Efficient scheme for experimental quantification of non-Markovianity in high-dimensional systems

The non-Markovianity is a prominent concept of the dynamics of open quantum systems, which is of fundamental importance in quantum mechanics and quantum information. Despite lots of efforts, the experimental measurement of non-Markovianity of an open system is still limited to very small systems. Presently, it is still impossible to experimentally quantify the non-Markovianity of high-dimensional systems with the widely used Breuer-Laine-Piilo trace distance measure. In this paper, we propose a method, combining experimental measurements and numerical calculations, that allow quantifying the non-Markovianity of an $N$-dimensional system only scaled as $N^2$, successfully avoiding the exponential scaling with the dimension of the open system in the current method. After the benchmark with a two-dimensional open system, we demonstrate the method in quantifying the non-Markovanity of a high-dimensional open quantum random walk system.

Figure 1

Comparing the trace distance directly measured from experiment and the results obtained from our method. The blue circles show the trace distance as function of time of the experimentally determined optimal state pair, by scanning the whole state space. The red squares show the trace distance of the optimal pair as function of time, obtained by four basis dynamics followed by numerical optimization, as described in the main text.

Figure 2

Cartoon of the three-step open quantum walk system, with $X=2$. The blue blocks denote the polarized beam splitter. This quantum walk system includes both the lattice (location of the photon) and the coin (polarization) which is an intrinsically high-dimensional system.

Figure 3

The trace-distances as functions of time for the optimal state pairs obtained using our method for $X=0$ (red line), 1 (blue line), and 2 (green line).