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The finite Fourier Transform and projective 2-designs

There are several approaches to define an eigenvector decomposition of the finite Fourier Transform, which is in some sense unique, and at best resembles the eigenstates of the quantum harmonic oscillator. A solution given by Balian and Itzykson in 1986 for prime dimensions d = 3 (mod 4) is revisited. It is shown, that by applying the Weyl-Heisenberg matrices to this eigenvector basis, a projective 2-design is generated.


Motivation: The Quantum Harmonic Oscillator
The position operator X and the impulse operator P are each defined on a dense subset of L 2 (R) via the equations (Xf )(x) = xf (x), and Pf )(x) = −i d dx f (x). Here we have set = 1. They fulfill the Canonical commutation relation [X, P] = iI. When we set all physical parameters to 1, the Hamiltonian for the quantum harmonic oscillator is H = 1 2 (X 2 + P 2 ). Its eigenvalues resp. energy levels are n + 1 2 with n = 0, 1, 2, . . .. The corresponding eigenstates are ψ n (x) = 1 √ 2 n n! √ π H n (x)e −x 2 2 with the Hermite polynomials H n (x) = (−1) n e x 2 d n dx n (e −x 2 ). All ψ n constitute also a particular choice of eigenstates for the Fourier-Transform F has the 4 eigenvalues ±1, ±i: (Fψ n )(x) = (−i) n ψ n (x).
To X and P a two-parameter strongly continuous group can be assigned, the so called Weyl-Heisenberg group of unitary operators W(r, s) := e i(rP+sX) = e − irs 2 e irP e isX .
With r, s ∈ R → W(r, s) they make up the unique unitary, irreducible, projective representation of the additive group R × R, [31].
Now we turn to the finite dimensional counterparts.

Weyl-Heisenberg Matrices and Fourier Matrix
Let d be the dimension of a finite-dimensional complex vector space.
We use throughout the paper the notation of bra's .| (row-vectors) and ket's |. (column-vectors). Let |e r , with r ∈ Z d be the standard basis. Addition of indices is always modulo d, so e.g. for bra's |e r+s := |e (r+s)(mod d) .
Let U|e r = e i2πr d |e r and V|e r = |e r+1 or in matrix form: , and for r, s ∈ Z d for odd d, resp. r, s ∈ Z 2d for even d The case of even d needs 2d × 2d matrices due to the phase added. We will look only at the odd case in the following.
These matrices generate the so-called finite Weyl-Heisenberg group, and make up the unique, irreducible, projective representation of the additive group The factor τ rs is the analogue of the factor e − irs 2 in the infinite dimensional case and simplifies calculations significantly (Appleby [2] In contrast to the infinite case, we don't have uniquely determined infinitesimal generators of the Weyl-Heisenberg group in finite dimensions (as X, P). Therefore we have also no uniquely defined finite counterpart of the quantum harmonic oscillator and its eigenstates. 1 Remark on the notation: Due to the fact that U and V can also be considered as generalizations of the Pauli Matrices Z, X, defined in dimension d = 2 (as rotations of the Bloch Sphere around the Z-and X-axis), in several papers these letters are used instead. We use the letters U and V as e.g. Schwinger [27] and in many subsequent physics papers, to emphasize the fact that we work on qudits (d ≥ 2) and not just qubits.
The letter W stands for Weyl, who brought the matrices above onstage in physics first [33]. Very often D for Displacement is used instead of W.
But we have at least a finite counterpart to the Fourier-Transform: The Fourier-Matrix (also called Finite or Discrete Fourier-Transformation or Schur- In matrix form As F 4 = I, the possible eigenvalues of F are ±1, ±i, as in the infinite case. The multiplicity of the eigenvalues as function of the dimension d is given by the following table (see [8]).
There is long and ongoing history to define an, in some sense unique, eigenvector decomposition of the Fourier Matrix.
The approaches are either starting with the continuous harmonic oscillator eigenstates (e.g. sampling at equidistant points), or focusing on algebraic methods.
Here we propose a characterization in terms of Quantum Designs, specifically projective 2-designs.

Projective 2-designs
be the corresponding projection matrices. There are many equivalent definitions, when these sets form a projective 2-design, see e.g. [10-12, 19, 25, 26, 32, 36], (historically) starting with the condition, that for any homogeneous polynomial f of degree 2, and the normed unitary invariant (Haar-)integral on the right side. This is equivalent to where Π sym is the orthogonal projection on the symmetric subspace of Another equivalent condition is, that the inequality becomes an equality. Well know examples are [10,11]: • Complete sets of MUBs -(exist for all d = prime power): N = d(d + 1) • Clifford group applied to any vector (for d = prime): In the last example the Clifford group for prime d is also an example of a socalled unitary 2-design. We only refer to projective 2-designs in this paper and skip the attribute projective frequently.

An example for d = 3 and some numerical search
The Fourier matrix for d=3 has the 3 eigenvalues ±1 and i, and the (up to phases) unique normed eigenvectors are We apply the d 2 Weyl-Heisenberg matrices to each of the d = 3 vectors and get a set of d 3 = 27 vectors {W (r,s) |ψ x : x = ±1, i and 0 ≤ r, s, ≤ 2}. These vectors form a 2-design! Actually |ψ i is also a fiducial vector for a SIC-POVM, so due to the additivity of the 2-design property the first 2 eigenvectors generate also a 2-design of 18 vectors.
Numerical search found no complete eigenvector basis of the Fourier matrix for each d = 4, 5, 6, that generates a 2-design like above. But for d = 7, 11 there were found in each case (seemingly unique) solutions.
For d = 5 there is a (unique) basis of 2 eigenvectors of eigenvalue 1, which together with the unique eigenvector for eigenvalue −1 generates a set of 3d 2 = 75 vectors, that form a 2-design.
In all these case the eigenvectors can be taken to be real-valued only.

Clifford Group
In the following section d is taken to be odd. 2 The Clifford group is defined as normalizer of the Weyl-Heisenberg group τ q W (r,s) , 0 ≤ q, r, s ≤ d − 1, in the group of unitary matrices U ∈ U (d)). It includes the Weyl-Heisenberg group itself as normal subgroup. We are interested here in the subgroup of U ∈ U (d), for which for all r, s ∈ Z d for some r ′ , s ′ ∈ Z d (with no additional phase involved!), and call it restricted Clifford group C(d). It is well known [2,4,10,11], that the restricted Clifford group is a projective representation of SL(2, Z d ) with the group homomorphism h defined via where U G is a representative, which is fixed up to an overall phase factor. The Fourier matrix is an element of the Clifford group:

The Subgroup of elements commuting with F
Let FC(d) ⊂ C(d) be the subgroup of elements of the restricted Clifford group, that commute with the Fourier matrix. In terms of SL(2, Z d ) this means If d is an odd prime then F SL(2, Z d ) is a cyclic group (see [6,7,9,14]) with order

Representations of C(d) and especially F C(d) in C d
We assume d to be an odd prime. An explicit representation of C(d) is given by Appleby [3] (see also [23]). We restrict to the subgroup F SL(2, Z d ), and get, except when β = 0 (which means in F SL(2, Z d ): α = ±1 → G = ±I).
The Weyl-Heisenberg matrices are an orthogonal basis of all matrices. Therefore we can also expand the elements above in this basis. An according representation (see [9], and Athanasiu, Floratos, Nicolis [7]) is (except the case α = 1, β = 0).
The overall phases factors e iθ rsp. e iη can be chosen such, that the presentation becomes de-projectivized (ordinary rsp. faithfull). We don't need this here, and set the factors to 1.

2-designs from the Fourier matrix
From here on we assume d to be prime with d = 4k + 3.
In this case equation (2), with e iη = 1, simplifies to the d matrices We get this form, when we set (mod d) These maps are well defined, as 4m 2 + 1 = 0 (mod d) has no solution (rsp. −1 is no quadratic residue) for d = 4k + 3.
It was observed by Balian and Itzykson ( [9], 1986) that for primes d = 4k + 3 these matrices provide a unique common orthogonal basis of eigenvectors of F. Up to a phase F corresponds to m = d−1 2 . See also [6,7,14].
They play also a role in the concept of MUB-balanced states [1,4,34].
Now we can state the central result of this paper Theorem 1. Let d = 4k + 3 be a prime. Let {|ψ i : 1 ≤ i ≤ d} be the unique common orthogonal basis of eigenvectors of the d matrices The Actually, in the following, we are going to work with the d corresponding projection matrices {P i := |ψ i ψ i | : 1 ≤ i ≤ d} on the eigenvectors. Let We are going to prove For this we need some preparation • In the next subsection we sum up some features of the Weyl-Heisenberg matrices and of their tensor products, which we use subsequently.
• In the subsection afterwards we expand the projection matrices P i on the eigenvectors in terms of the Weyl-Heisenberg matrices. We don't calculate them explicitly, but just show some of their properties, based on relations of the Weyl-Heisenberg matrices and the R m , which allow us • in the final subsection to prove the theorem.
The last equation describes the already mentioned orthogonality of the Weyl-Heisenberg matrices. Immediate consequence is (see also [35]) We furthermore notice the following formula This is a special case of SWAP = 1 d d−1 r=0 d−1 s=0 g (r,s) ⊗ g * (r,s) for any orthogonal matrix base {g (r,s) }, 0 ≤ r, s ≤ d − 1, were the standard norm g (r,s) 2 2 = d. See the paper by Siewert [29], were this relation is extensively exploited. An immediate consequence is (see also [24]: 6.2 Some properties of R m and the projection matrices P i Lemma 1. Let d = 4k + 3 be prime, and R m as in definition (3).
Proof. tr (R m ) = 1 follows directly, and further using (7), after short calculation Here we used, that for prime numbers d = 4k + 3 the quadratic Gauss sum are d−1 r=0 τ ar 2 = a d i √ d with the Legendre symbol a d = ±1, for a = 0.
Next we define auxiliary matrices X m as orthonormalization of the R m .
Lemma 2. Let d = 4k + 3 be prime, and Proof. The proof is straight forward using the definition and equations (10).
Next we derive an Ansatz for the projection matrices on the eigenvectors of R m via the auxiliary matrices X m . This helps us to prove some properties. Lemma 3. Let d = 4k + 3 be prime, and let be the projection matrices on the common eigenvectors of all R m in arbitrary order. Then for all 0 ≤ i ≤ d − 1 with and we have Proof. The matrix Λ = (λ im ) 0≤i≤d−1; 0≤m≤d−1 transfers between the d orthonormal matrices X m and the also orthonormal d matrices P i . Therefore it must be unitary, which we use below.
We insert X m rsp. R m in definition (13) and get By applying the trace on (13), and using tr(P i ) = 1 and equation (12) we get This proves (14) with the definition of p (r,s) i as in (15). Now for the properties of these coefficients. ) 0≤i≤d−1 is for any 0 ≤ r, s ≤ d − 1 the unitary transform of the vector (τ m(r 2 +s 2 ) ) 0≤m≤d−1 by Λ, and this vector has obviously squared norm (τ m(r 2 +s 2 ) ) 0≤m≤d−1 2 2 = d.

Proof of Theorem 1
We want to prove equation (5). Let When we insert P i , as described in Lemma 3, the sums reduce significantly, due to the equation (8) about sums of tensor-products of Weyl-Heisenberg matrices.
As a consequence we state Corollary 1. Let d = 4k + 3 be a prime. Let {P i := |ψ i ψ i | : 1 ≤ i ≤ d} be the projection matrices on the unique common orthogonal basis of eigenvectors of the d matrices R m as in Theorem 1 and P (k,l) i = W (k,l) P i W −1 (k,l) with 0 ≤ i, k, l ≤ d − 1. Then for any matrix ρ with tr(ρ) = 1 (e.g. density matrices) This result can be seen as counterpart of the equation for the d 2 projection matrices of a SIC 2-designs ρ = (d + 1) d 2 i=1 ρ i P i − I, with ρ i = 1 d tr(ρP i ) which (as "measurement in the sky") attracted a great deal of attention in the context of the QBism approach to quantum theory by Fuchs et al. [15,16].

Concluding remarks
One can try to generalize the construction given here to further dimensions d. The rank 1 projection matrices of SICs, as well as complete sets of MUBs are projective 2-designs, which, like the construction here, are covariant under the Weyl-Heisenberg groups. Are there more similar 2-designs?
Finally it would be interesting, if and how the 2-design property could possibly be related to the quantum harmonic oscillator.