On approximate real mutually unbiased bases in square dimension

Abstract

Construction of Mutually Unbiased Bases (MUBs) is a very challenging combinatorial problem in quantum information theory with several long standing open questions in this domain. With certain relaxations, the object Approximate Mutually Unbiased Bases (AMUBs) is defined in this context. In this paper we provide a method to construct upto $(\sqrt {d} + 1)$ many AMUBs in dimension d = q², where q is a positive integer. The result is particularly important when q ≡ 0 mod 4, as we obtain Approximate Real MUBs (ARMUBs) assuming the cases where a Hadamard matrix of order q exists. In this construction, we also characterize the inner product values between the elements of two different bases. In particular, when d is of the form (4x)² where x is a prime, we obtain $(\frac {\sqrt {d}}{4}+1)$ ARMUBs such that for any two vectors v₁,v₂ belonging to different bases, $|{\left \langle \left . v_{1} \right | v_{2} \right \rangle }| \leq \frac {4}{\sqrt {d}}$.