Deterministic dense coding and entanglement entropy

TL;DR

This paper analytically investigates deterministic dense coding with non-maximally entangled states, revealing that lower entanglement can support more messages, and confirms conjectures about minimal entanglement entropy in specific dimensions.

Contribution

It proves that less entangled states can enable more messages, confirms conjectures on minimal entanglement entropy, and constructs explicit local unitaries for dimensions 3 and 4.

Findings

Lower entanglement entropy can support more messages than higher entropy states.

Confirmed conjecture on minimal entanglement entropy for sending d + j messages in all dimensions.

Established the inequality c < d/K for K = d+1 and equality cases for K = d+2 or 2d-1.

Abstract

We present an analytical study of the standard two-party deterministic dense-coding protocol, under which communication of perfectly distinguishable messages takes place via a qudit from a pair of non-maximally entangled qudits in pure state |S>. Our results include the following: (i) We prove that it is possible for a state |S> with lower entanglement entropy to support the sending of a greater number of perfectly distinguishable messages than one with higher entanglement entropy, confirming a result suggested via numerical analysis in Mozes et al. [Phys. Rev. A 71 012311 (2005)]. (ii) By explicit construction of families of local unitary operators, we verify, for dimensions d = 3 and d=4, a conjecture of Mozes et al. about the minimum entanglement entropy that supports the sending of d + j messages, j = 2, ..., d-1; moreover, we show that the j=2 and j= d-1 cases of the conjecture are valid in all dimensions. (iii) Given that |S> allows the sending of K messages and has the square roof of c as its largest Schmidt coefficient, we show that the inequality c <= d/K, established by Wu et al. [ Phys. Rev. A 73, 042311 (2006)], must actually take the form c < d/K if K = d+1, while our constructions of local unitaries show that equality can be realized if K = d+2 or K = 2d-1.