Algorithms to prove the maximum number of MUBs in arbitrary dimension

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. Moreover, we present a heuristic approach to solve the semi-decision problem of determining if there are k MUBs in a given dimension d, complementing our theoretical results. In addition to these algorithmic contributions, we establish another result: the maximum number of MUBs in any dimension can be achieved using definable complex parameters, computable complex parameters, and other similar fields. This finding highlights the broader mathematical structure underpinning MUBs and has important implications for the understanding and computation of MUBs in various dimensions. Fil: Cifuentes, Santiago. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; Argentina Fil: Ciancaglini, Nicolás Atahualpa. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigación en Ciencias de la Computación. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigación en Ciencias de la Computación; Argentina