Not So SuperDense Coding - Deterministic Dense Coding with Partially Entangled States

The utilization of a $d$-level partially entangled state, shared by two parties wishing to communicate classical information without errors over a noiseless quantum channel, is discussed. We analytically construct deterministic dense coding schemes for certain classes of non-maximally entangled states, and numerically obtain schemes in the general case. We study the dependency of the information capacity of such schemes on the partially entangled state shared by the two parties. Surprisingly, for $d>2$ it is possible to have deterministic dense coding with less than one ebit. In this case the number of alphabet letters that can be communicated by a single particle, is between $d$ and 2d. In general we show that the alphabet size grows in "steps" with the possible values $d, d+1, ..., d^2-2$. We also find that states with less entanglement can have greater communication capacity than other more entangled states.