Let q=2^r, where r is a positive integer. We show that there is a q^2 x q^2 real orthogonal matrix D, say, of multiplicative order q+1 whose powers generate q+1 real pairwise mutually unbiased bases in R^{q^2}. This means that the scaled powers qD, qD^2, ..., qD^q are q different real Hadamard matrices. We also show that each diagonal entry of these q Hadamard matrices equals -1.
Let q=2^r, where r is a positive integer. We show that there is a q^2 x q^2 real orthogonal matrix D, say, of multiplicative order q+1 whose powers generate q+1 real pairwise mutually unbiased bases in R^{q^2}. This means that the scaled powers qD, qD^2, ..., qD^q are q different real Hadamard matrices. We also show that each diagonal entry of these q Hadamard matrices equals -1.
We prove the following two results relating real mutually unbiased bases and representations of finite groups of odd order. Let q be a power of 2 and r a positive integer. Then we can find a q2r×q2r real orthogonal matrix D, say, of multiplicative order q2r−1+1, whose q2r−1+1 powers D, \dots, Dq2r−1+1=I define q2r−1+1 mutually unbiased bases in Rq2r. Thus the scaled matrices qrD, \dots, qrDq2r−1 are q2r−1 different Hadamard matrices. When we take q=2, we achieve the maximum number of real mutually unbiased bases in dimension 22r using the elements of a cyclic group. We also prove the following. Let G be an arbitrary finite group of odd order 2k+1, where k≥3. Then G has a real representation R, say, of degree 22k−1 such that the elements R(σ), σ∈G, define ∣G∣ mutually unbiased bases in Rd, where d=22k−1. In addition, a group of order 5 defines five real mutually unbiased bases in R16 and a group of order 3 defines three real mutually unbiased bases in R4. Thus, an arbitrary group of odd order has a faithful representation by real scaled Hadamard matrices of 2-power size.
Let q be a power of 2. We show by representation theory that there exists a q x q unitary matrix of multiplicative order q+1 whose powers generate q+1 pairwise mutually unbiased base in C^q. When q is a power of an odd prime, there is a q x q unitary matrix of multiplicative order q+1 whose first (q+1)/2 powers generate (q+1)/2 pairwise mutually unbiased bases. We also show how the existence of these matrices implies the existence of a special type of orthogonal decomposition with respect to the Killing form of the special linear and symplectic Lie algebras.
In this note I point out that (1) pairs of real mutually unbiased bases (i.e., orthonormal bases of R^d) can only exist in dimensions 2 or d where d is a multiple of 4 and that (2) triples of real mutually unbiased bases can only exist for dimensions d that are also squares. For the case d=2^{2m} the maximal number of real MUBs is given by (d+2)/2 (this follows from known results on extremal euclidean line-sets). In the case d=s^2 where s is an even number that is not a power of 2, we have the lower bound N_MOLS(s)+2, where N_MOLS(s) denotes the maximal number of mutually orthogonal Latin square of order s, provided that there exists a Hadamard matrix of size s. It is not known how good this bound is. Moreover, I observe that the question of deciding if there always two real MUBs is equivalent to the Hadamard conjecture.
Mutually unbiased bases (MUBs) describe quantum measurements for which certainty in one basis gives uniform outcomes in the others. Their maximum number remains unknown in dimensions that are not powers of a prime. We introduce an ansatz that, in many dimensions, allows us to construct more MUBs than have previously been found. Each basis in the ansatz consists of a diagonal phase matrix applied to a fixed unitary, which is a tensor product of Fourier matrices and, optionally, any real Hadamard matrix. This construction yields five MUBs in dimension 12, six in dimensions 48, 96, and 192, seven in dimensions 36 and 108, and ten in dimension 648. These bases can then be extended to exceed the tensor product bound, asymptotically, in every eighteenth dimension. Moreover, we show that Paley's real Hadamard matrix can be used to construct q+1 bases in dimension d=q(q+1) for every prime power q≡3(mod4). This count grows as d and cannot be exceeded by tensor products of smaller sets.
On Construction of Approximate Real Mutually Unbiased Bases for an infinite class of dimensions d≡0mod4
It is known that real Mutually Unbiased Bases (MUBs) do not exist for any dimension d>2 which is not divisible by 4. Thus, the next combinatorial question is how one can construct Approximate Real MUBs (ARMUBs) in this direction with encouraging parameters. In this paper, for the first time, we show that it is possible to construct >⌈d⌉ many ARMUBs for certain odd dimensions d of the form d=(4n−t)s, t=1,2,3, where n is a natural number and s is an odd prime power. Our method exploits any available 4n×4n real Hadamard matrix H4n (conjectured to be true) and uses this to construct an orthogonal matrix Y4n−t of size (4n−t)×(4n−t), such that the absolute value of each entry varies a little from 4n−t1. In our construction, the absolute value of the inner product between any pair of basis vectors from two different ARMUBs will be ≤d1(1+O(d−41))<2, for proper choices of parameters, the class of dimensions d being infinitely large.
We investigate the number of real entries of an n×n complex Hadamard matrix. We analytically derive the numbers when n = 2, 3, 4, 6. In particular, the number can be any one of 0–22, 24, 25, 26, 30 for n = 6. We apply our result to the existence of four mutually unbiased bases in dimension six, which is a long-standing open problem in quantum physics and information. We show that if four MUBs containing the identity matrix exist then the real entries in any one of the remaining three matrices do not exceed 22.