On Weyl-Heisenberg orbits of equiangular lines

Transcription

1 J Algebr Comb ( : DOI /s On Weyl-Heisenberg orbits of equiangular lines Mahdad Khatirinejad Received: 9 May 2007 / Acceted: 25 October 2007 / Published online: 6 November 2007 Sringer Science+Business Media, LLC 2007 Abstract An element z CP d 1 is called fiducial if {gz : g G} is a set of lines with only one angle between each air, where G = Z d Z d is the one-dimensional finite Weyl-Heisenberg grou modulo its centre. We give a new characterization of fiducial vectors. Using this characterization, we show that the existence of almost flat fiducial vectors imlies the existence of certain cyclic difference sets. We also rove that the construction of fiducial vectors in rime dimensions 7 and 19 due to Aleby (J. Math. Phys. 46(5:052107, 2005 does not generalize to other rime dimensions (excet for ossibly a set with density zero. Finally, we use our new characterization to construct fiducial vectors in dimension 7 and 19 whose coordinates are real. Keywords Comlex equiangular lines Weyl-Heisenberg grou Fiducial vector SIC-POVM 1 Introduction One of the most challenging roblems in algebraic combinatorics is finding large sets of lines with few angles between the airs. In articular, the roblem of finding equiangular lines in real and comlex saces is still wide oen. In this aer, we only work with lines which lie in comlex sace C d, unless stated otherwise. A line in C d is an element in the comlex rojective sace CP d 1, the sace of one-dimensional comlex vector subsaces of C d. Each element in CP d 1 can be reresented by a unit vector u in C d. Note that such a reresentation is not unique since λu with λ =1 reresents the same line. In the rest of the aer, we only work with the unit vectors Research suorted by the Natural Sciences and Engineering Research Council of Canada (NSERC. M. Khatirinejad ( Deartment of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada mahdad@math.sfu.ca

2 334 J Algebr Comb ( : in C d to reresent a line. The cosine of the angle between the lines sanned by unit vectors u, v C d is defined as u v, the absolute value of their inner roduct. A set of lines in C d sanned by unit vectors v 1,...,v k is equiangular if there exists a constant α such that v i v j =α for every 1 i<j k. It is not hard to show that if X is a set of equiangular lines in C d then X d 2.If X =d 2 then X is called a maximum set of equiangular lines. If such a maximum set exists then we must have α = 1/ d + 1. To the best of our knowledge, such maximum sets are resented in [2, 10 13, 17]for 2 d 10 and d {12, 19}. In addition, it is claimed in [7] (with reference to rivate communication with Markus Grassl that such sets also exist for d {11, 13, 15}. Mostly, the roof that such given sets are equiangular is not ublished, as it may generally require ages of tedious algebra to give a comlete roof. Nevertheless, the roblem is still oen for a general d: Problem 1.1 For any ositive integer d, does there exist a set of d 2 equiangular lines in C d? Throughout the aer, let {e j : j Z d } be the standard basis for C d, and let ω be a d-th rimitive root of unity in C. The coordinates of a vector z C d are always indexed by Z d, the set of integers modulo d. When there is no confusion, we will write (z j instead of (z j j Zd to reresent z. The conjugate of an element x C is denoted by x and A denotes the conjugate transose of a comlex valued matrix A. The Pauli matrices for Z d are defined by their action on the standard basis as follows: X k : e j e j+k, Y k : e j ω jk e j, where k Z d. The grou GP(d = X j, Y k : j,k Z d is usually called the generalized Pauli grou or the one-dimensional finite Weyl-Heisenberg grou. Define H d = GP(d/Z(GP(d, where Z(G denotes the centre of G. Observe that the quotient grou H d = {X j Y k ωi d :j,k Z d } is isomorhic to Z d Z d, where I d denotes the identity matrix of order d. All of the known constructions of maximum sets of equiangular lines have the form {X j Y k z : j,k Z d } for some z C d (excet for the set of 64 lines in C 8 constructed by Hoggar [12], in which the grou Z 3 2 is used instead of Z 8. Definition 1.2 A vector z C d is called fiducial if {gz : g H d } is a set of equiangular lines. It is widely believed that for every d there exists a fiducial vector in C d (for examle see [2, 7, 9, 11, 13, 17]. The set of d 2 equiangular lines in C d has been discussed in different contexts. For examle, in quantum information theory it is a symmetric informationally comlete ositive oerator valued measure (SIC-POVM, which is comosed of d 2 rank-one oerators all of whose oerator inner roducts are equal. In

3 J Algebr Comb ( : the quantum information theory community, there has been notable interest to construct such SIC-POVMs in every dimension and most of the focus has been on SIC- POVMs which are invariant under the Weyl-Heisenberg grou (i.e. the SIC-POVMs that arise from fiducial vectors. Equiangular lines have several alications to quantum information such as quantum fingerrinting [15], quantum tomograhy [5] and quantum crytograhic rotocols [8]. They also lay a role in the Bayesian formulation of the quantum mechanics [5, 8] where they make nice standard quantum measurements. Equiangular lines have also been studied in the context of sherical codes and designs [6]. In Section 2 we will resent a new characterization of fiducial vectors (Theorem 2.2 which lays an essential role in the rest of the aer. In Sections 3, 4, and 5, we study three tyes of fiducial vectors which are closely related: almost flat, argument Legendre, and real Legendre. Using the new characterization, we show that the existence of almost flat fiducial vectors imlies the existence of certain cyclic difference sets. The fiducial vectors in dimensions 7 and 19 constructed by Aleby [2]are examles of the argument Legendre fiducial vectors. We rove that the construction of these vectors does not generalize to other rime dimensions (excet for ossibly a set with density zero. We also use our new characterization to construct fiducial vectors in dimension 7 and 19 whose coordinates are real. Finally, a concise fact regarding the non-existence of a general fiducial vector is discussed in Section 6. A note about the roofs: To sustain the flow of the aer we have moved the roofs of some of the theorems to the Aendix. 2 The characterizing identities One of the interesting roerties of a set of equiangular lines of the form {gz : g G} is that with certain assumtions on G, instead of checking the equality of ( G 2 inner roducts, one may only need to check G of them: Observation 2.1 Let G be any set of matrices that is closed under matrix multilication and the conjugate transose oerator. Assume G contains the identity matrix I and let z be a vector. Then {gz : g G} is a set of equiangular lines if and only if z gz =c for every g G \{I} and z z =1. Here c is the cosine of the common angle between the lines. Note that H d, introduced above, is closed under matrix multilication and the conjugate transose oerator and also contains the identity matrix. For a given dimension d, by definition, fiducial vectors form the set of all roots of a system of multivariate olynomials in z 0,...,z d 1, z 0,..., z d 1 over the field Q(ω. The following theorem shows that one may only consider Q instead of Q(ω and it gives a different characterization of fiducial vectors.

4 336 J Algebr Comb ( : Theorem 2.2 A vector z = (z j C d is fiducial if and only if for every (s, t Z d Z d the following identities hold: 0 for s,t 0, f s,t (z := z j z j+s z j+t z j+s+t = 1 d+1 for s 0, t= 0 or s = 0, t 0, j Z d 2 d+1 for s = t = 0. Proof For every k Z d we have X s Y k z = (z j+s ω k(j+s. Thus z X s Y k z = j Z d z j z j+s ω k(j+s. This imlies that z X s Y k z 2 = = j Z d z j z j+s ω k(j+s z j z j+s z j z j +sω k(j j j,j Z d = z j z j+s z j+t z j+t+s ω kt j Z d t Z d = f s (w k, j Z d z j z j +sω k(j +s where f s (x = d 1 t=0 f s,t(zx t. It follows that the vector z is fiducial if and only if f s (ω k 1 for (s, k = (0, 0, = (1 1 d+1 for (s, k (0, 0. Let d ={x C : x d = 1}.Fors = 0, the above identity holds if and only if f 0 (x 1/(d + 1 vanishes on d \{1} and f 0 (1 = 1, that is t Z d f 0,t (z = 1. For every s 0, identity (1 holds if and only if f s (x 1/(d + 1 vanishes on d. Since f s (x 1/(d + 1 is a olynomial of degree at most d 1, this is equivalent to the fact that f s (x is identically equal to zero. That is f s,t (z = 0 when t 0, and f s,0 (z = 1/(d + 1. Since f s,t (z = f t,s (z, by combining the above cases, we get the desired result. Remark. We have noted that the above theorem has also been discovered indeendently in [3], a few months after the original submission of this article. Remark. For every s,t Z d,wehavef s,t (z = f t,s (z = f s, t (z. Therefore the set Z d Z d in Theorem 2.2 can be relaced with the set: {(s, t Z d Z d : 0 s t d/2 }. As an immediate corollary, we get the following necessary conditions for a vector to be fiducial:

5 J Algebr Comb ( : Corollary 2.3 Let z = (r j e iθ j, where r j R and θ j [0, 2π, be a fiducial vector in C d. Then the following identities hold: 1 rj 2 d+1 for s Z d \{0}, r2 j+s = (2 j Z d 2 d+1 for s = 0. Remark Note that the set {X 0 Y k z : k Z d } is equiangular if and only if (2 holds, and also note that {X 0 Y k ωi d :k Z d } is a subgrou of H d isomorhic to Z d. Therefore, the equiangular condition test for this set is indeendent of the argument values of the coordinates of z and it is only deendent on their absolute values. A note on equiangular vectors A set of vectors in R d is called equiangular if the inner roduct between every two distinct vectors in the set is a constant. Note the difference between equiangular vectors and equiangular lines where we require the absolute value of the inner roducts to be a constant. Now, assume that z = (r j e iθ j, where r j R and θ j [0, 2π, is a fiducial vector in C d d+1. Also, let x = ( 2 r2 j. Then, using Corollary 2.3, one may observe that S ={x, X 1 x,...,x d 1 x} is a set of d equiangular vectors on the unit shere in R d with common angle 60. Note that the set S is unique u to a unitary transformation Q. This is because the matrix whose set of columns is S can be decomosed to QR, where Q is a unitary and R is an uer triangular matrix. In the following three sections, we will treat three tyes of fiducial vectors: almost flat, argument Legendre, and real Legendre. The argument Legendre fiducial vectors are almost flat and are discussed by Aleby [2] in secific dimensions. Since the real Legendre fiducial vectors have very similar roerties to the argument Legendre ones and also all of the coordinates of such vectors have the same argument, we have also investigated this class of fiducial vectors. 3 Almost flat fiducial vectors A vector in C d is called flat if all its coordinates have the same absolute value. It is roved [9] that there are at most d 2 d + 1 flat equiangular lines in C d (and there are exactly d 2 d + 1 such lines when d 1 is a rime ower. In articular, no flat fiducial vectors exist (this can also be concluded easily from Corollary 2.3. To take it one ste further, we say a vector is almost flat if it is flat excet for one coordinate. Aleby [2] constructed fiducial vectors in dimensions 7 and 19 which are almost flat. Using an eigenvalue argument, Roy [14] roved that for almost flat fiducial vectors in C d, the absolute values of the coordinates are determined in terms of d. We rovide an alternative roof of this fact here: Theorem 3.1 Let z be a fiducial vector in C d such that one coordinate of z has absolute value b, and all other coordinates have absolute value a. Then a 2 = 1 1/ d + 1 d, b 2 = 1 ± (d 1/ d + 1. d

6 338 J Algebr Comb ( : Proof Since z is a unit vector, we have b 2 + (d 1a 2 = 1. By letting s = 0in Corollary 2.3, we get b 4 + (d 1a 4 = 2/(d + 1. Solving for a 2 and b 2, we get the stated values. Remark. Note that (a 2,b 2 = ((1 + 1/ d + 1/d, (1 (d 1/ d + 1/d is only ossible for d 3, since we must have a 2,b 2 0. In fact, we can rove a stronger result. Namely, the existence of a cyclic difference set in Z d is a necessary condition for the existence of fiducial vectors in which the coordinates take exactly two distinct absolute values. Before stating the result, recall that a (d,k,λ-cyclic difference set is a set D ={α 1,...,α k } Z d such that each element in Z d \{0} can be reresented as a difference α i α j in exactly λ different ways (see for examle [4]. The roof of the following theorem is given in Aendix B. Theorem 3.2 Let a b be real numbers and k,d be integers such that 0 <k d/2. Let z = (z j C d be a fiducial vector such that k coordinates of z have absolute value b, and all other coordinates have absolute value a. Let D ={j Z d : z j =b}. Then ( ( a 2 = 1 k(d 1 1, b 2 = 1 (d k(d 1 1 ±, d (d + 1(d k d k(d + 1 and D forms a (d,k,λ-cyclic difference set in Z d. In articular, d 1 must divide k(k 1. Remark. Note that (a 2,b 2 = ((1 + k(d 1/(d + 1/(d k/d, (1 (d k(d 1/(d + 1/k/d is only ossible for d 2k Argument Legendre fiducial vectors Let denote a rime number. For every j Z let ( j denote the Legendre symbol, that is ( j is 0 (resectively 1 or 1 if j = 0 (resectively if j is a quadratic or a non-quadratic residue. Definition 4.1 We say a fiducial vector z = (z j C is argument Legendre (AL if there exist a,b,θ R such that { b if j = 0, z j = ae i(j θ if j 0. Note that AL fiducial vectors are almost flat. Thus, by Theorem 3.1 and its succeeding remark, if >3 then we must have a 2 = (1 1/ + 1/ and b 2 = (1 + ( 1/ + 1/. In fact, if an AL fiducial vector exists, then the value of θ is also determined in terms of :

7 J Algebr Comb ( : Proosition 4.2 Let >3 such that 3 (mod 4. If z C is an AL fiducial vector with arameters (a,b,θthen θ = { cos 1 (1/ for 3 (mod 8, cos 1 ( for 7 (mod 8. (3 The roof of this roosition is rather long and technical and is given in the Aendix. By using Theorem 2.2 it is easy to check that the vector ( 2/3,e iπ/3 / 6, e iπ/3 / 6 is an AL fiducial vector in C 3. The construction of AL fiducial vectors in C 7 and C 19 is given by Aleby in [2]. However, a roof that the given vectors are in fact fiducial is not given in his work. One may interret that the roofs are basic but require some extensive tedious algebra. We will give a short roof that Aleby s vectors [2] are fiducial: Theorem 4.3 AL fiducial vectors exist for = 7 and = 19. Proof Let f s,t (z be as defined in Theorem 2.2. Recall from Theorem 2.2 (and the remark following it that an AL fiducial vector z C with arameters a,b, and c = cos θ exists if and only if the system of equations f 0,0 (z 2/( + 1 = f 0,r (z 1/( + 1 = f s,t (z = 0, where 0 <r /2 and 0 <s t /2, (4 has a solution. If = 7, we may easily verify that f 0,0 (z = 6a 4 + b 4, f 0,1 (z = f 0,2 (z = f 0,3 (z = 5a 4 + 2a 2 b 2, f 1,1 (z = f 2,2 (z = f 3,3 (z = 4a 4 c 2 (4c a 2 b 2 + 2a 3 bc, f 1,2 (z = f 1,3 (z = f 2,3 (z = 4a 4 c 2 a 4 + 4a 3 bc(2c 2 1. It is therefore straightforward to check that the system (4 has a solution for = 7 when 4 2 a = 28, b = , c = cos θ = 2. (the values of a,b, and θ are taken from Theorem 3.1 and Proosition 4.2. Analogously for = 19, we have f 0,0 (z = 18a 4 + b 4, f 0,r (z = 17a 4 + 2a 2 b 2, f r,r (z = 16a 4 c 2 (2c a 2 b 2 + 2a 3 bc(4c 2 3, f r,2r (z = f r,7r (z = a 4 (16c 4 4c a 3 bc(2c 2 1, f r,3r (z = f r,4r (z = 4a 4 c 2 a 4 + 4a 3 bc(2c 2 1, for all r Z 19 such that 1 r 9. A straightforward evaluation at

8 340 J Algebr Comb ( : a = 190, b = , c = cos θ = shows that the system (4 has a solution for = , Using MAPLE, Roy[14] confirms (numerically that there are no other AL fiducial vectors for 400. We conjecture that AL fiducial vectors only exist for {3, 7, 19}. Excet for a set of rimes with density zero, we are able to confirm this conjecture: Theorem 4.4 There exists a set of rimes P with zero density (in the set of all rimes that are equal to 3 modulo 4 such that for all / P there exists no AL fiducial vector in C. In fact, in the roof of Theorem 4.4, we will see that P is the set of rimes >7 such that a( = 3 and 11 or 19 (mod 24, where a( denotes the trace of the Frobenius endomorhism of the ellitic curve y 2 = x(x + 1(x + 2(x + 3: a( = ( x(x + 1(x + 2(x + 3. x Z The fact that the set P has density zero follows from Theorem 20 in [16]. 5 Real fiducial vectors We say a fiducial vector z of dimension d is real if z R d. Examle 5.1 By Theorem 2.2, a vector z = (a,b,c R 3 C 3 is fiducial if and only if a 4 + b 4 + c 4 1/2 = a 2 b 2 + b 2 c 2 + c 2 a 2 1/4 = abc(a + b + c = 0. It is easy to see that (0, 1/ 2, 1/ 2 is a solution to this system and is thus a real fiducial vector. It seems that real fiducial vectors rarely exist as the assumtion z R d is rather strong. On the other hand, searching for such fiducial vectors should be easier since one needs to deal with less arameters. In fact, the number of unknown real arameters in a real fiducial vector in C d is only d comared to the number of unknown real arameters in a general fiducial vector in C d which is 2d 1 (we may always assume z 0 R. Desite this fact, we are able to find real fiducial vectors in dimension 7 and 19 where the coordinates only take 3 distinct values. It would be quite interesting to know whether real fiducial vectors in dimensions other than 3, 7, and 19 exist. In this section, we will discuss a secial tye of real fiducial vectors (called real Legendre which have similar characteristics to the argument Legendre fiducial vectors.

9 J Algebr Comb ( : Real Legendre fiducial vectors Let be a rime number. We call a fiducial vector z = (z j C real Legendre (RL if there exist a,b,c R such that a if j = 0, z j = b if ( j = 1, c if ( j = 1. In the next two theorems, we will show that real Legendre fiducial vectors exist for {7, 19}. Theorem 5.2 Let {±a} be the set of real roots of 56 x x 2 1 and let {±b,±c} be the set of real roots of 3136 x x x 4 56 x with a<0 <b,c. Then the RL vector z = (a,b,b,c,b,c,c R 7 is a fiducial vector in C 7. Proof By Theorem 2.2, the vector z R 7 is fiducial if and only if (a,b,c is a solution to the following system f 0,0 (z = a 4 + 3(b 4 + c 4 = 1/4, f 0,1 (z = f 0,2 (z = f 0,3 (z = a 2 (b 2 + c 2 + b 4 + c b 2 c 2 = 1/8, f 1,1 (z = f 2,2 (z = f 3,3 (z = (2a(b + c + b 2 + c 2 + bcbc = 0, f 1,2 (z = f 1,3 (z = f 2,3 (z = a 2 bc + a(b 3 + c 3 + bc(b + c 2 = 0. Let I = f 0,0 (z 1/4,f 0,1 (z 1/8,f 1,1 (z, f 1,2 (z. Using a comuter algebra system, such as MAPLE, we may find the Gröbner basis G c of I with resect to the ure lexicograhic monomial order induced by a>b>c. We observe that G c = {h(c, b f(c,a g(c}, where h(x = 3136 x x x 4 56 x 2 + 1, f(x= 8/23 x(784 x x x 2 14, g(x = 1/23 x(9408x x x Hence h(c I and we must have h(c = 0. Similarly, we get h(b = 0 and 56 a a 2 1 = 0. Now, since h(0 = 1 > 0 and h(1/4<0, we may assume c (0, 1/4. On the interval (0, 1/4, wehavef(x > 0 and g(x < 0. Therefore b = f(c > 0 and a = g(c < 0. Remark. Since every fiducial vector has unit length, we have added the olynomial a b c 2 1 to the set of generators of I to reduce the degree of the olynomials in the Gröbner bases. Theorem 5.3 Let {±a} be the set of real roots of 76 x x 2 5 and let {±b,±c} be the set of real roots of x x x x with b<0 < a,c. Then the RL vector in C 19 with arameters (a,b,cis a fiducial vector.

10 342 J Algebr Comb ( : Proof The given RL vector z R 19 is fiducial if and only if (a,b,c is a solution to the following system: f 0,0 (z = a 4 + 9(b 4 + c 4 = 1/10, f 0,1 (z = a 2 (b 2 + c 2 + 4(b 4 + c b 2 c 2 = 1/20, f 1,1 (z = abc(a + b + c + 2(b + c 2 (b 2 + c 2 = 0, f 1,2 (z = 2 abc(b + c + b b 3 c + 7 b 2 c bc 3 + c 4 = 0, f 1,3 (z = a(b + c(b 2 + c 2 + 5bc(b 2 + bc + c 2 = 0. The rest of the roof is similar to the roof of Theorem 5.2. Remark. As in dimension 7, adding the olynomial a b c 2 1tothesetof generators of I would reduce the degree of the olynomials in the Gröbner bases. The following result is an analogous version of Theorem 4.4 for RL vectors: Theorem 5.4 There exists a set of rimes P with zero density (in the set of all rimes that are equal to 3 modulo 4 such that for all / P there exists no RL fiducial vector in C. Since the roof of the above theorem is similar to that of Theorem 4.4 and only involves some basic algebra, we have omitted the roof. However, in the roof of the Theorems 4.4 and 5.4, one can see that the exact same set P satisfies the conditions of these two theorems. This strongly suggests that one may find a one to one corresondence between the set of AL fiducial vectors and the set of RL fiducial vectors. If such a transformation is found then the roof of one of the mentioned theorems can be skied. Now let us look at a crucial grou for which a fiducial vector is invariant. We will discuss briefly why such a transformation (if any cannot be in this grou. Let C(d denote the Clifford grou, the grou of all unitary oerations U which normalize the generalized Pauli grou GP(d, i.e.u GP(d U = GP(d. LetJ be the maing that mas (z j C d to ( z j.theextended Clifford grou is the grou consisting of C(d and all elements of the form JU, where U C(d. This grou is denoted by EC(d. The relevance of the (extended Clifford grou to the set of equiangular lines arising from GP(d has been discussed by several authors (for examle see [2, 10]. Note that if z is a fiducial vector and U EC(d then Uz is also a fiducial vector. Therefore EC(d lies in the automorhism grou of the set of fiducial vectors in C d. Aleby [2] roves that for odd d the grou EC(d modulo its centre I(d is isomorhic to the grou ESL(2, Z d (Z d Z d, where ESL(2, Z d denotes the grou of all 2 2 matrices over Z d with determinant equal to ±1 and denotes the wreath roduct. On age 17 in [2], Aleby also describes a method to find the stability grou of a given fiducial vector z C d,thesetofallu EC(d/I(d for which z is eigenvector. Alying the same method, it turns out that the stability grou of the RL fiducial vector in dimension 7 (from Theorem 5.2 isisomorhicto the order 3 subgrou generated by ( 30 [ 0 2, ( 3 ]. 0

11 J Algebr Comb ( : But the stability grou of the AL fiducial vector in dimension 7 (from Theorem 4.3 is isomorhic to the order 6 subgrou (see [2] generated by ( ( [, ], and therefore the two AL and RL fiducial vectors in dimension 7 do not belong to the same orbit under the action of the extended Clifford grou. Therefore, the transformation discussed in the revious aragrah cannot be found in EC(d. Similar argument shows that the two AL and RL fiducial vectors in dimension 19 do not belong to the same orbit under the action of the extended Clifford grou. In fact, the stability grou of the RL fiducial vector in dimension 19 (from Theorem 5.3 is isomorhic to the order 9 subgrou generated by ( ( [, ], whereas the stability grou of the AL fiducial vector in dimension 19 (from Theorem 4.3 is isomorhic to the order 18 subgrou (see [2] generated by ( ( [, ] Periodic fiducial vectors We say that a sequence (a j j Zd is eriodic if there exists Z d \{0} such that a j+ = a j for every j Z d. The smallest such is called the eriod of the sequence. Proosition 6.1 There exists no fiducial vector in C d such that the absolute values of its coordinates form a eriodic sequence. Proof Towards a contradiction assume that such a fiducial vector, namely z = (z j, exists. Let be the eriod of the sequence ( z j 2 j Zd and let d = k for some integer k>1. For j = , let R j = z j 2. It follows from Corollary 2.3 that 1 1 k(k+1 for 1 s 1, R j R j+s = j=0 2 k(k+1 for s = 0. We also know that 1 j=0 R j = 1/k. By substituting the above values in the identity ( j R j 2 = j R2 j + i j R ir j, we get 1 k 2 = 2 k(k ( 1 1 k(k + 1. Simlifying the above equation, we get k = 1, which is a contradiction.

12 344 J Algebr Comb ( : Acknowledgements The author thanks Petr Lisoněk for fruitful discussions and acknowledges Noam Elkies for his valuable comments. The author also thanks the referees for their useful remarks. Aendix Here we resent the technical, yet interesting, details that were skied throughout the aer. In Section A we rovide some necessary tools and state some roerties of the Legendre symbol which will be used in the roof of theorems resented in Section B. A Proerties of the Legendre symbol For every j Z recall that ( j denotes the Legendre symbol. The basic roerties of the Legendre symbol can be found in almost any introductory number theory textbook (for examle see [1]. Also recall that a( denotes the trace of the Frobenius endomorhism of the ellitic curve y 2 = x(x + 1(x + 2(x + 3: a( = ( j(j + 1(j + 2(j + 3. j Z The following lemma is quite straightforward, but we include it for the sake of comleteness: Lemma A.1 For every rime such that 3 (mod 4, we have (i ( j j Z = 0, (ii ( j(j+ɛ j Z = 1 for every fixed ɛ Z \{0}, (iii ( (j ɛj(j+ɛ j Z = 0 for every fixed ɛ Z. Proof Since 3 (mod 4 we have ( ( x = x. This immediately imlies (i and (iii. Since the Legendre symbol is multilicative we get ( j(j+ɛ j Z = ( 1+ɛj 1 ( j Z \{0} = 1 = 1by(i. In the next three lemmas, we count the number of fixed oints of certain mas on Z that involve the Legendre symbol ( j. The lemmas are very similar, but none of them quite imlies another one and therefore we have included them all. However, since the roofs are similar, we have only resented one of the roofs. The key idea in all of them is to count the number of elements of the set {j Z : ( ( j (, j+1 (,..., j+t 1 = C} for every given constant C { 1, 1} t. We will do this for t = 2, t = 3, and t = 4, resectively, in the next three lemmas. Lemma A.2 For every rime such that 3 (mod 4, and j Z, let ( ( j j + 1 κ(j =.

13 J Algebr Comb ( : Also let K(c = {j : κ(j = c}. Then K(0 = 1 2 ( 3, K(2 = 1 4 ( + 1, K( 2 = 1 4 ( 3. Proof For every α = (α 0,α 1 { 1, 1} { 1, 1},let k (α = 1 4 j Z \{0, 1} ( ( ( ( j j + 1 α α By alying Lemma A.1, we get ( ( ( ( k (α = ( α 0 α 1 α α = 2 α 0 α 1 + α 0 α 1. (5 On the other ( hand, for every δ { 1, 1} and x 0, the value of the exression (1/2 δ ( x + 1 is equal to 1 if ( x = δ and 0 otherwise. Therefore k (α = {j Z : ( j = α0, ( j+1 = α1 }. Hence K(0 = k (1, 1 + k ( 1, 1, K(2 = k (1, 1, and K( 2 = k ( 1, 1. The result follows immediately using (5. Lemma A.3 For every rime such that 3 (mod 4, and j Z, let ( ( j j 1 μ(j = 2 Also let M(c = {j : μ(j = c}. Then ( j ( 3 for 3 (mod 8, M(0 = ( 7 for 7 (mod 8, 1 4 M(2 = M( 2 = 1 ( 3, ( 3 for 3 (mod 8, M(4 = M( 4 = j Z \{0,±1} 1 8 ( + 1 for 7 (mod 8. Remark. Note that the equality k (α 0,α 1 = m (1,α 0,α 1 +m ( 1,α 0,α 1 does not necessarily hold, where m (β, α 0,α 1 = 1 ( ( ( ( ( ( j 1 j j + 1 β + 1 α α This is why Lemma A.2 can not be imlied from Lemma A.3.

14 346 J Algebr Comb ( : Lemma A.4 For every rime such that 3 (mod 4, and j Z, let ( ( ( ( j + 3 j + 2 j + 1 j ν(j = +. Also let N(c= {j : ν(j = c}. Then N(0 = 1 ( ( ( a( 2( + 1( + 1, 8 where N(2 = N( 2 = 1 ( 4 + a(, 4 N(4 = N( 4 = 1 ( ( ( a( + 2( + 1( + 1, 16 a( = ( j(j + 1(j + 2(j + 3. j Z B The roofs Proof of Theorem 3.2 By letting s = 0 in Corollary 2.3 and the fact that z is a unit vector, it follows that kb 2 + (d ka 2 = 1, kb 4 + (d ka 4 = 2/(d + 1. It is easy to see that if k = 0 then no a and b exist. Thus k 1 and d 2. Solving for a 2 and b 2, we get ( ( a 2 = 1 k(d 1 1, b 2 = 1 (d k(d 1 1 ±. d (d + 1(d k d k(d + 1 Hence (a 2 b 2 2 = d 1 ( k d 2 (d + 1 d k + d k d = k (d + 1(d kk. (6 Let N s (x, y = {i Z d : z i =x, z i+s =y}. Since there are k coordinates that have absolute value b and d k coordinates that have absolute value a, wehave N s (b, b + N s (b, a = k = N s (b, b + N s (a, b. (7 N s (a, a + N s (a, b = d k = N s (a, a + N s (b, a. (8 On the other hand, by Corollary 2.3 we have N s (a, aa 4 + N s (b, bb 4 + (N s (a, b + N s (b, aa 2 b 2 = 1 d + 1

15 J Algebr Comb ( : for every s Z d \{0}. Thus, by using identities (7 and (8, this can be rewritten as (N s (b, b k(a 2 b 2 2 = 1/(d + 1. Substituting identity (6 imlies that N s (b, b = k (d kk d 1 = k(k 1 d 1 := λ is indeendent of s. By definition of D and N s (b, b, this means that every s Z d \ {0} can be reresented as a difference of two distinct elements of D in exactly λ different ways. Proof of Proosition 4.2 Let c = cos θ, δ = + 1, and ψ = δ + 2. By Theorem 3.1, wehavea = 1/ δ(δ + 1 and b = ψ/ δ(δ + 1. By definition, we must have j Z z j z j+1 2 = 1/( + 1. Using Lemma A.2 and Lemma A.3, wemay rewrite this as (2bc + ( 1ac 2 a 2 + 4(1 c 2 (b ac 2 = 1/a 2 ( + 1. By making the above substitutions for a,b,, and δ and factoring out the non-zero terms, the revious exression can be written as ( ( (ψc 1 (ψ 2 2c + ψ (ψ 2 2(ψ 2 4c 2 + 2ψc+ ψ 2 6 = 0. (9 Let f s,t (z be as defined in Theorem 2.2. Using Lemma A.3 and Lemma A.4, we get { a 4 ( 3c 2 (2c a 3 b(4c 3 3c + a 2 b 2 for 3 (mod 8 f 1, 1 (z = a 4 c 2 ((2c c a 3 bc + a 2 b 2 for 7 (mod 8 Note that f 1, 1 (z = 0 by Theorem 2.2. As before, both olynomials on the left hand side can be factored in R[ψ]. After factoring out the non-zero terms, we get the following: If 3 (mod 8 then ( (ψc 1 2(ψ 2 4c 3 + 2ψc 2 (ψ 2 6c ψ = 0. (10 It is easy to check that for >3, the only common root of the equations (9 and (10 is c = 1/ψ. Therefore θ = cos 1 (1/ , as desired. If 7 (mod 8 then ( ( (ψ 2 2c + ψ 2(ψ 2 2c 3 2ψc 2 (ψ 2 4c + ψ = 0. (11 The only common root of the equations (9 and (11 isc = ψ/(ψ 2 2 and therefore we must have θ = cos 1 ( in this case. Remark. For >19 the quadratic factor in equation (9 is always ositive. Therefore equation (9 simlifies to (ψc 1 ( (ψ 2 2c + ψ = 0for>19. Proof of Theorem 4.4 Let z = (z j be an AL fiducial vector in C with arameters a,b, c = cos θ. We already know that such vector exists for = 3 and = 7. So

16 348 J Algebr Comb ( : assume >7. As in the roof of Proosition 4.2, by letting δ = + 1, and ψ = δ + 2 we get a = 1/ δ(δ + 1 and b = ψa. Also, by Proosition 4.2 we have c = 1/ψ if 3 (mod 8 and c = ψ/(ψ 2 2 if 7 (mod 8. Now, by letting q = a( and using Lemma A.3 and Lemma A.4 we get the following: If 23 (mod 24 then f 1,2 (z = ( + 2 qa 4 c 4 + 2(q 3a 4 c 2 + 4a 3 bc qa 4 and if 23 (mod 24 then f 1,2 (z = ( 6 qa 4 c 4 + 8a 3 bc 3 + 2(q + 1a 4 c 2 4a 3 bc qa 4. By Theorem 2.2, wemusthavef 1,2 (z = 0. Since a,b,c and can be described in terms of ψ, we can describe f 1,2 (z in terms of q and ψ. Solving for q, we get: 3 for 11 or 19 (mod 24 q = a( = ψ 2 (3ψ 2 8/(ψ for 23 (mod 24 ψ 2 (5ψ 2 24/(ψ for 7 (mod 24 For 23 (mod 24 we can easily check that q is not an integer (in fact, if = 24l + 23 and l>13 then 4 <q< 3. This is imossible since q = a( is an integer by definition. Similarly, the case 7 (mod 24 when 7 is excluded. Thus the set P ={ : is rime, 11 or 19 (mod 24, > 7, a(= 3} has the desired roerties. As mentioned before, the fact that the set P has density zero follows from Theorem 20 in [16]. Remark. By considering the identity f 1,4 (z = 0 in the revious theorem, we may further restrict the forbidden set P to its roer subset { P a ( = 3}, where a ( = ( j(j+1(j+4(j+5 j Z. Since this subset of P is still non-emty and has the same density as P, namely zero, we have skied the details. References 1. Andrews, G.E.: Number Theory. Dover, New York ( Aleby, D.M.: Symmetric informationally comlete-ositive oerator valued measures and the extended Clifford grou. J. Math. Phys. 46(5, ( Aleby, D.M., Dang, H.B., Fuchs, C.A.: Physical significance of symmetric informationallycomlete sets of quantum states. Prerint arxiv: v1 ( Baumert, L.D., Gordon, D.M.: On the existence of cyclic difference sets with small arameters. In: High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams. Fields Inst. Commun., vol. 41, Amer. Math. Soc., Providence ( Caves, C.M., Fuchs, C.A., Schack, R.: Unknown quantum states: the quantum de finetti reresentation. J. Math. Phys. 43, ( Delsarte, P., Goethals, J.M., Seidel, J.J.: Sherical codes and designs. Geom. Dedicata 6(3, (1977

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