Optimal state pairs for non-Markovian quantum dynamics

We study a recently proposed measure for the quantification of quantum non-Markovianity in the dynamics of open systems which is based on the exchange of information between the open system and its environment. This measure relates the degree of memory effects to certain optimal initial state pairs featuring a maximal flow of information from the environment back to the open system. We rigorously prove that the states of these optimal pairs must lie on the boundary of the space of physical states and that they must be orthogonal. This implies that quantum memory effects are maximal for states which are initially distinguishable with certainty, having a maximal information content. Moreover, we construct an explicit example which demonstrates that optimal quantum states need not be pure states.

Figure 1

Illustration of the decomposition (9) and of the corresponding time evolution given by Eq. (10).

Figure 2

Probability of the total increase of the trace distance for randomly drawn pairs of pure, orthogonal states for one period of oscillation of the decay rates. The arrow indicates the value of the maximal increase of the trace distance for the pair of states given in Eq. (35) including a mixed state. The sample size of pairs of pure orthogonal states is equal to ${10}^5$.