Quantum Non-Markovian Processes Break Conditional Past-Future Independence

For classical Markovian stochastic systems, past and future events become statistically independent when conditioned to a given state at the present time. Memory non-Markovian effects break this condition, inducing a nonvanishing conditional past-future correlation. Here, this classical memory indicator is extended to a quantum regime, which provides an operational definition of quantum non-Markovianity based on a minimal set of three time-ordered quantum system measurements and postselection. The detection of memory effects through the measurement scheme is univocally related to departures from Born-Markov and white noise approximations in quantum and classical environments, respectively.

Figure 1

Measurement scheme. At times $t_x<t_y<t_z$ an open system is subjected to three measurement processes whose random outcomes are $x\to y\to z$. A set of operators $\{{\mathrm{\Omega }}_x\}$, $\{{\mathrm{\Omega }}_y\}$, and $\{{\mathrm{\Omega }}_z\}$ define the measurement processes in a quantum regime. $\mathcal{E}$ and ${\mathcal{E}}^{\prime }$ are the system propagators between consecutive measurements.

Figure 2

Left panels: CPF correlation (17) for the spin bath model (12), with coupling $g_k=g/\sqrt{N}$. The parameters of the initial condition (13) are $a=1$, $b=0$, ${\alpha }_k={\beta }_k=1/2$, and $N=50$. Right panels: CPF correlation for the stochastic Hamiltonian model (19), with noise correlation $\overline{{\xi }_t{\xi }_{t^{\prime }}}=g^2\exp [-|t-t^{\prime }|/{\tau }_c]$. The system starts at the same initial condition. The parameters are ${\tau }_{c}g=100$. For ${\tau }_c\to \infty$, the left panels are recovered.