Cellular Automata-Based Methods for the Construction of Mutually Unbiased Bases

Andrés García Sandoval (), Cristian L. León Nuño and Ivan F. Valtierra Carranza ()
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Andrés García Sandoval: Departamento de Matemáticas, Universidad de Guadalajara, Revolución 1500, Guadalajara 44420, Mexico
Cristian L. León Nuño: Maestría en Ciencias en Matemáticas, Universidad de Guadalajara, Revolución 1500, Guadalajara 44420, Mexico
Ivan F. Valtierra Carranza: Departamento de Física, Universidad de Guadalajara, Revolución 1500, Guadalajara 44420, Mexico

Mathematics, 2025, vol. 13, issue 16, 1-15

Abstract: Mutually unbiased bases (MUBs) are essential tools in quantum information science, with applications in state tomography, quantum cryptography, and quantum error correction. In this work, we introduce a constructive framework for generating MUBs using linear bipermutive cellular automata (LBCAs). By leveraging the algebraic structure of generalized Pauli operators over finite fields, we show that disjoint families of LBCAs correspond to commuting sets of such operators (CSPOs), which, in turn, generate MUBs. This correspondence enables the systematic construction of complete or incomplete sets of MUBs, depending on the number of disjoint LBCAs available in a given dimension. We also provide algebraic conditions to verify disjointness and discuss how the finite dimensionality constrains MUB completeness. Our approach reinterprets classical combinatorial structures in a quantum setting, offering new computational pathways for exploring MUBs through discrete dynamical systems.

Keywords: mutually unbiased bases; cellular automata; generalized Pauli operators; finite fields; linear bipermutive rules (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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