Universality of finite-time disentanglement

In this paper we investigate how common the phenomenon of finite time disentanglement (FTD) is with respect to the set of quantum dynamics of bipartite quantum states with finite-dimensional Hilbert spaces. Considering a quantum dynamics from a general sense as just a continuous family of completely positive trace preserving maps (CPTP) (parametrized by the time variable) acting on the space of the bipartite systems, we conjecture that FTD happens for all dynamics but those when all maps of the family are induced by local unitary operations. We prove that this conjecture is valid for two important cases: (i) when all maps are induced by unitaries and (ii) for pairs of qubits, when all maps are unital. Moreover, we prove some general results about unitaries that preserve product states and about CPTP maps preserving pure states.

Figure 1

Pictorial representation of the set of quantum states when $\dim \left(\mathcal{H}\right)<\infty$.

Figure 2

Arrows represent how initial states are mapped to time-evolved ones: (a) flow directed towards a separable state and (b) flow directed towards an entangled one. In fact, we should stress that it is not always true that the whole family keeps fixed some $\rho$, i.e., ${\mathrm{\Lambda }}_t\left(\rho \right)=\rho \forall t\ge 0$ for some state $\rho$.