Entanglement-Optimal Trajectories of Many-Body Quantum Markov Processes

We develop a novel approach aimed at solving the equations of motion of open quantum many-body systems. It is based on a combination of generalized wave function trajectories and matrix product states. We introduce an adaptive quantum stochastic propagator, which minimizes the expected entanglement in the many-body quantum state, thus minimizing the computational cost of the matrix product state representation of each trajectory. We illustrate this approach on the example of a one-dimensional open Brownian circuit. We show that this model displays an entanglement phase transition between area and volume law when changing between different propagators and that our method autonomously finds an efficiently representable area law unraveling.

Figure 1

(a) Schematic setting of a continuously monitored quantum many-body system with adaptive measurements. (b) Propagation of a state from time $t$ into the future with two options for the stochastic propagator (color coded, gray vs green). The colored dashed lines are potential QTs for a given unraveling, and their stochastic averages are shown as solid lines. A QT (black) is generated randomly at each time step, corresponding to the stochastic propagator that gives the lowest expected entanglement (green).

Figure 2

(a) Schematic illustration of the EOQT-MPS algorithm. To propagate a state from $t$ to $t+dt$ we first apply a “stochastic layer”: We calculate the unraveling that minimizes the expected EE in the next time step and apply the corresponding stochastic propagator. This procedure is implemented sequentially for all jump operators. Then we implement a “deterministic layer,” propagating the system with $H_{\mathrm{sys}}$ via standard methods such as TEBD. (b) Time dependence of the excess EAEE, $\overline{E}-E_f$, of a continuously monitored Bell pair for the number measurement (blue), the homodyne unraveling with ${\phi }_j=0$ (red), and the EOQT (green, number of trajectories $N={10}^4$). Inset: The statistics of measurement choices of the EOQT algorithm. The error bars are denoted by gray filling.

Figure 3

EAEE for the open RBC. (a) EAEE profile in the long time limit for various unravelings. Solid lines denote homodyne unravelings with changing phase from ${\phi }_j=0$ (red) to ${\phi }_j=\pi /2$ (gray) in increments of $\pi /20$, and dashed blue and green lines correspond to the number and EOQT measurements. Insets: Time evolution of EAEE for various unravelings and measurement choices of the EOQT. Here $m=n=16$, $\chi =128$. (b) Half-chain EAEE for larger systems (with $\chi =512$). Depending on the homodyne phase, the ensembles display an area law at small ${\phi }_j$ and a volume law at ${\phi }_j\approx \pi /2$. The EOQT method results in an area-law EAEE close to the homodyne unraveling with ${\phi }_j=0$, whereas the number unraveling leads to an area law with larger EAEE. The phase diagram is shown in the inset. For all data in this figure we used $\alpha /\gamma =0.1$ ($\alpha =1$) and $N=200$. The statistical error bars lie within the lines’ thickness.