  In this paper, we explore the concept of Mutually Unbiased Bases (MUBs) in
discrete quantum systems. It is known that for dimensions $d$ that are powers
of prime numbers, there exists a set of up to $d+1$ bases that form an MUB set.
However, the maximum number of MUBs in dimensions that are not powers of prime
numbers is not known.
  To address this issue, we introduce three algorithms based on First-Order
Logic that can determine the maximum number of bases in an MUB set without
numerical approximation. Our algorithms can prove this result in finite time,
although the required time is impractical. Additionally, we present a heuristic
approach to solve the semi-decision problem of determining if there are $k$
MUBs in a given dimension $d$.
  As a byproduct of our research, we demonstrate that the maximum number of
MUBs in any dimension can be achieved with definable complex parameters,
computable complex parameters, and other similar fields.

Quantum Physics