Optimal dense coding with mixed state entanglement

I investigate dense coding with a general mixed state on the Hilbert space Cᵈ\⊗ Cᵈ 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 {\ a priori} probabilities, the capacity of dense coding is maximized. It is also proved that the optimal capacity of dense coding \χ * satisfies ER(\ρ)\≤ \χ *\≤ ER(\ρ )+log₂d, where ER(\ρ) is the relative entropy of entanglement of the shared entangled state.