Optimal dense coding with mixed state entanglement

Abstract

I investigate dense coding with a general mixed state on the Hilbert space $C^{d}\otimes C^{d}$ shared between a sender and receiver. The following result is proved. When the sender prepares the signal states by mutually orthogonal unitary transformations with equal {\it a priori} probabilities, the capacity of dense coding is maximized. It is also proved that the optimal capacity of dense coding $χ^{*}$ satisfies $E_{R}(ρ)\leq χ^{*}\leq E_{R}(ρ)+\log_{2}d$, where $E_{R}(ρ)$ is the relative entropy of entanglement of the shared entangled state.