Locally Unextendible Non-Maximally Entangled Basis

TL;DR

This paper introduces the concept of locally unextendible non-maximally entangled bases in quantum systems, analyzing their properties and implications for superdense coding, including how entanglement affects information transmission capacity.

Contribution

It defines the LUNMEB, explores its structure, and derives formulas for classical information capacity in superdense coding using non-maximally entangled states.

Findings

LUNMEB consists of d orthogonal vectors for non-maximally entangled states.

Maximum of (d-1)^2 orthogonal vectors for maximally entangled states in (d-1) dimensional subspace.

Number of classical bits transmitted is (1+p_0*d/(d-1)) log d, with p_0 as the smallest Schmidt coefficient.

Abstract

We introduce the concept of the locally unextendible non-maximally entangled basis (LUNMEB) in $H^d \bigotimes H^d$. It is shown that such a basis consists of $d$ orthogonal vectors for a non-maximally entangled state. However, there can be a maximum of $(d-1)^2$ orthogonal vectors for non-maximally entangled state if it is maximally entangled in $(d-1)$ dimensional subspace. Such a basis plays an important role in determining the number of classical bits that one can send in a superdense coding protocol using a non-maximally entangled state as a resource. By constructing appropriate POVM operators, we find that the number of classical bits one can transmit using a non-maximally entangled state as a resource is $(1+p_0\frac{d}{d-1})\log d$, where $p_0$ is the smallest Schmidt coefficient. However, when the state is maximally entangled in its subspace then one can send up to $2\log (d-1) $ bits. We also find that for $d= 3$, former may be more suitable for the superdense coding.