Phase boundaries in deterministic dense coding

TL;DR

This paper investigates the boundary conditions of entangled states in deterministic dense coding, revealing when the maximum Schmidt coefficient reaches its theoretical limit and how encoding operations influence this boundary.

Contribution

It provides new conditions showing when the bound on the largest Schmidt coefficient cannot be achieved, clarifying the shape of phase boundaries in dense coding.

Findings

Bound on the largest Schmidt coefficient cannot always be reached.

Unitary encoding can be strictly better than non-unitary encoding.

Insights into the shape of boundaries between different dense coding capabilities.

Abstract

We consider dense coding with partially entangled states on bipartite systems of dimension $d\times d$, studying the conditions under which a given number of messages, $N$, can be deterministically transmitted. It is known that the largest Schmidt coefficient, $\lambda_0$, must obey the bound $\lambda_0\le d/N$, and considerable empirical evidence points to the conclusion that there exist states satisfying $\lambda_0=d/N$ for every $d$ and $N$ except the special cases $N=d+1$ and $N=d^2-1$. We provide additional conditions under which this bound cannot be reached -- that is, when it must be that $\lambda_0<d/N$ -- yielding insight into the shapes of boundaries separating entangled states that allow $N$ messages from those that allow only $N-1$. We also show that these conclusions hold no matter what operations are used for the encoding, and in so doing, identify circumstances under which unitary encoding is strictly better than non-unitary.