Connecting the UMEB in $\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}$ with partial Hadamard matrices

TL;DR

This paper explores the relationship between unextendible maximally entangled bases (UMEB) and partial Hadamard matrices in complex tensor product spaces, revealing conditions for their existence and construction methods.

Contribution

It establishes a connection between UMEB and partial Hadamard matrices, providing new existence results and construction techniques for UMEB in various dimensions.

Findings

Certain partial Hadamard matrices cannot extend to complete ones.

Almost all dimensions admit UMEB except specific prime-related cases.

Constructs of UMEB are possible in tensor spaces with dimensions multiple of 3, 5, and 7.

Abstract

We study the unextendible maximally entangled bases (UMEB) in $\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}$ and connect it with the partial Hadamard matrix. Firstly, we show that for a given special UMEB in $\mathbb{C}^{d}\bigotimes\mathbb{C}^{d}$, there is a partial Hadamard matrix can not extend to a complete Hadamard matrix in $\mathbb{C}^{d}$. As a corollary, any $(d-1)\times d$ partial Hadamard matrix can extend to a complete Hadamard matrix. Then we obtain that for any $d$ there is an UMEB except $d=p\ \text{or}\ 2p$, where $p\equiv 3\mod 4$ and $p$ is a prime. Finally, we argue that there exist different kinds of constructions of UMEB in $\mathbb{C}^{nd}\bigotimes\mathbb{C}^{nd}$ for any $n\in \mathbb{N}$ and $d=3\times5 \times7$.