Biangular Gabor frames and Zauner's conjecture

TL;DR

This paper explores biangular Gabor frames as a new approach to Zauner's conjecture, aiming to establish the existence of equiangular tight frames in complex dimensions without relying solely on explicit constructions.

Contribution

It introduces biangular Gabor frames as an alternative framework that could potentially lead to a proof of Zauner's conjecture independent of current constructive methods.

Findings

Proposes biangular Gabor frames as a promising approach.

Connects the problem to Stark conjectures.

Suggests potential for an unconditional proof.

Abstract

Two decades ago, Zauner conjectured that for every dimension $d$, there exists an equiangular tight frame consisting of $d^2$ vectors in $\mathbb{C}^d$. Most progress to date explicitly constructs the promised frame in various dimensions, and it now appears that a constructive proof of Zauner's conjecture may require progress on the Stark conjectures. In this paper, we propose an alternative approach involving biangular Gabor frames that may eventually lead to an unconditional non-constructive proof of Zauner's conjecture.

Equations

$$A\Vert x{\Vert }_2^2\le \sum _{j=1}^n|⟨x,f_j⟩{|}^2\le B\Vert x{\Vert }_2^2.$$

$$(Tv)(j)=v(j-1),(Mv)(j)={\omega }^j\cdot v(j),v\in {\mathbb{C}}^d.$$

$$\frac{1}{d}\sum _{k,\ell =0}^{d-1}|⟨v,M^{\ell }{T}^{k}v⟩{|}^4\ge \frac{2}{d+1},$$

$$d\Vert v{\Vert }_2^4=\sum _{k,\ell =0}^{d-1}|⟨v,M^{\ell }{T}^{k}v⟩{|}^2=\Vert v{\Vert }_2^4+(d-1)\alpha +(d^2-d)\beta ,$$

$$f(t):=\frac{1}{\sqrt{d}}\cdot e^{2\pi i{t}^3/d},t\in \{0,\dots ,d-1\},$$

$$\alpha (t):=|⟨v(t),Tv(t)⟩{|}^2,\beta (t):=|⟨v(t),MTv(t)⟩{|}^2,t\in [0,1].$$

$$\Delta (t):=\beta (t)-\alpha (t)=\frac{1-\alpha (t)}{d}-\alpha (t)=\frac{1-(d+1)\alpha (t)}{d}.$$

$$4x^2=\alpha ,4y^2=\beta ,(1-x^2-y^2{)}^2=\beta .$$

Full text

1 Introduction

Let $F=\{f_{j}\}_{j=1}^{n}$ denote a finite sequence of vectors in $\mathbb{C}^{d}$. We say $F$ is a frame for $\mathbb{C}^{d}$ if there exist $A,B>0$ such that for every $x\in\mathbb{C}^{d}$, it holds that

[TABLE]

We say $F$ is tight if one may take $A=B$, and we say $F$ is unit norm if $\|f_{j}\|_{2}=1$ for every $j$. Finally, we say a unit norm $F$ is equiangular if there exists $\alpha\geq 0$ such that $|\langle f_{j},f_{j^{\prime}}\rangle|^{2}=\alpha$ whenever $j\neq j^{\prime}$. The lines spanned by the vectors in an equiangular tight frame (ETF) happen to form an optimal packing of points in projective space, as they achieve equality in the so-called Welch bound [1]. As an artifact of this optimality, ETFs enjoy applications in compressed sensing [2], digital fingerprinting [3], multiple description coding [4], and quantum state tomography [5]. The Gerzon bound [6] states that there exists an equiangular tight frame of $n$ vectors in $\mathbb{C}^{d}$ only if $n\leq d^{2}$. Zauner conjectured in his doctoral thesis[7] that for every dimension $d$, there exists an equiangular tight frame that saturates the Gerzon bound. Such an equiangular tight frame is also known as a symmetric informationally complete positive operator-valued measure (SIC).

In the sequel, we identify $\mathbb{C}^{d}$ with the space of complex-valued functions over $\mathbb{Z}_{d}:=\mathbb{Z}/d\mathbb{Z}$. Put $\omega:=e^{2\pi i/d}$, and define the translation and modulation operators $T,M\colon\mathbb{C}^{d}\to\mathbb{C}^{d}$ by

[TABLE]

It is straightforward to verify that $G(v):=\{M^{\ell}T^{k}v\}_{k,\ell=0}^{d-1}$ is a tight frame with frame bound $d\|v\|^{2}$ for every choice of $v\neq 0$. We refer to $G(v)$ as the Gabor frame generated by $v$. When $G(v)$ is equiangular, we say that $v$ is a fiducial vector. In his conjecture, Zauner actually predicted the existence of a fiducial vector (of a particular form) in $\mathbb{C}^{d}$ for every $d$. As a consequence of the theory of projective $t$-designs [8], it holds that

[TABLE]

with equality precisely when $v$ is a fiducial vector. As such, one can hunt for fiducial vectors by numerically minimizing the left-hand side of (1), and in fact, this approach has been used to identify putative fiducial vectors to machine precision for every $d\leq 151$, and for a handful of larger dimensions [9]. We say “putative fiducial vectors” because it is possible (albeit unlikely) that there is no solution to the defining system of polynomials that resides in a neighborhood of the numerical solution; a guarantee to the contrary would require a version of the Łojasiewicz inequality [10] with explicit constants.

In order to identify honest fiducial vectors, one is inclined to solve the defining system of polynomials, and to this end, solutions have been obtained by Gröbner basis calculation [11]. However, calculating a Gröbner basis for even modest polynomial systems requires a substantial amount of memory and runtime, and so progress with this approach quickly stalled. Interestingly, the resulting fiducial vectors exhibit some predictable field structure [12], and these observations have been leveraged to systematically promote numerical solutions to exact solutions [13] in dimensions that are much too large for Gröbner basis calculation. At this point, a new bottleneck has emerged: the naive description length of fiducial vectors grows quickly with the dimension. Case in point, the exact coordinates of one fiducial in dimension $d=48$ “occupies almost a thousand A4 pages (font size 9 and narrow margins)” [13]. Figure 1 illustrates that the description length appears to scale like $d^{4}$. Presumably, these coordinates enjoy a more compact description in some other representation. For example, the number fields in which the known fiducial coordinates reside are conjectured to be generated by Stark units [14]. Recently, Kopp [15] leveraged Stark units to formulate a conjectural construction of fiducial vectors in prime dimensions $d\equiv 2\bmod 3$, and this construction produced the first known exact solution in dimension $d=23$.

Overall, the community appears to be converging towards a constructive proof of Zauner’s conjecture that is conditional on the Stark conjectures. As an unconditional alternative, one might entertain the possibility of a non-constructive proof. One idea along these lines, posed by Peter Shor on MathOverflow [18], is to leverage some sort of geometric fixed point theorem; sadly, no progress in this direction has been made public. In this paper, we propose another possible route towards a non-constructive proof. In particular, we relax the set of equiangular Gabor frames to a set of biangular Gabor frames. This larger set includes well-known constructions of mutually unbiased bases. We observe that this set is frequently one-dimensional, which opens up the possibility of a proof of Zauner’s conjecture by the intermediate value theorem.

2 The proposed approach

In this section, we outline an approach to prove Zauner’s conjecture using the intermediate value theorem. We say $G(v)$ is biangular if there exists $\alpha$ and $\beta$ such that

• (i)

$|\langle v,T^{k}v\rangle|^{2}=\alpha$ for every $k\in\{1,\ldots,d-1\}$, and

• (ii)

$|\langle v,M^{\ell}T^{k}v\rangle|^{2}=\beta$ for every $k\in\{0,\ldots,d-1\}$ and $\ell\in\{1,\ldots,d-1\}$.

In this case, we can be more precise by saying that $G(v)$ is $(\alpha,\beta)$-biangular. We note that the angle parameters $\alpha$ and $\beta$ depend on one another:

Lemma 1

If $G(v)$ is an $(\alpha,\beta)$-biangular Gabor frame for $\mathbb{C}^{d}$, then $\alpha+d\beta=\|v\|_{2}^{4}$.

Proof 2.1**.**

By tightness, we have

[TABLE]

and so rearranging gives the result.

It is helpful to consider a few examples of biangular Gabor frames:

Example 2.2**.**

• (a)

Let $\mathbf{1}$ denote the all-ones vector in $\mathbb{C}^{d}$. Then $G(\mathbf{1})$ is biangular.

• (b)

Suppose $G(v)$ is a $(0,1/d)$-biangular Gabor frame in $\mathbb{C}^{d}$. Then each $\{M^{\ell}T^{k}v\}_{k=0}^{d-1}$ is an orthonormal basis, and together, these bases are mutually unbiased **[19]**. For example, if $d\geq 5$ is prime and $v$ is the Fourier transform of the corresponding Alltop sequence **[20]**

[TABLE]

then $G(v)$ is $(0,1/d)$-biangular. We refer to such $G(v)$ as Gabor MUBs.

• (c)

If $G(v)$ is equiangular, then $G(v)$ is biangular with $\alpha=\beta$.

• (d)

If $G(v)$ is biangular, then $G(cv)$ is also biangular for every $c\in\mathbb{C}^{\times}$.

Let $B_{d}$ denote the real algebraic variety of $v\in\mathbb{C}^{d}$ for which $G(v)$ is biangular. Perhaps surprisingly, we observe that $B_{d}/\mathbb{C}^{\times}$ is at times one-dimensional even though $B_{d}$ is defined by $\Omega(d^{2})$ polynomials over $2d$ real variables. We suspect that this feature can be leveraged to prove the existence of SICs. For example, the following result allows us to promote MUBs to SICs:

Lemma 2**.**

Suppose there exists a Gabor MUB in $\mathbb{C}^{d}$ and $B_{d}$ is path-connected. Then there exists a SIC in $\mathbb{C}^{d}$.

Proof 2.3**.**

Select $v_{0}$ such that $G(v_{0})$ is an MUB, put $v_{1}=\frac{1}{\sqrt{d}}\mathbf{1}$. Then by path-connectivity, there exists a parameterized curve $v\colon[0,1]\to\mathbb{C}^{d}$ such that $v(0)=v_{0}$, $v(1)=v_{1}$, and $G(v(t))$ is biangular for every $t\in(0,1)$. Without loss of generality, it holds that $\|v(t)\|_{2}=1$ for every $t$. Define $\alpha,\beta\colon[0,1]\to\mathbb{R}$ such that

[TABLE]

By Lemma 1, it holds that

[TABLE]

Considering $\alpha(0)=0$ and $\alpha(1)=1$, then the continuous function $\Delta\colon[0,1]\to\mathbb{R}$ satisfies $\Delta(0)=1/d>0$ and $\Delta(1)=-1<0$. The intermediate value theorem then guarantees the existence of $t^{\star}\in(0,1)$ such that $\Delta(t^{\star})=0$, i.e., $\alpha(t^{\star})=\beta(t^{\star})$. As such, $G(v(t^{\star}))$ is equiangular, i.e., the claimed SIC.

Importantly, Gabor MUBs (unlike SICs) are known to exist in infinitely many dimensions. The bottleneck of applying Lemma 2 is demonstrating path-connectivity. The following provides a sufficient condition to this end:

Lemma 3**.**

If $C_{d}:=\{v\in B_{d}:v(0)=1\}$ is path-connected, then $B_{d}$ is path-connected.

Proof 2.4**.**

Suppose $C_{d}$ is path-connected, and for each $j\in\mathbb{Z}_{d}$, denote $S_{j}:=\{v\in B_{d}:v(j)=1\}$ so that $S_{0}=C_{d}$. Then by symmetry, every $S_{j}$ is path-connected. To see that $B_{d}$ is also path-connected, pick any $v_{0},v_{1}\in B_{d}$. For each $k\in\{0,1\}$, we have that $v_{k}$ is nonzero by assumption, and so one of its coordinates is nonzero, say, coordinate $j_{k}\in\mathbb{Z}_{d}$. Let $c_{k}\colon[0,1]\to\mathbb{C}^{\times}$ denote any parameterized curve in $\mathbb{C}^{\times}$ from $c_{k}(0)=1$ to $c_{k}(1)=v_{k}(j_{k})$. Then $v_{k}(t):=v_{k}/c_{k}(t)$ is a curve in $B_{d}$ such that $v_{k}(0)=v_{k}$ and $v_{k}(1)\in S_{j_{k}}$. As such, we can traverse from $v_{0}$ to $v_{0}/v_{0}(j_{0})$ along $v_{0}(\cdot)$, and then from $v_{0}/v_{0}(j_{0})$ to $\mathbf{1}$ by the path-connectivity of $S_{j_{0}}$, and then from $\mathbf{1}$ to $v_{1}/v_{1}(j_{1})$ by the path-connectivity of $S_{j_{1}}$, and then from $v_{1}/v_{1}(j_{1})$ to $v_{1}$ along the reversal of $v_{1}(\cdot)$.

As a proof of concept, we leverage the above results to prove the (well-known) existence of a SIC in $\mathbb{C}^{2}$. (Importantly, our proof is non-constructive, unlike the usual proof.)

Corollary 4**.**

There exists a SIC in $\mathbb{C}^{2}$.

Proof 2.5**.**

Put $u=(1,(1+\sqrt{2})i)$. It is straightforward to verify that $G(u/\|u\|_{2})$ is an MUB. We will demonstrate that $C_{2}$ is path-connected so that the result follows from Lemmas 2 and 3. To this end, note that $v\in C_{2}$ if and only if there exist $x,y,\alpha,\beta\in\mathbb{R}$ such that $v=(1,x+yi)$ and

[TABLE]

In other words, $C_{2}$ is the set of all $(1,x+yi)$ such that $x^{2}+(y\pm 1)^{2}=2$. Geometrically, this is the union of two circles of radius $\sqrt{2}$ centered at $(0,\pm 1)$; see Figure 1 for an illustration. Since these circles intersect, it follows that $C_{2}$ is path-connected, as desired.

To prove Zauner’s conjecture, we would need to replicate this non-constructive proof technique in every dimension. This suggests the following:

Problem 2.6**.**

For which dimensions $d$ is $B_{d}$ is path-connected?

There has already been some work to prove path-connectivity of certain varieties of frames. Most work along these lines has focused on the variety of unit norm tight frames. Initial work [21] leveraged so-called eigensteps [22] to construct explicit paths that demonstrate path-connectivity, whereas a more recent treatment [23] exploits technology from symplectic geometry to obtain a non-constructive proof. It would be interesting if similar technology could be applied to tackle Problem 2.6.

Next, MUBs are only known to exist in prime power dimensions, and so we would need to improve Lemma 2 before we can hope to prove Zauner’s conjecture. In fact, “Gabor MUB” in Lemma 2 can be replaced by any biangular Gabor frame with appropriately small $\alpha$, suggesting the following:

Problem 2.7**.**

For every $d$, find $v\in\mathbb{C}^{d}$ such that both $\|v\|_{2}=1$ and $G(v)$ is $(\alpha,\beta)$-biangular with $\alpha<\frac{1}{d+1}$.

Considering the successful instance of Gabor MUBs, it seems reasonable to suspect that Problem 2.7 can be solved in closed form, even though the $\alpha=\frac{1}{d+1}$ case of SICs has resisted such a solution. Finally, note that we do not require all of $B_{d}$ to be path-connected, as it suffices to find $v_{0}$ and $v_{1}$ for which there exist $\alpha_{0}$ and $\alpha_{1}$ such that

• (i)

$G(v_{j})$ is $(\alpha_{j},\frac{1}{d}(1-\alpha_{j}))$-biangular for each $j\in\{0,1\}$,

• (ii)

$v_{0}$ and $v_{1}$ are path-connected in $B_{d}$, and

• (iii)

$\alpha_{0}<\frac{1}{d+1}<\alpha_{1}$.

Of course, it is likely easier to solve Problem 2.6.

To illustrate our observation that biangular Gabor frames enjoy path-connectivity, we run a simple numerical experiment: For each $d\in\{2,4,5\}$, we consider the numerical fiducial reported by Scott and Grassl [11] (when $d=3$, the variety of SIC fiducials is already interesting). Call this vector $v_{0}$. We slightly perturb this fiducial and then locally minimize the sum of the squares of the polynomials that define the variety of biangular Gabor seed vectors. This produces a new point $v_{1}$ on the variety. Next, we locally minimize from the perturbation $v_{j}+c\cdot\frac{v_{j}-v_{j-1}}{\|v_{j}-v_{j-1}\|_{2}}$ to obtain $v_{j+1}$ (with $j=1$), and we iterate this procedure to identify a sequence of points on the variety. (Here, $c$ is a small constant.) The results of this experiment are illustrated in Figure 2.

3 Discussion

This paper proposed a new approach to tackle Zauner’s conjecture. Specifically, we relax the set of SICs to a larger set of biangular Gabor frames, which appear to form a path-connected variety. This feature could very well allow for a non-constructive proof of Zuaner’s conjecture, and we isolate Problems 2.6 and 2.7 as steps towards this end. In addition, it would also be interesting to leverage the variety of biangular Gabor frames to facilitate the search for numerical SICs. We leave these investigations for future work.

ACKNOWLEDGMENTS

The authors thank John Jasper and Hans Parshall for commenting on a draft of this paper. MM and DGM were partially supported by AFOSR FA9550-18-1-0107. DGM was also supported by NSF DMS 1829955 and the Simons Institute of the Theory of Computing.