Worcester Polytechnic Institute DigitalCommons@WPI Mathematical Sciences Faculty Publications Department of Mathematical Sciences 010 On the Equivalence Between Real Mutually Unbiase Bases an a Certain Class of Association Schemes Nicholas LeCompte Worcester Polytechnic Institute, nl@wpi.eu William J. Martin Worcester Polytechnic Institute, martin@wpi.eu William Owens Worcester Polytechnic Institute, wilco@wpi.eu Follow this an aitional works at: https://igitalcommons.wpi.eu/mathematicalsciences-pubs Part of the Discrete Mathematics an Combinatorics Commons Suggeste Citation LeCompte, N., Martin, W. J., & Owens, W. (010). On the Equivalence Between Real Mutually Unbiase Bases an a Certain Class of Association Schemes. European Journal of Combinatorics, 31(6), 1499-151. http://x.oi.org/http://x.oi.org/10.1016/ j.ejc.009.11.014 This Article is brought to you for free an open access by the Department of Mathematical Sciences at DigitalCommons@WPI. It has been accepte for inclusion in Mathematical Sciences Faculty Publications by an authorize aministrator of DigitalCommons@WPI. For more information, please contact akgol@wpi.eu.
On the equivalence between real mutually unbiase bases an a certain class of association schemes Nicholas LeCompte, William J. Martin, William Owens Department of Mathematical Sciences Worcester Polytechnic Institute Worcester, MA {nl,martin,wilco}@wpi.eu Submitte August 18, 008. Final revision March, 009. Abstract Mutually unbiase bases (MUBs) in complex vector spaces play several important roles in quantum information theory. At present, even the most elementary questions concerning the maximum number of such bases in a given imension an their construction remain open. In an attempt to unerstan the complex case better, some authors have also consiere real MUBs, mutually unbiase bases in real vector spaces. The main results of this paper establish an equivalence between sets of real mutually unbiase bases an 4-class cometric association schemes which are both Q-bipartite an Q-antipoal. We then explore the consequences of this equivalence, constructing new cometric association schemes an escribing a potential metho for the construction of sets of real MUBs. 1 Introuction In quantum information theory, one important challenge is to construct mutually (i.e., pairwise) unbiase bases in complex vector spaces C. A pair of unitary bases for C are unbiase with respect to one another if, in the change-of-basis matrix from one basis to the other, all entries have the same magnitue. While much progress has been mae an various connections to combinatorics have emerge (see, e.g., [15, ]), much remains to be one. In an effort to better unerstan the problem, several authors [5, 0] have recently propose the stuy of real mutually unbiase bases. While the moifie problem seems to be of a somewhat ifferent nature, there are some similarities to the problem in complex space an the stuy of real MUBs seems interesting on its own. Our goal is to show that this latter 1
problem is equivalent to the stuy of a certain class of association schemes whose characterization is implicit in the work of Delsarte [10] in 1973. Inee, this connection is alreay evient in recent work on an important special case by Bannai an his co-authors [1, 4]. In the next section, we provie a very brief review of association schemes, giving all the efinitions necessary for the statement of our results. Section 3, base on the paper [5] of Boykin, et al., summarizes what is currently known about sets of real MUBs. The main results of the paper are presente in Section 4 an the implications of these results are explore in Section 5. Finally, in the appenix, we inclue all parameters of the unerlying association scheme that we stuy. Cometric association schemes We begin with a review of the basic efinitions concerning cometric association schemes. The reaer is referre to [3] or [6] for backgroun material. A (symmetric) association scheme (X, R) consists of a finite set X of size v an a set R of binary relations on X satisfying (i) R = {R 0,..., R D } is a partition of X X; (ii) R 0 is the ientity relation; (iii) R i = R i for each i; (iv) there exist integers p k ij such that {c X : (a, c) R i an (c, b) R j } = p k ij whenever (a, b) R k, for each i, j, k {0,..., D}. As usual, we efine A i to be the matrix with rows an columns inexe by X with (a, b)- entry equal to one if (a, b) R i an zero otherwise. In this way, we obtain a collection of v v symmetric 01-matrices A = {A 0, A 1,..., A D } such that: (i ) D i=0 A i = J where J is the all 1 s matrix, (ii ) A 0 is the ientity matrix; (iii ) A i = A i for each i; (iv ) the set A forms a basis for a commutative matrix algebra A calle the Bose-Mesner algebra. Since no two matrices in A have a nonzero entry in the same location, the Bose-Mesner algebra is also close uner entrywise (or Schur) multiplication, enote. The matrices A may be simultaneously iagonalize; there are D + 1 maximal common eigenspaces for A known as the eigenspaces of the scheme, an it follows from elementary
linear algebra that the primitive iempotents E 0, E 1,..., E D representing orthogonal projection onto these eigenspaces form another basis for A. If we let P ji enote the eigenvalue of A i on the j th eigenspace of the scheme, i.e., P ji satisfies A i E j = P ji E j, then the (D + 1) (D + 1) matrix P containing P ji as its entry in the j th row, i th column is calle the first eigenmatrix of the association scheme. The secon eigenmatrix Q of the scheme is efine as Q = vp 1 (so that E j = 1 v i Q ija i ) but also satisfies a secon orthogonality relation. If v i enotes the valency of the relation R i (i.e., the common row sum of the matrix A i ) an m j enotes the imension of the j th eigenspace (i.e., the rank of E j ), then we have, for all i an j, v i Q ij = m j P ji (.1) (Equation (3), [6, p46]). Since we have E j = 1 v i Q ija i, the entry in row a, column b of E j is Q ij /v whenever (a, b) R i. This is also the value of the stanar inner prouct of column a an column b of the same matrix E j. An association scheme is metric (or P -polynomial) if there is an orering R 0, R 1,..., R D on the relations so that, for each i, A i may be expresse as a matrix polynomial of egree exactly i in A 1. Such an orering is calle a P -polynomial orering. Delsarte [10] showe that metric association schemes, with specifie P -polynomial orering, are in one-to-one corresponence with istance-regular graphs (see [6, Prop..7.1] or [3, Prop. III.1.1]). By analogy, an association scheme is sai to be cometric (or Q-polynomial) if there is an orering E 0, E 1,..., E D on the primitive iempotents so that, for each j, E j may be expresse as a polynomial of egree exactly j applie entrywise to the values in E 1. Such an orering is calle a Q-polynomial orering. It is becoming conventional to specify the parameters of a D-class cometric association scheme (X, R) by its Krein array ι (X, R) = {b 0, b 1,..., b D 1; c 1, c,..., c D} where b j := q j 1,j+1 (0 j < D) an c j := q j 1,j 1 (1 j D). All parameters may be recovere from these few. For example, m 1 = b 0, c 1 = 1 an the parameters a j := q j 1,j (0 j D) satisfy c j + a j + b j = m 1 for 0 j D where, by convention, we efine c 0 = b D = 0. A cometric association scheme (X, R) is Q-bipartite if q k ij = 0 whenever i + j + k is o. Suzuki [18] points out that this is equivalent to a j = 0 for j = 1,,..., D. In [16], this is also shown to be equivalent to the conition Q D i,1 = Q i,1 for 0 i D. For a bipartite istance-regular graph, the first column of the matrix P is symmetric about the origin; for a Q-bipartite cometric scheme, the first column of matrix Q has this property. 3
A cometric association scheme (X, R) is Q-antipoal if q D k i,d j = qk i,j whenever j + k D. Suzuki [18] points out that this is equivalent to b j = c D j for for all j = 0, 1,,..., D 1 except possibly j = D. Throughout this paper, we use the natural orering of the relations: we re-label relations R 0,..., R D if necessary so that Q 0,1 > Q 1,1 > > Q D,1. With this orering, it is shown in [16] that the Q-antipoal conition is equivalent to the conition { m D i even; Q i,d = 1 i o. An antipoal istance-regular graph has the property that the graph (X, R D ) is a union of complete graphs of size v D + 1. A Q-antipoal cometric scheme has the property that the graph (X, R 0 R R e ) is a union of m D + 1 complete graphs where e = D. An association scheme is sai to be imprimitive if some graph (X, R i ) (1 i D) is isconnecte. Equivalently, the scheme is isconnecte if some E j (1 j D) has repeate columns. It is well-known that an imprimitive istance-regular graph is either bipartite or antipoal or both. Theorem.1 (Suzuki [18]). If (X, R) is a D-class imprimitive cometric association scheme with D 6, then (X, R) is either Q-bipartite or Q-antipoal or both. We remark that it is likely that this result hols for D = 6 as well, but one exceptional series of parameter sets remains to be rule out. The vertex set of a Q-antipoal association scheme amits a natural partition X = X 1 X X w into subsets of size v/w such that, for even i, each ege of the graph (X, R i ) lies within some X j an, for each o i, each ege of the graph (X, R i ) has enpoints in istinct cells X j of this partition. In [16], a Dismantlability Theorem is prove which establishes that, for any Y X which is expressible as a union of some subcollection of the X j the D relations restricte to Y inuce a cometric subscheme of this association scheme. 3 Real mutually unbiase bases Let B 1, B,..., B w be w orthonormal bases for R. We say that these bases are mutually unbiase if, whenever i j, the expansion of any element of B i in terms of basis B j has all coefficients of equal magnitue. That is, a, b = ± 1 whenever a an b are chosen from istinct bases among B 1,..., B w. 4
Example 3.1. The 4-cell is a regular polytope in R 4 with vertex set {±e i : 1 i 4} {(w, x, y, z) : w, x, y, z {1/, 1/} }. This correspons naturally to a set of 3 real mutually unbiase bases in R 4 by taking one vector from each parallel pair among the twelve pairs of epenent vectors in the above set. With this coorinatization, the bases may be taken to be B 1 = {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)}, {( ) ( ) ( ) ( 1 B =, 1, 1, 1 1,, 1, 1, 1 1,, 1, 1 1, 1,, 1, 1, 1 {( B 3 = 1 ) ( ) ( ) (, 1, 1, 1 1,, 1, 1, 1 1,, 1, 1, 1 1,, 1, 1, 1 )}, )}. It is no coincience that these 4 vectors also etermine a 4-class cometric association scheme. For, let M enote the maximum number of real mutually unbiase bases in R. Also, let N n enote the maximum number of mutually orthogonal latin square (MOLS) of sie n. The following theorem summarizes what is currently known about the numbers M. Theorem 3. (Various authors). Let 3. Then (i) [11] M + 1; (ii) [7] if = 4 k for some integer k, then M = + 1; (iii) [5] if is not ivisible by four, then M = 1; (iv) [5] M if an only if there is a Haamar matrix of sie ; (v) [5] M 3 if an only if there exist Haamar matrices H 1, H, H 3 of sie satisfying H 1 H = H 3 (vi) [5] if is not a square, then M ; (vii) [5] if /4 is an o square, then M 3; (viii) [0] if there exists a Haamar matrix of sie n =, then M N n +. Finally, M =. Some remarks are in orer here. The paper [11] of Delsarte, Goethals an Seiel gives bouns on sets of lines through the origin with few angles. One of these bouns the secon example in Table I of [11] applies irectly to the situation at han. When is a power of four ( at least sixteen), there is a construction achieving the boun in part (i) of the theorem base on Kerock coes. This configuration is implicit in [7] but the best source for the explicit set of vectors in Eucliean space is [8]. It is wiely believe that Haamar matrices exist of sie n for all n ivisible by four. The state of the art regaring the values N n is summarize in [9, III.3.6]. 5
4 The equivalence In this section, we establish an equivalence between 4-class cometric association schemes which are both Q-bipartite an Q-antipoal, on the one han, an collections of real mutually unbiase bases, on the other. Let B 1, B,..., B w be w mutually unbiase bases in R an set X = ±B 1 ±B ±B w. Then X = w an any pair of vectors from X have inner prouct belonging to the set { A = σ 0 := 1, σ 1 := 1, σ := 0, σ 3 := 1 }, σ 4 := 1. For a, b X an 0 i, j 4, set p i,j (a, b) := {c X : a, c = σ i, c, b = σ j }. We aim to show that p i,j (a, b) is inepenent of the choice of a an b, but epens only on i, j an a, b. This result generalizes a result of Bannai, et al. (see [4],[1]). Theorem 4.1. Let B 1, B,..., B w be w mutually unbiase bases in R an let X be efine as above. Then relations R 0,..., R 4 given by R i = {(a, b) X X : a, b = σ i } form a Q-bipartite, Q-antipoal cometric association scheme on X with intersection numbers L i = [p k ij] k,j given by L 0 = I, 0 (w 1) 0 0 0 1 + (w ) 1 (w ) 0 L 1 = 0 (w 1) 0 (w 1) 0 0 (w ) 1 + (w ) 1, L = 0 0 0 (w 1) 0 0 0 0 (w 1) 0 0 (w ) 1 + (w ) 1 L 3 = 0 (w 1) 0 (w 1) 0 1 + (w ) 1 (w ) 0, L 4 = 0 (w 1) 0 0 0 0 0 ( 1) 0 0 0 1 0 1 0 1 0 ( ) 0 1 0 1 0 1 0 0 0 ( 1) 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 Proof. In most cases, it is straightforwar to verify that a given intersection number is well-efine. For L, this follows from the observation that each set ±B i is the set of vertices of an orthoplex (or cross polytope ). Moreover, the value of p 1 1, for instance, is obtaine by noting that, for a, b = σ 1, the map c c on {c X : a, c = 0} is a bijection between {c X : a, c = 0, c, b = σ 1 } 6.,
an {c X : a, c = 0, c, b = σ 3 }. Such consierations establish that all p k ij are well-efine except possibly the eight quantities p 1 11, p 1 13, p 3 11, p 3 13, p 1 31, p 1 33, p 3 31, p 3 33. Without assuming that we have an association scheme, we fin that p 11 (a, b) + p 13 (a, b) = (w ) (4.1) p 31 (a, b) + p 33 (a, b) = (w ) (4.) p 31 (a, b) = p 13 (a, b) (4.3) p 11 (a, b) = p 13 (a, b) (4.4) whenever a an b are chosen from istinct extene bases among the ±B h. So it suffices to prove that p 11 (a, b) oes not epen on the choice of a an b provie (a, b) R 1. To o so, we apply Lemma 7.3 in [1]. Since each orthoplex ±B h is a spherical 3-esign in R, the union of the w of them is also a spherical 3-esign. So we can take i = j = 1 (since 1 + 1 3) in Lemma 7.3 of [1] to obtain the linear equation 4 which, for (a, b) R 1, reuces to 4 h=0 l=0 a, b σ h σ l p hl (a, b) = X. 1 p 11(a, b) 1 p 13(a, b) 1 p 31(a, b) + 1 p 33(a, b) + 4 = k. So, applying the above ientifications, we obtain the linear system p 11 (a, b) + p 13 (a, b) = (k ) (4.5) p 11 (a, b) p 13 (a, b) = (k ) (4.6) which has a unique solution, inepenent of the choice of a an b, simply provie (a, b) R 1. Now it is straightforwar to verify that this association scheme is both Q-bipartite an Q-antipoal. Since R 4 consists of the pairs (a, a) for a X, we have an imprimitive scheme an this can only be a Q-bipartite system of imprimitivity, by Suzuki s Theorem. But we also have the partition of X into the orthoplexes ±B h, which are of size at least four (provie > 1); so the scheme is Q-antipoal as well. Our next result gives the reverse implication. Theorem 4.. Let (X, R) be a cometric 4-class association scheme which is both Q-antipoal an Q-bipartite an let E 1 enote the first primitive iempotent in a Q-polynomial orering for (X, R). Set = rank E 1. Write E 1 = X UU 7
for some X matrix U with orthogonal columns all having the same norm X /. Then all rows of U are unit vectors in R an, for each row a of U, we have that a is also a row of U. Let Y S 1 be constructe by choosing arbitrarily one vector from each such parallel pair of rows of U. Then Y is naturally partitione into a collection of w = X / real mutually unbiase bases in R. Proof. We apply basic facts about imprimitive cometric schemes first observe in [16]. Let Q be the matrix of ual eigenvalues Q ij (0 i, j 4) of (X, R). Then, uner the natural orering of the relations, the secon column of Q (with entries Q i1 ) is symmetric about zero an we have Q 01 = m 1 > Q 11 > Q 1 = 0 > Q 31 = Q 11 > Q 41 = m 1. Since the entries in this column are all istinct, we can ientify the elements of X with the columns of E 1 or, equivalently, with the rows r a of matrix U in such a way that (a, b) R i precisely when r a, r b = Q i1 /m 1. Since our association scheme is Q-antipoal, the relation a b r a, r b {1, 0, 1} is an equivalence relation on X. Ientifying pairs a, b with (a, b) R 4 yiels a -class quotient scheme, a strongly regular graph. Since our 4-class scheme is Q-antipoal, this graph must also be imprimitive. So it is a complete multipartite graph wk for some integers w an satisfying w = 1 X. The secon eigenmatrix for this strongly regular graph is Q = 1 w( 1) w 1 1 0 1 1 w w 1 Stanar properties of imprimitive schemes inform us that this matrix must appear as a submatrix of the secon eigenmatrix Q of our 4-class scheme in fact, the matrix Q with each of its last two rows uplicate, gives us columns 0, an 4 of Q. So our 4-class scheme has secon eigenmatrix of the following form: Q =. 1 m 1 w( 1) w m 1 w 1 1 Q 11 0 Q 11 1 1 0 w 0 w 1 1 Q 11 0 Q 11 1 1 m 1 w( 1) m 3 w 1 (Here, we have use stanar ientities such as [6, Lemma..1].) Since we have assume a Q-polynomial orering on the eigenspaces, we have the three-term recurrence Q i1 = m 1 + q 11Q i (0 i 4). (Since our scheme is Q-bipartite, we have q 1 11 = 0.) Taking i = first gives q 11 = m 1 /w; next, take i = 0 to fin m 1 =. Finally take i = 1 to establish Q 11 =. Thus the X = w 8.
rows of the matrix U efine in the statement of the theorem have pairwise inner proucts 1, 0, 1 for vectors in the same equivalence class, an ± 1 for vectors chosen from istinct equivalence classes. Since each equivalence class has size, we may choose one vector from each parallel pair of rows of U an obtain w mutually unbiase bases in R. The above two results are summarize in the following Theorem 4.3. Let w an be integers with w,. Then there exist w real mutually unbiase bases in R if an only if there exists a cometric 4-class association scheme on w vertices which is both Q-bipartite an Q-antipoal with Q-antipoal classes of size. 5 Applications of the main results In view of the above results, every construction an every boun for real mutually unbiase bases gives rise to constructions an non-existence results for 4-class cometric association schemes which are both Q-bipartite an Q-antipoal. As observe by [5] an other authors, any pair of MUBs in R is equivalent to a Haamar matrix. In this case, the unerlying association scheme is not only Q-polynomial, but P -polynomial as well; these are the Haamar graphs. The construction of Wocjan an Beth [0] gives infinitely many new cometric association schemes with Krein arrays {, 1, } (w 1) w, 1; 1, w, 1, whenever there is a Haamar matrix of sie n := an w N n +. The current state of affairs, regaring the optimal value of w for a given imension, is summarize in Theorem 3.. For >, only w = 1 is possible unless is a multiple of four. (In this case, we have a trivial strongly regular graph.) For a mutiple of four an 10, we have the following ranges for the maximum value of w: 4 8 1 16 0 4 8 3 36 40 44 48 5 56 60 w 3 9-3 64 68 7 76 80 84 88 9 96 100 104 108 11 116 10 w 33-3 So there are only a few open questions for these small parameter sets. Less is known about the optimal value of w for imensions of the form = 16s, where s is not a power of two; for example it is not known if the absolute boun w + 1 can be achieve for any other than a power of four. Some key small values to consier are = 144, 400, 576, 784, 196 an 1600. For example, in R 144, the construction of Wocjan an Beth gives w = 7 real MUBs but the best upper boun we have is w 73. In [16], an infinite family of cometric 4-class schemes is constructe by taking the extene Q-bipartite oubles of the Cameron-Seiel schemes; these 3-class cometric schemes were foun in [7] as linke systems of symmetric esigns. Using Theorem 4., this gives us 9
+ 1 MUBs in R for = 4 k, k. Of course, this is the same configuration as the one given in [8]. Bannai an co-authors [1, 4] were the first to realize that this configuration of MUBs gives rise to a cometric association scheme. But the Dismantlability Theorem in [16] tells us that any subcollection of the Q-antipoal classes in this association scheme also inuce a cometric association scheme which is again both Q-bipartite an Q-antipoal. The configuration of MUBs which one obtains from these schemes are just those obtaine from the extremal example by eleting some subcollection of bases. In [17], Mathon conucte an exhaustive stuy of linke systems of (16, 6, ) symmetric esigns. Via the extene Q-bipartite ouble construction an Theorem 4. above, these give rise to various configurations of maximal MUBs in R 16 with less than the optimal number of bases. (The optimal value of nine bases is achievable only by the Cameron-Seiel construction.) A translation association scheme [6, p65] is an association scheme which amits an abelian group acting regularly on its vertices. In [6, p45], Brouwer, et al. point out that a 4-class P -polynomial association scheme which is both antipoal an bipartite is equivalent to a symmetric (m, µ)-net [6, p18]: a set P of points an a set L of lines with the properties (i) any point lies on m lines; (ii) any line meets m points; (iii) any two points are joine by either µ or zero lines; (iv) any two lines meet in either µ or zero points; an (v) the configuration is non-egenerate. We simply observe that, since every translation scheme gives rise to a ual association scheme on its characters an the ual of a P -polynomial association scheme is Q-polynomial, with imprimitivity properties mapping over naturally, every symmetric (m, µ)-net which is a translation scheme gives rise in this way to a set of mutually unbiase bases in real space. It is an open question as to whether any non-trivial examples exist. 5.1 Arrangements of Haamar matrices As a precursor to our next theorem, we partition the ajacency matrices of our scheme accoring to the two imprimitivity systems iscusse above. Suppose we have w mutually unbiase bases in R. The rows an columns of the ajacency matrices shall be inexe so that elements in each Q-antipoal class are groupe together, an pairs in Q-bipartite classes correspon to consecutive row/column labels l 1, l. Put geometrically, we inex by grouping the vectors in each of the w extene bases ±B i together, an inex each parallel pair ±b consecutively. As usual, A 0 is the ientity matrix of size w. Now A 1 encoes the relation (a, b) R 1 a, b = 1 an therefore has the form A 1 = 0 N 1, N 1,w N,1 0 N,w...... N w,1 N w, 0, 10
where each N i,j is a 01-matrix compose of blocks each equal to [ 1 0 0 1 ] or [ 0 1 1 0 ]. By the symmetry of A 1, Ni,j T = N j,i. Since A 3 escribes the relation corresponing to pairs with inner prouct 1, we have 0 J N 1, J N 1,w J N,1 0 J N,w A 3 =......, J N w,1 J N w, 0 where J is the all-ones matrix of size. Next, A encoes the orthogonality relation among these vectors, an is hence a block iagonal matrix with blocks of size, an blocks of the form J I J on the iagonal; A = I w (J I J ). Finally, A 4 escribes the relation of 1 cosine, an is a block iagonal matrix with w blocks of the form [ 0 1 1 0 ] on the iagonal. Theorem 5.1. There exist w mutually unbiase bases in R if an only if there exist ( w Haamar matrices of size (say H i,j, 1 i < j w), satisfying H i,j H j,k = H i,k for each triple i, j, k of istinct values from {1,..., w} where write H j,i = Hi,j T for j > i. Proof. For the proof in the forwar irection, we will make use of a simple linear transformation φ mapping matrices to real numbers. Define φ ([ α β γ δ ]) 1 = (α + δ β γ) an note that when at least one of M or N takes the form [ a b a b ], we not only have φ(m +N) = φ(m) + φ(n) but also φ(mn) = φ(m)φ(n). We exten φ to map matrices M = [m r,s ] of size w w to matrices of size w w in the natural way: the (k, l)-entry of the resulting matrix φ(m) is φ ([ m k 1,l 1 m k 1,l ]) m k,l 1 m k,l. Let our collection {B i : 1 i w} of MUBs be given an consier the association scheme etermine by any two of the bases, B i an B j. From above, this association scheme has first ajacency matrix of the form A 1 = [ 0 N N 0 where we have use the abbreviations N = N i,j an N = N j,i. Using the intersection numbers compute in Theorem 4.1, we have that A 1 = I + [ I + A = ] I ) J 0 0 I + (J I ) J giving NN = N N = I + (J I ) J. Applying φ to both sies of this equation gives φ(n)φ(n) = I. Clearly, since each block of N is either I or J I, each entry of φ(n) is ±1. So for each i j, the matrix H i,j := φ(n i,j ) is a Haamar matrix. ], ) 11
We use the same iea to establish H i,j H j,k = H i,k for any three istinct inices i, j, k {1,..., w}. Again applying Theorem 4.1, we obtain a 4-class cometric association scheme with three Q-antipoal classes an, again using the above conventions for orering the rows an columns, we have A 1 = 0 N i,j N i,k N j,i 0 N j,k N k,i N k,j 0, A 3 = 0 J N i,j J N i,k J N j,i 0 J N j,k J N k,i J N k,j 0 with A an A 4 as above being all zero off the iagonal blocks. From Theorem 4.1, we have now A 1 = A 0 + + A 1 + A + A 3. Consier some noniagonal block of both sies of this equation, say block (i, k). We fin Applying φ to both sies gives N i,j N j,k = N i,k + J. H i,j H j,k = H i,k, as esire. Since this hols for any choice of istinct inices i, j an k, the forwar implication of the theorem is now establishe. Now we reverse the construction. Suppose we are given ( w ) Haamar matrices of orer, {H i,j : 1 i < j w}, enjoying the property H i,j H j,k = H i,k (5.1) whenever i < j < k. Defining H j,i := Hi,j for j > i, one easily verifies that Equation (5.1) now hols whenever i, j an k are istinct elements of {1,..., w}. We blow up each H i,j in the obvious way to a 01-matrix N i,j by mapping ψ : 1 [ 1 0 0 1 ] an ψ : 1 [ 0 1 1 0 ]. Now we efine 0 N 1, N 1,w 0 J N 1, J N 1,w N,1 0 N,w A 1 =......, A J N,1 0 J N,w 3 =......, N w,1 N w, 0 J N w,1 J N w, 0 efining A 0, A an A 4 in the obvious way as above. Since H j,i H i,j = I, we have N j,i N i,j = I + (J I J ). The secon term on the right arises from consiering the off-iagonal entries of H j,i H i,j, which is a sum of 1 s an 1 s, an hence uner ψ map to J. Similar consierations 1,
give N i,j N j,k = N i,k + + J for any three istinct i, j an k. Using these facts, we expan the various blocks ( N j,i N i,j = (w 1) I + ) (J I J ) i j an of A 1 to obtain ( N i,j N j,k = (w ) Ni,k + + ) J j i,k A 1 = (w 1)A 0 + + (w )A 1 + (w 1)A + (w )A 3. In the same manner, one may routinely very the remaining equations A i A j = k pk ija k, thus concluing the proof that we have an association scheme with the same parameters as in the statement of Theorem 4.1. Then Theorem 4. gives the esire result. Before giving the next corollary, we introuce some convenient terminology. Fix a imension an consier a set system {C j : j I} where I is some inex set an each C j consists of vectors in R. (We will have only ±1-vectors in our setting.) In this system, a cohesive triple refers to a set of three vectors u, v, u v all belonging to the same C j. A fole triple refers to a set of three vectors u, v, u v with u an v belonging to the same C k an u v an element of some C j, j k. Finally, a split triple refers to a set of three vectors u, v, u v all belonging to ifferent sets C j in this set system. We will call a ±1 vector z balance if the number entries equal to 1 is congruent to zero moulo four an semibalance if the number entries equal to 1 is congruent to two moulo four. Corollary 5.. Suppose = 16s for some o integer s an suppose {B 1,..., B w } is a collection of mutually unbiase bases in R. Fix any i, 1 i w, an for each j i let C j enote the set of columns of the matrix H i,j constructe in Theorem 5.1. With respect to the set system {C j : j i}, the following hol true: (i) each C j contains istinct vectors an the sets {C j : j i} are pairwise isjoint; (ii) if u, v, u v is a cohesive triple in C j, then every other vector in C j is balance an every vector in any C k with k j is semibalance; (iii) if u, v, u v is a fole triple with u v in C j, then every other vector in C j is balance an every vector in any C k with k j is semibalance; 13
(iv) if u, v, u v is a split triple in {C j : j i}, then every other vector in the same C j as either u, v or u v is semibalance an every vector in any other C k is balance. Proof. By Theorem 5.1, the matrices H i,j are all Haamar matrices an therefore can have no repeate columns. If j k an yet C j an C k have a vector in common, then some column of H i,j is equal to some column of H i,k for simplicity, let us suppose this is the first column in each case. Then we have H j,i H i,k = Hi,jH i,k = 4sH j,k an yet the (1, 1)-entry of this prouct is equal to = 16s, a contraiction. This establishes (i). For parts (ii) (iv), we will use the Sylvester matrix [ ] [ ] [ ] 1 1 1 1 1 1 M =. 1 1 1 1 1 1 If t = [t 1, t, t 3, t 4, t 5, t 6, t 7, t 8 ] is any vector an we write x = [x 000, x 001, x 010, x 011, x 100, x 101, x 110, x 111 ], then the linear system Mx = t has unique solution x = 1 Mt with each entry of 8 x having the form x = t 1 ± t ± ± t 8. 8 Now suppose that z, u, v an u v are all members of j i C j. We consier the system of equations z, z = t 1 := 16s, u, u v = t, v, u v = t 3, u, v = t 4, z, 1 = t 5, z, v = t 6, z, u = t 7, z, u v = t 8 where t, t 3, t 4, t 6, t 7, t 8 {4s, 0, 4s} an t 5 = 16s σ, σ being the number of 1 s in the vector z. Now let x 000 enote the number of coorinate positions h where z h = u h = v h = 1, an similarly let x 001,..., x 111 count coorinate positions h with the respectively properties so that x 001 : z h = 1, u h = 1, v h = 1 x 010 : z h = 1, u h = 1, v h = 1 x 011 : z h = 1, u h = 1, v h = 1 x 100 : z h = 1, u h = 1, v h = 1 x 101 : z h = 1, u h = 1, v h = 1 x 110 : z h = 1, u h = 1, v h = 1 x 111 : z h = 1, u h = 1, v h = 1 x 000 + x 001 + x 010 + x 011 + x 100 + x 101 + x 110 + x 111 = 16s. Then, with x as in the previous paragraph, the eight inner proucts above yiel the linear system Mx = t. The fact that each x... must be an integer forces the integer t + t 3 + t 4 + t 5 + t 6 + t 7 + t 8 14
to be ivisible by eight. Now if u, v, u v is a cohesive triple, then t = t 3 = t 4 = 0. If z belongs to the same cell C j as these vectors, then t 6 = t 7 = t 8 = 0 as well an our linear system is solve to give x 001 = 16s + 16s σ ; 8 so z must be balance. On the other han, if z belongs to any other set C k, we have t 6 + t 7 + t 8 { 1s, 4s, 4s, 1s} an z must be semibalance. The other cases are hanle in a similar manner. Without going into etails, we remark that this result has strong implications for constructions of large sets of real MUBs in such imensions, ruling out many configurations for two istinct triples of the form u, v, u v. Acknowlegments The authors woul like to thank Eiichi Bannai, Ewin van Dam an Chris Gosil for helpful iscussions regaring relate material. During a portion of this work, all three authors were fune by NSA grant number H9830-07-1-005, WJM as PI an NL an WO as summer research assistants. We are grateful to the US National Security Agency for this support. References [1] K. Abukhalikov, E. Bannai, S. Sua. Association schemes relate to universally optimal configurations, Kerock coes an extremal Eucliean line-sets. J. Combin. Theory, Ser. A 116 (009), 434-448. [] M. Aschbacher, A. M. Chils an P. Wocjan. Limitations of nice mutually unbiase bases. J. Alg. Combin. 5 (no. ) (007), 111 13. [3] E. Bannai an T. Ito. Algebraic Combinatorics I: Association Schemes. Benjamin- Cummings, Menlo Park (1984). [4] E. Bannai an E. Bannai. On antipoal spherical t-esigns of egree s with t s 3. To appear, J. Comb. Inf. Syst. Sci., special volume honoring the 75th birthay Prof. D. K. Ray-Chauhuri. [5] P. O. Boykin, M. Sitharam, M. Tarifi an P. Wocjan. Real mutually unbiase bases. Preprint. arxiv:quant-ph/05004v [math.co] (revise version ate Feb. 1, 008) [6] A. E. Brouwer, A. M. Cohen an A. Neumaier. Distance-Regular Graphs. Springer- Verlag, Berlin (1989). 15
[7] P. J. Cameron an J. J. Seiel. Quaratic forms over GF (). Proc. Koninkl. Neerl. Akaemie van Wetenschappen, Series A, Vol. 76 (Inag. Math., 35) (1973), 1 8. [8] A. R. Calerbank, W. M. Kantor, J. J. Seiel, N. J. A. Sloane. Z 4 -Kerock coes, orthogonal spreas, an extremal Eucliean line-sets. Proc. Lonon Math. Soc. 75, no. 3 (1997), 436 480. [9] C. J. Colbourn an J. H. Dinitz, es. The CRC Hanbook of Combinatorial Designs ( n e.) CRC Press, Boca Raton, 006. [10] P. Delsarte. An algebraic approach to the association schemes of coing theory. Philips Res. Reports Suppl. 10 (1973). [11] P. Delsarte, J.-M. Goethals, J. J. Seiel. Bouns for systems of lines an Jacobi polynomials. Philips Res. Rpts. 30 (1975), 91-105. [1] P. Delsarte, J.-M. Goethals, J. J. Seiel. Spherical coes an esigns. Geom. De. 6 (1977), 363-388. [13] C. D. Gosil. Algebraic Combinatorics. Chapman an Hall, New York (1993). [14] A. Roy an C. Gosil. Equiangular lines, mutually unbiase bases, an spin moels. In press, Europ. J. Combin., 008. (Appeare as quant-ph/07000000, 007.) [15] A Klappenecker an M. Rötteler. Constructions of mutually unbiase bases. pp. 137 144. in: Finite Fiels an Appl., Proceeings of 7th International Conference, Fq7, Toulouse, France, May 003. [16] W. J. Martin, M. Muzychuk an J. Willifor. Imprimitive cometric association schemes: constructions an analysis. J. Algebraic Combin. 5 (007), 399-415. [17] R. Mathon. The systems of linke -(16, 6, ) esigns. Ars Combin. 11 (1981), 131 148. [18] H. Suzuki. Imprimitive Q-polynomial association schemes. J. Alg. Combin. 7 (1998), no., 165 180. [19] H. Suzuki. Association schemes with multiple Q-polynomial structures. J. Alg. Combin. 7 (1998), no., 181 196. [0] P. Wocjan an T. Beth. New constructions of mutually unbiase bases in square imensions. Quantum Information an Computation 5 (005), no., 93 101. 16
Appenix: Remaining parameters of the association scheme In this section, we give the eigenmatrices an Krein parameters for a 4-class Q-bipartite Q-antipoal association scheme. (The intersection numbers are given in the statement of Theorem 4.1.) The only free parameters are w an where, as above, the vertices of the scheme correspon to w real MUBs in imension (so that X = w). P = L 1 = 1 (w 1) ( 1) (w 1) 1 1 (w 1) 0 (w 1) 1 1 0 0 1 1 (w 1) 0 (w 1) 1 1 ( 1) 1 L 3 = 0 0 0 0 1 0 1 0 0 0 /w 0 (w 1)/w 0 0 0 1 0 1 0 0 0 0, Q =, L = 0 0 0 (w 1) 0 0 0 ( 1)(w 1) 0 w 1 0 w (w 1) 0 w (w 1) 0 1 0 ( 1)(w 1) 0 w 0 0 (w ) 0 Clearly all Krein conitions are satisfie for, w. 1 w( 1) (w 1) w 1 1 0 1 1 0 w 0 w 1 1 0 1 1 w( 1) (w 1) w 1 0 0 w( 1) 0 0 0 1 0 ( 1)(w 1) 0 1 0 w( ) 0 w 1 0 1 0 ( 1)(w 1) 0 0 0 w( 1) 0 0, L 4 = 0 0 0 0 w 1 0 0 0 w 1 0 0 0 w 1 0 0 0 1 0 w 0 1 0 0 0 w,., 17