Quantum decoherence for multi-photon entangled states

We investigate quantum decoherence of the multi-photon entangled state vertical bar psi Nm > = N-m[cos gamma vertical bar N - M >(1)vertical bar M >(2) + e(i theta m) sin gamma vertical bar m > N-1 - m >(2)], When the entangled channel vertical bar psi Nm > is embedded in an environment, the channel decoheres and becomes a mixed state governed by a master equation. We calculate the linear entropy and the relative entropy of entanglement, which describe the mixedness and the amount of entanglement for the mixed state, respectively. We show that quantum decoherence weakens the amount of entanglement and enhances the mixedness with the time evolution. It is indicated that the relative entropy of entanglement depends on not only the initial entanglement angle and the decohering parameter, but also the number of photons in each mode. In particular, we End that the decohering speed depends on the number-difference of photons in the two modes. The larger the number-difference of photons is, the higher the decohering speed.