New bounds for Mutually unbiased maximally entangled bases in Cd ⊗ Ckd XiaoyaCheng1 ∗ YunShang1 2 †

1 Institute of Mathematics, AMSS, CAS, Beijing 100190, P.R.China 2NCMIS, AMSS, CAS, Beijing 100190, P.R.China Abstract. Mutually unbiased bases which is also maximally entangled bases is called mutually unbiased maximally entangled bases (MUMEBs). We study the construction of MUMEBs in bipartite system. In detail, we construct 2(pa − 1) MUMEBs in Cd ⊗ Cd by properties of Gauss sums for arbitrary odd d. It improves the known lower bound pa − 1 for odd d. Furthermore, we construct MUMEBs in Cd ⊗ Ckd for general k ≥ 2 and odd d. We get the similar lower bounds as k, b are both single prime powers. Particularly, when k is a square number, by using mutually orthogonal Latin squares, we can construct more MUMEBs in Cd ⊗ Ckd. Keywords: mutually unbiased bases, maximally entangled states, , mutually orthogonal Latin squares

1 Introduction etc., all of which depend on an explicit set of maximal- 0 ly entangled states. A basis B of d ⊗ d is called a The notion of mutually unbiased bases was first intro- C C maximally entangled basis (MEB) if it consists of dd0 duced by Schwinger [1] in 1960. Two orthogonal bases d d d maximally entangled states. Let A = {B1, B2,..., Bm} B1 = {|φii}i=1 and B2 = {|ψji}j=1 of C are called d d0 be a set of orthonormal MEBs in C ⊗ C . We call A mutually unbiased if |hφ |ψ i| = √1 , (1 ≤ i, j ≤ d). I- i j d a set of mutually unbiased maximally entangled bases vanovic found the first application for mutually unbiased (MUMEBs) if every pair in {B1, B2,..., Bm} is mutually bases (MUBs) in the problem of quantum state determi- unbiased. Let M(d, d0) be the maximal cardinality of any nation. Since the outcome is random when a measure- d d0 set of MUMEBs in C ⊗ C . ment is made in a basis unbiased to that in which the Recently, many researches focused on the construction state was prepared. Mutually unbiased bases are used in of MUMEBs [17, 18, 19]. In [18], Tao et al. proved many protocols such as codes that M(2, 4) ≥ 5 and M(2, 6) ≥ 3. For d = d0 and [5], quantum state tomography [6, 7], quantum key dis- every integer d ≥ 2, in [17], Liu et al. had construct- tribution [9], cryptographic protocols [9, 10], mean king a1 ed p1 − 1 mutually unbiased maximally entangled bases problem [11], quantum teleportation and superdense cod- a1 (MUMEBs), where p1 is the minimal prime power di- ing [12, 13, 14]. viding d. Furthermore, in [20], Xu obtained that if d is How many mutually unbiased bases N(d) one can find an integer power of a , then it is possible for arbitrary d is still an open problem, where N(d) de- to find 2(d − 1) mutually unbiased maximally entangled notes the maximum number of mutually unbiased bases bases (MUMEBs). Besides, Xu constructed MUMEBs in in the d-dimensional d. In general, for d kd C C ⊗ C for k be another prime power and proved that a1 every integer d ≥ 2, it is proved that p1 + 1 ≤ N(d) ≤ M(d, kd) ≥ min{k, M(d, d)}. a1 as a1 as d + 1 for d = p1 . . . ps with p1 ≤ · · · ≤ ps . And However, up to now we know very little about its re- m N(d) = p + 1 = d + 1 when d is an integer power of search. The problem to find the lower bound on M(d, kd) a prime number [7]. There are many different methods for more general d and k (not simply prime powers) re- to construct MUBs. By Weil sums over finite fields and mains unsolved. Next, we will mainly study the problem exponential sums over Galois rings, Klappenecker et al. and obtain some results. [15] studied MUBs for odd prime power d = pa, p ≥ 3 and even prime power d = 2m respectively. In addition, 2 Preliminaries in 2004, Wocjan et al. [16] showed that for d = s2 the 1 In this paper, we will focus on the lower bounds of number of N(d) is greater than s 14.8 for all s but finitely many exceptions, and this bound is better than the pre- M(d, kd) for more general k and odd d, by constructing d kd vious one in many non-prime-power cases. But if d is a MUMEBs in bipartite system C ⊗ C . composite number, the value of N(d) is still unknown. Firstly, we provide a basic criterion and some facts: Furthermore, when one considers MUBs in tensor s- Proposition 1. [20] For U and V in Ukd(C), ΨU and d kd paces, the problem becomes more interesting and com- ΨV in C ⊗ C are mutually unbiased if and only if plicated. Durt [4] presented how to use the constructed 1 mutually unbiased bases in quantum-informatics applica- X λ(rξ)w(r,j),(r+η,l) = √ , (1) tions, such as entanglement swapping, state tomography, r∈R k ∗[email protected] for all ξ, η ∈ R and j, l = 1, . . . , k, where U †V = †[email protected] (w(r,j),(s,l)), (r, j), (s, l) ∈ R × { 1, . . . , k }. Fact 1. [20] We have the following statements: [20, Section 5], we construct MUMEBs in Cd ⊗ Ckd. But unlike [20], we do not need to restrict d and k to be prime ∗ (1) For any a, b ∈ R , ΨU(a) and ΨU(b) are mutually powers. Therefore we obtain the MUMEBs in d ⊗ kd ∗ C C unbiased, provided that a − b ∈ R . for general k and odd d. 0 0 0 a0 ∗ For each t = 1, . . . , l, if p is odd, let q = (p ) t and (2) For any a, b ∈ R , ΨV (a) and ΨV (b) are mutually t t t ∗ for j ∈ 0 , define unbiased, provided that a − b ∈ R . t Fqt 2 ! ∗ T / (jtm +mn) (t) 1 Fq0 Fp0 (3) For any a ∈ R , ΨId and ΨV (a) are mutually unbi- t t B = ζ 0 ; jt p p ased, provided that q0 t t (m,n)∈ 2 Fq0 t √ 0 ∗ X 2 ∗ if pt = 2, define 2 ∈ R and λ satisfies λ(cr ) = d ∀c ∈ R . r∈R   (t) 1 (jt+2n)m B = √ ζ ∀jt ∈ Ta0 , (2) jt a0 4 t 2 t (m,n)∈T 2 a0 ∗ t (4) For any a, b ∈ R , ΨU(a) and ΨV (b) are mutually a0 where T 0 is a set of 2 t element in the Galois ring unbiased, provided λ satisfies (2). at 0 GR(4, at) (see [20] for detailed definitions). By the prop- Where the constructions of ΨU and ΨV refer to Xu’s erties of Gauss sums and Galois rings, one can check that paper [20]. (t) B ∈ U 0 ( ) and the absolute value of each entry in jt qt C Then we give our main results. † (t) (t) p 0 B B equals to 1/ q for any two distinct j , i ∈ 0 . jt it t t t Fqt Fix an injection νt : q0 −→ q0 for each t > 1 and 3 Main results F 1 F t define d d 3.1 Construction of MUMEBs in C ⊗ C (1) (2) (l) 0 Bj = Bj ⊗ B ⊗ ... ⊗ B ∈ Uk(C), j = 1, . . . , q1 We restrict ourself to the case d ⊗ d, i.e., k = 1. ν2(j) νl(j) C C (3) By using properties of Gauss sums and generalising both We also write B = I. By the property of matrix tensor the methods presented in [17] and [20], for k = 1 and 0 product, then (A ⊗ B)(C ⊗ D) = AC ⊗ BD, one has a1 as a1 as d be odd (write d = p1 . . . ps with p1 ≤ · · · ≤ ps † Bj ∈ Uk(C) and the absolute value of each entry in Bj Bi where p1, p2, . . . , ps are odd distinct primes), we obtain √ 0 a1 equals to 1/ k for any two distinct j, i ∈ { 0, 1, . . . , q }. M(d, d) ≥ 2(p1 −1). The result proved that the bound of 1 Xu’s still holds for any odd d, i.e., The following theorem. Theorem 3. Let d be an odd number. Write d = a1 as a1 as a1 p1 . . . ps with p1 ≤ · · · ≤ ps . Suppose k ≥ 2 and write Theorem 2. Let d be an odd number. Write d = p ... 0 0 0 0 1 0 a1 0 al 0 a1 0 al as a1 as a1 k = (p1) ... (pl) with (p1) ≤ · · · ≤ (pl) . Then ps with p1 ≤ · · · ≤ ps . Then M(d, d) ≥ 2(p1 − 1). d d 0 That is, there exists a set of MUMEBs in ⊗ of size 0 a1 C C M(d, kd) ≥ min{(p1) + 1,M(d, d)} a1 2(p1 − 1). 0 0 a1 a1 a1 as ≥ min{(p1) + 1, 2(p1 − 1)}. Proof. If d = p1 . . . ps is odd, let R and λ be in line with Liu’s constructions [17]. Since qt −1 ≥ q1 −1 for all t > 1, where q = pat , we can fix an injection ι : ∗ −→ ∗ t t t Fq1 Fqt Proof. Let d be an odd number. Without loss of general- 0 a0 for each t > 1. Define ity, suppose that (p ) 1 +1 ≤ M(d, d), since otherwise we 1 0 0 a1 0 ∗ ∗ can prove the result similarly. Let n = (p1) = q1 and S = { (u, ι2(u), . . . , ιs(u)) ∈ R | u ∈ } . Fq1 {Bj|j = 0, 1, . . . , n} be defined as before. Since n + 1 ≤ M(d, d), by Theorem 2 there exist U ,U ,...,U dis- Clearly, S is a subset of R∗ such that a − b ∈ R∗ 0 1 n tinct matrices in U ( ) such that Ψ , Ψ ,..., Ψ are for all a 6= b ∈ S. It follows from Fact 1 (1) that d C U0 U1 Un MUMEBs. For any 0 ≤ t ≤ n, C = B ⊗ U is a unitary { Ψ | a ∈ S } is a set of MUMEBs and from Fact 1 (2) t t t U(a) matrix. The following prove that these MEBs {Ψ }n that { Ψ | a ∈ S } is also a set of MUMEBs. More- Ci i=1 V (a) are mutually unbiased i.e., for any 0 ≤ t < t0 ≤ n, the over, by using properties of Gauss sums we can prove the † 0 matrix Ct Ct0 satisfies (1). Let ξ, η ∈ Fd , 0 ≤ t < t ≤ n assumption (2) holds for the choice of R and λ. It fol- † † and Bt Bt0 = (bi,j), Ut Ut0 = (ui,j). Then lows from Fact 1 (4) that ΨU(a) and ΨV (a) are mutually unbiased for any a ∈ R∗. X † X 0 In summary, { ΨU(a) | a ∈ S } ∪ { ΨV (a) | a ∈ S } is a λ(rξ)(Ct Ct )(r,i),(r+η,j) = λ(rξ)bi,jur,r+η set of MUMEBs. In particular, the size of this set is r∈R r∈R a1 2 | S |= 2(p1 − 1). X Thus the proof is compete. = bi,j λ(rξ)ur,r+η r∈R d kd 3.2 Construction of MUMEBs in C ⊗ C for X general k ≥ 2 = |bi,j| λ(rξ)ur,r+η

Let d be odd and k ≥ 2, then we have the decom- r∈R 0 0 0 0 0 a1 0 al 0 a1 0 al 1 position k = (p1) ... (pl) with (p1) ≤ · · · ≤ (pl) , = √ , 0 where each pt, t = 1, 2 . . . , l is distinct prime. In line with k d kd 0 2e1 0 2el 3.3 Construction of MUMEBs in C ⊗ C with (1) k ≥ 2 is square number, i.e., k = (p1) ... (pl) = 2 0 2e1 0 2el k being a square number x with (p1) ≤ · · · ≤ (pl) . In the previous subsections, we obtain a bound for (a) x ≡ 2 (mod 4). Then the minimal prime pow- M(d, kd) for general k. Now we consider it for some spe- er dividing x is 2. Thus Theorem 3 gives cial k. Since Latin square is also a useful tool in charac- M(d, kd) ≥ min{5,M(d, d)}. By Beth’s re- terizing MUB problem (c.f. [21]), we continue to consider sult [22] we know that NMOLS(x) ≥ 6 for whether mutually orthogonal Latin square (MOLS)(c.f. x ≥ 76. Therefore, Theorem 6 gives M(d, kd) ≥ [16]) is helpful to improve the value of M(d, kd). By using min{8,M(d, d)} for k = x2, x ≥ 76. results on MOLSs in [16], It turns out that if k is a square 2 2 2 0 a0 number, the bound for M(d, kd) can be improved. Let (b) k = 26 √= 2 × 13 . We have (p1) 1 + 1 = 5 and N ( k = 26)+2 ≥ 6 where N (26) ≥ 4 NMOLS(x) denote the maximum cardinality of any set MOLS MOLS of mutually orthogonal Latin squares (MOLS) of order x (c.f. [16]). It follows from Theorem 3 that (see [16]). M(d, kd) ≥ min{5,M(d, d)} , and by Theorem 6, then M(d, kd) ≥ min{6,M(d, d)}. Hence Lemma 4. [See [16]] The existence of w MOLS is equiv- Theorem 6 is better. alent to the existence of a (n, x)-net with n = w + 2. √ 1 (c) l ≥ 35. Then we have NMOLS( k)+2 ≥ k 29.6 ≥ 0 0 2el 0 a Lemma 5. [See [16]] Let {m11, . . . , m1x, m21, . . . , m2x, (p1) + 1 = (p1) 1 + 1. . . . , mn1, . . . , mnx} be a (n, x)−net and H an arbitrary 2 a1 an generalized Hadamard matrix of size x (all its entries (2) k is not square, but k = 26 p1 . . . pn where pi ≥ 3. 2 ∗ For instance, k = 26 × 5. have modulus one and HH = xIx ). Then the n set- s for b = 1, . . . , n In the previous remark, we exhibit some examples to 1 compare the two lower bounds given by Theorems 3 and L := {√ (h ↑ m ) | l = 1, . . . , x, i = 1, . . . , x} b x l bi 6. In all cases, these two theorems together give the combined lower bound: are n MUBs for the Hilbert space Ck. a1 Corollary 7. Let d be an odd number. Write d = p1 ... a1 Theorem 6. Let d be an odd number. Write d = p ... as a1 as 0 2e1 1 ps with p1 ≤ · · · ≤ ps . Suppose that k = (p1) ... as a1 as 2 p with p ≤ · · · ≤ p . Suppose k = x is a square 0 2el 0 2e1 0 2el s 1 s (pl) is a square number with (p1) ≤ · · · ≤ (pl) . number. Then Then √ √ 0 M(d, kd) ≥ min{NMOLS( k) + 2,M(d, d)}, (4) 0 a1 M(d, kd) ≥ min{max{NMOLS( k)+2, (p ) +1},M(d, d)}. √ √ 1 1/29.6 where NMOLS( k) + 2 ≥ k for all k but finitely many exceptions. Proof. Using the same discussion as in the proof Remark 3. If d = 2m, our bounds of M(d, kd) still hold. of Theorem 3 and replacing B ,...,B 0 by L ,L ,..., 0 q1 1 2 The proof is same as the argument except that we shall m LNMOLS(x)+2, we obtain the bound (4). For the lower use the 2(2 − 1) MUMEBs constructed in [20]. bound on NMOLS, we refer to [16]. Now, we compare the bounds obtained by Latin square 4 Conclusion method with reducing prime power method, we obtain many interesting results. In some cases, we find by Latin In this paper, we study the constructions of MUMEBs d kd square, we can reach greater bounds than reducing into in bipartite system C ⊗ C for general k and odd d. prime power problem. First, by using properties of Gauss sums, we construct 2(pa − 1) MUMEBs in Cd ⊗ Cd for arbitrary odd d. It Remark 2. In the following cases, Theorem 3 is better improves the known lower bound pa − 1 for odd d and than Theorem 6: it also generalizes the lower bound 2(pa − 1) for d being a single prime power. Then, we construct MUMEBs in (1) Obviously, k is not square, but is an odd number or d ⊗ kd for general k ≥ 2 and odd d. We get the similar a prime power. C C lower bounds as k, b are both single prime powers. At (2) k = p2e, where p is an arbitrary prime and e ≥ 1. last, when k is a square number, by using mutually or- Theorem 6 gives M(d, kd) ≥ min{pe + 1,M(d, d)}, thogonal Latin squares, we can construct more MUMEBs but Theorem 3 gives M(d, kd) ≥ min{p2e + in Cd ⊗ Ckd, and obtain greater lower bounds than re- 1,M(d, d)}. ducing the problem into prime power dimension in some cases. Certainly, the above bounds of M(d, kd) still hold 2 4 (3) x = 76,√ then N(k = x ) ≥ 2 + 1 = 17 and for d = 2m. In the future work, we will consider the NMOLS( k = 76) ≥ 6 by [16]. Theorem 6 gives construction problem of MUMEBs in bipartite system M(d, kd) ≥ min{8,M(d, d)}, but Theorem 3 gives d kd C ⊗ C for general d. M(d, kd) ≥ min{17,M(d, d)}. However, in some cases, Theorem 6 is better than The- orem 3: References [17] J. Liu, M. Yang, and K. Feng. Mutually unbiased maximally entangled bases in d ⊗ d. Quant. Inf. [1] J. Schwinger. Unitary operator bases. Proc.Nat. A- C C Proc., 16(6):159, 2017. cad. U.S.A., 46:570–579, 1960. [18] Y. Tao, H. Nan, J. Zhang, and S. Fei. Mutually un- [2] A. Vourdas. Quantum systems with finite hilbert s- biased maximally entangled bases in d× kd. Quant. pace. Rep.Prog. Phys., 67(3):267, 2004. C C Inf. Proc., 14(6):2291–2300, 2015. [3] G. Bj¨ork,A. B. Klimov, and L. L. Sanchez-Soto. The [19] J. Zhang, Y. Tao, H. Nan, S. Fei. 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