GHZ- the full Wigner function of an atom. By taking this composite type entanglement remains in the state. approach there is much insight to be gained, from properties of Figure 4b considers this equatorial slice taking measurement light–matter interaction to other phenomena such as spin–orbit with the Wigner function parity operator. However, this may not coupling. In this section we will focus on the former application always be the most desirable choice to look for these oscillations, as it more directly applies to use in quantum technologies. since as the number of qubits increases, the amplitude of the There are many ways phase-space methods can be used oscillations steadily decreases. Alternatively, one can try different in discerning correlations that arise in the interaction be- ⨂choices of generalized parity operator, such as in ref. [64] where tween continuous- and discrete-variable systems. Like in Equa- 휎𝗓 was taken, as can be seen in Figure 4f. Experimental results tion (155) for two qudits, it is usual to simply consider the for taking the equatorial slice using this parity is shown for 10, 12, marginal functions for each system individually, tracing out ei- 14, 16, 17, and 18 qubits in Figure 4f, showing good agreement ther the discrete-variable system or the continuous-variable sys- between theory and experiment. tem. This is also implicitly the case in measuring the Wigner We note that the use of the 휎𝗓 operator as a parity operator function in qubit experiments when using quantum states of does not satisfy the Stratonovich–Weyl correspondence, requir- light to manipulate qubits. In theory, the qubit state should not ing some extra assumptions for informational completeness, see have any correlations with the optical state, however there will Equation (62) and the discussion around it. However, if we are always be some leakage, interpreted as decoherence in the qubit. measuring the state of a quantum system it is safe to make the On the other side, it is typical to just consider the optical state relevant assumptions—that the operator is trace 1. The kernel in a light-matter interaction. An important example is the gener- 휎𝗓 휃 휙 = 1 휃 휙 휎𝗓 1 휃 휙 † ( , ) U2 ( , ) U2 ( , ) can then be considered as a gen- ation of an optical Schrödinger cat state. In such experiments, it eralized observable, creating a characteristic function that has a is normal to entangle light with a two-level atom, by doing this Hermitian kernel, where the Pauli operators 휎𝗑, 휎𝗒,and휎𝗓 are one can then split the light into the superposition of two coher- yielded by setting (휃, 휙) = (−휋∕2, 0), (휃, 휙) = (−휋∕2, −휋∕2), and ent states, generating a Schrödinger cat state.[47] At the point of (휃, 휙) = (0, 0), respectively. This allows a direct comparison to ver- measurement of the optical state, the goal is to produce separa- ification and fidelity estimation protocols, for example, refs. [76, ble atomic and optical states, and therefore minimizing decoher- 147] and the fidelity estimation results in Section 4.1. ence in the Wigner function. Like in the previous case, there may The experiments from ref. [64] were performed using a be some correlations that exist between the qubit and the field, 20-qubit machine that generated entangled states by em- therefore it may be more desirable to have an idea of the correla- ploying a single-axis twisting Hamiltonian.[78] The single-axis tions that exist between the two systems. twisting Hamiltonian also creates different multi-component Different examples of how one can plot such hybrid states are Schrödinger cat states throughout evolution before reaching the shown in Figure 5. We present some theoretical methods dis- bipartite GHZ state. Figure 4d,e shows snapshots of different played in Figure 5. We will then discuss how some of these meth- points in the evolution in the Q function representation. Show- ods have been implemented experimentally and the future pos- ing the Q function for a spin coherent state and a spin squeezed sibilities of measurement of hybrid quantum systems. state, followed by a five-, four-, three-, then two-component We will first give examples of the results when taking the atomic Schrödinger cat state. These figures show the power of the marginal Wigner functions, calculated Q function to demonstrate the presence of spin coherent states, 2휋 휋 however it is lacking in the ability to show the level of quantum in- 훼 = 1 훼 휃 휙 휃 휃 휙 W( ) 휋 ∫ ∫ W( , , )sin d d (156) terferences between states. In the last panel, the authors show the 4 0 0 Wigner function for the five-component Schrödinger cat state, ∞ where the presence of negative values between each spin coher- 1 W(휃, 휙) = W(훼, 휃, 휙)d2훼 (157) ent state demonstrates the presence of quantum interference and 2휋 ∫−∞ therefore entanglement. For the separable states, |0⟩|↑⟩ and |0⟩|↓⟩,showninFig- ure 5a,b, the marginal functions are as expected, in both cases 4.5. Hybrid Continuous- Discrete-Variable Quantum Systems the HW Wigner function is the vacuum state and the qubit is in the | ↑⟩ and | ↓⟩ state, respectively. Figure 5c show√ the marginal We now look into ways to measure the phase-space distribution of Wigner functions for the state (|0⟩|↓⟩ + |1⟩|↑⟩ )∕ 2–ahybrid states that are hybridizations of continuous- and discrete-variable analog of a Bell state. As a result, the quantum correlations are systems. Many experiments and processes that consider the state non-local and the marginal Wigner functions are mixed states. of qubits also implicitly involve the presence of quantum light. Where the qubit state is maximally mixed, which can be seen in Likewise, many optical experiments also involve the interaction Figure 5c(ii) with a constant value over the whole distribution. with some kind of atom, whether or not artificial. This interac- Note that we have chosen an alternative projection to display tion is the foundation of quantum electrodynamics and circuit the qubit state. We have used the Lambert azimuthal equal-area quantum electrodynamics. projection,[150] in order to display the qubit completely. It is simi- Conversely, the continuous-variable Wigner function can serve lar to a stereographic projection with the north pole in the center, as the position and momentum of an atom, so creating a compos- however the south pole is now projected to a unit circle, rather ite function of discrete a continuous variables may be interpreted than to infinity. The function is then distributed around the disk as the atomic state depending on its position and momentum. to preserve the area—which is an important property when con- This was shown useful to consider in ref. [148] when looking at sidering probability distributions. This results in the equator of

Figure 5. Examples of hybrid continuous and discrete variable systems in phase space. We show various ways to display different states in phase space. | ⟩|↑⟩ | ⟩|↑⟩ (a) shows different representations√ of 0 . Likewise (b) shows different representations of 0 . The last state, which is shown in (c), is a hybrid | ⟩|↓⟩ + | ⟩|↑⟩ ∕ Bell state ( 0 1 ) 2. For each state there are five different plots. In each (i) we show the marginal function for the HW Wigner√ function, with colorbar values 휂 = 2, similarly in (ii) is the marginal Stratonovich Wigner function for the qubit where the colorbar values are 휂 = (1 + 3)∕2. We can see that for the two separable states in (a) and (b) that the two marginal functions are pure states, conversely we see in (c) that the entangled state produces mixed states in the marginal functions, where the qubit has a uniform value over the distribution. Note that the qubit has been plotted in a Lambert azimuthal area preserving map, see text for details. In (iii) we have plotted the density matrix for the qubit, where in each entry we show the HW Wigner function. Similarly in (iv) we have a composition of the discrete Weyl function where in each entry the HW Wigner function has been plotted, note that this corresponds to taking Pauli measurements of the qubit and then plotting the HW Wigner function, as a result the marginal HW 휂 = Wigner function is appears as the identity measurement of the qubit. In both (iii)√ and (iv) the colorbar values are 2. Finally in (v) we show a hybrid representation developed in refs. [148, 149] which have colorbar values 휂 = 1 + 3.

√ the sphere existing on the concentric circle with radius 1∕ 2. For decoherence in quantum systems by coupling a large atomic state a more in-depth explanation of this projection for use in phase- to an optical mode. However, this lacks the consideration of the space methods see the text and appendices of ref. [65]. non-local correlations between the two systems, and any entan- The Lambert azimuthal equal-area projection was also taken gled state will result in Wigner functions similar to Figure 5c(i) in ref. [81] where the authors also considered marginal Wigner and (ii). function for both subsystems. The work in ref. [81] considered the To get a sense of these non-local correlations a hybridization Tavis–Cummings interaction[79] between an optical cavity and an approach is needed. One approach has been to hybridize a atomic j = 5∕2 state initially in an atomic Schrödinger cat state. density matrix for a qubit with the Wigner function for the ﬁeld × By considering the Wigner functions of both systems, it is then mode. This results in a 2 2 grid, representing the entries of the insightful to visualize the transfer of the Schrödinger cat states qubit density matrix, where on each entry of the density matrix between the two systems. Demonstrating how one can overcome the corresponding Wigner function for the ﬁeld mode is shown.

We can label this Wigner function w(i, j), where i, j ∈ {0, 1}, ure 5b(v). Like with the Wootters Wigner function, when there such that are non-local correlations between the two systems, we see a de- [( ) ] pendence between the degrees of freedom. Where in Figure 5c(v) | w(i, j) = Tr |i⟩⟨j| ⊗ Π(훼) 휌 (158) we can see a radial dependence on the state of the qubit. All these methods provide different snapshots into such cou- where 휌 is the state for the full system. Such an approach re- pled systems and in combination build up an understanding veals much more about the non-local correlations between the into the of these types of non-local correlations. We also see cer- two systems, and can be found in works such as refs. [151, 152]. tain overlap between different methods of plotting such systems, Examples of this approach are shown in Figure 5(iii). Note that which further allows us to get a sense of the overall picture of the real value has been taken, which does not make a√ difference coupled quantum systems in phase space. for the first two states. However, (|0⟩|↓⟩ + |1⟩|↑⟩ )∕ 2 has an imaginary component on the off-diagonal entries that has been 5. Conclusions and Outlook ignored. Since we’re concerned with eigenstates of 휎𝗓, the local terms are shown in diagonal entries and the non-local correla- In this review, we have explored the phase-space formulation of tions can be seen in the off-diagonal entries, which is clearly seen quantum mechanics, considering how different systems can be in Figure 5c(iii). A similar approach was taken in ref. [153] where represented in phase space. Our focus was providing the frame- one can generate a nonclassicality quasi-probability matrix to ex- work to represent discrete-variable systems in phase space, in plore. particular the different ways single- and multi-qubit states can Alternatively one could take Pauli measurements of the qubit, be considered. and then calculate the HW Wigner function in that basis, the We provided the general framework to generate an informa- resulting distribution is shown in Figure 5d and can be consid- tionally complete phase-space distribution from any arbitrary op- ered as a hybridization of the discrete Weyl function with the HW erator, as well as how to transform between any of the phase- Wigner function, where space representations and how the dynamics of quantum sys- [( ) ] tems can be modelled in phase space. This was followed by ex-  훼 =  ⊗ Π 훼 휌 (z, x, ) Tr 2(z, x) ( ) (159) amples of discrete systems in phase space, with a focus on the construction on qubit and multi-qubit states. We have provided Similarly to the density matrix approach this produces a 2 × 2 a link between between the Stratonovich kernel and the Woot- grid, where this time the elements correspond to Wigner func- ters kernel for a qubit state. Given the resulting structure of a tion, yielding a joint quasi-probability distribution. For each state, simplex inside a sphere, this provides a path into alternative con- we take an identity measurement, this is equivalent to taking the structions of discrete Wigner function for qudits and symmetric marginal HW Wigner function, where the equivalence can be informationally complete states, where we can consider an N2- considered with Figure 5c(i). dimensional simplex inside the SU(N) N2 − 1-dimensional sur- Such a method has gained much interested in practice, where face. these exact states were considered in ref. [154], where the au- From the construction on Wigner functions for discrete quan- thors took measurements in the Pauli basis at different points tum states, we then explored how such distributions can be use- in position-momentum phase space. By building up measure- ful for applications to quantum technologies. Exploring how well- ment statistics each point in phase space was then represented known metrics in quantum information can be written in the lan- by the expectation value of the given Pauli operator at this point guage of phase space, as well as methods to use them in experi- in phase space. Similar results that consider entanglement be- mental settings. There are still more metrics than can be trans- tween a qubit and a Schrödinger cat states have also shown how lated into phase space, and perhaps even more that are peculiar this method is useful for characterizing entanglement.[155,156] to the phase-space representation, such as the negative volume Alternatively we can consider a different method to hybridize of the Wigner function. Wigner functions for the two systems. One could hybridize the We then considered direct measurement of quantum systems Wootters Wigner function with the HW Wigner function in in phase space, beginning with the phase-space reconstruction a similar way to coupling the density matrix, as was done in with single-qubit states before treating more complicated com- ref. [157]. However here we want to consider hybridizing the posite systems. Where we first looked at multi-qubit states, where Stratonovich–Kernel Wigner function with the the HW Wigner the dimension of the Hilbert space increases as 2N , whereas the function.[148,149] At each point 훼̇ we can plot the Wigner function degrees of freedom for a phase-space distribution only scale as for the qubit, producing a lattice of Wigner functions where 2N for N qubits. Therefore such methods can provide compu- [( ) ] tational simplifications in larger systems – especially if we are 훼̇ 휃 휙 = Π 훼̇ ⊗ Π1 휃 휙 휌 W( , , ) Tr ( ) 2( , ) (160) only interested in finding certain types of correlations within a larger state. We can then add transparency to each of the spheres according We then moved onto the current work of hybrid quantum sys- to the maximum value of the Wigner function at that point 훼̇ , tems. Although the distributions considered here and simple 훼̇ 휃 휙 max휃,휙 W( , , ), resulting in an envelope over the distribution. single-qubit states these provide an outline into how come can Examples of this are given in Figure 5a–c(v), where for the sep- adapt phase-space methods for composite systems with hetero- arable states we can see the envelope of the vacuum state. At ev- geneous degrees of freedom. We note that there is also some ex- ery discrete point in position and momentum phase space we ploration into how this can be extended to consider multi-qubit then see the | ↑⟩ state, in Figure 5a(v), and the | ↓⟩ state in Fig- states coupled to a field mode via a Tavis-Cummings coupling in

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Russell Rundle is currently a Research Associate at the University of Bristol, UK, where he researches verification methods for quantum technologies. Russell received his Ph.D. in 2020 from Loughbor- ough University where he researched quantum state visualization, verification, and validation via phase-space methods. His current research focusses on the verification and certification of quantum states and processes as part of the EPSRC Hub in Quantum Computing and Simulation.

Mark Everitt is a Senior Lecturer in quantum control and the director of studies at Loughborough Uni- versity, UK, and member of the Wigner initiative. He received his Ph.D. from Sussex University in 2000 where he researched quantum dynamics and measurement of single quantum objects. His research interests include phase-space methods, quantum systems engineering, open quantum systems, quantum chaos, and foundations of quantum mechanics, in particular the quantum to classical tran- sition.