A Note on Locally Unextendible Non-Maximally Entangled Basis

TL;DR

This paper examines the properties of locally unextendible non-maximally entangled bases (LUNMEB), identifies errors in previous proofs, provides a counterexample for dimension four, and fully solves the case for dimension two.

Contribution

The paper corrects previous misconceptions about LUNMEB, offers a counterexample in four dimensions, and completely characterizes the case for two dimensions.

Findings

Previous proof claiming at most d vectors in LUNMEB is incorrect.

Counterexample with 5 vectors in 4-dimensional case disproves the claim.

Complete solution provided for the 2-dimensional case.

Abstract

We study the locally unextendible non-maximally entangled basis (LUNMEB) in $H^{d}\bigotimes H^{d}$. We point out that there exists an error in the proof of the main result of LUNMEB [Quant. Inf. Comput. 12, 0271(2012)], which claims that there are at most $d$ orthogonal vectors in a LUNMEB, constructed from a given non-maximally entangled state. We show that both the proof and the main result are not correct in general. We present a counter example for $d=4$, in which five orthogonal vectors from a specific non-maximally entangled state are constructed. Besides, we completely solve the problem of LUNMEB for the case of $d=2$.