Factorization law for two lower bounds of concurrence

Entanglement, one of the important features of quantum systems, which does not exist classically, has been known as a key resource for some quantum computation and information processes. But the entanglement of a system changes due to its unavoidable interactions with environment. To study the entanglement changes, one needs to make use of an entanglement measure in order to specify the entanglement amount of a system. Unfortunately, most of the measures having been proposed for quantification of entanglement cannot be computed in general, and because of this, many lower and upper bounds, which can be computed easily, have been introduced for these entanglement measures. Using these bounds, one can estimate the amount of entanglement. In Ref. [1], Konrad et al. have provided a factorization law for concurrence, which is one of the remarkable entanglement measures. They have shown that the concurrence of a two-qubit state, when one of its qubits goes through an arbitrary quantum channel, is equal to the product of its initial concurrence and concurrence of the maximally entangled state undergoing the effect of the same quantum channel. Then Li et al. [2] have shown that the generalization of the preceding factorization law to arbitrary dimensional bipartite states only leads to an upper bound for the concurrence of the system. If, besides this upper bound, we have a lower bound obeying a similar factorization law, then we can make better use of this useful dynamical property. So, it will be valuable to seek such entanglement lower bounds. In Sec. II, we introduce the concurrence and some of its lower bounds. Next, in Sec. III, we briefly review the results of Refs. [1,2]. Then, in Secs. IV and V, we investigate the factorization property of the lower bounds introduced in Sec. II. In Sec. VI, as an application, we discuss an example. Finally, we give some conclusions in Sec. VII.