Entanglement increase from local interactions and not completely positive maps

Abstract

Simple examples are constructed that show the entanglement of two qubits being both increased and decreased by interactions on just one of them. One of the two qubits interacts with a third qubit, a control, that is never entangled or correlated with either of the two entangled qubits and is never entangled, but becomes correlated, with the system of those two qubits. The two entangled qubits do not interact, but their state can change from maximally entangled to separable or from separable to maximally entangled. Similar changes for the two qubits are made with a swap operation between one of the qubits and a control; then there are compensating changes of entanglement that involve the control. When the entanglement increases, the map that describes the change of the state of the two entangled qubits is not completely positive. Combination of two independent interactions that individually give exponential decay of the entanglement can cause the entanglement to not decay exponentially but, instead, go to zero at a finite time.

Adapted projection operator technique for the treatment of initial correlations

“…importantly, for ρ α = ρ E , the functions in Eqs. (30) and (31) reduce to the usual covariance functions of the environmental interaction operators with respect to the reference state ρ E , that is,…”

“…As a consequence, the dynamics of open quantum systems in the presence of initial correlations with the environment has been the object of intense study, even though a general convenient treatment of such dynamics is still missing. Mostly, the investigation has been focused on the possibility to define reduced maps at the level of the set of states of the open system only and in that case to extend the CPTP property to this scenario [26][27][28][29][30][31][32][33][34][35][36][37][38][39]. Furthermore, it was shown that specific behaviors of distinguishability quantifiers among quantum states, which can be tomographically reconstructed, can be traced back to the presence [40][41][42][43][44][45] or even to the classical or quantum nature [46,47] of initial correlations.…”

Finite-Time Destruction of Entanglement and Non-Locality by Environmental Influences

“…Additionally, it was shown that for a large class of states including Werner states and pure states, a state may satisfy the two-qubit CHSH inequalities in finite time, even when ESD does not take place and the time for complete disentanglement is infinite. Jordan, Shaji, and Sudarshan have discussed the effects of "non-" completely positive maps and their expression as local interactions that may increase or decrease the amount of entanglement present [63]. Furthermore, they discussed how noise influences that each separately disentangle asymptotically in time, when combined, may in finite time destroy all entanglement.…”

Adapted projection operator technique for the treatment of initial correlations

“…In addition, the full understanding of the role of the correlations, and possibly of their quantum or classical nature, in the evolution of open quantum systems should indeed include the analysis of those correlations that are present between the system and the environment at the initial time, thus complementing the related studies on the correlations built up by the dynamics [21][22][23][24][25] As a consequence, the dynamics of open quantum systems in the presence of initial correlations with the environment has been the object of intense study, even though a general convenient treatment of such dynamics is still missing. Mostly, the investigation has been focused on the possibility to define reduced maps at the level of the set of states of the open system only, and, in case, to extend the CPTP property to this scenario [26][27][28][29][30][31][32][33][34][35][36][37][38][39]. What is more, it was shown that specific behaviors of distinguishability quantifiers among quantum states, which can be tomographically reconstructed, can be traced back to the presence [40][41][42][43][44][45] or even to the classical or quantum nature [46,47] of initial correlations.…”