Complete correlation characteristic (Weyl) functions for any quantum ...

Complete correlation characteristic (Weyl) functions for any quantum system or ensemble R. P. Rundle,1, 2 Todd Tilma,3, 1 V. M. Dwyer,1, 2 R. F. Bishop,4, 1 and M. J. Everitt1, ∗ 1

arXiv:1708.03814v1 [quant-ph] 12 Aug 2017

Quantum Systems Engineering Research Group, Department of Physics, Loughborough University, Leicestershire LE11 3TU, United Kingdom 2 The Wolfson School, Loughborough University 3 Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8550, Japan 4 School of Physics and Astronomy, Schuster Building, The University of Manchester, Manchester, M13 9PL, United Kingdom (Dated: Tuesday 15th August, 2017) The Weyl function in quantum mechanics is usually introduced as a Fourier transform of the Wigner function. The Weyl function hence plays a secondary role to the Wigner function. Nevertheless, it finds application as a method of identifying non-classical correlations in quantum states. Here, by treating it as a primary object in its own right, we show that it is possible to define a continuous Weyl function for discrete systems that is a direct analog of the Weyl function for continuous systems parameterized in terms of position and momentum. We show that it is possible to define an informationally complete characteristic Weyl-like function for any quantum system. We also show that our characteristic function shares many properties and features in common with the usual Weyl function and we provide examples for spin Schr¨ odinger cat states in orbital angular momentum and an ensemble of spins.

I.

INTRODUCTION

The field of quantum physics is undergoing rapid expansion, not only in such high-profile applications as those promised by quantum information technologies, but also in such foundational areas as quantum thermodynamics. Wigner was motivated by the latter context in his seminal work “On the Quantum Correction For Thermodynamic Equilibrium” [1], where he defined the function that now takes his name. However, the original Wigner function, and its extensions [2–10], are now also finding great utility in the former context. As to the latter context, there has been some very interesting recent work in the quantum thermodynamics of correlated systems (see, e.g., [11–14]). Here quantum correlations, being of a fundamentally different nature to their classical counterparts, play a central role. The Wigner function plays the role of a quantum version of a probability density function that has been extended to include quantum correlations, which it does in a very intuitive way. In classical physics and information processing, the Fourier transform of the probability density function, which is given a number of different names but always characterizes correlations, plays a very important role. The quantum version of this function is the Weyl function that is the Fourier transform of the Wigner function. This function is defined for continuous systems parameterized in terms of position and momentum (such as the harmonic oscillator). Here we introduce an informationally complete generalization of the Weyl function that can be defined for any system or set of systems. This function is then a complete quantum correlation characteristic func-

∗

[email protected]

tion that will find utility in all areas of quantum physics, from thermodynamics to computing, where it is important to understand correlations. To support this claim we briefly review some of the applications of Fourier transforms of probability density functions under their various pseudonyms. To begin, a characteristic function may be used to define completely the probability distribution of a real random variable. Indeed, it always exists, even if a probability density function does not. If a probability density function does exist, then the characteristic function can be given as the Fourier transform of the probability distribution function. The application of these Fourier transforms has found a wide range of uses in different fields, including thermodynamics, signal processing, and fluid mechanics, where these functions can also reveal properties such as static structure, compressibility, entropy, pressure, free energy and microscopic structure. As already mentioned, for quantum systems the characteristic function [15] (here the Weyl function) can be found by taking the Fourier transform of the Wigner function, which for spin systems has been attempted in [16] from a spin Wigner function created from the trace of a density matrix with delta functions. Characteristic functions have also been widely used in signal processing, where the Fourier transform of the time-frequency Wigner function generates the socalled ambiguity function [17, 18]. This ambiguity function is often reduced to produce an autocorrelation function by taking the slice where the Doppler shift is zero, a reminder of the original signal processing applications in Radar. A generalized quantum analog of the ambiguity function, valid for systems of qubits and more general qudits is hence likely to be of great utility in quantum information. Our aim here is to provide just such a generalization, by treating the ordinary Weyl function as a primary object in its own right, and by showing explic-

2 itly how the notion of Fourier transformation, which links it to the ordinary Wigner function, can be suitably extended to achieve that goal in practice. In so doing we will also make strong connections with recent work [2, 3] on extending the standard Wigner function to spin and more general qudit systems. II.

A REVIEW OF WIGNER AND WEYL FUNCTIONS

We first recall that the Wigner function, Wρ (z) ≡ Wρ (q, p), is a special case of the more general WeylWigner transformation that describes how to transform a Hilbert space operator Aˆ to a classical phase-space function WA (z) ≡ WA (q, p) [19–23],   Z +∞ Q −ipQ/~ Q WA (q, p) ≡ dQ q + Aˆ q − e , (1) 2 2 −∞ √ where z ≡ (q + ip)/ 2. We regain the standard Wigner function Wρ (q, p) by replacing Aˆ with the density operator ρˆ. Because ρˆ is Hermitian Wρ (q, p) is real-valued. It has been observed [24, 25] that Eq. (1) can be recast in the form, h i ˆ Wρ (z) = 2 Tr ρˆ Π(z) , (2) using the expectation value of a displaced parity operator, ˆ ˆ Π ˆD ˆ † (z), Π(z) ≡ D(z)

(3)

ˆ where D(z) is just the Weyl displacement operator, ˆ D(z) ≡ exp[(zˆ a† − z ∗ a ˆ)/~],

(4)

defined ˆ ≡ (ˆ q+ √ in terms of the annihilation operator, a iˆ p)/ 2, (ˆ q and pˆ are the usual position and momentum operators) and where [24] ˆ ≡ exp(iπˆ Π a† a ˆ) =

∞ X

n

(−1) |ni hn| ,

(5)

n=0

is the usual parity operator (written here in terms of Fock states of the harmonic oscillator). An alternative form for the displacement operator, easily derived from the definitions of z and a ˆ, is ˆ ˆ p) = exp[i(pˆ D(z) ≡ D(q, q − q pˆ)/~].

(6)

The density operator itself can be fully reconstructed from its Wigner transform by inverting Eq. (2) to give Z 2 ˆ ρˆ = d2 z Wρ (z)Π(z), (7) π where d2 z ≡ d(|Re z|) d(|Im z|) and the integral is over the entire complex z-plane. Displacement operators are

nice because they are naturally related to coherent states ˆ |zi via action on the vacuum state |0i: D(z) |0i = |zi [26, 27]. Furthermore, by using displacement operators, we get parity operators that have a familiar quanˆ |zi = |−zi [24, 25]. tum mechanical property: Π Recently, we have shown that an expression in the same form can be used to generate Wigner functions for spin systems [2, 3]. Specifically, the displacement operator is replaced by a generalized rotation operator and the usual parity replaced by an appropriate, generalised, parity (The conditions that need to be met were given by Stratonovich and Weyl - see [19, 28] for more details). This has led us to ask whether we could take a similar approach for the counterpart Weyl functions. The usual Weyl function is the two-dimensional Fourier transform of the Wigner function, Z ˜ ρ (z) ≡ d2 z Wρ (z) eiz·Z/~ , W (8) where√z · Z ≡ (Re z)(Re Z) + (Im z)(Im Z). If we define Z ≡ 2(−P + iQ), so that z · Z = −P q + Qp, we can also obtain from Eq. (1) the explicit form for the Weyl ˜ ρ (z) ≡ W ˜ ρ (Q, P ), function, W +∞

  Q −iqP/~ Q dq q + ρˆ q − e , 2 2 −∞ (9) which bears a remarkable similarity to Eq. (1). Clearly, q and P are dual variables in the Fourier transform sense, and the same is true for p and Q. The variables Q and P may also be interpreted as position and momentum increments, with the Weyl function accordingly interpreted as a generalized correlation function [29, 30]. It is interesting to note that the Weyl function can also be written as [24] ˜ ρ (Q, P ) = π~ W

Z

h i ˜ ρ (Z) = π Tr ρˆ D ˆ (iZ/2) , W

(10)

which is just the analog of Eq. (2) for its Wigner counterpart. Indeed, Eq. (10) may simply be taken as the definition of the Weyl function. From the explicit form of Eq. (4), we see immediately that Eq. (10) may be equivalently written as h i ˜ ρ (Q, P ) = π Tr ρˆ D(−Q, ˆ W −P ) . (11) The original density operator can be recovered by inverting Eq. (10) to yield Z 1 ˜ ρ (Z)D ˆ (−iZ/2) , ρˆ = d2 Z W (12) (2π)2 where the integral is over the entire complex Z-plane. It is clear that Eq. (12) is just the analog of Eq. (7) for its Wigner transform counterpart. Furthermore, it is easy √ to see from the definition of Z ≡ 2(−P + iQ) and the

3 explicit form of Eq. (4) that an alternate form for Eq. (12) is ρˆ =

1 2π 2

Z

+∞

Z

+∞

˜ ρ (Q, P )D(Q, ˆ dP W P ).

dQ −∞

(13)

−∞

Here, the normalization is modified from that in Eq. (12) √ by the factors of 2 in the definition of Z. To create a Wigner function, we require both notions of displacement and parity. It is interesting to note that a Weyl function instead requires the dual notions of displacement and Fourier transformation. For continuous ˆ is the square of the infinite systems the parity operator Π ˆ = Fˆ 2 , so there is a deep connecFourier operator Fˆ , Π tion between the two. From Eq. (5) the Fourier operator may be explicitly defined as ∞
X Fˆ ≡ exp iπˆ a† a ˆ/2 = in |ni hn| .

(14)

III. FIRST STEPS TO A GENERALIZED WEYL FUNCTION: ORBITAL ANGULAR MOMENTUM

Considering the case of SU(2) first, we begin by looking at the generalized rotation operator proposed by Arecchi [32] and expanded on by Perelomov [33],   ˆ R(ξ) = exp ξ Jˆ+ − ξ ∗ Jˆ− . (17) for ξ = θe−iϕ /2, where ϕ is the azimuthal angle, θ is the ordinate, and Jˆ± ≡ Jˆx ± iJˆy . An equivalent form of Eq. (17) is easily seen to be h  i ˆ ˆ R(ξ) ≡ R(ϕ, θ) = exp iθ − sin ϕJˆx + cos ϕJˆy . (18) The connection with our past work [2, 3] can be found by noting that this rotation operator can be written in terms of an Euler angle decomposition that was employed there, according to

n=0

ˆ ˆ (ϕ, θ, −ϕ), R(ϕ, θ) = U

The defintion of Eq. (14) immediately implies that Fa ˆ† F † = i a ˆ† and F a ˆF † = −i a ˆ. For an arbitrary funcˆ ˆF ≡ tion T, we may define its Fourier transform as T † ˆ ˆ ˆ F TF . From this we get ˆ = T(ˆ ˆ q , pˆ) ⇐⇒ T ˆ F = T(ˆ ˆ p, −ˆ T q ).

(15)

It is clear from Eq. (15) that Fourier transformation is equivalent to a rotation of the q − p plane about an axis perpendicular to it by an angle of π/2. Indeed, this simple observation will provide us with the key to generalize the notion of the Weyl transform, as we discuss in more detail below [31]. We note in particular that the ˆ appearance of the operators D(±iZ/2) in Eq. (10) and Eq. (12) arises from the fact that they are simply the ˆ Fourier transforms of the respective operators D(±Z/2), 

ˆ D(±Z/2)



ˆ = D(±iZ/2).

(16)

F

While defining parity and Fourier operators for continuous infinite systems is relatively straightforward, when one attempts to define corresponding Wigner and Weyl function for spins, which are continuous but discrete systems, their counterpart notions in this new context are now not so easily or uniquely defined. Instead, we are left with a bewildering array of possible choices and starting points. However, an intuitive starting point for constructing a suitably extended Weyl transform is to follow our previous methodology [2, 3] for extending the notion of Wigner functions, and choose a suitable rotation operator for the extended Weyl function to replace the ˆ (iZ/ 2) that appear in Eq. (10). fundamental operators D As noted before, our motivation here is to treat Eq. (10) as the definition of the standard Weyl function, which we now wish to extend and generalize.

(19)

where       ˆ (ϕ, θ, Φ) = exp iJˆz ϕ exp iJˆy θ exp iJˆz Φ U

(20)

is the full rotation operator, and the angles are restricted to the ranges 0 ≤ θ ≤ π, 0 ≤ ϕ ≤ 2π, 0 ≤ Φ ≤ 4π. The similarity in form between Eq. (4) and Eq. (17) is now immediately visible and leads us to define a transformed ˜ rotation operator R(ξ) as follows,   ˜ ˆ R(ξ) ≡ R(iξ) = exp iξ Jˆ+ + iξ ∗ Jˆ− . (21) This is our initial idea for a Weyl operator. We note that it takes the explicit form, h  i ˜ ˜ R(ξ) ≡ R(ϕ, θ) = exp iθ cos ϕJˆx + sin ϕJˆy . (22) We see that it simply represents a rotation of the angle ϕ in Eq. (18), ϕ → ϕ − π/2 or, equivalently, by the replacements Jˆx → −Jˆy and Jˆy → Jˆx in Eq. (18). This is precisely what one would obtain by a rotation about the Jˆz axis by an angle −π/2. It is thus the clear analog of Fourier transformation for the standard Weyl transform, which we have shown above is equivalent to a rotation of the q − p plane about an axis perpendicular to it by an angle of π/2. Note that an equivalent procedure to produce the same operator would be to use Fourier operators, ˜ ˆ Fˆ † , R(ξ) = Fˆj R(ξ) j

(23)

where Fˆj is a suitably modified and truncated form of the Fourier operator in Eq. (14), given explicitly for SU(2) as Fˆj ≡

j X m=−j

im+j |j; mi hj; m| ,

(24)

4 ˜ Wρ (Z)

Wigner function

˜ ρ (Z) W

ϑ

(a)

(b) ˜ Wρ (Z)(ϕ, θ, −ϕ)

Spin Wigner function

(d)

˜| |W

(c) π

˜ ρ (Z)(ϕ, θ, −ϕ) Projected W

(e)

0

(f)

FIG. 1. Here we show (top row: a-c) the coherent superposition (Schr¨ odinger cat) state of three macroscopically distinct coherent states: (a) is the Wigner function and (b-c) are different visualisations of the Weyl function. In the bottom row (d-f) we show a spin coherent state version of the state shown in the top row (a-c). These are a macroscopically distinct coherent superposition of spin coherent states (a spin Schr¨ odinger cat) on the sphere where j = 40. Each of the “cats” in this state has been created by applying the operator in Eq. (20) to the lowest weighted state |j; −ji, it’s position with relation to the south pole is determined by the θ rotation, here θ = π/5, as j increases the value of θ will need to decrease to form the same analogue of a cat state seen in a continuous system, and thus in the stereographic projection, the spin coherent Schr¨ odinger states at θ = π/5, ϕ = 2πn/3 (n = 0, 1, 2 for the three cats), will appear to get further away from each other. Inset next to each sphere in (d) and (e) is the corresponding stereographic (Riemann) projection of the lower hemisphere onto a circle in Euclidean space, with the boundary at the equator. Here (d) shows the spin Wigner function and (e) shows the spin Weyl function. Figure (c) shows the same Weyl function as (b) and (f) shows the Riemann projected half sphere of the inset in (e). Both (e) and (f) contain contain both magnitude (intensity) and phase (colour) information for the complex valued Weyl functions as shown by the inset color wheel.

a diagonal operator in the usual angular momentum basis where kets |j; mi are the eigenstates of the operators J2 and Jz . If we now define transformed operators ˆtFj ≡ FˆjˆtFˆj† , we can investigate the corresponding transforms of the angular momentum operators. Trivially, we have that Fˆj Jˆz Fˆj† = Jˆz . Furthermore, for any operators of the general form j X

ˆt± ≡

t± j;m |j; m ± 1i hj; m| ,

(25)

m=−j

which clearly include the operators Jˆ± , it is easy to prove that (ˆt± )Fj = ±i ˆt± . From these relations and the definitions of Jˆ± ≡ Jˆx ± iJˆy , one may readily prove the relations,

to what would be found using Eq. (10) for analogous states in continuous systems, it has the problem that it is not informationally complete (i.e., it is not an invertible transform). Thus, for example, both the highestand lowest-weighted angular-momentum states, and any statistical mixture of the two, produce the same Weyl function. The solution is to be found by returning to Eq. (19), as all three Euler angles are needed to provide a full Weyl function. If we now take the full rotation operator of Eq. (20) and perform the same transform as before with the Fourier operators ˜ (ϕ, θ, Φ) = Fˆj U ˆ (ϕ, θ, Φ)Fˆ † U j

(27)

Fˆj Jˆz Fˆj† = Jˆz , (26)

˜ we produce a Weyl operator that is equivalent to R(ξ) in Eq. (21) when Φ = −ϕ. Its explicit form is thus given as the standard SU(2) rotation operator for spin-j systems       ˜ (ϕ, θ, Φ) = exp iJˆz ϕ exp iJˆx θ exp iJˆz Φ . (28) U

which are exactly those required. From this one might be tempted to define a h initially i ˜ Weyl function as Tr ρˆ R(ξ) . While this looks promising at first sight, and produces similar-looking functions

With this Weyl operator, the full Weyl function can now be defined as h i ˜ ρ (ϕ, θ, Φ) = Tr ρˆ U ˜ (ϕ, θ, Φ) . W (29)

Fˆj Jˆx Fˆj† = −Jˆy ,

Fˆj Jˆy Fˆj† = Jˆx ,

5 ˜ ρ is restricted to be a twoIf we set Φ = −ϕ then W dimensional function with properties in line with those that we would expect of a Weyl function. Nevertheless, when this slice is taken, there is a doubling of states for every Weyl function, which is also why the use of Eq. (23) as the Weyl operator is not informationally complete. This is due to the nature of the limit j → ∞ for discrete SU(2) systems. As the limit is reached, the size of the surface increases without bound, which in turn leads to the distance from the pole to the equator also tending to infinity. At this limit we effectively have two harmonic oscillators, one on each hemisphere of the sphere. For example, any coherent state displaced by θ from the lowest weighted state − 2j is then equivalent to the highest  weighted state 2j displaced by −θ. These doubled states can now be differentiated by the imaginary value of the Weyl function when varying the value of Φ. Furthermore, this spin Weyl function is invertible via the explicit transform

(a)

(b)

0.8

0 ϑ -0.8 ˜| |W

Wρ (c)

π

0 W ˜ ρ = |W ˜ |eiϑ

(d)

2

0

-2

ρˆ =

(2j + 1) 16π 2

Z 0

2π

Z 0

π

Z

4π

˜ ρ (Ω)U ˜ † (Ω), dΩ W

(30)

0

where Ω ≡ {ϕ, θ, Φ}, dΩ ≡ sin θ dϕdθ dΦ, and the integral ranges are from Eq. (20). We can see the direct relation to Eq. (12) and Eq. (13); the only difference is that now we have a normalization given by the dimension of the system (2j + 1) divided by the SU(2) Bloch sphere invariant volume generated by Eq. (20). We note that although we define the range of integration as 0 ≤ θ ≤ π (and also use this range when plotting a stereographic projection), for visualization onto a sphere of the Φ = −ϕ slice we take 0 ≤ θ ≤ 2π, examples of which are shown in both Fig. 1 (e) and Fig. 2 (b) and (d). When doubling the θ range there is a mirroring of the absolute value Weyl function between the bottom and top hemispheres. We can see this mirroring in Fig. 1 (e) where part of the mirrored Weyl function is visible at the top of the Weyl sphere. For large values of j, it can then be useful also to display the Wigner and Weyl functions as stereographic projections. Taking the limit of the projection to be the equator, we can choose either hemisphere depending on the choice made for the lowest-weighted state. We can now build on the specific case in Eq. (29) to construct a more general Weyl function for systems which are more complex.

IV.

GENERALIZATION TO SU(N )

Please note that this section is mathematically dense and understanding the contents is reliant on a reasonably advanced level of group theory. The result is however a straightforward (i.e., formulaic) procedure for calculating the Weyl function generating kernel for any single quantum system. To begin, we follow the discussions set out originally in [35–39]. Start with M = 2j, where j is our spin quan-

FIG. 2. Here we show (a-b) the superposition state for a spin- 25 spin coherent Schr¨ odinger cat state [34], given by 5  5 
√ − + / 2 and in (c-d) the five-qubit GHZ state 2 2 √ (|00000i + |11111i) / 2 where we have taken the equal angle slice ϕi = ϕ, θi = θ . Figures (a) and (c) show the spherical plot for the the spin Wigner functions where blue is positive and red in negative; (b) and (d) give the spin Weyl functions spherical plots for the slice Φ = −ϕ, where the phase is given by colour according to the colour wheel in the centre of the figure. The absolute value is shown by saturation, so that the Weyl function is white when the value at that point is zero. Note that we have extended the range when mapping the function onto the sphere, so that 0 ≤ θ ≤ 2π.

tum number. M defines the dimension d of the representation, hence the system size for a N -level system is given as   N +M −1 d≡ . (31) M Next, we denote the SU(N ) generators [38–40] by the set ˆ N,M (k)} where k = 1, 2, . . . , N 2 − 1. {Λ

(32)

This set is made up of off-diagonal generators simiˆ {1} (a, b) and Λ ˆ {2} (a, b) for a, b = lar to Jx and Jy : Λ N,M N,M 1, 2, 3, . . . , N ; a < b, and diagonal generators similar to ˆ {3} (c) for 1 ≤ c ≤ N − 1. A detailed procedure Jz : Λ N,M to construct these matrices and their properties is given through the following procedure from [38]: 1. Define a general spin-j basis |m1 , m2 , . . . , mN i PN where M = k=1 mk and j can have any integer or half-integer value.

6 2. Define the following three operators: Jba |m1 , m2 , . . . , ma , . . . , mb , . . . , mN i = p (ma + 1)mb |m1 , m2 , . . . , ma + 1, . . . , mb − 1, . . . , mN i for 1 ≤ a < b ≤ N , Jba |m1 , m2 , . . . , mb , . . . , ma , . . . , mN i = p ma (mb + 1) |m1 , m2 , . . . , ma − 1, . . . , mb + 1, . . . , mN i for 1 ≤ b < a ≤ N , and,

X a

V

V

where Y

KSU(N ) =

Y

Ker(y, j(z)),

(36)

N ≥z≥2 2≤y≤z

s Jaa |m1 , m2 , . . . , ma , . . . , mb , . . . , mN i =

The volume of the manifold defined by Eq. (33) does not depend on the dimension of the representation M [36]. As such, the volume is (from [35, 37]) generated by integrating the Haar measure of SU(N ) Z Z VSU(N ) = dVSU(N ) = KSU(N ) dθdφdΦ (35)

2 × a(a + 1)

  sin(2θ1+j(z) ) Ker(y, j(z)) = cos(θ(y−1)+j(z) )2y−3 sin(θ(y−1)+j(z) )  cos(θ 2z−3 (z−1)+j(z) ) sin(θ(z−1)+j(z) )

 mk − ama+1 |m1 , m2 , . . . , ma , . . . , mb , . . . , mN i

k=1

for 1 ≤ a ≤ N − 1. 3. For a, b = 1, 2, 3, . . . , N ; a < b: ˆ {1} (a, b) ≡ J a + J b , Λ b a N,M ˆ {2} (a, b) ≡ −i(Jba − Jab ), Λ N,M

depending on whether y = 2 (top), 2 < y < z (middle), or y = z (bottom), and j(z) is from equation (33). The ranges for the integrations V can be found in Appendix C of [35]. Extending the definition of the Fourier operator given in Eq. (24) we now define a Fourier operator for any representation of SU(N ) as a d×d diagonal matrix whose elements are given explicitly by Fi,i = (−i)i

ˆ {3} (c) ≡ Jcc for c = 1, 2, . . . , N − 1. Λ N,M

Fd ≡

d−1 X

Fi,i |ii hi|

(37)

i=0

When d = N , i. e. M = 1, the representation is fundamental and the generators above reduce to the generalized Gell-Mann matrices {λk } for SU(N ) [40, 41]. ˆ N,M (k)}, we can employ the Given the generators {Λ parametrization given in [35] for SU(N ). Using this M of an M repparametrization, a SU(N ) operator UN resentation can be written as  Y  Y M M UN (θ, φ, Φ) = AM (y, j(z)) BN , (33) N N ≥z≥2 2≤y≤z

where h i ˆ {3} AM N (y, j(z)) = exp iΛN,M (1)φ(y−1)+j(z) h i ˆ {2} (1, y)θ(y−1)+j(z) , × exp iΛ N,M and

where Fd (Fd )† = 1ld , and do the following transformation † M M ˜N U (θ, φ, Φ) = Fd UN (θ, φ, Φ) (Fd ) .

It can be shown that for all M Z M ˜N U (θ, φ, Φ)dVSU(N ) = 0,

M BN =

h

{3}

i

ˆ exp iΛ N,M (c)Φ(N (N −1)/2)+c .

1≤c≤N −1

(39)

V

and, due to the fact that the eigenvalues of Fd are {−i, 1, i, −1} and thus form a cyclic group of order 4 under multiplication, we get for all a < b the following cyclic chart (here for M = 1): [42] ˆ {1} (a, b) (Fd )† = Λ ˆ {2} (a, b) b − 1 ≡ a (mod 4) Fd Λ N,1 N,1 {2} † ˆ (a, b) (Fd ) = −Λ ˆ {1} (a, b) Fd Λ N,1 N,1 ˆ {1} (a, b) (Fd )† = −Λ ˆ {1} (a, b) b − 2 ≡ a (mod 4) Fd Λ N,1 N,1 ˆ {2} (a, b) (Fd )† = −Λ ˆ {2} (a, b) Fd Λ N,1

Y

(38)

N,1

ˆ {1} (a, b) (Fd )† = −Λ ˆ {2} (a, b) b − 3 ≡ a (mod 4) Fd Λ N,1 N,1 {2} † ˆ (a, b) (Fd ) = Λ ˆ {1} (a, b) Fd Λ N,1 N,1 ˆ {1} (a, b) (Fd )† = Λ ˆ {1} (a, b) b − 4 ≡ a (mod 4) Fd Λ N,1 N,1 ˆ {2} (a, b) (Fd )† = Λ ˆ {2} (a, b) Fd Λ

PN −z Here j(z) = 0 for z = N and j(z) = i=1 (N − i) for M z 6= N . Note that the BN term is generated by the Cartan sub-algebra of the M representation of SU(N ). One can easily show that

as well as the obvious for all 1 ≤ c ≤ N − 1 regardless of representation size

M M UN (θ, φ, Φ)UN (θ, φ, Φ)† = 1ld ,

ˆ {3} (c) (Fd )† = Λ ˆ {3} (c). Fd Λ N,M N,M

(34)

M and from [39] it is clear that UN (θ, φ, Φ) generates generalized spin coherent states |θ, φi via action on the lowest weighted state |0i.

N,1

N,1

(40)

˜ M (θ, φ, Φ) is the With the above, our proposal is that U N generalized Weyl operator for general SU(N ) spin systems of size d.

7 Using the above, the generalised spin Weyl function can now be defined as ˜ ρ (θ, φ, Φ) = Tr[ρ U ˜ M (θ, φ, Φ)] W N

(41)

where ρ=

Z

d VSU(N )

VI. TRANSFORMING BETWEEN THE WIGNER AND WEYL FORMALISM

Returning to the Wigner function, our earlier work [2] shows how to write a spin Wigner function h i in a similar 0 ˆ d (Ω0 ) such that form as Eq. (2), Wρ (Ω ) = Tr ρ ∆

˜ ρ (θ, φ, Φ)U ˜ M (θ, φ, Φ)† (42) dVSU(N ) W N

ˆ † (Ω0 ) ˆ 0 )Π ˆ d (Ω0 ) = 1 U(Ω ˆ dU ∆ d

V

and Z

where d = ˜ ρ (θ, φ, Φ) = 0 dVSU(N ) W

Pl

i

di from Eq. (31),

(43)

V

(48)

0

U(Ω ) =

to go along with Eq. (39).

l O

0

0

0

Mi UN (θi , φi , Φi ), i

(49)

i=1

and V.

COMPOSITE SYSTEMS

ˆd = Π We are now in a position to look at collections of qudit states of various spin representations. If we define ˜ U(Ω) =

l O

˜ ji (ϕi , θi , Φi ) U Ni

(44)

i=1

˜ Mi is given in Eq. (38) then where U Ni h i ˜ρ (Ω) = Tr ρˆ U(Ω) ˜ W

(45)

Pl where now Ω ≡ i=1 {θi , φi , Φi } For example, it is clear that when l = 1 and N = 2 we get back Eq. (29). Furthermore, the corresponding transform back to the density matrix (and, by extension, any Hilbert space operator) is given as Z 1 † ˜ ρ (Ω)U(Ω) ˜ ρˆ = dΩ W (46) VΩ V Ql S where dΩ ≡ I=1 dVSU(Ni ) and V = V defines the normalization, l

Y di 1 ≡ , VΩ V i=1 SU(Ni )

(47)

To recover the SU(2) case given in Eq. (30) one only needs to again define l = 1 and N = 2 to yield VSU(2) = 16π 2 and d = 2j + 1. Comparing the different Weyl functions for two comparable states, we can see in Fig. 2 (a-b), the superposition of the highest and lowest weighted states for an SU(2), spin- 25 spin-coherent Schr¨ odinger cat state [34] using Eq. (29) and in Fig. 2 (c-d) the maximally entangled five-qubit GHZ state, using l = 5, N = 2 and j = 21 in Eq. (44) as the Weyl operator. In the latter, we take the equal angle slice ϕi = ϕ, θi = θ, and Φi = Φ = −ϕ. The similarity between the two cases is striking.

d X

{3}

ˆ (i) β[i]Λ d,1

(50)

i=1 {3}

ˆ (i) being the various Cartan sub-algebra with the Λ d,1 elements of the Lie algebra of su(d) in the fundamental representation [43]. Now, Eq. (50) is a generalization of the displaced parity given in [2] that is based on observations from [3] for product states and from [8] wherein a given symmetric sub-space Wigner operator √ 2j L 0 0 0 0 2 π X X j K(θ , φ ) = √ Y ∗ (θ , φ )TˆLM 2j + 1 L=0 M =−L LM s j 2L + 1 X j ˆ TLM = C jn |j, ni hj, m| (51) 2j + 1 m,n=−j jm,LM can be easily shown to be equivalent to our formalism (we will expand on this point in a future work) with l = 1, M = 2j, and N = 2 ˆ 0 )K(0, 0)U ˆ † (Ω0 ). ˆ [2j+1] (Ω0 ) ≡ U(Ω ∆

(52)

∗ Here, YLM are the conjugated spherical harmonics and jn Cjm,LM are Clebsch-Gordan coefficients which couple two representations of spin j and L to a total spin j. ˆ [2j+1] (Ω0 ) is equal to K(θ0 , −φ0 ) It is easy to show that ∆ for all j and that K(0, 0), since it is a diagonal Hermitian matrix, can be decomposed into a linear sum of the Cartan sub-algebra (and Identity) of the su(2j + 1) Lie algebra in its fundamental representation:

K(0, 0) ≡

2j+1 X

{3}

ˆ β[i]Λ 2j+1,1 (i).

(53)

i=1

Given the above, and recalling that the inverse of the Wigner function is Z 0 0 ˆ d (Ω0 ), ρˆ = dΩ Wρ (Ω )∆ (54) Ω0

8 we can see that the Weyl function can be written as ˜ ρ (Ω) = Tr[ρ U(Ω)] ˜ W Z   0 0 0 d ˆ ˜ = Tr dΩ Wρ (Ω )∆ (Ω ) U(Ω) Ω0 Z h i 0 0 ˆ d (Ω0 ) U(Ω) ˜ = dΩ Wρ (Ω )Tr ∆ (55)

But here, for the Weyl, our dΩ is the product Haar measure for SU(d) given in Eq. (35). The interesting thing to note is that we can use the latter Haar measure instead of the former dVCP Ni −1 when calculating the Wigner functions defined in [2, 3] without any loss of information.

Ω0

Using Eq. (37) and Eq. (44) we can define new operators 0

D(Ω ) ≡

l O

!† 0 0 0 ji UN (θi , φi , Φi ) i

i=1 0

D(Ω )† ≡

l O

l O

VII.

Fdi

i=1 †

(Fdi )

i=1

l O

! 0

0

0

ji UN (θi , φi , Φi ) i

(56)

i=1

allowing us to generate, in what might be thought of as a generalization of the Fourier transform, Z h i 0 0 ˜ ˆ d (Ω0 ) U(Ω) ˜ Wρ (Ω) = dΩ Wρ (Ω )Tr ∆ 0 ZΩ i h 0 0 ˆ d Υ(Ω, Ω0 ) (57) = dΩ Wρ (Ω )Tr Π Ω0

where we have defined 0

0

0

Υ(Ω, Ω ) ≡ D(Ω )U(Ω)D(Ω )† .

(58)

We now can easily transform between the Weyl and Wigner formalisms for any spin state or collection of spin states. Furthermore, we have a way to now look at displaced parity similar to that done in [3] in terms of more 0 general Weyl displacement operators Υ(Ω, Ω ). With Eq. (57) in hand it is important to recognize 0 that, similar to [2], our dΩ is the product integral kernel for the complex projective space in d − 1 dimensions (CP d−1 ), 0

dΩ ≡

l Y

dVCP Ni −1 .

CONCLUSIONS

(59)

We have presented a definition of a correlation characteristic function that is a natural generalization of the Weyl function. We have shown that it can be defined for any quantum system or composite system. Unlike recent generalization of the Wigner function which needed subtle extensions of parity [2, 3] the characteristic function we present here has no such complication. This leads us to speculate that in some sense these characteristic functions may be more fundamental than the Wigner function. In any case, both are informationally complete, and connected by a simple integral transform. Based on the wide use of characteristic functions in other disciplines it is our view that the extension to the Weyl function that we present here will find great utility in quantum physics, information, thermodynamics and beyond.

ACKNOWLEDGMENTS

MJE would like to thank Kae Nemoto, John Samson and Andrew Archer for interesting and informative discussions. TT notes that this work was supported in part by JSPS KAKENHI (C) Grant Number JP17K05569. Note RPR and TT contributed equally to this work.

i=1

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