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 $> \lceil \sqrt{d} \rceil$ 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 $4n \times 4n$ real Hadamard matrix $H_{4n}$ (conjectured to be true) and uses this to construct an orthogonal matrix ${Y}_{4n-t}$ of size $(4n - t) \times (4n - t)$, such that the absolute value of each entry varies a little from $\frac{1}{\sqrt{4n-t}}$. In our construction, the absolute value of the inner product between any pair of basis vectors from two different ARMUBs will be $\leq \frac{1}{\sqrt{d}}(1 + O(d^{-\frac{1}{4}})) < 2$, for proper choices of parameters, the class of dimensions $d$ being infinitely large.
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.
Construction of a large class of Mutually Unbiased Bases (MUBs) for non-prime power composite dimensions (d=k×s) is a long standing open problem, which leads to different construction methods for the class Approximate MUBs (AMUBs) by relaxing the criterion that the absolute value of the dot product between two vectors chosen from different bases should be ≤dβ. In this chapter, we consider a more general class of AMUBs (ARMUBs, considering the real ones too), compared to our earlier work in [Cryptography and Communications, 14(3): 527--549, 2022]. We note that the quality of AMUBs (ARMUBs) constructed using RBD(X,A) with ∣X∣=d, critically depends on the parameters, ∣s−k∣, μ (maximum number of elements common between any pair of blocks), and the set of block sizes. We present the construction of O(d) many β-AMUBs for composite d when ∣s−k∣<d, using RBDs having block sizes approximately d, such that ∣⟨ψil∣ψjm⟩∣≤dβ where β=1+2d∣s−k∣+O(d−1)≤2. Moreover, if real Hadamard matrix of order k or s exists, then one can construct at least N(k)+1 (or N(s)+1) many β-ARMUBs for dimension d, with β≤2−2d∣s−k∣+O(d−1)<2, where N(w) is the number of MOLS(w). This improves and generalizes some of our previous results for ARMUBs from two points, viz., the real cases are now extended to complex ones too. The earlier efforts use some existing RBDs, whereas here we consider new instances of RBDs that provide better results. Similar to the earlier cases, the AMUBs (ARMUBs) constructed using RBDs are in general very sparse, where the sparsity (ε) is 1−O(d−21).
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.
Mutually Unbiased bases has various application in quantum information procession and coding theory. There can be maximum d + 1 MUBs in C^d and d/2 +1 MUBs in R^d. But , over R^d MUBs are known to be non existent when d is odd and for most of the other even d there are mostly 3 Real MUBs. In case of C^d the construction for complete set of MUBs are known for only Prime Power dimension. Thus in general large set of MUBs are not known, particularly for composite dimensions which are not of the form of prime powers. Because of this, there are many constructions of Approximate version of MUBs. In this paper we make an attempt to define certain measures to characterise the AMUBs. Our construction of measures derives its inspiration from the applications of MUBs, and based on them, we define certain quantifiable measures, which are can be computed and gives estimates of how close the Approximate MUBs are to the MUBs. We use geometric interpretation, projective design features of MUBs and applications like Optimal State determination and Entropic Uncertainty of MUBs. We show generic relationship between these measures and show that it can be evaluated for APMUBs without known the exact construction details, thereby showing that definition of APMUB is sufficient completely characterise it. We also evaluate these measure for an interesting class of AMUBs called Weak MUBs and certain AMUBs constructed using RBDs.
An infinite family of strongly unextendible mutually unbiased bases in C22h
A set of b mutually unbiased bases (MUBs) in Cd (for d>1) comprises bd vectors in Cd, partitioned into b orthogonal bases for Cd such that the pairwise angle between all vectors from distinct bases is arccos(1/d). The largest number μ(d) of MUBs that can exist in Cd is at most d+1, but constructions attaining this bound are known only when d is a prime power. A set of b MUBs in Cd that cannot be enlarged, even by the first vector of a potential (b+1)-th MUB, is called strongly unextendible. Until now, only one infinite family of dimensions d containing b(d) strongly unextendible MUBs in Cd satisfying b(d)<μ(d) was known, this family, due to Szántó, is asymptotically "large" in the sense that b(d)/μ(d)→1 as d→∞. However, the existence of 2m−1+1 strongly unextendible MUBs in C2m for each integer m>1 has been conjectured by Mandayam et al. We prove their conjecture for all even values of m, using only elementary linear algebra. The existence of this "small" new infinite family suggests, contrary to widespread belief, that μ(d) for non-prime-powers d might be significantly larger than the size of particular unextendible sets.
A set of k orthonormal bases of Cd is called mutually unbiased if ∣⟨e,f⟩∣2=1/d whenever e and f are basis vectors in distinct bases. A natural question is for which pairs (d,k) there exist~k mutually unbiased bases in dimension d. The (well-known) upper bound k≤d+1 is attained when~d is a power of a prime. For all other dimensions it is an open problem whether the bound can be attained. Navascués, Pironio, and Acín showed how to reformulate the existence question in terms of the existence of a certain C∗-algebra. This naturally leads to a noncommutative polynomial optimization problem and an associated hierarchy of semidefinite programs. The problem has a symmetry coming from the wreath product of Sd and Sk. We exploit this symmetry (analytically) to reduce the size of the semidefinite programs making them (numerically) tractable. A key step is a novel explicit decomposition of the Sd≀Sk-module C([d]×[k])t into irreducible modules. We present numerical results for small d,k and low levels of the hierarchy. In particular, we obtain sum-of-squares proofs for the (well-known) fact that there do not exist d+2 mutually unbiased bases in dimensions~d=2,3,4,5,6,7,8. Moreover, our numerical results indicate that a sum-of-squares refutation, in the above-mentioned framework, of the existence of more than 3 MUBs in dimension 6 requires polynomials of total degree at least~12.
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.