CIRCULANT MATRICES, GAUSS SUMS AND MUTUALLY UNBIASED BASES, I. THE PRIME NUMBER CASE

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1 arxiv: v1 [math-ph] 30 Oct 2007 CIRCULANT MATRICES, GAUSS SUMS AND MUTUALLY UNBIASED BASES, I. THE PRIME NUMBER CASE Monique Combescure November 2, 2018 Abstract In this paper, we consier the problem of Mutually Unbiase Bases in prime imension. It is known to provie exactly +1 mutually unbiase bases. We revisit this problem using a class of circulant matrices. The constructive proof of a set of +1 mutually unbiase bases follows, together with a set of properties of Gauss sums, an of bi-unimoular sequences. 1 INTRODUCTION Mutually Unbiase Bases (MUB s) are a set {B 0,...,B N } of orthonormal bases of C such that the scalar prouct in C of any vector in B j with any vector in B k, j k is of moulus 1/2. Starting from the natural base B 0 consiting of vectors v 1 = (1,0,...0), v 2 = (0,1,0,...,0),...,v = (0,0,...,1), it is known that this problem reuces to fin N unitary Haamar matrices P j such that Pj P k is also a unitary Haamar matrix j k. (A unitary matrix is Haamar if all its entries are of moulus 1/2 ). The problem has been solve in prime power imension = p n, p,n N,p prime, an yiels exactly +1 MUB s which is the maximum number of MUB s ([2] an references herein containe). If is factorizable in p m 1 1 p m with p i p j prime numbers, it is known that one has at least N = min p m i i [5]. Inthispaper weshow thatinprimeimension = p, thediscretefouriertransform F togetherwithasuitablecirculantmatrixc allowtoconstruct asetof+1mub s. In aition this construction allows to obtain, as a by-prouct, a set of properties 1

2 2 of Gauss sums of the following form : ( [ 2iπ lk(k +1) exp +jk]) 2 =, j F,l F coprime with, 3 (1.1) (F is the fiel of resiues moulo ). A irect proof of this property is given in [12]. Similar results on generalize Gauss Sums appear in [1]. The efinition of F an of circulant matrices is given below. The natural role playe by circulant matrices in this context is a new result. The circulant unitary matrices are known to be in one-to-one corresponence with the bi-unimoular sequences c = (c 1,c 2,...,c ) [4], namely sequences such that c j = (Fc) j = 1, where F is the iscrete Fourier transform. Not surprisingly Gauss sums appear naturally in this context, since suitable Gauss sequences are examples of bi-unimoular sequences. At the en of this paper, we consier the case of non-prime imension, an show that Gauss sums properties can be euce in the o an in the even imension cases. In a forthcoming work we shall consier the case of prime power imensions an show that the theory of block-circulant matrices with circulant blocks solve the MUB problem in that case. A matrix is Haamar if all its entries have equal mouli [7], an H H = 1l Definition 1.1 A matrix H is Haamar if H j,k is constant j,k = 1,...,, an H H = 1l. We call H a unitary Haamar matrix if H j,k = 1/2, j,k = 1,...,, an Hj,l H l,k = δ j,k l=1 Definition 1.2 A matrixc is callecirculant [6], an enotecirc(c 0,c 1,...,c 1 ), if all its rows an columns are successive circular permutations of the first. It is of the form C = c 0 c 1.. c 1 c 1 c 0.. c c 1 c 2.. c 0 Proposition 1.3 (i) The set C of all circulant matrices is a commutative algebra. (ii) C is a subset of normal matrices (ii) Let V = circ(0,0,...,1). Clearly V = 1l. Then C is circular if an only if

3 3 it commutes with V, an one has for any sequence c = (c 1,...,c ) C, C = circ(c 1,...,c ) : C = P c (V) = c 0 1l+c 1 V +...+c 1 V 1 where P c is the polynomial P c (x) = c k x k Proof : See [6] For the prouct of two circulant matrices C, C (that therefore commute with V), one has VCC = CVC = CC V which establishes that CC is inee circulant. Moreover it is well-known that there is a close link between the circulant matrices an the iscrete Fourier transform. Namely the latter iagonalizes all the circulant matrices. The Discrete Fourier Transform is efine by the following unitary matrix F with matrix elements : ( ) 2iπjk F j,k = 1/2 exp, j,k = 0,1,..., 1 (1.2) Proposition 1.4 (i) The circulant matrix V = circ(0, 0,..., 1) is such that F VF = U iag(1,ω,ω 2,...,ω 1 ) where ω = exp ( ) 2iπ (ii) Let C = circ(c 0,c 1,...,c 1 ) be a circulant matrix. Then (1.3) F CF = iag(ĉ 0,ĉ 1,...,ĉ ( 1) ) where ĉ l = 1 c k ω kl (1.4) Proof : (i) It is enough to check that v k, the k-th column vector of F which has components (v k ) j = ωjk, j,k = 0,1,..., 1 is eigenvector of V with eigenvalue ω k, which is immeiate since (Vv k ) j = ωk(j+1) = ω k (v k ) j

4 4 (ii) One has (Proposition 1.3(ii)) C = c k V k Thus with F CF = c k (F VF) k = c k U k = iag( 0,..., 1 ) l = c k ω lk = ĉ l Lemma 1.5 For any k N, we enote by [k] the rest of the ivision of k by. Given any sequence c = (c 0,...,c 1 ) C its autocorrelation function obeys c k c [j+k] = ĉ l 2 ω jl where the Fourier transform ĉ of c has been efine in (1.4). since Proof : For any j = 0,..., 1, one has ĉ l 2 ω jl = 1 l=0 = k,k =0 l=0 1 c k c k l=0 l=0 ω jl 1 c k ω lk c k ω k l ω l(k k j) = ω l(k j k) = δ k,[j+k] 1 1 l=0 k =0 c [j+k] c k It is known ([4], [13]) that circulant unitary Haamar matrices are in one-toone corresponance with bi-unimoular sequences c = (c 0,c 1,...,c 1 ). Definition 1.6 A sequence c = (c 0,c 1,...,c 1 ) is calle bi-unimoular if one has c j = ĉ j = 1, j = 1,...,, where ĉ j is efine by (1.4). Proposition 1.7 Let (c 0,...,c 1 ) be a bi-unimoular sequence. Then the circulant matrix C = 1/2 circ(c 0,c 1,...,c 1 ) is an unitary Haamar matrix.

5 5 Proof : This is a stanar if an only if statement. One uses Lemma 1.5 : c k c [j+k] = ĉ l 2 ω jl But since ĉ l = 1, l = 0,..., 1, the RHS is simply δ j,0, an therefore l=0 c k c [k+j] = δ j,0 which proves the unitarity of the circulant Haamar matrix C. In all that follows we call inifferently F or P 0 the iscrete Fourier transform. In [8], the authors introuce for any imension being the power of a prime number a set of operators calle rotation operators which can be viewe as circulant matrices (this property is however not put forwar explicitely by the authors). In this paper, restricting ourselves to the prime number case, we show that these operatorscanbeusetoefineasetof+1mutuallyunbiasebasesinimension. Mutually Unbiase bases are extensively stuie in the framework of Quantum Information Theory. They are efine as follows : Definition 1.8 A set {B 1,B 2,...,B m } of orthonormal bases of C is calle MUB if for any vector b (k) j B k an any b (k ) j B k one has b (k) j b (k ) j = 1/2, k k = 1,...,m, j,j = 1,..., where the ot represents the Hermitian scalar prouct in C. Remark 1.9 It is trivial to show that if the orthonormal bases B k are the column vectors of an unitary matrix A k, then the property that must satisfy the A k s in orer that {1l,B 1,...,B m } are MUB s is that all A k, k = 1,...,m an A k A k, k k = 1,...,m are unitary Haamar matrices. Namely if u j, v k are column vectors for unitary matrices A, A respectively, then u j v k = (A A ) j,k Thus unitary Haamar matrices play a major role in the MUB problem. It is known that the maximum number of MUB s in any imension is + 1, an that this number is attaine if = p m, p being a prime number. In this paper, restricting ourselves to m = 1, we revisit the proof of this property, using circulant matrices introuce by [8]. We then show that it implies beautiful properties of Gauss sums, namely the following ([12]) :

6 6 Proposition 1.10 Let 3 be an o number. Then k = 1,..., 1 coprime with the sequences ( ( )) iπkj(j +1) g (k) := exp (1.5) are bi-unimoular. Thus (1.1) hols true.,..., 1 This property will appear as a subprouct of our stuy of MUB s for a prime number 3 via the circulant matrices metho. As stresse above, the link between circulant matrices an bi-unimoular sequences is well establishe. What is new here is the fact that the MUB problem via a circulant matrix metho allows to recover the bi-unimoularity of Gauss sequences. The crucial role playe by the Gauss sequence is ue to the crucial role playe by the Discrete Fourier Transform (or in other therms the Fourier-Vanermone matrices) in the MUB problem for prime numbers. Let us introuce it now explicitely. It is known since Schwinger ([11]) that a simple toolbox of unitary matrices sometimes refere to as generalize Pauli matrices U, V allows to fin MUB s. U, V generate the iscrete Weyl-Heisenberg group [14]. Denote by ω the primitive root of unity (1.3). The matrix U is simply U = iag(1,ω,ω 2,...,ω 1 ) which generalizes the Pauli matrix σ z to imensions higher than two. The matrix V generalizes σ x : V = circ(0,0,...,1) = Then one has the following result : Theorem 1.11 (i) The U, V matrices obey the ω-commutation rule : VU = ωuv (ii) The Discrete Fourier Transform matrix P 0 = F efine by (1.2), namely P 0 = 1 1 ω ω 2.. ω 1 1 ω 2 ω 4.. ω 2( 1) ω 1 ω 2( 1).. ω ( 1)( 1)

7 7 iagonalizes V, namely (iii) One has P 4 0 = 1l V = P 0 UP 0 = P 0 U P 0 (ii) is simply Proposition 1.4(i). For the proof of(iii) reminiscent to the properties of continuous Fourier transform, it is enough to check that P0 2 = W = Thus W = 1l in imension = 2, an W 2 = 1l, 3. In [5] we have shown that for o one can a to the general toolbox of unitary Schwinger matrices U, V a iagonal matrix D of the form D = iag(1,ω,ω 3,...,ω k(k+1)/2,...,1) so that the MUB problem for oprime imension reuces to properties of U, V, D, an that certain properties of quaratic Gauss sums follow as a by-prouct. 2 THE =2 CASE We have U = σ z an V = σ x, σ z, σ x being the usual Pauli matrices. Since UV = σ y, fining MUB s in imension = 2 amounts to iagonalize σ x, σ y. One has : σ x = P 0 σ z P 0 with P 1 is circulant. σ y = P 1 σ z P 1 P 0 = 1 ( ) 1 1, P = 1 ( ) 1 i 2 i 1 Proposition 2.1 The set {1l,P 0,P 1 } efines three MUB s in imension 2. Since the matrices P 0, P 1 are trivially unitary Haamar matrices, it is enough to check that P0 P 1 is itself a unitary Haamar matrix, which hols true since ( ) P0 P 1 = eiπ/ i i

8 8 3 THE PRIME DIMENSION 3 being prime, we enote by F the Galois fiel of integers mo. Let us recall the efinition of the rotation operator of[8], which, as alreay stresse in nothing but a circulant matrix in the o prime imension. Definition 3.1 Define R as an unitary operator commuting with V an iagonalizing VU. Proposition 3.2 (i) R is a circulant matrix. (ii) R k is also circulant k Z. This follows from Proposition 1.3. Therefore we are le to consier a subclass of circulant matrices that are unitary. They must satisfy : k = 0,..., 1 c k = 1/2 an k = 1,..., 1 1 c jc [ k+j] = 0 (orthogonality conition). Now it remains to show that such a matrix R exists. In [5] we have constructe a unitary matrix P 1 that iagonalizes VU. It is efine as P 1 = D 1 P 0 for any o integer (not necessarily prime). We have establishe the following result : Proposition 3.3 (i) For any o integer, the matrix P 0 P 1 is a unitary Haamar matrix. (ii) For 3 o integer, let P k := D k P 0. Then P 0P k is a unitary Haamar matrix for all k coprime with. (iii) The matrix D efine above is such that TrD k =, k F coprime with For the simple proof of this result see [5]. (iii) is a simple consequence of (i) an(ii). Namely the eigenvector of V U belonging to the eigenvalue 1 has components v j = 1/2 ω j(j+1) 2 Since P0P 1 is unitary Haamar matrix, the element of P0P 1 of the first row an first column is simply 1 1 ω j(j+1) 2

9 9 an since its moulus must be 1/2 we obtain (iii) for k = 1. The proof for any k coprime with can be obtaine similarly using (ii). The problem is that P 1 is not circulant. However the circulant matrix R is obtaine from P 1 by multiplying the kth column vector of P 1 by a phase which is ω k(k 1) 2 This operation preserves the fact that it is unitary an that it iagonalizes VU. We thus have : VU = P 1 UP 1 = RUR (3.6) Proposition 3.4 Let R be the matrix : It is a unitary Haamar matrix. R = 1/2 circ (1,ω 1,ω 3,...,ω k(k+1)/2,...,1) Proof : By construction it is a unitary Haamar matrix, since P 1 is. To prove the fact that it is circulant, it is enough to know that thus (P 1 ) j,k = 1/2 ω jk j(j+1) 2 R jk = 1/2 ω jk j(j+1) 2 k(k 1) 2 = 1/2 ω (j k)(j k+1) 2 (3.7) since j(j +1) k(k 1) (j k)(j k +1) jk = Thus all column vectors are obtaine from the first by the circularity property. Furthermore the R k have the property that they iagonalize V k U, k = 1,..., 1: Theorem 3.5 (i) R = αp 0 DP 0 where α := 1/2 1 ω k(k+1)/2 is a complex number of moulus 1. (ii) R = α 1l where 1l enotes the unity matrix. (iii) R k U(R ) k = V k U, k = 0,..., The proof is extremely simple : (i) We have shown that c j = ω j(j+1) 2 is a bi-unimoular sequence, thus α is a complex number of moulus one. Moreover from Proposition 1.4 (ii), the unitary matrix R = 1/2 circ(c j ) is such that ˆR = P 0RP 0 = iag(ĉ k )

10 10 But since Therefore ĉ k = 1 ω jk j(j+1) 2 = αω k(k+1) 2 ω j(j+1) 2 = ω (j+k)(j+k+1) 2 = ˆR = αiag(ω k(k+1) 2 ) = αd ω jk j(j+1) k(k+1) 2 2 (ii) is simply a consequence of (i) since D = 1l. (iii) is obtaine by recurrence. Namely it is true for k = 1 by (3.6). For k 2 one has : R k U(R ) k = RV k 1 UR = V k 1 RUR since R commutes with V. But V k 1 RUR = V k 1 (VU) = V k U There is a irect link between the matrices P k that iagonalize VU k an the R k that iagonalize V k U : Theorem 3.6 α being the complex number of moulus one efine above, we have for any k = 0,..., 1 P k = α k P 0 R k P 2 0 Proof : In [5] we have proven that P k = D k P 0 iagonalizes VU k. But one has so the result follows immeiately. D k = D k = α k P 0 R k P 0 Corollary 3.7 For any k = 1,..., 1, R k is a unitary Haamar circulant matrix when 3 is prime. Proof : R k is circulant an unitary since R is. Therefore we have only to check that it is Haamar. We have : R k = α k P 2 0 P 0 P kp 2 0 But we recall that P 2 0 equals the permutation matrix W. Thus all matrix elements of R k equal, up to a phase, some matrix elements of P 0 P k. But we have establishe in [5] that the matrix P 0 P k is unitary Haamar k = 1,..., 1, thus all its matrix elements are equal in moulus to 1/2. This completes the proof. Now we show how this property of the matrix R reflects itself in Gauss sums properties.

11 11 Proposition 3.8 (i) Let be an o prime. Then for any k = 1,2,..., 1 R k is an unitary Haamar matrix if an only if one has ( ) iπ exp [kj2 +j(k +2m)] =, m = +1,..., 1 (ii) Uner the same conitions R k is a unitary Haamar matrix if an only if k 1 ( ) iπ exp k j2 +(k +2m)j = k Proof : Since R k = α k P 0 D k P 0, the matrix elements of Rk are (R k ) m,l = αk ω j(m l)+kj(j+1) 2 Since R k is circulant, it is unitary Haamar matrix if an only if the matrix elements of the first column (l=0) are of moulus 1/2, thus if an only if ( ) iπ exp [kj2 +j(k +2m)] = which proves (i). (ii) Using the reciprocity theorem for Gauss sums ([3]), we have for all integers a,b with ac 0 an ac+ even that the quantity obeys S(a,b,) = a S(a,b,) := 1/2 ( ) πi exp (aj2 +bj) ( ) πi exp 4 [sgn(a) b2 /a] S(, b,a) Thus S(a,b,) = if an only if S(, b,a) = a. Applying if to a = k = 1,..., 1 coprime with an b = 2m+k, an taking the complex conjugate yiels the result. Namely a + b = k + k +2m is even for all k = 0,..., 1 since is o. Corollary 3.9 Let 3 be a prime number. Then for any k = 1,2,..., 1 the sequences ) g (k) := (ω kj(j+1) 2,..., 1 are bi-unimoular.

12 12 Remark 3.10 This property is known, but has an extension in the non prime o imensions. See next section. It isknown that theiagonalizationofvu k, k = 0,..., 1 provies aset of+1 MUB s for a prime p. Here we show that the same is true with the iagonalization of V k U, k = 0,..., 1. Theorem 3.11 Let 2 be a prime imension. Then the orthonormal bases efine by the unitary matrices 1l,P 0,R,R 2,...,R 1 provie a set of +1 MUB s. Proof : For = 2 this has been alreay proven in Section 2. For 3 (thus o, since it is prime), it is enough to check that : (i) P 0,R k, k = 1,..., 1 are unitary Haamar matrices, together with (ii) P 0R k, k = 1,..., 1 an (R k ) R k, 1 k < k 1. Since(i)hasbeenalreayestablishe, itremainstoshow(ii). SinceR k = α k P 0 D k P 0 we have P 0R k = α k D k P 0 which is trivially an unitary Haamar matrix (since P 0 is, α is of moulus 1 an D is iagonal an unitary). For (R ) k R k it is trivial since (R ) k R k = R k k which is unitary Haamar for any k k, k,k = 1,..., 1. 4 THE CASE OF ARBITRARY ODD DIMEN- SION We have shown in [5] that for any o imension 3 the matrices P 0P k is an unitary Haamar matrix provie k is co-prime with. This can be transfere to a similar property for the matrix R k, an therefore to the bi-unimoularity of the sequence g (k) efine in (1.5). Proposition 4.1 Let 3 be an o integer, an k be any number coprime with. Then (i) R k is an unitary Haamar matrix. (ii) The sequence g (k) is bi-unimoular. (iii) Both properies are equivalent. This implies Proposition Theorem 4.2 Let be o an k > 2 be the smallest ivisor of. Then the orthonormal bases efine by the unitary matrices { 1l,F,R,R 2,...,R k 1} provie a set of k +1 MUB s in imension.

13 5 THE CASE OF ARBITRARY EVEN DIMEN- SION Let be even an ω = exp ( ) 2iπ We enote ω 1/2 = e iπ/. One efines the Discrete Fourier Transform F as usually. The theory ofcirculant matrices isalso pertinent for even imensions 4. Namely in that case the matrix D = iag(1,ω 1/2,...,ω k2 /2,...,ω 1/2 ) has been shown ([5]) to be such that the unitary Haamar matrix P 1 = D F iagonalizes V U. But the circulant matrix R obtaine by multiplying the k-th column vector of P 1 by ω k2 /2 also iagonalizes VU : Proposition 5.1 The circulant matrix whose matrix elements are R j,k = 1 ω (j k)2 /2 iagonalizes VU an is such that F R is an unitary Haamar matrix. 13 thus Proof : (P 1 ) j,k = 1 ω j2 2 +jk R j,k = 1 ω (j k)2 2 R j,k onlyepensonj k thusiscirculant(anunitary). Thereforeitisiagonalize by F, namely there exists an unitary iagonal matrix D such that F RF = D Again this implies that F R is an unitary Haamar matrix. Corollary 5.2 (i) The orthonormal bases efine by the unitary matrices {1l, F, R} provie a set of 3 MUB s in arbitrary even imension. (ii) One has the following property of quaratic Gauss sums for even : ( ) ik 2 π exp =

14 14 Remark 5.3 R 2 is circulant, unitary, but not Haamar. Thus it oes not help to fin more than 3 MUB s in even imensions. In imensions = 2 n, another metho is necessary to prove that there exists +1 MUB s. Acknowlegments : It is a pleasure to thank B. Helffer for his interest in this work, an to Bahman Saffari for interesting iscussions an for proviing to me reference [12]. References [1] Albouy O., Kibler M., SU 2 nonstanar bases : the case of mutually unbiase bases, Symmetry, Integrability an Geometry : Methos an Applications, (2007) [2] Banyopahyay S., Boykin P.O., Roychowhury V., Vatan F., A new proof of the existence of mutually unbiase bases, Algorithmica, 34, , (2002) [3] Bernt B. C., Evans R. J., Williams K. S., Gauss an Jacobi Sums, Canaian Mathematical Society Series of Monographs an Avance Texts, Vol 21, Wiley, (1998) [4] Björck G. Saffari B, New classes of finite unimoular sequences with unimoular Fourier transforms. Circulant Haamar matrices with complex entries, C. R. Aca. Sci. Paris, 320 Serie 1, (1995), [5] Combescure M. The Mutually Unbiase Bases Revisite, Contemporary Mathematics, (2007), to appear [6] Davis P. J., Circulant matrices, Wiley, (1979) [7] Haamar J., Résolution une question relative aux éterminants, Bull. Sci. Math. 17, (1893) [8] Klimov A. B., Muñoz C., Romero J. L., Geometrical approach to the iscrete Wigner function, arxiv:quant-ph/ , (2006)

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