A sufficient and necessary condition for superdense coding of quantum states

Recently, Harrow et al. [Phys. Rev. Lett. 92, 187901 (2004)] gave a method for preparing an arbitrary quantum state with high success probability by physically transmitting some qubits, and by consuming a maximally entangled state, together with exhausting some shared random bits. In this paper, we discover that some states are impossible to be perfectly prepared by Alice and Bob initially sharing those entangled states that are superposed by the ground states, as the states to be prepared. In particular, we present a sufficient and necessary condition for the states being enabled to be exactly prepared with probability one, in terms of the initial entangled states (maybe nonmaximally) superposed by the ground states. In contrast, if the initially shared entanglement is maximal, then the probabilities for preparing these quantum states are smaller than one. Furthermore, the lower bound on the probability for preparing some states are derived.