On Two Ways to Look for Mutually Unbiased Bases

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 C(d) to the search for d(d+1) vectors in C(d*d) satisfying constraint relations is clarified. Symmetric informationally complete positive-operator-valued measures are briefly discussed in a similar vein.


Introduction
The concept of mutually unbiased bases (MUBs) plays an important role in finite-dimensional quantum mechanics and quantum information (for more details, see [1][2][3][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 C d (endowed with an inner product denoted 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 C d 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 C d can be approached via the search for d(d + 1) vectors in C d 2 satisfying constraint relations. It is the main aim of this note to make the results in [5] more precise and to show that the two approaches (looking for d + 1 MUBs in C d or for d(d + 1) vectors in C d 2 ) are entirely equivalent. The central results are presented in Sections 2 and 3. In Section 4, parallel developments for the search of 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.
Reciprocally, should we find d(d + 1) vectors w(aα) in C d 2 , of components w pq (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 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 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 Therefore, we can construct a complete set {B a : 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 M aα matrix with the eigenvalue 1. This establishes a link with the above-mentioned diagonalization procedure.

A parallel problem
The present work takes its origin in [6] where some similar developments were achieved for the search of a SIC POVM. Symmetric informationally complete positive-operator-valued measures play an important role in quantum information. Their existence in arbitrary dimension is still the object of numerous studies (see for instance [7] where I is the identity operator. The search for such a SIC POVM amounts to find d 2 vectors |Φ x in C d satisfying and with x, y = 1, 2, . . . , d 2 .
The P x operator can be developed as so that the determination of d 2 operators P x (or d 2 vectors |Φ x ) is equivalent to the determination of d 2 vectors v(x), of components v pq (x), in C d 2 . In the spirit of the preceding sections, we have the following result.

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 C d and for equiangular vectors in C d 2 (cf. [8]). Equiangular lines in C d correspond to while equiangular vectors in C d 2 correspond to where the w(aα) · w(bβ) inner product in C d 2 is defined as 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 to know if 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 M aα satisfying the conditions in Propositions 1 and 2 would mean that d + 1 MUBs do not exist in C d . 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 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][10][11] and references therein for an extensive list of related works).
Similar remarks apply to SIC POVMs. The existence problem of SIC POVMs in arbitrary dimension 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 C d correspond to and the equiangular vectors in C d 2 to v(x) · v(y) = 1 d + 1 for x = y (30) where the v(x) · v(y) inner product in C d 2 is defined as The parallel between MUBs and SIC POVM characterized by the couples of equations (26)-(29), (27)-(30) and (28)-(31) should be noted. These matters shall be the subject of a future work.