Eigenstate thermalization and representative states on subsystems

We consider a quantum system $A\cup B$ made up of degrees of freedom that can be partitioned into spatially disjoint regions $A$ and $B$. When the full system is in a pure state in which regions $A$ and $B$ are entangled, the quantum mechanics of region $A$ described without reference to its complement is traditionally assumed to require a reduced density matrix on $A$. While this is certainly true as an exact matter, we argue that under many interesting circumstances expectation values of typical operators anywhere inside $A$ can be computed from a suitable pure state on $A$ alone, with a controlled error. We use insights from quantum statistical mechanics—specifically the eigenstate thermalization hypothesis (ETH)—to argue for the existence of such “representative states.”

Figure 1

(a) $\langle {\psi }_A\left|c_i^{†}{c}_i\right|{\psi }_A\rangle$ plotted against the position $i$ for 100 000 randomly picked representative states $|{\psi }_A\rangle$ in a dimerized free-fermion system of linear dimensions $L=256,L_A=128$ and temperature $T=0$. The thick, red (grey) line denotes the canonical average. The error is maximum for boundary operators. Inset: Same results for a system at temperature $T=1$. In this case there is no discernible difference in the variance between boundary and bulk operators consistent with the volume law. (b) Standard deviation of $\langle {\psi }_A\left|c_i^{†}{c}_i\right|{\psi }_A\rangle$ for $i$ at the boundary at $T=0$ (blue circles), $i$ at the boundary at $T=1$ (green squares), and $i$ deep in the bulk at $T=1$ (red stars) plotted against system size. The best-fit lines confirm the $\sqrt{\frac{1}{L^{d-1}}}$ scaling of the error for boundary operators at $T=0$, and the $\sqrt{\frac{1}{L^d}}$ scaling for both boundary and bulk operators at finite $T$. (c) Standard deviation of $\langle {\psi }_A\left|c_i^{†}{c}_i\right|{\psi }_A\rangle$ as a function of position $i$, showing exponential decay with distance from the boundary.

Figure 2

Level spacing ratio statistics of $H_E$ for the Rokhsar-Kivelson state (green crosses) (9) compared to the Poisson (red dashed line) and GOE (black solid line) distributions. The statistics clearly look GOE consistent with a nonintegrable $H_E$. This is to be contrasted with the Poissonian statistics of the integrable transfer matrix (blue dots) $T_{{\sigma }_i,{\sigma }_j}$ in (11). $r$ refers to the ratio of subsequent level spacings, and $P\left(r\right)$ is the probability of obtaining a given $r$. The GOE form is derived in Ref. [21].