Acta Polytechnica doi:10.14311/AP.2014.54.0124 Acta Polytechnica 54(2):124–126, 2014 © Czech Technical University in Prague, 2014 available online at http://ojs.cvut.cz/ojs/index.php/ap ON TWO WAYS TO LOOK FOR MUTUALLY UNBIASED BASES Maurice R. Kibler Université de Lyon, Université Claude Bernard Lyon 1 et CNRS/IN2P3, Institut de Physique Nucléaire, 4 rue Enrico Fermi, 69622 Villeurbanne, France correspondence: kibler@ipnl.in2p3.fr Abstract. Two equivalent ways of looking for mutually unbiased bases are discussed in this note. The passage from the search for d + 1 mutually unbiased bases in Cd to the search for d(d + 1) vectors in Cd 2 satisfying constraint relations is clarified. Symmetric informationally complete positive-operator-valued measures are briefly discussed in a similar vein. Keywords: finite-dimensional quantum mechanics, quantum information, MUBs, SIC POVMs, equiangular lines, equiangular vectors. 1. Introduction The concept of mutually unbiased bases (MUBs) plays an important role in finite-dimensional quantum me- chanics and quantum information (for more details, see [1–4] and references therein). Let us recall that two orthonormal bases {|aα〉 : α = 0, 1, . . . ,d − 1} and {|bβ〉 : β = 0, 1, . . . ,d− 1} in the d-dimensional Hilbert space Cd (endowed with an inner product de- noted as 〈 | 〉) are said to be unbiased if the modulus of the inner product 〈aα|bβ〉 of any vector |bβ〉 with any vector |aα〉 is equal to 1/ √ d. It is known that the maximum number of MUBs in Cd is d + 1 and that this number is reached when d is a power of a prime integer. In the case where d is not a prime integer, it is not known if one can construct d + 1 MUBs (see [4] for a review). In a recent paper [5], it was discussed how the search for d + 1 mutually unbiased bases in Cd can be approached via the search for d(d + 1) vectors in Cd 2 satisfying constraint relations. The main aim of this note is to make the results in [5] more precise and to show that the two approaches (looking for d + 1 MUBs in Cd or for d(d+ 1) vectors in Cd 2 ) are entirely equivalent. The central results are presented in Sec- tions 2 and 3. In Section 4, parallel developments for the search for a symmetric informationally complete positive-operator-valued measure (SIC POVM) are considered in the framework of similar approaches. Some concluding remarks are given in the last section. 2. The two approaches It was shown in [5] how the problem of finding d + 1 MUBs in Cd, i.e., d + 1 bases Ba = {|aα〉 : α = 0, 1, . . . ,d− 1} (1) satisfying |〈aα|bβ〉| = δα,βδa,b + 1 √ d (1 − δa,b) (2) can be transformed into the problem of finding d(d+1) vectors w(aα) in Cd 2 , of components wpq(aα), satis- fying wpq(aα) = wqp(aα), p,q ∈ Z/dZ (3) d−1∑ p=0 wpp(aα) = 1 (4) and d−1∑ p=0 d−1∑ q=0 wpq(aα)wpq(bβ) = δα,βδa,b + 1 d (1−δa,b) (5) with a,b = 0, 1, . . . ,d and α,β = 0, 1, . . . ,d − 1 in (1)–(5). (In this paper, the bar denotes complex con- jugation.) This result was described by Proposition 1 in [5]. In fact, the equivalence of the two approaches (in Cd and Cd 2 ) requires that each component wpq(aα) be factorized as wpq(aα) = ωp(aα)ωq(aα) (6) for a = 0, 1, . . . ,d and α = 0, 1, . . . ,d− 1, a condition satisfied by the example given in [5]. The factorization of wpq(aα) follows from the fact that the operator Πaα defined in [5] is a projection operator. The introduction of (6) in (3), (4) and (5) leads to some simplifications. First, (6) implies the hermiticity condition (3). Second, by introducing (6) into (4) and (5), we obtain d−1∑ p=0 |ωp(aα)|2 = 1 (7) and∣∣∣∣∣ d−1∑ p=0 ωp(aα)ωp(bβ) ∣∣∣∣∣ 2 = δα,βδa,b + 1 d (1 − δa,b) (8) respectively. It is clear that (7) follows from (8) with a = b and α = β. Therefore, (3) and (7) are redundant 124 http://dx.doi.org/10.14311/AP.2014.54.0124 http://ojs.cvut.cz/ojs/index.php/ap vol. 54 no. 2/2014 On Two Ways to Look for Mutually Unbiased Bases in view of (5) and (6). As a consequence, Proposition 1 in [5] can be precised and reformulated in the following way. Proposition 1. For d ≥ 2, finding d + 1 MUBs in Cd (if they exist) is equivalent to finding d(d + 1) vectors w(aα) in Cd 2 , of components wpq(aα) such that d−1∑ p=0 d−1∑ q=0 wpq(aα)wpq(bβ) = δα,βδa,b + 1 d (1−δa,b) (9) and wpq(aα) = ωp(aα)ωq(aα), p,q ∈ Z/dZ (10) where a,b = 0, 1, . . . ,d and α,β = 0, 1, . . . ,d− 1. This result can be transcribed in matrix form. Therefore, we have the following proposition. Proposition 2. For d ≥ 2, finding d + 1 MUBs in Cd (if they exist) is equivalent to finding d(d + 1) matrices Maα of dimension d, with elements (Maα)pq = ωp(aα)ωq(aα), p,q ∈ Z/dZ (11) and satisfying the trace relations Tr (MaαMbβ) = δα,βδa,b + 1 d (1 − δa,b) (12) where a,b = 0, 1, . . . ,d and α,β = 0, 1, . . . ,d− 1. 3. Equivalence Suppose that we have a complete set {Ba : a = 0, 1, . . . ,d} of d + 1 MUBs in Cd, i.e., d(d + 1) vectors |aα〉 satisfying (2), then we can find d(d + 1) vectors w(aα) in Cd 2 , of components wpq(aα), satisfying (9) and (10). This can be achieved by introducing the projection operators Πaα = |aα〉〈aα| (13) where a = 0, 1, . . . ,d and α = 0, 1, . . . ,d − 1. In fact, it is sufficient to develop Πaα in terms of the Epq generators of the GL(d,C) complex Lie group; the coefficients of the development are nothing but the wpq(aα) complex numbers satisfying (9) and (10), see [5] for more details. Reciprocally, should we find d(d + 1) vectors w(aα) in Cd 2 , of components wpq(aα), satisfying (9) and (10), then we could construct d(d+1) vectors |aα〉 satisfying (2). This can be done by means of a diagonalization procedure of the matrices Maα = d−1∑ p=0 d−1∑ q=0 wpq(aα)Epq (14) where a = 0, 1, . . . ,d and α = 0, 1, . . . ,d − 1. An alternative and more simple way to obtain the |aα〉 vectors from the w(aα) vectors is as follows. Equation (8) leads to∣∣∣∣∣ d−1∑ p=0 ωp(aα)ωp(bβ) ∣∣∣∣∣ = δα,βδa,b + 1√d(1 − δa,b) (15) to be compared with (2). Then, the |aα〉 vectors can be constructed once the w(aα) vectors are known. The solution, in matrix form, is |aα〉 =   ω0(aα) ω1(aα) ... ωd−1(aα)   (16) a = 0, 1, . . . ,d α = 0, 1, . . . ,d− 1 (17) Therefore, we can construct a complete set {Ba : a = 0, 1, . . . ,d} of d + 1 MUBs from the knowledge of d(d + 1) vectors w(aα). Note that, for fixed a and α, the |aα〉 vector is an eigenvector of the Maα matrix with eigenvalue 1. This establishes a link with the above-mentioned diagonalization procedure. 4. A parallel problem The present work takes its origin in [6], where some similar developments were achieved in the search of a SIC POVM. Symmetric informationally complete positive-operator-valued measures play an important role in quantum information. Their existence in arbi- trary dimension is still the object of numerous studies (see for instance [7]). A SIC POVM in dimension d can be defined as a set of d2 nonnegative operators Px = |Φx〉〈Φx| acting on Cd and satisfying 1 d d2∑ x=1 Px = I (18) and Tr (PxPy) = dδx,y + 1 d + 1 (19) where I is the identity operator. The search for such a SIC POVM amounts to find d2 vectors |Φx〉 in Cd satisfying 1 d d2∑ x=1 |Φx〉〈Φx| = I (20) and |〈Φx|Φy〉| = √ dδx,y + 1 d + 1 (21) with x,y = 1, 2, . . . ,d2. The Px operator can be developed as Px = d−1∑ p=0 d−1∑ q=0 vpq(x)Epq (22) so that the determination of d2 operators Px (or d2 vectors |Φx〉) is equivalent to the determination of d2 vectors v(x), of components vpq(x), in Cd 2 . In the spirit of the preceding sections, we have the following result. 125 Maurice R. Kibler Acta Polytechnica Proposition 3. For d ≥ 2, finding a SIC POVM in Cd (if it exists) is equivalent to finding d2 vectors v(x) in Cd 2 , of components vpq(x) such that 1 d d2∑ x=1 vpq(x) = δp,q, p,q ∈ Z/dZ (23) d−1∑ p=0 d−1∑ q=0 vpq(x)vpq(y) = dδx,y + 1 d + 1 (24) and vpq(x) = νp(x)νq(x), p,q ∈ Z/dZ (25) where x,y = 1, 2, . . . ,d2. 5. Concluding remarks The equivalence discussed in this work of the two ways of looking at MUBs amounts in some sense to the equivalence between the search for equiangular lines in Cd and for equiangular vectors in Cd 2 (cf. [8]). Equiangular lines in Cd correspond to |〈aα|bβ〉| = 1 √ d for a 6= b (26) while equiangular vectors in Cd 2 correspond to w(aα) · w(bβ) = 1 d for a 6= b (27) where the w(aα)·w(bβ) inner product in Cd 2 is defined as w(aα) · w(bβ) = d−1∑ p=0 d−1∑ q=0 wpq(aα)wpq(bβ) (28) Observe that the modulus disappears and the 1/ √ d factor is replaced by 1/d when passing from (26) to (27). It was questioned in [5] if the equiangular vectors approach can shed light on the still unsolved question whether one can find d + 1 MUBs when d is not a (strictly positive) power of a prime integer. In the case where d is not a power of a prime, the impossibility of finding d(d+1) vectors w(aα) or d(d+1) matrices Maα satisfying the conditions in Propositions 1 and 2 would mean that d + 1 MUBs do not exist in Cd. However, it is hard to know if one approach is better than the other. It is the hope of the author that the equiangular vectors approach can be tested in the d = 6 case for which one knows only three MUBs instead of d+ 1 = 7 in spite of numerous numerical studies (see [9–11] and references therein for an extensive list of related works). Similar remarks apply to SIC POVMs. The ex- istence problem of SIC POVMs in arbitrary dimen- sion is still unsolved although SIC POVMs have been constructed in every dimension d ≤ 67 (see [7] and references therein). For SIC POVMs, the equiangular lines in Cd correspond to |〈Φx|Φy〉| = 1 √ d + 1 for x 6= y (29) and the equiangular vectors in Cd 2 to v(x) · v(y) = 1 d + 1 for x 6= y (30) where the v(x) · v(y) inner product in Cd 2 is defined as v(x) · v(y) = d−1∑ p=0 d−1∑ q=0 vpq(x)vpq(y) (31) The parallel between MUBs and SIC POVM charac- terized by the couples of equations (26)-(29), (27)-(30) and (28)-(31) should be noted. These matters will be the subject of future work. Acknowledgements The material contained in the present note was planned to be presented at the eleventh edition of the workshop Analytic and Algebraic Methods in Physics (AAMP XI). Unfortunately, the author was unable to participate in AAMP XI. He is greatly indebted to Miloslav Znojil for suggesting that he submits this work to Acta Polytechnica. References [1] A. Vourdas. Quantum systems with finite Hilbert space. Rep Prog Phys 67(3):267–320, 2004. doi: 10.1088/0034-4885/67/3/R03 [2] J. Tolar, G. Chadzitaskos. 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Int J Quantum Inf 10(5):1–11, 2012. doi: 10.1142/S0219749912500566 126 http://dx.doi.org/10.1088/0034-4885/67/3/R03 http://dx.doi.org/10.1088/1751-8113/42/24/245306 http://dx.doi.org/10.1088/1751-8113/42/35/353001 http://dx.doi.org/10.1142/S0219749910006502 http://dx.doi.org/10.3390/e15051726 http://dx.doi.org/10.1007/s10946-007-0032-5 http://arxiv.org/abs/1312.0555v1 http://dx.doi.org/10.1016/j.ejc.2008.01.002 http://dx.doi.org/10.1016/j.physleta.2007.04.059 http://dx.doi.org/10.1103/PhysRevA.79.052316 http://dx.doi.org/10.1142/S0219749912500566 Acta Polytechnica 54(2):124–126, 2014 1 Introduction 2 The two approaches 3 Equivalence 4 A parallel problem 5 Concluding remarks Acknowledgements References