Evolution equation of entanglement for bipartite systems

Abstract

We explore how entanglement of a bipartite system evolves when one subsystem undergoes the action of an arbitrary noisy channel. It is found that the dynamics of entanglement of such system is determined by the channel's action on the maximally entangled state, which includes as a special case the results for two-qubit systems ͓Konrad et al., Nat. Phys. 4, 99 ͑2008͔͒. In particular, for multiqubit or qubit-qudit systems, we get a general factorization law for the evolution equation of entanglement, with one qubit being subject to a noisy channel.

Entanglement evolution in multipartite cavity-reservoir systems under local unitary operations

“…Some earlier studies also pointed out this fact that even a very weakly dissipative environment can disentangle the quantum system in a finite time [4][5][6][7]. The ESD phenomenon has recently received a lot of attentions [8][9][10][11][12][13][14][15] (see also a review paper [16] and references therein), and, experimentally, it has been detected in photon [17] and atom systems [18].…”

Rank-dependant factorization of entanglement evolution

“…Because a general four-qubit mixed state density matrix has rank 16, it is difficult to generate low-rank density matrices even from the maximally entangled states. Therefore, very few numerical results have been obtained to confirm the factorizations (15)- (16) for an arbitrary local channels and only for initial maximally entangled states (12) and (13).…”

“…If, however, the final state density matrix has rank four, the concurrences C 123|4 and C 12|34 factorize differently. The rank four density matrix can be obtained, for example, by the action of three PF channels and single BF channel on the initial GHZ state (12) or by action of four PF channels on W state (13). In these cases we have…”

“…It was shown, in particular, that the entanglement evolution of an arbitrary pure two-qubit state can be completely described by two factors, which are given by the initial entanglement of the pure state and the entanglement dynamics of the maximally entangled state. Subsequently, this result has been extended to the cases of high-dimensional bipartite systems [11][12][13], mixed initial states [14] and multiqubit systems [15]. All these results, however, were obtained under assumption that just one subsystem of the entangled system undergoes the action of an environmental channel (i.e.…”

“…In spite of the above limitations, concurrence (1) has been shown to be quite powerful measure of entanglement [9,12,14]. Using the bipartite concurrence Li et al [26] formulated an analytical lower bound for multiqubit concurrence, which is given by a squared sum of the bipartite concurrences computed for all possible bi-partitioning of the multiqubit system.…”