Constructions of Unextendible Maximally Entangled Bases in \(\mathbb {C}^{d}\otimes \mathbb {C}^{d^{\prime}}\)

TL;DR

This paper introduces a systematic method for constructing unextendible maximally entangled bases (UMEBs) in bipartite quantum systems of dimensions \\(d \\) and \\(d' \\), providing explicit examples and a general framework for various parameters.

Contribution

It presents a new operational method to construct UMEBs with specific sizes and extends the construction to a broader class of bipartite systems.

Findings

Constructed UMEBs in \\(\mathbb{C}^5 \otimes \mathbb{C}^6\) and \\(\mathbb{C}^5 \otimes \mathbb{C}^{12}\).

Developed a systematic construction method for UMEBs with size \\(d(d'-r)\).

Provided explicit examples of UMEBs in \\(\mathbb{C}^3 \otimes \mathbb{C}^{10}\).

Abstract

We study unextendible maximally entangled bases (UMEBs) in \(\mathbb {C}^{d}\otimes \mathbb {C}^{d^{\prime}}\) ($d<d'$). An operational method to construct UMEBs containing $d(d^{\prime}-1)$ maximally entangled vectors is established, and two UMEBs in \(\mathbb {C}^{5}\otimes \mathbb {C}^{6}\) and \(\mathbb {C}^{5}\otimes \mathbb {C}^{12}\) are given as examples. Furthermore, a systematic way of constructing UMEBs containing $d(d^{\prime}-r)$ maximally entangled vectors in \(\mathbb {C}^{d}\otimes \mathbb {C}^{d^{\prime}}\) is presented for $r=1,2,\cdots, d-1$. Correspondingly, two UMEBs in \(\mathbb {C}^{3}\otimes \mathbb {C}^{10}\) are obtained.