Generalized trace distance approach to quantum non-Markovianity and detection of initial correlations

A measure of quantum non-Markovianity for an open system dynamics, based on revivals of the distinguishability between system states, has been introduced in the literature using the trace distance as quantifier for distinguishability. Recently it has been suggested to use as measure for the distinguishability of quantum states the trace norm of Helstrom matrices, given by weighted differences of statistical operators. Here we show that this approach, which generalizes the original one, is consistent with the interpretation of information flow between the system and its environment associated to the original definition. To this aim we prove a bound on the growth of the external information, that is information which cannot be accessed by performing measurements on the system only, as quantified by means of the Helstrom matrix. We further demonstrate by means of example that it is of relevance in generalizing schemes for the local detection of initial correlations based on the increase of internal information. Finally we exploit this viewpoint to show the optimality of a previously introduced strategy for the local detection of quantum correlations.

Figure 1

Cartoon of the two states one-shot discrimination procedure: Alice prepares a quantum system in the states ${\rho }^1\left({\rho }^2\right)$ with probabilities $p_1\left(p_2\right)$, whereas “1” and “2” are the possible results of Bob's measurement. Increasing the number of possible results of the measurement does not lead to an improved distinguishability of the prepared states.

Figure 2

Probability of detection of initial correlations for different values of $\{\alpha ,\beta \}$, for 40 and 30 equally spaced values of $p_1$ and $\lambda$, respectively. The black line is the result shown in [36] (Fig. 2 therein), where the authors considered the trace distance to detect initial correlations, i.e., $p_1=0.5$. We see that, after the threshold value $\lambda \gtrsim 0.4$, highlighted by a red line, initial correlations are detected only choosing $p_1>1/2$ and hence using our approach.

Figure 3

Behavior of the trace norm of (13) for $p_1=0.5$ and $p_1=0.6$, for different values of $\lambda$, as functions of $\alpha$ and $t$. On the left side, related to the trace distance case $p_1=0.5$, no increase of distinguishability is detected for $\lambda \gtrsim 0.4$, while the trace norm with $p_1=0.6$ shows an increase also for larger values of $\lambda$. The red line separates a positive witness of initial correlations from a failure of the method, for $p_1=0.5$ and $p_1=0.6$.

Figure 4

In green (gray) we plot the increase of internal information ${\mathcal{I}}_{\mathrm{int}}\left(t\right)-{\mathcal{I}}_{\mathrm{int}}\left(0\right)$, while the meshed surface is the bound (22), as a function of $\left|\alpha \right|\in [0,1]$ and $p_1\in [0,1]$. Note that the maximum increase is given by the unbiased case $p_1=1/2$, that is highlighted by a black line, where the bound is also saturated: in this situation, the information contained in the quantum correlations is completely transferred, via the cnot, into the system. No initial quantum correlations are detected for $p_1<1/3$, as indicated by the red line.