Optimal dense coding with mixed state entanglement

I investigate dense coding with a general mixed state on the Hilbert space C^d ⊗ 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 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) ≤ χ* ≤ E_R(ρ) + log₂ d, where E_R(ρ) is the relative entropy of entanglement of the shared entangled state.