Mutually unbiased maximally entangled bases from difference matrices

TL;DR

This paper introduces a novel method using difference matrices to construct mutually unbiased bases with maximally entangled states in bipartite quantum systems, covering prime power and certain composite dimensions.

Contribution

It presents new constructions of mutually unbiased bases using difference matrices, including for non-prime power dimensions, advancing quantum information theory.

Findings

Constructed $q$ mutually unbiased bases in prime power dimensions.

Improved lower bounds for maximally entangled bases in composite dimensions.

Established constructions for prime and prime power dimensions with explicit bases.

Abstract

Based on maximally entangled states, we explore the constructions of mutually unbiased bases in bipartite quantum systems. We present a new way to construct mutually unbiased bases by difference matrices in the theory of combinatorial designs. In particular, we establish $q$ mutually unbiased bases with $q-1$ maximally entangled bases and one product basis in $\mathbb{C}^q\otimes \mathbb{C}^q$ for arbitrary prime power $q$. In addition, we construct maximally entangled bases for dimension of composite numbers of non-prime power, such as five maximally entangled bases in $\mathbb{C}^{12}\otimes \mathbb{C}^{12}$ and $\mathbb{C}^{21}\otimes\mathbb{C}^{21}$, which improve the known lower bounds for $d=3m$, with $(3,m)=1$ in $\mathbb{C}^{d}\otimes \mathbb{C}^{d}$. Furthermore, we construct $p+1$ mutually unbiased bases with $p$ maximally entangled bases and one product basis in $\mathbb{C}^p\otimes \mathbb{C}^{p^2}$ for arbitrary prime number $p$.