Optimal dense coding with thermal entangled states

We study optimal dense coding with thermal entangled states of a two-qubit Heisenberg XX chain and a two-qutrit system. For a two-qubit Heisenberg XX chain, the dense coding capacity is a function of temperature and external magnetic field. Only in the case of an external magnetic field being less than the coupling constant, the optimal dense coding can be realized with thermal entangled states. For a two-qutrit system, we consider the dense coding capacity taking into account of nonlinear coupling constant and an external magnetic field. We find that the nonlinear coupling constant must be less than 0 for dense coding. For the two models, we give the conditions that the parameters of the models have to satisfy a valid dense coding.