AN APPLICATION OF POSITIVE DEFINITE FUNCTIONS TO THE PROBLEM OF MUBS Mathematics Subject Classification. Primary 43A35, Secondary

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1 AN APPLICATION OF POSITIVE DEFINITE FUNCTIONS TO THE PROBLEM OF MUBS MIHAIL N. KOLOUNTZAKIS, MÁTÉ MATOLCSI, AND MIHÁLY WEINER Abstract. We present a new approach to the problem of mutually unbiase bases (MUBs), base on positive efinite functions on the unitary group. The metho provies a new proof of the fact that there are at most + 1 MUBs in C, an it may also lea to a proof of non-existence of complete systems of MUBs in imension 6 via a conjecture algebraic ientity Mathematics Subject Classification. Primary 43A35, Seconary 15A30, 05B10 Keywors an phrases. Mutually unbiase bases, positive efinite functions, unitary group 1. Introuction In this paper we present a new approach to the problem of mutually unbiase bases (MUBs) in C. Our approach has been motivate by two recent results in the literature. First, in [22] one of the present authors escribe how the Fourier analytic formulation of Delsarte s LP boun can be applie to the problem of MUBs. Secon, in [25, Theorem 2] F. M. Oliveira Filho an F. Vallentin prove a general optimization boun which can be viewe as a generalization of Delsarte s LP boun to non-commutative settings (an they applie the theorem to packing problems in Eucliean spaces). As the MUB-problem is essentially a problem over the unitary group, it is natural to combine the two ieas above. Here we present another version of the non-commutative Delsarte scheme in the spirit of [22, Lemma 2.1]. Our formulation in Theorem 2.3 below escribes a less general setting than [25, Theorem 2], but it makes use of the unerlying group structure an is very M. Kolountzakis was partially supporte by grant No 4725 of the University of Crete. M. Matolcsi was supporte by the ERC-AG an by NKFIH- OTKA Grant No. K104206, M. Weiner was supporte by the ERC-AG QUEST Quantum Algebraic Structures an Moels an by NKFIH-OTKA Grant No. K

2 2 M. N. KOLOUNTZAKIS, M. MATOLCSI, AND M. WEINER convenient for applications. It fits the MUB-problem naturally, an leas us to consier positive efinite functions on the unitary group. The paper is organize as follows. In the Introuction we recall some basic notions an results concerning mutually unbiase bases (MUBs). In Section 2 we escribe a non-commutative version of Delsarte s scheme in Theorem 2.3. We believe that this general scheme will be useful for several other applications, too. We then apply the metho in Theorem 2.4 to give a new proof of the fact that there are at most + 1 MUBs in C. While the result itself has been prove by other methos, we believe that this approach is particularly suite for the MUB-problem an may lea to non-existence proofs in the future. In particular, in Section 3 we speculate on how the non-existence of complete systems of MUBs coul be prove in imension 6 via an algebraic ientity conjecture in [23]. Throughout the paper we follow the convention that inner proucts are linear in the first variable an conjugate linear in the secon. Recall that two orthonormal bases in C, A = {e 1,..., e } an B = {f 1,..., f } are calle unbiase if for every 1 j, k, e j, f k = 1. A collection B 1,... B m of orthonormal bases is sai to be (pairwise) mutually unbiase if any two of them are unbiase. What is the maximal number of mutually unbiase bases (MUBs) in C? This problem has its origins in quantum information theory, an has receive consierable attention over the past ecaes (see e.g. [14] for a recent comprehensive survey on MUBs). The following upper boun is well-known (see e.g. [1, 3, 31]): Theorem 1.1. The number of mutually unbiase bases in C is less than or equal to + 1. We will give a new proof of this fact in Theorem 2.4 below. Another important result concerns the existence of complete systems of MUBs in prime-power imensions (see e.g. [1, 11, 12, 18, 21, 31]). Theorem 1.2. A collection of + 1 mutually unbiase bases (calle a complete system of MUBs) exists (an can be constructe explicitly) if the imension is a prime or a prime-power. However, if the imension = p α p α k k is not a prime-power, very little is known about the maximal number of MUBs. By a tensor prouct construction it is easy to see that there are at least p α j j + 1 MUBs in C where p α j j is the smallest of the prime-power ivisors of

3 POSITIVE DEFINITE FUNCTIONS AND MUBS 3. One coul be tempte to conjecture the maximal number of MUBs always equals p α j j + 1, but this is alreay known to be false: for some specific square imensions = s 2 a construction of [30] yiels more MUBs than p α j j + 1 (the construction is base on orthogonal Latin squares). Another important phenomenon, prove in [29], is that the maximal number of MUBs cannot be exactly (it is either + 1 or strictly less than ). The following basic problem remains open for all non-primepower imensions: Problem 1.3. Does a complete system of + 1 mutually unbiase bases exist in C if is not a prime-power? For = 6 it is wiely believe among researchers that the answer is negative, an the maximal number of MUBs is 3. The proof still elues us, however, espite consierable efforts over the past ecae [3, 4, 5, 6, 19]. On the one han, some infinite families of MUBtriplets in C 6 have been constructe [19, 32]. On the other han, numerical evience strongly suggests that there exist no MUB-quartets [5, 6, 8, 16, 32]. For non-primepower imensions other than 6 we are not aware of any well foune conjectures as to the exact maximal number of MUBs. It will also be important to recall the relationship between mutually unbiase bases an complex Haamar matrices. A matrix H is calle a complex Haamar matrix if all its entries have moulus 1 an 1 H is unitary. Given a collection of MUBs B 1,..., B m we may regar the bases as unitary matrices U 1,..., U m (with respect to some fixe orthonormal basis), an the conition of the bases being pairwise unbiase amounts to Ui U j being a complex Haamar matrix scale 1 by a factor of for all i j. That is, Ui U j is a unitary matrix (which is of course automatic) whose entries are all of absolute value 1. A complete classification of MUBs up to imension 5 (see [7]) is base on the classification of complex Haamar matrices (see [17]). However, the classification of complex Haamar matrices in imension 6 is still out of reach espite recent efforts [2, 20, 24, 27, 28]. In this paper we will use the above connection of MUBs to complex Haamar matrices. In particular, we will escribe a Delsarte scheme for non-commutative groups in Theorem 2.3, an apply it on the unitary group U() to the MUB-problem in Theorem 2.4.

4 4 M. N. KOLOUNTZAKIS, M. MATOLCSI, AND M. WEINER 2. Mutually unbiase bases an a non-commutative Delsarte scheme In this section we escribe a non-commutative version of Delsarte s scheme, an show how the problem of mutually unbiase bases fit into this scheme. The commutative analogue was escribe in [22]. Let G be a compact group, the group operation being multiplication an the unit element being enote by 1. We will enote the normalize Haar measure on G by µ. Let a symmetric subset A = A 1 G, 1 A, be given. We think of A as the forbien set. We woul like to etermine the maximal carinality of a set B = {b 1,... b m } G such that all the quotients b 1 j b k A c {1} (in other wors, all quotients avoi the forbien set A). When G is commutative, some wellknown examples of this general scheme are present in coing theory [13], sphere-packings [9], an sets avoiing square ifferences in number theory [26]. We will iscuss the non-commutative case here. Recall that the convolution of f, g L 1 (G) is efine by f g(x) = f(y)g(y 1 x)µ(y). Recall also the notion of positive efinite functions on G. A function h : G C is calle positive efinite, if for any m an any collection u 1,..., u m G, an c 1,..., c m C we have m i,j=1 h(u 1 i u j )c i c j 0. When h is continuous, the following characterization is well-known. Lemma 2.1. (cf. [15, Proposition 3.35]) If G is a compact group, an h : G C is a continuous function, the following are equivalent. (i) h is of positive type, i.e. (1) ( f f)h 0 for all functions f L 2 (G) (here f(x) = f(x 1 )) (ii) h is positive efinite This statement is fully containe in the more general Proposition 3.35 in [15]. In fact, for compact groups Proposition 3.35 in [15] shows that instea of L 2 (G) the smaller class of continuous functions C(G) or the wier class of absolute integrable functions L 1 (G) coul also be taken in (i). All these cases are equivalent, but for us it will be convenient to use L 2 (G) in the sequel. (It is also worth mentioning here that if h is of positive type then it is automatically equal to a continuous function almost everywhere but we will not nee this fact in this paper.)

5 POSITIVE DEFINITE FUNCTIONS AND MUBS 5 We formulate another important property of positive efinite functions. Lemma 2.2. Let G be a compact group an µ the normalize Haar measure on G. If h : G C is a continuous positive efinite function then α = hµ 0, an for any α G 0 α the function h α 0 is also positive efinite. In other wors, for any m an any collection u 1,..., u m G an c 1,..., c m C we have m m (2) h(u 1 i u j )c i c j α c i 2. i,j=1 Proof. Let f L 2 (G) an efine a linear operator H : L 2 (G) L 2 (G) by (Hf)(x) = h(x 1 y)f(y) µ(y). As h is assume to be positive efinite, H is positive self-ajoint. Also, writing 1 for the constant one function on G we have i=1 H1 = α1, H1, 1 = α 0. Let us use the notation β = f(y)µ(y). We have the orthogonal ecomposition f = β1 + f 0, where f 0 1. Using the invariance of the Haar measure an exchanging the orer of integration we have Hf, 1 = (Hf)(x), 1(x)µ(x) = h(x)µ(x) f(y)µ(y) = αβ Therefore, αβ = Hf, 1 = H(β1 + f 0 ), 1 = αβ + Hf 0, 1, an hence Hf 0, 1 = 0. To show that h α is positive efinite we nee to check that Hf, f β 2 α 0, for all f L 2 (G). We have Hf, f = βα1 + Hf 0, β1 + f 0 = β 2 α + Hf 0, f 0 since f 0 1 an Hf 0 1. Hence Hf, f β 2 α = Hf 0, f 0 0.

6 6 M. N. KOLOUNTZAKIS, M. MATOLCSI, AND M. WEINER After these preliminaries we can escribe the non-commutative analogue of Delsarte s LP boun. (To the best of our knowlege the commutative version was first introuce by Delsarte in connection with binary coes with prescribe Hamming istance [13]. Another formulation of the non-commutative version is given in [25]). Theorem 2.3. (Non-commutative Delsarte scheme for compact groups) Let G be a compact group, µ the normalize Haar measure, an let A = A 1 G, 1 A, be given. Assume that there exists a positive efinite function h : G R such that h(x) 0 for all x A c, an hµ > 0. Then for any B = {b 1,... b m } G such that b 1 j b k A c {1} the carinality of B is boune by B h(1) hµ. Proof. Consier (3) S = On the one han, u,v B h(u 1 v). (4) S h(1) B, since all the terms u v are non-positive by assumption. On the other han, applying (2) with α = hµ, u, v B an c u = c v = 1, we get (5) S α B 2. Comparing the two estimates (5), (4) we obtain B h(1) hµ. The function h in the Theorem above is usually calle a witness function. We will now escribe how the problem of mutually unbiase bases fits into this scheme. Consier the group U() of unitary matrices, being given with respect to some fixe orthonormal basis of C. Consier the set CH of complex Haamar matrices. Following the notation of the Delsarte scheme above efine A c = 1 CH U(), i.e. let the complement of the forbien set be the set of scale complex Haamar matrices. Then the maximal number of MUBs in C is exactly the maximal carinality of a set B = {b 1,... b m } U() such that all the quotients b 1 j b k A c {1}. After fining an appropriate witness function we can now give a new proof of the fact the number of MUBs in C cannot excee + 1.

7 POSITIVE DEFINITE FUNCTIONS AND MUBS 7 Theorem 2.4. The function h(z) = 1 + i,j=1 z i,j 4 (where Z = (z i,j ) i,j=1 U()) is positive efinite on U(), with h(1) = 1 an h = 1. Consequently, the number of MUBs in imension cannot +1 excee + 1. Proof. Consier the function h 0 (Z) = i,j=1 z i,j 4. First we prove that h 0 is positive efinite. For this, recall that the Hilbert-Schmit inner prouct of matrices is efine as X, Y HS = Tr (XY ), an for any vector v in a finite imensional Hilbert space H the (scale) projection operator P v is efine as P v u = u, v v. For any two vectors u, v H we have u, v 2 = Tr P u P v. Also, recall that the inner prouct on H H is given by u 1 u 2, v 1 v 2 = u 1, v 1 u 2, v 2. Let U 1,..., U m be unitary matrices, c 1,..., c m C, an let {e 1,..., e } be the orthonormal basis with respect to which the matrices in U() are given. Then (6) U r U t e j, e k 4 = U t e j, U r e k 4 = U t e j U t e j, U r e k U r e k 2 = Tr P Ute j U te j P Ure k U re k. Therefore, with the notation Q t = m j=1 P U te j U te j we have (7) h(u r U t ) = j,k U r U t e j, e k 4 = Tr Q t Q r. (8) Finally, as esire. m h(ur U t )c r c t = r,t=1 m c t Q t 2 HS 0, It is known [10] that the integral of h 0 on U() is 2. By applying +1 Lemma 2.2 to h 0 with α 0 = 1 < h 0 we get that h is also positive efinite. Note also that h vanishes on the set 1 CH of scale complex t=1 Haamar matrices, h(1) = 1, an h = = Therefore, Theorem 2.3 implies that the number of MUBs in C is less than or equal to h(1) h = + 1. We remark here that one coul consier the witness functions h β = h 0 β for any 1 β 2. All these functions satisfy the conitions of +1 Theorem 2.3. However, an easy calculation shows that the best boun is achieve for β = 1.

8 8 M. N. KOLOUNTZAKIS, M. MATOLCSI, AND M. WEINER 3. Dimension 6 The function h(z) = 1 + i,j=1 z i,j 4 in Theorem 2.4 was a fairly natural caniate, as it vanishes on the set of (scale) complex Haamar matrices 1 CH, for any. Other such caniates are h k (Z) = 1 + n k 2 i,j=1 z i,j 2k for any k 2, but they give worse upper bouns than h. Furthermore, Theorem 1.2 implies that the result of Theorem 2.4 is sharp whenever is a prime-power, an hence we cannot hope to construct better witness functions than h, in general. However, let us examine the situation more closely in imension = 6, an iscuss why we hope that the non-existence of a complete system of MUBs coul be prove by this metho. For = 6 we have other functions which are conjecture to vanish on 1 CH. Namely, Conjecture 2.3 in [23] provies a selection of such functions. Let (9) m 1 (Z) = 6 z π(1),j z π(2),j z π(3),j z π(4),j z π(5),j z π(6),j, π S 6 j=1 where S 6 enotes the permutation group on 6 elements. Also, let m 2 (Z) = m 1 (Z ). Then m 1 an m 2 are real-value (because each term appears with its conjugate), an they are conjecture to vanish on 1 CH. Furthermore, as the inner sum in (9) is conjecture to be zero for all π S 6, we may even multiply each term with ( 1) sgn π, if we wish. This leas to other possible choices of m 1 an m 2. We remark here that in higher even imensions = 8, 10,... the corresponing expressions o not vanish on the set of complex Haamar matrices. Therefore, the algebraic expression (9) is special to imension 6, an provies some further natural caniates of witness functions for the MUB-problem. Namely, let m(z) = F (m 1 (Z), m 2 (Z)), where F (a, b) is a symmetric non-negative polynomial such that F (0, 0) = 0 (e.g. F (a, b) = (a + b) 2, a 2 b 2, etc.). In such a case m(i) = 0, an m(z)µ > 0. Therefore, if for any ε > 0 the function Z U() h(z) + εm(z) is positive efinite, we get a better boun than in Theorem 2.4, an obtain that the number of MUBs in imension 6 is strictly less than 7, i.e. a complete system of MUBs oes not exist. The question is whether a suitable choice of F an ε exist. This leas us to the following general problem.

9 POSITIVE DEFINITE FUNCTIONS AND MUBS 9 Problem 3.1. Given a polynomial function f(z) of the matrix elements z i,j an their conjugates z i,j, what is a necessary an sufficient conition for f to be positive efinite on the unitary group U()? Finally, it woul also be interesting to fin any analogue of Conjecture 2.3 in [23] for any imensions other than = 6. Acknowlegement The authors are grateful to the referee for his/her insightful comments which have improve the presentation of the manuscript. References [1] S. Banyopahyay, P. O. Boykin, V. Roychowhury & F. Vatan, A New Proof for the Existence of Mutually Unbiase Bases. Algorithmica 34 (2002), [2] K. Beauchamp & R. Nicoara, Orthogonal maximal Abelian -subalgebras of the 6 6 matrices. Linear Algebra Appl. 428 (2008), [3] I. Bengtsson, W. Bruza, Å. Ericsson, J.-A. Larsson, W. Taej & K. Życzkowski, Mutually unbiase bases an Haamar matrices of orer six. J. Math. Phys. 48 (2007), [4] P. O. Boykin, M. Sitharam, P. H.Tiep, & P. Wocjan, Mutually unbiase bases an orthogonal ecompositions of Lie algebras. Quantum Inf. Comput. 7 (2007), [5] S. Brierley & S. Weigert, Maximal sets of mutually unbiase quantum states in imension six. Phys. Rev. A (3) 78 (2008), [6] S. Brierley & S. Weigert, Constructing Mutually Unbiase Bases in Dimension Six. Phys. Rev. A (3) 79 (2009), [7] S. Brierley, S. Weigert & I. Bengtsson, All Mutually Unbiase Bases in Dimensions Two to Five. Quantum Information an Computing 10 (2010), [8] P. Butterley & W. Hall, Numerical evience for the maximum number of mutually unbiase bases in imension six. Physics Letters A 369 (2007) 5-8. [9] H. Cohn & N. Elkies, New upper bouns on sphere packings I. Ann. of Math. (2) 157 (2003), [10] B. Collins & P. Sniay Integration with respect to the Haar measure on unitary, orthogonal an symplectic group. Commun. Math. Phys. 264 (2006), [11] M. Combescure, Circulant matrices, Gauss sums an mutually unbiase bases. I. The prime number case. Cubo 11 (2009), [12] M. Combescure, Block-circulant matrices with circulant blocks, Weil sums, an mutually unbiase bases. II. The prime power case. J. Math. Phys. 50 (2009), [13] P. Delsarte, Bouns for unrestricte coes, by linear programming. Philips Res. Rep. 27 (1972),

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11 POSITIVE DEFINITE FUNCTIONS AND MUBS 11 M. N. K.: Department of Mathematics an Applie Mathematics, University of Crete, Voutes Campus, Heraklion, Greece. aress: M. M.: Buapest University of Technology an Economics (BME), H-1111, Egry J. u. 1, Buapest, Hungary (also at Alfré Rényi Institute of Mathematics, Hungarian Acaemy of Sciences, H-1053, Realtanoa u 13-15, Buapest, Hungary) aress: matomate@renyi.hu M. W.: Buapest University of Technology an Economics (BME), H-1111, Egry J. u. 1, Buapest, Hungary aress: mweiner@renyi.hu