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New Bounds for Mutually Unbiased Maximally Entangled Bases in C D

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New Bounds for Mutually Unbiased Maximally Entangled Bases in C D New bounds for Mutually unbiased maximally entangled bases in Cd ⊗ Ckd XiaoyaCheng1 ∗ YunShang1 2 y 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, Pauli matrices, 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 = fB1; B2;:::; Bmg B1 = fjφiigi=1 and B2 = fj jigj=1 of C are called d d0 be a set of orthonormal MEBs in C ⊗ C . We call A mutually unbiased if jhφ j ij = p1 ; (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 fB1; B2;:::; Bmg 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 quantum error correction 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 prime number, 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 Hilbert space 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) ≥ minfk; M(d; d)g. 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) = p ; (1) tions, such as entanglement swapping, state tomography, r2R k ∗[email protected] for all ξ; η 2 R and j; l = 1; : : : ; k; where U yV = [email protected] (w(r;j);(s;l)), (r; j); (s; l) 2 R × f 1; : : : ; k g. 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 2 R , ΨU(a) and ΨU(b) are mutually powers. Therefore we obtain the MUMEBs in d ⊗ kd ∗ C C unbiased, provided that a − b 2 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 2 R , ΨV (a) and ΨV (b) are mutually t t t ∗ for j 2 0 , define unbiased, provided that a − b 2 R . t Fqt 2 ! ∗ T = (jtm +mn) (t) 1 Fq0 Fp0 (3) For any a 2 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 2 Fq0 t p 0 ∗ X 2 ∗ if pt = 2, define 2 2 R and λ satisfies λ(cr ) = d 8c 2 R : r2R (t) 1 (jt+2n)m B = p ζ 8jt 2 Ta0 ; (2) jt a0 4 t 2 t (m;n)2T 2 a0 ∗ t (4) For any a; b 2 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 2 U 0 ( ) and the absolute value of each entry in jt qt C Then we give our main results. y (t) (t) p 0 B B equals to 1= q for any two distinct j ; i 2 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 2 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 y Bj 2 Uk(C) and the absolute value of each entry in Bj Bi where p1; p2; : : : ; ps are odd distinct primes), we obtain p 0 a1 equals to 1= k for any two distinct j; i 2 f 0; 1; : : : ; q g. 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) ≥ minf(p1) + 1;M(d; d)g a1 2(p1 − 1). 0 0 a1 a1 a1 as ≥ minf(p1) + 1; 2(p1 − 1)g: 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.
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