Novel Constructions of Mutually Unbiased Tripartite Absolutely Maximally Entangled Bases

TL;DR

This paper introduces a novel method for constructing mutually unbiased tripartite absolutely maximally entangled bases using Latin squares, providing explicit examples and generalizations for various dimensions.

Contribution

The authors develop a new technique leveraging Latin squares to construct mutually unbiased tripartite maximally entangled bases, extending previous methods to more general dimensions.

Findings

Explicit constructions in $ ext{C}^3 ext{C}^3 ext{C}^3$

Constructions in $ ext{C}^2 ext{C}^2 ext{C}^4$

Generalization to $ ext{C}^{d_1} ext{C}^{d_2} ext{C}^{d_1 d_2}$

Abstract

We develop a new technique to construct mutually unbiased tripartite absolutely maximally entangled bases. We first explore the tripartite absolutely maximally entangled bases and mutually unbiased bases in $\mathbb{C}^{d} \otimes \mathbb{C}^{d} \otimes \mathbb{C}^{d}$ based on mutually orthogonal Latin squares. Then we generalize the approach to the case of $\mathbb{C}^{d_{1}} \otimes \mathbb{C}^{d_{2}} \otimes \mathbb{C}^{d_{1}d_{2}}$ by mutually weak orthogonal Latin squares. The concise direct constructions of mutually unbiased tripartite absolutely maximally entangled bases are remarkably presented with generality. Detailed examples in $\mathbb{C}^{3} \otimes \mathbb{C}^{3} \otimes \mathbb{C}^{3},$ $\mathbb{C}^{2} \otimes \mathbb{C}^{2} \otimes \mathbb{C}^{4}$ and $\mathbb{C}^{2} \otimes \mathbb{C}^{5} \otimes \mathbb{C}^{10}$ are provided to illustrate the advantages of our approach.