Generalized spin bases for quantum chemistry and quantum information

TL;DR

This paper introduces a unified method for constructing generalized spin bases applicable to quantum chemistry and quantum information, leveraging group symmetries and generalized Pauli matrices, with solutions for prime-dimensional Hilbert spaces.

Contribution

It presents a novel approach using generalized Pauli matrices for creating mutually unbiased bases in prime dimensions, linking symmetry groups with quantum information structures.

Findings

Complete solution for mutually unbiased bases when dimension is prime

Framework for extending to prime power dimensions

Connection established between symmetry groups and quantum bases

Abstract

Symmetry adapted bases in quantum chemistry and bases adapted to quantum information share a common characteristics: both of them are constructed from subspaces of the representation space of the group SO(3) or its double group (i.e., spinor group) SU(2). We exploit this fact for generating spin bases of relevance for quantum systems with cyclic symmetry and equally well for quantum information and quantum computation. Our approach is based on the use of generalized Pauli matrices arising from a polar decomposition of SU(2). This approach leads to a complete solution for the construction of mutually unbiased bases in the case where the dimension d of the considered Hilbert subspace is a prime number. We also give the starting point for studying the case where d is the power of a prime number. A connection of this work with the unitary group U(d) and the Pauli group is brielly underlined.