Stochastic Matrix Product States

The concept of stochastic matrix product states is introduced and a natural form for the states is derived. This allows us to define the analogue of Schmidt coefficients for steady states of nonequilibrium stochastic processes. We discuss a new measure for correlations which is analogous to entanglement entropy, the entropy cost $S_C$, and show that this measure quantifies the bond dimension needed to represent a steady state as a matrix product state. We illustrate these concepts on the basis of the asymmetric exclusion process.

Figure 1

Pictorial representation of the natural SMPS decomposition. Image (a) can be seen as the graphical representation of Eq. (5). From a given source $P_{\lambda }$, the correlations are distributed via the two channels on the left and right. The normalizing factor is included in the $A$’s. In (b) the analogy to the quantum MPS becomes evident for the decomposition as given in Eq. (7). The probabilities in the matrices $P^{[k]}$ are analogous to the singular values which arise upon a Schmidt decomposition of the quantum state [20].

Figure 2

(a) The mutual information of a chain length $N=20$ for different inflow parameters $\alpha$ and $\beta$ has been calculated numerically. (b) Entropy cost for a representation with $D^{\max }=21$. Note the different scales of the two plots. The resulting plots clearly reflect the underlying phase diagram of the ASEP [17], both for the mutual information and the approximate entropy cost. The entropy cost as well as the actual mutual information drop to zero along the mean-field line $\alpha +\beta =1$. The mutual information is consistently low throughout the diagram, explaining why mean-field approaches have given such good results [1].

Figure 3

(a) Monte Carlo simulations of the mutual information for different $\alpha$ and $\beta$ values. The system size was varied from $N=2,\dots ,180$. The chain was cut in the middle, the length $L$ of the individual blocks are given by half of the total system size. The plot shows the exponential of the mutual information $\exp I(A:B)$. The simulations suggest that the mutual information grows only sublogarithmically along the coexistence line $\alpha =\beta$, $\alpha +\beta \le 1$. (b) The logarithm of the probability distribution $\{p_{\lambda }\}$ plotted for different values of $\alpha$ and $\beta$. The dimension of the matrices $\mathcal{E}$ and $\mathcal{D}$ is $D^{\max }=21$, corresponding to a chain of length $N=20$. The distributions decay superexponentially. For all $\lambda \ge 11$, $p_{\lambda }=0$.