Positive Tensor Network Approach for Simulating Open Quantum Many-Body Systems

Open quantum many-body systems play an important role in quantum optics and condensed matter physics, and capture phenomena like transport, the interplay between Hamiltonian and incoherent dynamics, and topological order generated by dissipation. We introduce a versatile and practical method to numerically simulate one-dimensional open quantum many-body dynamics using tensor networks. It is based on representing mixed quantum states in a locally purified form, which guarantees that positivity is preserved at all times. Moreover, the approximation error is controlled with respect to the trace norm. Hence, this scheme overcomes various obstacles of the known numerical open-system evolution schemes. To exemplify the functioning of the approach, we study both stationary states and transient dissipative behavior, for various open quantum systems ranging from few to many bodies.

Figure 1

Markov dynamics of a quantum spin chain on the level of local tensors. Panel (a) shows the relationship between a density matrix $\rho$ in the MPO representation (top) and the locally purified tensor network (bottom) with tensors $A_l$, physical dimension $d$, bond dimension $D^{\prime }$, and Kraus dimension $K$. (b) Action of a local channel $T$ acting exclusively on lattice site 2 on the level of the MPO and on the level of the locally purified tensors. In the latter, the Kraus rank $k_2$ of the quantum channel $T$ is joined together with $K$. (c) Compression schemes for the bond and Kraus dimension of a local tensor via singular value decompositions (SVD). (d) Locally purified evolution of a time step ${\mathit{e}}^{\tau \mathcal{L}}$ for a nearest neighbor Hamiltonian and on-site Lindblad operators. We only show three of the five Trotter-Suzuki layers from Eq. (3).

Figure 2

Main: excitation populations of the four sites (see the main text) in the coupled spin-cavity model (yellow, cavity; violet, spin; dashed, right; straight, left), here for $\gamma =0.05$, ${\alpha }_1={\alpha }_2=0.48$, ${\alpha }_{CC}=-1.0$, ${\omega }_C={\omega }_S=1.0$, as well as their sum $\mathcal{N}$ (green line). The latter is nicely fitted by an exponential, with decay rate ${\gamma }_{\mathrm{fit}}=0.04997\pm 8\times {10}^{-5}$. Inset: comparison of the locally purified evolution, here for bond dimension $D=40$ and Kraus dimension $K=40$, with the exact Liouville evolution: infidelity $\mathcal{I}$ (blue line) and relative Hilbert-Schmidt distance (red line). Infidelities are estimated to be numerically reliable above ${10}^{-7}$ (nonshaded area).

Figure 3

Comparison of simulated steady state (points) with analytical results (lines) from Ref. [9] for the $XXZ$ model with edge driving of $\gamma =1$ in several parameter regimes. Green squares, $\mathrm{\Delta }=0.5$; red dots, $\mathrm{\Delta }=1.0$; cyan triangles, $\mathrm{\Delta }=1.5$. Top: local magnetization in the $z$ direction $⟨{\sigma }_j^z⟩$ as a function of the site $j$, for a chain of length $N=100$. Bottom: time averaged steady state spin current $I_j=2\mathrm{Im}⟨{\sigma }_j^{+}{\sigma }_{j+1}^{-}⟩$ at the chain center $j=N/2$, as a function of the total chain length $N$.