We consider a bipartite quantum system S (including parties A and B), interacting with an environment E through a localized quantum dynamics FSE . We call a quantum dynamics FSE localized if, e.g., the party A is isolated from the environment and only B interacts with the environment: FSE = idA ⊗ FBE, where idA is the identity map on the part A and FBE is a completely positive (CP) map on the both B and E. We will show that the reduced dynamics of the system is also localized as ES = idA ⊗ĒB, whereĒB is a CP map on B, if and only if the initial state of the system-environment is a Markov state. We then generalize this result to the two following cases: when both A and B interact with a same environment, and when each party interacts with its local environment.
Measurement-induced nonlocality is a measure of nonlocalty introduced by Luo and Fu [Phys. Rev. Lett 106, 120401 (2011)]. In this paper, we study the problem of evaluation of Measurementinduced nonlocality (MIN) for an arbitrary m×n dimensional bipartite density matrix ρ for the case where one of its reduced density matrix, ρ a , is degenerate (the nondegenerate case was explained in the preceding reference). Suppose that, in general, ρ a has d degenerate subspaces with dimension mi(mi ≤ m, i = 1, 2, ..., d). We show that according to the degeneracy of ρ a , if we expand ρ in a suitable basis, the evaluation of MIN for an m × n dimensional state ρ, is degraded to finding the MIN in the mi × n dimensional subspaces of state ρ. This method can reduce the calculations in the evaluation of MIN. Moreover, for an arbitrary m × n state ρ for which mi ≤ 2, our method leads to the exact value of the MIN. Also, we obtain an upper bound for MIN which can improve the ones introduced in the above mentioned reference. In the final, we explain the evaluation of MIN for 3 × n dimensional states in details.
We study the dynamics of two lower bounds of concurrence in bipartite quantum systems when one party goes through an arbitrary channel. We show that these lower bounds obey the factorization law similar to that of [Konrad et al., Nat. Phys. 4, 99 (2008)]. We also, discuss the application of this property, in an example.
Measurement-induced nonlocality is a measure of nonlocalty introduced by Luo and Fu [Phys. Rev. Lett \textbf{106}, 120401 (2011)]. In this paper, we study the problem of evaluation of Measurement-induced nonlocality (MIN) for an arbitrary $m\times n$ dimensional bipartite density matrix $\rho$ for the case where one of its reduced density matrix, $\rho^{a}$, is degenerate (the nondegenerate case was explained in the preceding reference). Suppose that, in general, $\rho^{a}$ has $d$ degenerate subspaces with dimension $m_{i} (m_{i} \leq m , i=1, 2, ..., d)$. We show that according to the degeneracy of $\rho^{a}$, if we expand $\rho$ in a suitable basis, the evaluation of MIN for an $m\times n$ dimensional state $\rho$, is degraded to finding the MIN in the $m_{i}\times n$ dimensional subspaces of state $\rho$. This method can reduce the calculations in the evaluation of MIN. Moreover, for an arbitrary $m\times n$ state $\rho$ for which $m_{i}\leq 2$, our method leads to the exact value of the MIN. Also, we obtain an upper bound for MIN which can improve the ones introduced in the above mentioned reference. Finally, we explain the evaluation of MIN for $3\times n$ dimensional states in details.
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