Noise-tailored Constructions for Spin Wigner Function Kernels

The effective use of noisy intermediate-scale quantum devices requires error mitigation to improve the accuracy of sampled measurement distributions. The more accurately the effects of noise on these distributions can be modeled, the more closely error mitigation will be able to approach theoretical bounds. The characterisation of noisy quantum channels and the inference of their effects on general observables are challenging problems, but in many cases a change in representation can greatly simplify the analysis. Here, we investigate spin Wigner functions for multi-qudit systems. We generalise previous kernel constructions, capturing the effects of several probabilistic unitary noise models in few parameters.

The effective use of noisy intermediate-scale quantum devices requires error mitigation to improve the accuracy of sampled measurement distributions.The more accurately the effects of noise on these distributions can be modeled, the more closely error mitigation will be able to approach theoretical bounds.The characterisation of noisy quantum channels and the inference of their effects on general observables are challenging problems, but in many cases a change in representation can greatly simplify the analysis.Here, we investigate spin Wigner functions for multi-qudit systems.We generalise previous kernel constructions, capturing the effects of several probabilistic unitary noise models in few parameters.

I. INTRODUCTION
Quantum technologies promise increased capabilities in a number of areas, such as metrology [1], cryptography [2,3], physical simulations [4], and computational fluid dynamics [5].However, to see an advantage in many of these applications requires that the quantum device operate fault tolerantly and at a large scale [6], a challenge that is unlikely to be met with current architectures within the next five to ten years.Recently, noisy intermediate-scale quantum (NISQ) devices [7] have received significant attention, with much effort expended to show near-term quantum advantage.Several works [8,9] have claimed evidence of such advantage for certain highly particular problems, but the utility of these devices remains uncertain [10,11].
Error mitigation methods [12] have been developed to increase the accuracy of sampled quantities on NISQ quantum devices constrained in qubit number and circuit depth.The characterisation of noise and its effect on a quantum state is however itself a difficult problem [13].As a result, we often find either that the noise is symmetrized (and increased) via twirling, or that error mitigation techniques such as probabilistic error cancellation [14] or zero-noise extrapolation, which may in general involve some subtlety [15], are justified empirically rather than from first principles at the device level.
Common features of many noise channels are high levels of symmetry and the dominance of low energy components, and these properties recommend a signalsprocessing perspective of filters and window functions.In the case of the harmonic oscillator, we observe that noise processes are often fruitfully decomposed [16,17] as either displacements, rotations, or steps in photon number, and we would like to explore similar methods of decomposition for multi-qubit systems.Spin Wigner functions [18,19] have been developed as finite-dimensional analogues of the well-known quasi-probability distribution for the harmonic oscillator, and have previously been proposed as a visualisation tool and for verifying the * m.hanks@imperial.ac.uk preparation of quantum states [20,21].Moreover, the reference frames of more discrete variants of phase-space quasi-probability distributions [22] have been exploited for Clifford circuit simulation.We find in this paper that spin Wigner functions, with their significant freedom in parameterisation, can also provide a convenient representation for the treatment of probabilistic unitary noise.
The paper is laid out as follows: Section II reviews spin Wigner functions and higher-dimensional spherical harmonics.Section III explores the requirements of general spin Wigner function kernels, laying out a general form in terms of the spherical harmonics and discussing several example constructions.Section IV turns these constructions toward the key question of noise, discussing how the form of the kernel is altered when parameterising operations occur only over a restricted subgroup or subalgebra.Section V discusses the potential for insights in error mitigation, rescaling expectation values according to the form of the noise distribution.Finally, we conclude in Section VI by discussing the implications of our results, their scope, and possible next steps.

A. Spin Wigner Functions
The spin Wigner function [18,19], is a complete representation of an N -dimensional quantum state, defined as a kernel transformation from the density matrix ρ satisfying several key properties: 1.It must be an informationally complete quasiprobability distribution on the hyper-sphere, realvalued with a self-conjugate kernel operator.
2. Rotations of the spherical coordinates ξ correspond to unitary transformations of the kernel ∆ (ξ).
We reproduce the governing equations for these properties explicitly in Appendix A.
Diagonalising a mixed-state density matrix, ρ = ψ p ψ |ψ ψ|, the linearity of the trace allows us to similarly decompose the Wigner function as (1) If transitions between the states |ψ ψ| are in the group of kernel rotations defining our parameterisation, then the cyclic property of the trace allows us to further write with R ψ a coordinate rotation.In such a case, the mixture ρ is represented in the spin-Wigner representation by a convex sum over rotations of the Wigner function of some reference state.We note that this is only possible for probabilistic-unitary noise channels.Where a noise process can be represented as a convex combination of coordinate rotations, we can express it in terms of a convolution where we take g (x) → W ψ0 (x) and let the function f (y) return the probability density for an erroneous coordinate rotation y −1 .

B. Hyper-Spherical Harmonics
Real, square-integrable functions f : S p−1 → R on the (p − 1)-sphere (including the spin Wigner functions) can be expressed as sums over spherical harmonics [23]: A complete, orthonormal set of degree-n spherical harmonics has elements, and harmonics of distinct degrees are also orthogonal, giving us the coefficient expression Many functions will have a maximum degree n max , the bandwidth, above which all coefficients c n,j are zero.Following the convolution of two functions f and g, we observe due to orthogonality that there can be no surviving terms for harmonics with degree greater than As discussed in Appendix B, convolution may be treated independently for each degree n: Further, in certain cases we will be interested in noise distributions that are invariant under rotations about some reference point ξ 0 .For functions of this type, the standard convolution theorem applies (also Appendix B), such that we may take the simple element-wise product over coefficients c n,j .On S 2 , this subclass of distributions corresponds to those harmonic expansions containing only zonal harmonic functions.
Having introduced the key structures with which we will be working, we turn next to the question of how we can parameterise the spin Wigner function kernel, ∆ (ξ).

III. PARAMETERISATION VIA HARMONICS
Prior to using the spin Wigner representation, we must choose a kernel ∆ (ξ) and its coordinates ξ.Accounting for normalisation and a global phase, any pure, Ndimensional quantum state undergoing quantum operations in SU(N ) is completely characterised by 2 (N − 1) angles, which contain all the information of the N − 1 complex numbers in the standard state representation.It is however possible to construct informationally-complete Wigner functions using fewer than 2 (N − 1) parameters.For instance, the well-known Wigner function for the harmonic oscillator is capable of representing arbitrary rotations among the first N Fock states, and yet is fully parameterised with only two variables that may be expressed in angles as q = tanh (θ/2) and p = tanh (φ/2).
In general, the fewer parameters in ξ, the more complex structures will appear in the Wigner function (i.e. the higher the degree of the maximum harmonic).As one potential choice, consider following the final example from [19] and choosing the kernel where ΛN 2 −1 is the final, diagonal element of the generalised Gell-Mann matrices, under the standard construction.Parameterising the spin-Wigner function by a maximal independent set of 2 (N − 1) angles ξ = {θ 1 , φ 1 , . . ., θ N −1 , φ N −1 }, it then becomes equal (as noted in [19]) to the coherent-state based Wigner function of [24], which simply quantifies the projection of the density matrix onto the spin coherent state |ξ [25].These spin coherent states therefore form the only structures of consequence.While informationally complete, |ξ contains a number of parameters exponential in the qubit number, and an alternative qubit-local parameterisation with only 2 log 2 (N ) angles is described in [20].This alternative is more convenient for the study of qubit-local noise, and we build upon it in Section III B and use it in several of the examples to follow.We would, however, like to be able to tailor our parameterisation to various other noise models, and so we now turn to identify more general requirements of a valid parameterisation.

A. Requirements of Alternative Parameterisations
We can construct valid kernels from any complete, orthogonal, Hermitian basis { Ôi } for the traceless Hermitian matrices: (n,j) is some complete set of discrete, orthogonal functions over the indices (n, j).
There are three kernel requirements implied by the Stratonovich-Weyl relations, as summarised in Appendix A. Kernels in the form of Equation (11) satisfy the first requirement trivially, and the second by fixing C ∆ to achieve normalisation.The third requirement depends on how we define the relationships between operators and spherical coordinates.Since where R (n) are orthogonal matrices and g (i) (n,k) also form discrete orthogonal functions by the completeness of f (i) (n,j) .A rotation of coordinates R must therefore correspond to a unitary operation implementing the transformation For unspecified harmonic bases Y n,j , the only way to guarantee this property is to support the entire orbit of the unitary transformation from the operator Ôi on harmonics of the same degree n.We note that the unitary operations corresponding to coordinate rotations need not explore the entire state space, and in general need not even form a group in their own right.We next consider the structures of two known parameterisations, expressing them in the form of Equation (11) to show that they are equivalent.

Example 1: Kernel as Displaced Parity Operator
Our first example, introduced in [19] and described above in Equation (10), has the kernel with |ξ a spin coherent state and coefficients A, B chosen to preserve normalisation.Now, noting that [24] expanding the real, square-integrable function ξ| Λi |ξ in harmonics we have (n,j) again drawn from some complete set of discrete orthogonal functions, or with Dn,j defined to be The kernel is now which, with the identification Λi → Ôi takes the form of Equation (11).

Example 2: Generalised Displacement Operators
In [18] the authors introduced a kernel construction based on harmonic-weighted spin coherent states: with normalising constant C ∆ and operator basis We briefly recap why this construction satisfies the Stratonovich-Weyl conditions in Appendix C.
Because we have chosen to use real-valued spherical harmonics, the generalised displacement operators used as an operator basis here are Hermitian, and can therefore be expanded in any of the common bases for Hermitian operators.Choosing the generalised Gell-Mann matrices gives us directly Tr Dn,j Λi Y n,j (ξ), (22) which has the form we require on taking the substitution The orthogonality of the functions f (n,j) can be verified by exploiting the second orthogonality relation for the spherical harmonics in Equation ( 6 As described in Appendix C, this construction (and therefore also the previous one) avoids the problem of choosing functions f (i) (n,j) consistently across harmonics of different degrees by defining the coordinates of the harmonic functions and the spin coherent states to be covariant in the definition of Dn,j .This reduces coordinate transformations at all degrees to a change of coordinate for a single coherent state |ξ .As we will see in the sections to follow, when transformations of the kernel do not span the full quantum state space these co-variant states need not span the full space of SU(N ) spin coherent states.

B. Tensor Product Structures
When we come to consider noise models in Section IV, we will need to be able to define coefficients f (i) (n,j) consistent with unitary transformations restricted to operations induced by the noise itself.In many cases, this noise will operate locally on some subsystem, and in such cases we can induce a tensor product structure on the kernel that allows us to factorise f (i) (n,j) and solve for the factors independently in spaces of smaller dimension.
Take as an example the qubit-local parameterisation of [20]: This can be generalised straightforwardly to subsystems of any dimension with where again coefficients A and B are chosen to preserve normalisation, and |N is the lowest-weight state.
Distributing Û (ξ) across the tensor product structure of the separable state with |ξ k a spin coherent state for subsystem k.Expanding these states in lambda-matrices Λ (k) i k for each individual subsystem k and extending the index for the operator basis i → i 1 , . . ., i m , we then have where harmonics Y (k) n,j have been labelled with the subsystem index to indicate that they are defined over the restricted spherical space defined by coordinates ξ k The kernel is now which, with the identifications takes the form of Equation ( 11) (harmonics not decomposing in this product structure have zero-coefficients).

IV. EXAMPLE NOISE MODELS
We have now seen how the spin Wigner function may be parameterised using different operator bases and tensor product structures.In this section, we turn to consider several noise models and construct kernels to efficiently represent their influence on the quantum state.
A. Noise in a Lie Subgroup Suppose that operations generated by noise exist in a Lie subgroup (say, H) of U(N ), generated by an algebra over Hermitian elements { Ĥk } such that This algebra is clearly closed under Hermitian conjugation and so, borrowing from representation theory as applied in quantum error correction [26], the operator Hilbert space is isomorphic to such that for all k, L(A k ) forms an irreducible representation for the action of the algebra on A k , the action on B k corresponds to the identity, and Now, let {â k,i } and { bk,j } be bases of traceless Hermitian operators over A k and B k respectively, {ĉ (m) kl } the basis elements over the full Hilbert space that allow transitions between the k and l sub-spaces, and { dk } the basis elements inducing a relative phase between subspace k and all others.These operators give us a basis in which to expand the kernel.As described in Appendix D, decompose this expansion in noise and residual coordinates ξ and η respectively.This gives Here Q dk (η) are orthogonal functions in the residual coordinates η, since the operators dk are invariant under the action of the noise algebra.
Though Equation 34 appears quite unwieldy in its cur-rent form, an explicit, orthogonal expansion in terms of harmonics Y n,j ( ξ) is required only within groups of operators equivalent up to the action of the noise-algebra.Further, in practice we will only reference the coordinates η when integrating over the full space.These observa-tions allow the kernel to be replaced for our purposes with the simpler effective expression where and lex(i, j, k, . ..) returns the position in any lexicographic order over the argument-indices.To satisfy the orthogonality requirements of the coefficients f (n,l) , we may apply the generalised displacement operators of [18] (described in Equation ( 21)), but now with states defined in the reduced space of noise-induced rotations.

Example 1: Single Qubit Dephasing
As an example, consider single-qubit dephasing.In this case, the noise may be expressed as a classical distribution over evolution operators which clearly form a Lie subgroup.The Hilbert space decomposes as the tensor-sum of two one-dimensional spaces (the eigenstates of σz ), and the effective kernel becomes where the coefficients are derived from the circular space of states |θ = (e −iθ , e iθ ) T and harmonics sin(nθ), cos(nθ) as n,j Tr Dn,j σx Y n,j (θ) ∝ dφ δ(θ − φ) φ|σ x |φ In analogy with the properties of the spherical harmonics and spin coherent states, we observe that The anomalous case θ|σ z |θ = 0 does not cause an issue here, as we are only interested in generating orthogonal coefficients for σx and σy .We make the identifications There are no explicit âk,i or bk,j operators in this expression.This is a common feature when the noise forms a maximally commuting subgroup, since all generators may be simultaneously diagonalised and the structure collapses to

Example 3: Charge-preserving Transformations
Next we consider the group of transformations generated by operators of the form Since the factor terms in Û no longer commute, we no longer have that dim(A k ) = 1.The conservation of charge implied by the Jaynes-Cummings-like operator Ĵ(1) + nonetheless imposes a block-diagonal structure on the state space.Denoting the dimensions of the first and second subsytem d 1 and d 2 respectively, and the charge number J, the subspace A J has dimension ± act as ladder operators for a onedimensional chain of states, in the absence of Ĵ(1/2) z the subspaces A J with the same dimension could be identified.The multiplicity of dimension D J would then define the dimension of a corresponding invariant tensor product space B DJ .However, the inclusion of operators Ĵ(1/2) z breaks this additional symmetry.The charge is preserved within each subspace, so that the action of e iφ( Ĵ (1)   z + Ĵ(2) z ) on subspace A J is to add a global phase e iφJ .We can therefore make the substitution e iφ2 Ĵ (2)   z → e iφ2J e −iφ2 Ĵ (1)  z .Next, observing that Ĵ(1) + generate the group SU(2) in A J , within this subspace we use the SU(2) spin coherent states in D J dimensions [25], e ijφ2 e i(DJ −j−1)φ1 sin j (θ) cos DJ −j−1 (θ) The total state space is then generated by the action of Û on the highest-weight composite state The special cases J ∈ {0, d 1 + d 2 } span the kernel of the exchange operator, and are left invariant up to respective global phases.
Although the SU(2) spin coherent states in each subspace give respective resolutions of unity on integration, we cannot apply the method of [18] to the full space as described in Section III A 2. Constructing a pseudodisplacement operator Dn,j = dξY n,j (ξ)|ξ ξ|, does not allow us to construct orthogonal coefficients for all elements of a complete basis of traceless Hermitian operators in N = d 1 • d 2 dimensions.Nonetheless, we may identify subsets of basis operators for which orthogonal coefficients can be constructed.These correspond to the parenthetic sums in Equation (35), and include for instance the X-and Y-type lambda matrices for a given transition, which are related to one another through the phase rotations e iφ1 Ĵ (1)   z and e iφ2 Ĵ(2) z .Additionally, since we can have dim(A J ) > 1, there is now many-to-one relationship between the irreducible subspaces A k for the state space and for the basis operators.The particular orthogonality requirements for this construction are discussed further in Appendix E.

B. Noise Inducing only Limited Entanglement
When considering noisy gates that can generate small amounts of entanglement, consider the algebra of entangling operations generated in the small-angle limit, In this regime, and allowing conjugation by arbitrary local rotations, we now have operators of the form Û (η 1,2,3 , φ1,2,3 , θ 12 , θ 23 ) = where Ĉ(kj) X is the controlled-NOT gate from qubit k to qubit j.These operators are not closed under composition and so no longer form a group.This is a necessary restriction as arbitrary quantum circuits can be generated from the composition of one-and two-qubit gates.This section is the first where we have had to restrict the set of transformations so that they do not form a group.
Following the method of Vidal [27], consider a quantum state expressed recursively in the Schmidt basis λ [1]  α1 Φ [1]   α1 Φ [2,3] Γ [1]i1 α1 λ [1]  α1 Γ [2]i2 α1α2 λ [2]  α2 Γ [3]i3 α2 |i 1 i 2 i 3 . ( Here χ = 2 is the maximal Schmidt rank chosen to restrict the space of allowable states, λ [j] αj are the Schmidt coefficients for the jth round of decomposition, and Γ [j]ij α k αj are the coefficients of the computational basis states for qubit j within the Schmidt basis vector of index (α k , α j ).Before taking the sum over α j , the tensor Γ [1]i1 α1 λ [1]  α1 Γ [2]i2 α1α2 λ [2]  α2 Γ [3]i3 α2 (56) provides a convenient representation for studying the action of Û (η 1,2,3 , φ1,2,3 , θ 12 , θ 23 ) on the state space.Vectorising this tensor, we observe that the Schmidt coefficient pair λ [j] αj ∈{0,1} (for each j) is equivalent to introducing an entangled pair (57) across the jth partition, and that the sum over α j is equivalent to projecting this pair onto the fixed state of zero relative phase in the Fourier basis.Equation 54 contains only CNOT entangling gates, which may be generated with local operations and the consumption of a maximally-entangled state Let the density matrix corresponding to this Bell state be denoted ρΦ + .Transform an initial n-qubit density matrix ρ according to and observables of interest according to The action of the noise is now completely described within a tensor product coordinate representation of the kind described in Section III B, where the local qudit systems are composed of a single computational qubit alongside one ancillary qubit from each of up to two Bell pairs.The dimension of this coordinate representation continues to scale linearly with the number of computational qubits, but also increases now with the maximum allowable Schmidt rank.

V. ERROR-MITIGATION VIA RESCALED EXPECTATION VALUES
The space of observable expectation values, their joint numerical range [28] is a convex (N 2 − 1)-dimensional region with axes corresponding to Hermitian-operator basis elements and a boundary shape governed by their commutation relations.As noise affects a quantum state, it distorts and shrinks this region, altering measured values.In [29], the authors note that the simplicity of global depolarising noise allows the distortion to be inverted, correcting the mean at the expense of increasing the sample variance.Decomposing Ô in a basis of measurement operators [30] could potentially provide similar expressions for a wider range of noise models.For general probabilistic unitary noise, however, the expectation value becomes an arbitrary mixture over the basis elements, and can therefore not be reconstructed without knowledge of the full state ρ.The Stratonovich-Weyl conditions (Appendix A) require that the expectation value of a basis operator can be expressed as In the spin Wigner function picture Equation (60) becomes with f (y • ξ 0 ) a probability density function for erroneous rotations as described in Section II.
From the form for the kernel in Equation 34we see that, just as expected from the operator representation, the number of additional basis operators that must be measured to rescale the expectation value corresponds to the size of the equivalence class of such operators under the action of the noise operator algebra.While we could therefore work directly in terms of these operator bases, the spin Wigner function picture becomes convenient when the noise may be expressed concisely in terms of harmonics Y n,j rather than in terms of basis operators.In particular, as we note in Appendix B, when a noise distribution f (y • ξ 0 ) is symmetric with respect to rotations about some principal axis, its convolution may be applied element-wise.Such symmetry indicates that the noise has a depolarising structure over some subspace (all rotations in SU(N − 1) about this axis have equal probability), and as a result the expectation values for basis operators with support in this subspace remain independent.
The simplest examples are cases of pure depolarising channels.In the global depolarising model, with probability 1 − p the state is left unchanged, while with probability p the state is reduced to the maximally mixed state.The maximally mixed state consists only of the zerothorder harmonic, so that the effect of depolarising noise is to re-scale all other harmonic coefficients by (1 − p).For this model the parameterisation of the Wigner function is irrelevant.For local depolarising noise the coefficients associated with all non-trivial basis operators or harmonics for qudit k are reduced by a factor p k .This leads to exponential decay in expectation values according to the number of qudits in the support of each basis operator.Labelling the set of qubits in the support of operator Ôi by S Ôi , the scale factor for the coefficient of Ôi is Beyond depolarising noise, the next level of complexity arises from noise depending on a single angular variable of zero mean in each of several tensor-product factor spaces.This is the case for the first two examples of Section IV A. Since the standard convolution theorem applies to functions on the circle, we have that the corresponding circular harmonics are reduced element-wise, and the operator coefficients therefore decay independently of one another.Re-scaling operator expectation values becomes a matter of determining this rate of decay from the coefficients f (i) (n,j) of each of the basis elements in our observable of interest.
As a brief example of a tractable circular noise process, consider again the exchange interaction discussed in Section IV A 3. If we remove the dephasing terms and assume that the rate of exchange is state independent, then within each constant-excitation subspace the smallangle exchange operator is represented by a tridiagonal Toeplitz matrix with eigenvalues [31] For larger angles this implies Now, just as in Section IV A 1 we construct the states in the eigenbasis of the exchange operator, and use the circular harmonics sin(nθ), cos(nθ) to define pseudodisplacement operators Dn,f∈{sin,cos} = dθ f (nθ)|θ θ|.
From these we derive coefficients for sets of traceless Hermitian basis elements related to one another through the action of the exchange operator.With the kernel thus defined, and supposing the noise distribution is Gaussian for simplicity, the Fourier transform would return a predictable Gaussian shape in the frequency basis, and this could be used to re-weight the circular-harmonic coefficients of W Ô (θ).

VI. DISCUSSION
In this work we have explored spin Wigner function parameterisation to efficiently represent and/or decompose the effects of noise on a quantum state or operator.Our results are built around the key observation that, since the noise operator algebra adopts the block diagonal structure represented in Equation 32, the spin coherent state construction pioneered by Brif and Mann [18] may be separated for each independent subspace.Using spin coherent states defined over smaller subspaces allows us to maintain the Stratonovich-Weyl conditions while reducing the number of parameters required to represent probabilistic unitary noise.In this manner we unify the description of several proposed kernels in the literature under a general form in Equation 11, which allows the substitution of any traceless Hermitian operator basis.We describe explicit efficient kernel constructions for several noise processes that are local or slightly-entangling, or which satisfy specific rotational symmetries.
We expect the additional freedom we have introduced in the choice of parameterisation to find application in the error mitigation problem of inverting noise maps to correct observable expectation values [29].Near term quantum systems often display noise that is either highly biased [32] or that quickly approaches a depolarising channel as circuit depth increases [33].Though we find reflections of several important properties that also appear in the operator-basis picture, we conjecture that spin Wigner functions could be useful in identifying depolarised subspaces in inhomogeneous circuits, for which the inverse map simplifies.
We identify circular and rotationally symmetric noise distributions as particularly convenient for the rescaling of harmonic coefficients.A typical application of zeronoise extrapolation [15] approximates circuit noise as a single-parameter function, and we believe it could be an interesting question to explore the relationship between the accuracy of zero-noise extrapolation and symmetries in the noise distribution of the kind discussed in this work.
There are a number of directions that could be pursued as extensions of this initial study.For instance, though we have focused on the generalised Gell-Mann matrices as an operator basis, different bases may be more or less convenient for the description of different noise processes.We also leave to future work the question of other conditions under which the element-wise convolution theorem might be applied, or for which the more general degreewise convolution discussed in Appendix B might be further restricted.Finally, kernel constructions exploiting a tensor product structure are capable of expressing correlated rotations between subsystems, and so another area of interest will be the construction of logical spin Wigner functions for encoded quantum states, perhaps built on a modular subsystem decomposition of the kind introduced in [34].
Taking η k → ξ 0 , we find that each integral in the convolution is equivalent to Rp Y n,j (y) P n ( yξ 0 , xξ 0 ) dy, (B10) which yields a simple multiple of Y n,j (x), because for any spherical harmonic Y n (ξ) [23], The standard convolution theorem therefore applies.For functions of this type, the choice of basis Y n,j is irrelevant; we always have c n,j = a n Ωp−1 N (p,n) , ∀j.On S 2 , these distributions correspond to harmonic expansions containing only 'zonal' harmonic functions.On S 1 , all functions obey the standard convolution theorem.

Appendix C: The Harmonic Displacement Kernel Construction
In [18], a quite general construction for the spin Wigner function kernel ∆(ξ) was proposed in terms of spin coherent states and spherical harmonics.In this paper we use an alternative convention for the spherical harmonics that leaves them real rather than complex, so we briefly recap the construction below under this convention.This leads to a few slight differences, such as the displacementequivalent operators Dn,j becoming Hermitian.
Let G be the group SU(N ) of operations on a quantum state, and H the isometry group SU(N − 1) leaving a single representative state vector |ψ 0 (which we will take to be the highest-weight state) invariant.The coset space X = G/H defines the coordinate space for the spin coherent states |ξ [25], and is isomorphic to the (2N − 1)-sphere: The spherical harmonics Y n,j (ξ) have been discussed in Section II.Define the following operators: which are simply the coherent states weighted by a spherical harmonic.The claim is that the kernel Y n,j (ξ) Dn,j , with some constant normalising factor C ∆ , satisfies the Stratonovich-Weyl conditions outlined in Appendix A: As the harmonics are real, it is straightforward to see that the kernel is Hermitian (as indeed are the Dn,j operators).Integrating over the coordinates, normality is satisfied as for some constant C ∆ (via the resolution of unity for the spin coherent states).For the final condition of the kernel, covariance, we have Now, for any rotation g of coordinates on the spherical coset space we have that [23] Y n,j (g for some orthogonal matrix [C lj ] depending on g.Further expanding the kernel, this give us   |η η| .

(C2)
The summation now appearing inside the integral corresponds to the transposed (i.e.inverse) transformation matrix.Noting that the integral is over the Haar measure, we obtain so that the covariance requirement is satisfied.
(n,j) Y n,j (ξ) .(D1) Our goal will be to decompose this term to extract the dependence on a subset ω of the angles in ξ (and we will denote the residual set of angles η = ξ/ω).We first note that the spherical harmonics Y n,j are homogeneous polynomials of degree n, and we may therefore write where |ω| is the dimension of the space spanned by the angles ω, g n−k (η) and h k (ω) are real homogeneous polynomials of respective degrees n − k and k, and the range of l is determined by the number of distinct monomial terms of degree k over |ω| elements (the stars-and-bars expression).Since Y n,j (ξ) is real and square-integrable and the h k,l (ω) are linearly independent monomial terms of maximum degree n, h k,l (ω) are also real and square integrable.They may be therefore be expressed again as linear combinations of spherical harmonics of degree k: The g n−k,l (η) may also be decomposed in spherical harmonics, and making the replacement n This is the form of the decomposition used in Equation (34).We know that the coefficients of distinct operators are orthogonal on integration over all angular parameters.This implies that  In Section IV A 3 we introduced pseudo-displacement operators based on the tensor sum of SU(2) spin coherent states of varying dimension, noting that coefficient expansions in these operators could only be taken within the parenthetical sums of Equation (34), rather than over the full set of basis operators.In this appendix, we confirm that for the pseudo-displacement operators constructed from Equation (51), the orthogonality relations [24] dξ ξ| Λi |ξ ξ| Λk |ξ ∝ δ ik , (E1) now only hold within restricted subsets { Λk }.

General Orthogonality Relations
For G a compact group, π α (g ∈ G) a complete set of irreducible representations with dimensions d α and φ α v,w (g) = v, π α (g)w the matrix coefficients for representation α, the Schur orthogonality relations tell us For SU (2), the decomposition of this tensor product structure into irreducible subspaces is known and is given by Wigner D-matrices of linearly increasing dimension, with elements weighted by the Clebsch-Gordon coefficients [35]: ξ ⊗ U However, operators in different irreducible subspaces, and transition operators between them, are not guaranteed to be orthogonal under the action of U ξ .The range of integration is over the full group SU(K), including the isotropy subgroup leaving the initial state |N invariant.
Appendix D: A Noise-Restricted Kernel Consider a single term from Equation (11), Ôi n,j f

)
Looking back at the initial Equation (D1), we now have Ôi Y k,m (ω), (D4) and we may re-arrange the order of summation to obtain Ôi j) g n−k,l (η)   Y k,m (ω).(D5) D7) are discrete orthogonal functions when summed over k and m.
Appendix E: Orthogonality Relations for aTensor-Sum of Spin Coherent States