Quasiorthogonality of Commutative Algebras, Complex Hadamard Matrices, and Mutually Unbiased Measurements
Sooyeong Kim, David Kribs, Edison Lozano, Rajesh Pereira, Sarah Plosker
TL;DR
The paper develops a matrix‑theoretic framework to quantify quasiorthogonality between unital subalgebras of M_n, with a focus on commutative MASAs arising from MUBs and MUMs. It introduces a block‑index matrix X and a derived matrix Y to express the quasiorthogonality measure Q as a Frobenius norm, and proves that commutative algebras are quasiorthogonal exactly when X is quasiable, with Hadamard‑aligned cases attaining Q=1. The authors extend these ideas to general non‑commutative algebras via a gamma‑term that captures cross‑block interactions and separating vectors, and provide a rich set of results linking quasiorthogonality to group algebras, Latin squares, and MASA structure. They further connect these concepts to MUMs and approximate notions, establishing a bridge to quantum privacy and superdense coding. Overall, the work yields computable criteria for quasiorthogonality, illuminates structural links between commutative and non‑commutative settings, and highlights practical implications for quantum information tasks and privacy frameworks.
Abstract
We deepen the theory of quasiorthogonal and approximately quasiorthogonal operator algebras through an analysis of the commutative algebra case. We give a new approach to calculate the measure of orthogonality between two such subalgebras of matrices, based on a matrix-theoretic notion we introduce that has a connection to complex Hadamard matrices. We also show how this new tool can yield significant information on the general non-commutative case. We finish by considering quasiorthogonality for the important subclass of commutative algebras that arise from mutually unbiased bases (MUBs) and mutually unbiased measurements (MUMs) in quantum information theory. We present a number of examples throughout the work, including a subclass that arises from group algebras and Latin squares.