Towards exact algorithmic proofs of maximal mutually unbiased bases sets in arbitrary integer dimension

TL;DR

This paper introduces three logic-based algorithms to exactly determine the maximum number of mutually unbiased bases in any dimension, including non-prime power dimensions, advancing understanding in quantum information theory.

Contribution

It presents novel algorithms based on First-Order Logic for exact proofs of MUB sets in arbitrary dimensions, addressing a longstanding open problem.

Findings

Algorithms can prove maximum MUB sets in finite time

Maximum MUBs can be achieved with definable and computable complex parameters

Provides heuristic methods for semi-decision problems in MUB determination

Abstract

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.