Deterministic dense coding and entanglement entropy
P. S. Bourdon, E. Gerjuoy, J. P. McDonald, and H. T. Williams

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 $|\psi\rangle$. Our results include the following: (i) We prove that it is possible for a state $\ket{\psi}$ 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 {\it et al.}\ [Phys.\ Rev. A {\bf 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 {\it et al.}\ about the minimum entanglement entropy that supports the sending of $d + j$ messages, $2 \le j \le d-1$; moreover, we show that the $j=2$ and $j= d-1$ cases of the conjecture are valid in all dimensions. (iii) Given that $\ket{\psi}$ allows the sending of $K$ messages and has $\sqrt{\lambda_0}$ as its largest Schmidt coefficient, we show that the inequality $\lambda_0 \le d/K$, established by Wu {\it et al.}\ [ Phys.\ Rev.\ A {\bf 73}, 042311 (2006)], must actually take the form $\lambda_0 &lt d/K$ if $K = d+1$, while our constructions of local unitaries show that equality can be realized if $K = d+2$ or $K = 2d-1$.

© 2008 The American Physical Society.