Mutually unbiased maximally entangled bases in $\mathbb{C}^d\otimes\mathbb{C}^d$

TL;DR

This paper introduces a generalized method for constructing mutually unbiased maximally entangled bases in bipartite quantum systems using commutative rings, expanding the known set of such bases especially for composite dimensions.

Contribution

It generalizes existing construction methods by employing commutative rings with $d$ elements, enabling the creation of multiple MUMEB's in systems where the dimension factors into primes.

Findings

Constructed $p_1^{a_1}-1$ MUMEB's for certain composite dimensions.

Extended the construction framework to include direct sums of finite fields.

Provided explicit examples for dimensions with prime power factors.

Abstract

We study mutually unbiased maximally entangled bases (MUMEB's) in bipartite system $\mathbb{C}^d\otimes\mathbb{C}^d (d \geq 3)$. We generalize the method to construct MUMEB's given in [16], by using any commutative ring $R$ with $d$ elements and generic character of $(R,+)$ instead of $\mathbb{Z}_d=\mathbb{Z}/d\mathbb{Z}$. Particularly, if $d=p_1^{a_1}p_2^{a_2}\ldots p_s^{a_s}$ where $p_1, \ldots, p_s$ are distinct primes and $3\leq p_1^{a_1}\leq\cdots\leq p_s^{a_s}$, we present $p_1^{a_1}-1$ MUMEB's in $\mathbb{C}^d\otimes\mathbb{C}^d$ by taking $R=\mathbb{F}_{p_1^{a_1}}\oplus\cdots\oplus\mathbb{F}_{p_s^{a_s}}$, direct sum of finite fields (Theorem 3.3).