We present a rigorous proof of an interesting boundary effect of deterministic dense coding first observed by S. Mozes, J. Oppenheim, and B. Reznik [Phys. Rev. A 71, 012311 (2005)]. Namely, it is shown that $d^2-1$ cannot be the maximal alphabet size of any isometric deterministic dense coding schemes utilizing $d$-level partial entanglement.

• Received 20 January 2006

DOI:https://doi.org/10.1103/PhysRevA.73.034307