Combinatorial & Algebraic Approaches in Analyzing Mutually Unbiased Bases (MUBs) and Their Approximations

Abstract

Mutually Unbiased Bases (MUBs) are an important concept in quantum information theory. Two orthonormal bases in a $d$-dimensional complex Hilbert space $\mathbb{C}^d$ are said to be mutually unbiased if the absolute value of the inner product between any pair of vectors, one from each basis, is $1/\sqrt{d}$. A set of $r$ orthonormal bases is called mutually unbiased if every pair of bases in the set is unbiased. It is known that at most $d + 1$ mutually unbiased bases can exist in $\mathbb{C}^d$, and a set achieving this bound is termed a \emph{complete sets of MUBs} in $\mathbb{C}^d$. We can construct $d+1$ MUBs if $d$ is a power of a prime. However, the existence of a complete sets of MUBs in composite dimensions that are not a power of a prime remains a longstanding open problem. In such cases, the (naive) known general lower bound for the number of MUBs is $p^r + 1$, where $p^r$ is the smallest prime power dividing $d$. Consequently, we do not have more than seven MUBs and fewer than three MUBs in dimension six. Zauner (1999) conjectured that there are no more than three MUBs in $\mathbb{C}^6$.

This thesis is mainly structured into two parts. The first part focuses on the existence and extendibility of MUBs. We study this part from the perspective of combinatorics and Algebraic geometry. We study the extendibility of MUB triplets in $\mathbb{C}^6$, which are product bases. We provide a combinatorial proof for the form of a unitary matrix under specific conditions. Subsequently, we demonstrate that the unitary matrix, which can be viewed as the fourth MUB of triplets of MUBs in $\mathbb{C}^6$, is not possible. Additionally, we have studied the extendibility of a given set of MUBs from the point of view of algebraic geometry and commutative algebra. We investigated the ideals of the affine algebraic variety derived from a given set of $k$ MUBs in any generic dimension $d$ $(k\leq d)$. We established a few notable results related to the complete intersection of ideals. Also, we prove that there is a one-to-one correspondence between the MUBs and the maximal commuting classes (bases) of orthogonal normal matrices in $\mathbb{C}^{d}$. It means that for $m$ MUBs in $\mathbb{C}^d$, there are $m$ commuting classes, each consisting of $d$ commuting orthogonal normal matrices. The existence of maximal commuting basis for $\mathcal M_d({\mathbb{C}})$ ensures the complete sets of MUBs in $\mathcal M_d({\mathbb{C}})$.

The second part addresses the combinatorial construction of MUBs and approximate real MUBs (ARMUBs) in certain specific dimensions. In particular, we focus on the construction of Approximate Real Mutually Unbiased Bases (ARMUBs) for dimensions that are not divisible by four. We show that it is possible to construct $\geq \lceil \sqrt{d} \rceil$ 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. Also, we consider the parametrisation of MUBs to understand the degrees of freedom, and this can help explore various choices of MUBs so that one can explore several classes of them for multiple applications in quantum information. For dimension $d=s^2$, we present the construction of affine-parametric classes with $MOLS(s)+2$ many MUBs, where $MOLS(s)$ is the number of Mutually Orthogonal Latin Squares of dimension $s$. If $s$ is a power of a prime, then $MOLS(s) = s-1$, and the number of MUBs will be $s+1$. Considering the first one to be the identity matrix, in our construction, each of the rest $MOLS(s)+1$ MUBs will have at least $s(s-1)$ free parameters, so that a global unitary operation cannot absorb.