Biangular Gabor frames and Zauner's conjecture
BBiangular Gabor frames and Zauner’s conjecture
Mark Magsino and Dustin G. MixonDepartment of Mathematics, The Ohio State University, Columbus, OH 43210
ABSTRACT
Two decades ago, Zauner conjectured that for every dimension d , there exists an equiangular tight frame consist-ing of d vectors in C d . Most progress to date explicitly constructs the promised frame in various dimensions, andit 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 toan unconditional non-constructive proof of Zauner’s conjecture.
1. INTRODUCTION
Let F = { f j } nj =1 denote a finite sequence of vectors in C d . We say F is a frame for C d if there exist A, B > x ∈ C d , it holds that A (cid:107) x (cid:107) ≤ n (cid:88) j =1 |(cid:104) x, f j (cid:105)| ≤ B (cid:107) x (cid:107) . We say F is tight if one may take A = B , and we say F is unit norm if (cid:107) f j (cid:107) = 1 for every j . Finally, wesay a unit norm F is equiangular if there exists α ≥ |(cid:104) f j , f j (cid:48) (cid:105)| = α whenever j (cid:54) = j (cid:48) . The linesspanned by the vectors in an equiangular tight frame (ETF) happen to form an optimal packing of points inprojective space, as they achieve equality in the so-called Welch bound. As an artifact of this optimality, ETFsenjoy applications in compressed sensing, digital fingerprinting, multiple description coding, and quantumstate tomography. The Gerzon bound states that there exists an equiangular tight frame of n vectors in C d only if n ≤ d . Zauner conjectured in his doctoral thesis that for every dimension d , there exists an equiangulartight frame that saturates the Gerzon bound. Such an equiangular tight frame is also known as a symmetricinformationally complete positive operator-valued measure (SIC) .In the sequel, we identify C d with the space of complex-valued functions over Z d := Z /d Z . Put ω := e πi/d ,and define the translation and modulation operators T, M : C d → C d by( T v )( j ) = v ( j − , ( M v )( j ) = ω j · v ( j ) , v ∈ C d . It is straightforward to verify that G ( v ) := { M (cid:96) T k v } d − k,(cid:96) =0 is a tight frame with frame bound d (cid:107) v (cid:107) for everychoice of v (cid:54) = 0. We refer to G ( v ) as the Gabor frame generated by v . When G ( v ) is equiangular, we saythat v is a fiducial vector . In his conjecture, Zauner actually predicted the existence of a fiducial vector (of aparticular form) in C d for every d . As a consequence of the theory of projective t -designs, it holds that1 d d − (cid:88) k,(cid:96) =0 |(cid:104) v, M (cid:96) T k v (cid:105)| ≥ d + 1 , (1)with equality precisely when v is a fiducial vector. As such, one can hunt for fiducial vectors by numericallyminimizing the left-hand side of (1), and in fact, this approach has been used to identify putative fiducial vectorsto machine precision for every d ≤ We say “putative fiducialvectors” because it is possible (albeit unlikely) that there is no solution to the defining system of polynomialsthat resides in a neighborhood of the numerical solution; a guarantee to the contrary would require a version ofthe (cid:32)Lojasiewicz inequality with explicit constants. Send correspondence to [email protected] and [email protected] a r X i v : . [ m a t h . M G ] A ug -3 -2 -1 0 1 2 3-3-2-10123 Figure 1. (left)
Description lengths of naive expressions of fiducial vectors of SICs. For each fiducial reported, count the number of characters used to describe the coordinates of the fiducial vector, and plot the results. Blue dotscorrespond to solutions obtained by Gr¨obner basis calculation, and red dots correspond to solutions obtained frompromoting numerical solutions. The horizontal axis corresponds to the dimension d , while the vertical axis denotes thedescription length. The dotted line plots the curve d . This illustrates that the naive representation of fiducial coordinatesin terms of radicals is inefficient, and so a more compact representation (in terms of Stark units, say) is necessary beforeone can find a constructive proof of Zauner’s conjecture. (right) Illustration of the proposed method. The set of ( x, y )for which G ((1 , x + yi )) is biangular equals the union of two intersecting circles, plotted in black. The blue dot at( x, y ) = (0 , √
2) corresponds to a Gabor MUB, while the red dot at ( x, y ) = (1 ,
0) corresponds to the trivial biangularGabor frame G ( ). The angles α and β satisfy α < β in the former case and α > β in the latter case. As such, for anycurve connecting these two points, there exists a point ( x (cid:63) , y (cid:63) ) at which α = β , i.e., G ((1 , x (cid:63) + y (cid:63) i )) is a SIC. Such a pointis plotted in green. In order to identify honest fiducial vectors, one is inclined to solve the defining system of polynomials, andto this end, solutions have been obtained by Gr¨obner basis calculation. However, calculating a Gr¨obner basisfor even modest polynomial systems requires a substantial amount of memory and runtime, and so progresswith this approach quickly stalled. Interestingly, the resulting fiducial vectors exhibit some predictable fieldstructure, and these observations have been leveraged to systematically promote numerical solutions to exactsolutions in dimensions that are much too large for Gr¨obner basis calculation. At this point, a new bottleneckhas 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 andnarrow margins)”. Figure 1 illustrates that the description length appears to scale like d . Presumably, thesecoordinates enjoy a more compact description in some other representation. For example, the number fields inwhich the known fiducial coordinates reside are conjectured to be generated by Stark units. Recently, Kopp leveraged Stark units to formulate a conjectural construction of fiducial vectors in prime dimensions d ≡ d = 23.Overall, the community appears to be converging towards a constructive proof of Zauner’s conjecture thatis conditional on the Stark conjectures. As an unconditional alternative, one might entertain the possibility of anon-constructive proof. One idea along these lines, posed by Peter Shor on MathOverflow, is to leverage somesort of geometric fixed point theorem; sadly, no progress in this direction has been made public. In this paper, wepropose another possible route towards a non-constructive proof. In particular, we relax the set of equiangularGabor frames to a set of biangular Gabor frames. This larger set includes well-known constructions of mutuallyunbiased bases . We observe that this set is frequently one-dimensional, which opens up the possibility of a proofof Zauner’s conjecture by the intermediate value theorem. . THE PROPOSED APPROACH
In this section, we outline an approach to prove Zauner’s conjecture using the intermediate value theorem. Wesay G ( v ) is biangular if there exists α and β such that(i) |(cid:104) v, T k v (cid:105)| = α for every k ∈ { , . . . , d − } , and(ii) |(cid:104) v, M (cid:96) T k v (cid:105)| = β for every k ∈ { , . . . , d − } and (cid:96) ∈ { , . . . , d − } .In this case, we can be more precise by saying that G ( v ) is ( α, β )-biangular. We note that the angle parameters α and β depend on one another: Lemma 1. If G ( v ) is an ( α, β ) -biangular Gabor frame for C d , then α + dβ = (cid:107) v (cid:107) .Proof . By tightness, we have d (cid:107) v (cid:107) = d − (cid:88) k,(cid:96) =0 |(cid:104) v, M (cid:96) T k v (cid:105)| = (cid:107) v (cid:107) + ( d − α + ( d − d ) β, and so rearranging gives the result.It is helpful to consider a few examples of biangular Gabor frames: Example 2. (a) Let denote the all-ones vector in C d . Then G ( ) is biangular.(b) Suppose G ( v ) is a (0 , /d ) -biangular Gabor frame in C d . Then each { M (cid:96) T k v } d − k =0 is an orthonormal basis,and together, these bases are mutually unbiased. For example, if d ≥ is prime and v is the Fouriertransform of the corresponding Alltop sequence f ( t ) := 1 √ d · e πit /d , t ∈ { , . . . , d − } , then G ( v ) is (0 , /d ) -biangular. We refer to such G ( v ) as Gabor MUBs .(c) If G ( v ) is equiangular, then G ( v ) is biangular with α = β .(d) If G ( v ) is biangular, then G ( cv ) is also biangular for every c ∈ C × . Let B d denote the real algebraic variety of v ∈ C d for which G ( v ) is biangular. Perhaps surprisingly, weobserve that B d / C × is at times one-dimensional even though B d is defined by Ω( d ) polynomials over 2 d realvariables. We suspect that this feature can be leveraged to prove the existence of SICs. For example, thefollowing result allows us to promote MUBs to SICs: Lemma 3.
Suppose there exists a Gabor MUB in C d and B d is path-connected. Then there exists a SIC in C d .Proof . Select v such that G ( v ) is an MUB, put v = √ d . Then by path-connectivity, there exists aparameterized curve v : [0 , → C d such that v (0) = v , v (1) = v , and G ( v ( t )) is biangular for every t ∈ (0 , (cid:107) v ( t ) (cid:107) = 1 for every t . Define α, β : [0 , → R such that α ( t ) := |(cid:104) v ( t ) , T v ( t ) (cid:105)| , β ( t ) := |(cid:104) v ( t ) , M T v ( t ) (cid:105)| , t ∈ [0 , . By Lemma 1, it holds that ∆( t ) := β ( t ) − α ( t ) = 1 − α ( t ) d − α ( t ) = 1 − ( d + 1) α ( t ) d . Considering α (0) = 0 and α (1) = 1, then the continuous function ∆ : [0 , → R satisfies ∆(0) = 1 /d > − <
0. The intermediate value theorem then guarantees the existence of t (cid:63) ∈ (0 ,
1) such that ∆( t (cid:63) ) = 0,i.e., α ( t (cid:63) ) = β ( t (cid:63) ). As such, G ( v ( t (cid:63) )) is equiangular, i.e., the claimed SIC.mportantly, Gabor MUBs (unlike SICs) are known to exist in infinitely many dimensions. The bottleneck ofapplying Lemma 3 is demonstrating path-connectivity. The following provides a sufficient condition to this end: Lemma 4. If C d := { v ∈ B d : v (0) = 1 } is path-connected, then B d is path-connected.Proof . Suppose C d is path-connected, and for each j ∈ Z d , denote S j := { v ∈ B d : v ( j ) = 1 } so that S = C d .Then by symmetry, every S j is path-connected. To see that B d is also path-connected, pick any v , v ∈ B d .For each k ∈ { , } , we have that v k is nonzero by assumption, and so one of its coordinates is nonzero, say,coordinate j k ∈ Z d . Let c k : [0 , → C × denote any parameterized curve in C × 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) ∈ S j k . As such, we can traverse from v to v /v ( j ) along v ( · ), and then from v /v ( j ) to by the path-connectivity of S j , and then from to v /v ( j ) by the path-connectivity of S j , and then from v /v ( j ) to v along the reversal of v ( · ).As a proof of concept, we leverage the above results to prove the (well-known) existence of a SIC in C .(Importantly, our proof is non-constructive, unlike the usual proof.) Corollary 5.
There exists a SIC in C .Proof . Put u = (1 , (1 + √ i ). It is straightforward to verify that G ( u/ (cid:107) u (cid:107) ) is an MUB. We will demonstratethat C is path-connected so that the result follows from Lemmas 3 and 4. To this end, note that v ∈ C if andonly if there exist x, y, α, β ∈ R such that v = (1 , x + yi ) and4 x = α, y = β, (1 − x − y ) = β. In other words, C is the set of all (1 , x + yi ) such that x + ( y ± = 2. Geometrically, this is the union of twocircles of radius √ , ± C is path-connected, as desired.To prove Zauner’s conjecture, we would need to replicate this non-constructive proof technique in everydimension. This suggests the following: Problem 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 alongthese lines has focused on the variety of unit norm tight frames. Initial work leveraged so-called eigensteps to construct explicit paths that demonstrate path-connectivity, whereas a more recent treatment exploitstechnology from symplectic geometry to obtain a non-constructive proof. It would be interesting if similartechnology could be applied to tackle Problem 6.Next, MUBs are only known to exist in prime power dimensions, and so we would need to improve Lemma 3before we can hope to prove Zauner’s conjecture. In fact, “Gabor MUB” in Lemma 3 can be replaced by anybiangular Gabor frame with appropriately small α , suggesting the following: Problem 7.
For every d , find v ∈ C d such that both (cid:107) v (cid:107) = 1 and G ( v ) is ( α, β ) -biangular with α < d +1 . Considering the successful instance of Gabor MUBs, it seems reasonable to suspect that Problem 7 can besolved in closed form, even though the α = d +1 case of SICs has resisted such a solution. Finally, note that wedo not require all of B d to be path-connected, as it suffices to find v and v for which there exist α and α such that(i) G ( v j ) is ( α j , d (1 − α j ))-biangular for each j ∈ { , } ,(ii) v and v are path-connected in B d , and(iii) α < d +1 < α .Of course, it is likely easier to solve Problem 6.To illustrate our observation that biangular Gabor frames enjoy path-connectivity, we run a simple numericalexperiment: For each d ∈ { , , } , we consider the numerical fiducial reported by Scott and Grassl (when d = 3, the variety of SIC fiducials is already interesting). Call this vector v . We slightly perturb this fiducialand then locally minimize the sum of the squares of the polynomials that define the variety of biangular Gabor Figure 2. Numerical experiments to illustrate path-connectivity in the variety of biangular Gabor frames. (left)
As acontrol, we first consider the case of d = 2, which we have already treated in the proof of Corollary 5. In blue, we plotthe set of all ( x, y ) for which (1 , x + yi ) generates a biangular Gabor frame. The red dot in this set corresponds to thenumerical fiducial reported by Scott and Grassl. We then traverse the variety using the numerical scheme discussed atthe end of Section 2; we plot the corresponding trajectory in black. In the display below, we plot how ( α, β ) evolve overthis trajectory (we compute these angles after normalizing v to have unit norm). The angles start at ( , ), correspondingto the SIC, then pass through (0 , ), corresponding to an MUB, and then finally approach (1 , d = 4 (middle) and d = 5 (right) , fixing v (0) = 1, plotting the trajectoryof (Re v (1) , Im v (1)) above, and then plotting the angles ( α, β ) below. Unlike the d = 2 case, we do not have analyticexpressions for the variety in these cases. seed vectors. This produces a new point v on the variety. Next, we locally minimize from the perturbation v j + c · v j − v j − (cid:107) v j − v j − (cid:107) to obtain v j +1 (with j = 1), and we iterate this procedure to identify a sequence of points onthe 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 alarger set of biangular Gabor frames, which appear to form a path-connected variety. This feature could verywell allow for a non-constructive proof of Zuaner’s conjecture, and we isolate Problems 6 and 7 as steps towardsthis end. In addition, it would also be interesting to leverage the variety of biangular Gabor frames to facilitatethe search for numerical SICs. We leave these investigations for future work.
CKNOWLEDGMENTS
The authors thank John Jasper and Hans Parshall for commenting on a draft of this paper. MM and DGMwere partially supported by AFOSR FA9550-18-1-0107. DGM was also supported by NSF DMS 1829955 andthe Simons Institute of the Theory of Computing.
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