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A family of Weyl–Wigner transforms for discrete variables defined in a finite-dimensional Hilbert space

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We study the Weyl–Wigner transform in the case of discrete variables defined in a Hilbert space of finite prime-number dimensionality N. We define a family of Weyl–Wigner transforms as function of a phase parameter. We show that it is only for a specific value of the parameter that all the properties we have examined have a parallel with the case of continuous variables defined in an infinite-dimensional Hilbert space. A geometrical interpretation is briefly discussed.

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One of the authors (PAM) acknowledges support by DGAPA, under Contract No. IN109014. He also acknowledges the kind hospitality of the Physics Department of the Technion, where this investigation was initiated.

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Correspondence to Pier A. Mello.


Appendix A: Schwinger operators

We consider an N-dimensional Hilbert space spanned by N orthonormal states \(|q\rangle \), with \(q=0,1, \ldots ,(N-1)\), and subject to the periodic condition \(|q+N\rangle =|q\rangle \); they are designated as the “reference basis”, or “computational basis” of the space. We introduce the unitary operators \(\hat{X}\) and \(\hat{Z}\) [29], which fulfill the periodic condition \( \hat{X}^N = \hat{Z}^N = \hat{\mathbb {I}} \) (\(\hat{\mathbb {I}}\) being the unit operator) and are defined by their action on the states of the reference basis as

$$\begin{aligned} \hat{Z}|q\rangle= & {} \omega ^q|q\rangle , \quad \omega =e^{2 \pi i/N}, \end{aligned}$$
$$\begin{aligned} \hat{X}|q\rangle= & {} |q+1\rangle , \end{aligned}$$

leading to the commutation relation

$$\begin{aligned} \hat{Z}\hat{X}=\omega \hat{X}\hat{Z} . \end{aligned}$$

The two operators \(\hat{Z}\) and \(\hat{X}\) form a complete algebraic set [29], so that any operator defined in our N-dimensional Hilbert space can be written as a function of \(\hat{Z}\) and \(\hat{X}\).

We introduce the Hermitian operators \(\hat{p}\) and \(\hat{q}\) through the equations [10, 24, 25, 32, 37]

$$\begin{aligned} \hat{X}= & {} \omega ^{-\hat{p}} = e^{-\frac{2\pi i}{N}\hat{p}}, \end{aligned}$$
$$\begin{aligned} \hat{Z}= & {} \omega ^{\hat{q}} =e^{\frac{2\pi i}{N}\hat{q}}. \end{aligned}$$

As \(\hat{X}\) performs translations in the variable q and \(\hat{Z}\) in the variable p, we designate \(\hat{p}\) and \(\hat{q}\) as “momentum-like” and “position-like” operators, respectively.

What we defined as the reference basis can thus be considered as the “position basis”. With (41) and definitions (42), (43), the commutator of \(\hat{q}\) and \(\hat{p}\) reduces, in the continuous limit [32, 36, 37], to the standard one, \([\hat{q},\hat{p}]=i\).

Appendix B: MUB

The operators \(\hat{X}\hat{Z}^b\), \(b=0, \cdots N-1\) define N of the \(N+1\) MUB, [see Eqs. (44), (45) below], while the operator \(\hat{Z}\) defines the reference basis. The operator \(\hat{X} \hat{Z}^{b}\) possesses N eigenvectors, denoted by \(|m,b\rangle \) (see Ref. [19, 20] and Eqs. (10), (11) of Ref. [24])

$$\begin{aligned} \hat{X}\hat{Z}^{b} |m,b\rangle= & {} \omega ^m |m;b\rangle ; \;\;\;\;\; b,m = 0,1, \cdots , N-1 , \end{aligned}$$
$$\begin{aligned} |m;b\rangle= & {} \frac{1}{\sqrt{N}}\sum _{q=0}^{N-1}\omega ^{\frac{b}{2}q(q-1)-qm} |q\rangle . \end{aligned}$$

Here, \(|q\rangle \) (\(q=0, \ldots , N-1\)) denote the N states of the reference basis. The states with \(b=0\) are eigenstates of \(\hat{p}\)

$$\begin{aligned} |m;0\rangle= & {} \frac{1}{\sqrt{N}}\sum _{q=0}^{N-1} e^{\frac{2\pi i}{N}(N-m)q} |q\rangle , \end{aligned}$$
$$\begin{aligned}= & {} |p=-m=(N-m) \mathrm{Mod}[N]\rangle . \end{aligned}$$

Appendix C: Orthogonality of the operators \(\hat{P}^{(c)}_{qp}\), Eq. (17)

We show the orthogonality, as given by Eq. (17), of the operators \(\hat{P}^{(c)}_{q,p}\) defined in Eq. (6). We first note the following two features of the lines described in the text.

  1. 1.

    We assume the phase c to be fixed; then a line is defined by the pair of numbers qp. Lines \(\hat{P}^{(c)}_{q,p}, \hat{P}^{(c)}_{q',p'}\) may be either identical, which means that \(p=p'\) and \(q=q'\), or distinct, meaning that either

    1. (i)

      \(q \ne q'\), \(p=p'\), or

    2. (ii)

      \(q=q'\), \(p\ne p'\), or

    3. (iii)

      \(q\ne q'\) and \(p \ne p'\).

    If two lines are distinct, then they have one point in common, i.e. there exists (one) b wherein the two lines intersect. Every line has one common “point” with every other line. If two lines have two points in common, they are identical.

    Let line 1 be given by (qp) and line 2 by \((q',p')\). We have the possibilities:

    (i, iii) The lines differ if \(q\ne q'\). Here both cases \(p=p'\) or \(p\ne p'\) imply \(b=\frac{p-p'}{q-q'}\) (\(b=0\) for the first case)—a unique value in either case.

    (ii) If the lines differ via \(p\ne p'\) but \(q=q'\), the common point is at \(b=\ddot{0}\).

    The uniqueness of the solutions implies that distinct lines cannot have more than one point in common.

  2. 2.

    All the “points” are MUB projectors , i.e.

    $$\begin{aligned} {\mathbb {P}}^{(c)}(q,p;b)= |M^{(c)}_{q,p}(b); b \rangle \langle M^{(c)}_{q,p}(b); b| . \end{aligned}$$

    For \(b\ne b'\) (including \(b=\ddot{0}\)) we have

    $$\begin{aligned} \mathrm{Tr} [ {\mathbb {P}}^{(c)}(q,p;b){\mathbb {P}}^{(c)}(q,p;b') ] =\frac{1}{N} . \end{aligned}$$

The actual proof of orthogonality involves computing the LHS of Eq. (17), giving

$$\begin{aligned} \mathrm{Tr} [ \hat{P}^{(c)}_{q,p}\hat{P}^{(c)}_{q',p'} ]= & {} \sum _{b=\ddot{0}}^{N-1}\mathrm{Tr}[{\mathbb {P}}^{(c)}(q,p;b) {\mathbb {P}}^{(c)}(q',p';b)] +\sum _{b\ne b'}\mathrm{Tr}[{\mathbb {P}}^{(c)}(q,p;b){\mathbb {P}}^{(c)}(q',p';b') ] \nonumber \\&-\sum _{b=\ddot{0}}^{N-1}\mathrm{Tr}[{\mathbb {P}}^{(c)}(q,p;b)] -\sum _{b=\ddot{0}}^{N-1}\mathrm{Tr}[{\mathbb {P}}^{(c)}(q',p';b)] +\mathrm{Tr}\; {\mathbb {I}}, \end{aligned}$$
$$\begin{aligned}\equiv & {} A+B+C+D+E . \end{aligned}$$

Calculation of A The first term, A, involves a sum over \(N+1\) traces of the product of two projectors, both in the same basis b.

  1. (i)

    If \(q=q', p=p'\), the lines are identical: then the sum reduces to \(N+1\) traces of projectors, each giving 1, the result thus being \(N+1\).

  2. (ii)

    If the lines are distinct, they have only one common point at b; we thus have the trace of a projector, giving 1. All other N terms involve the trace of a product of two orthogonal projectors and do not contribute. Thus

    $$\begin{aligned} A=\sum _b Tr {\mathbb {P}}^{(c)}(q,p;b){\mathbb {P}}^{(c)}(q',p';b) = (N+1) \delta _{qq'}\delta _{pp'} +1\cdot (1- \delta _{qq'}\delta _{pp'}) = 1+ N \delta _{qq'}\delta _{pp'} . \end{aligned}$$

    Calculation of B The second term involves traces of projectors of distinct bases. Since these are MUB projectors, each term gives 1 / N. There are \(N(N+1)\) terms in the sum, giving \(N(N+1)/N=N+1\):

    $$\begin{aligned} B=\sum _{b\ne b'}Tr {\mathbb {P}}^{(c)}(q,p;b){\mathbb {P}}^{(c)}(q',p';b')=N+1. \end{aligned}$$

    Calculation of C and D The third and fourth terms, C and D, involve traces of projectors each multiplied by unity. There are \(N+1\) terms, each giving 1. We thus find

    $$\begin{aligned} C=-\sum _b \mathrm{Tr} {\mathbb {P}}^{(c)}(q,p;b) = D=-\sum _b Tr {\mathbb {P}}^{(c)}(q',p';b') =-(N+1). \end{aligned}$$

    Calculation of E The last term, E, involves the trace of unity, giving N.

    Adding up the five terms, we finally obtain:

    $$\begin{aligned} \mathrm{Tr} \left[ \hat{P}^{(c)}_{q,p}\hat{P}^{(c)}_{q',p'}\right] =N\delta _{q,q'}\delta _{p,p'}. \end{aligned}$$

Appendix D: Derivation of Eq. (36)

The definition of WWT, Eq. (35), is based on the notion that the Weyl operators [1]

$$\begin{aligned} \mathbb {U} =\mathrm{e}^{i(\beta \hat{p} +\alpha \hat{q})} \end{aligned}$$

form a complete and orthogonal operator basis [5, 6, 34]. For the purpose of comparing the continuous and discrete cases in Sect. 3, we introduce the alternative set of complete and orthogonal operators

$$\begin{aligned} \mathbb {V} = \mathrm{e}^{i\beta \hat{p}} \; \mathrm{e}^{i\alpha \hat{q}} , \end{aligned}$$

and express the WWT in terms of them.

Making the change of variables \(v'=-v\) and relabelling \(v'\) again as v, we obtain, from Eq. (35)

$$\begin{aligned} W_{\hat{A}}(q,p) = \frac{1}{2\pi } \mathrm{Tr}\int _{-\infty }^{\infty } \int _{-\infty }^{\infty } \hat{A} \mathrm{e}^{-i(u\hat{q} - v\hat{p})} \mathrm{e}^{i(uq - vp)} \mathrm{d}u \mathrm{d}v. \end{aligned}$$

We use polar coordinates

$$\begin{aligned} u = r \cos \theta , \quad v = r \sin \theta \end{aligned}$$

and write

$$\begin{aligned} W_{\hat{A}}(q,p)= & {} \mathrm{Tr}\int _{0}^{\infty } r \mathrm{d}r \int _{0}^{2 \pi } \mathrm{d} \theta \; \hat{A} \mathrm{e}^{-i r \sin \theta (- \hat{p} + \cot \theta \; \hat{q} )} \nonumber \\&\times \mathrm{e}^{i r \sin \theta (- p + \cot \theta \; q)} \end{aligned}$$
$$\begin{aligned}= & {} \mathrm{Tr}\int _{0}^{\infty } r \mathrm{d}r \int _{0}^{2 \pi } \mathrm{d} \theta \; \hat{A} [\mathrm{e}^{i(-\hat{p} + \cot \theta \; \hat{q})} ]^{-r \sin \theta } \nonumber \\&\times \mathrm{e}^{i r \sin \theta (- p + \cot \theta \; q)}. \end{aligned}$$

Recalling the BCH identity

$$\begin{aligned} \mathrm{e}^{i(\alpha \hat{q} + \beta \hat{p})} = \mathrm{e}^{-\frac{i}{2}\alpha \beta } \mathrm{e}^{i \beta \hat{p}} \mathrm{e}^{i \alpha \hat{q}} , \end{aligned}$$

and choosing \(\beta = -1\), \(\alpha = \cot \theta \), we can write

$$\begin{aligned}&\mathrm{e}^{i(-\hat{p} + \cot \theta \; \hat{q})} = \mathrm{e}^{\frac{i}{2}\cot \theta } \mathrm{e}^{-i \hat{p}} \;\mathrm{e}^{i \cot \theta \; \hat{q}} , \end{aligned}$$

so (61) takes the form of Eq. (36) given in the text.

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Mann, A., Mello, P.A. & Revzen, M. A family of Weyl–Wigner transforms for discrete variables defined in a finite-dimensional Hilbert space. Quantum Stud.: Math. Found. 4, 89–101 (2017).

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