Figure 1

(Color online) Top: schematic of the matrix-product decomposition of a four-site lattice. The circles indicate sites, $i$, with an applied tensor, ${\Gamma }^{\left[i\right]}$, and the diamonds denote bonds which carry Schmidt coefficients, ${\lambda }^{\left[i\right]}$, on the $i\text{th}$ bond. The shading indicates a decomposition into left and right subsystems described by a renormalized basis $|l_{\alpha }⟩$ and $|r_{\alpha }⟩$ of eigenvectors of the respective reduced density matrices. Bottom: schematic depicting an overlap of two matrix-product states. The top line corresponds to a state $A$ (e.g., an ansatz state) in the matrix-product representation while the bottom line corresponds to a second state $E$ (e.g., an exact state).Reuse & Permissions

Figure 2

(Color online) Indistinguishability $I_{N/2}$ plotted as a function of magnetic field $h$ for the one-dimensional transverse quantum Ising model for several different system sizes, $N$, and two different ansatz states, $A=\text{F}$ (ferromagnetic, $h_{\text{ref}}=0$) and $A=\text{P}$ (paramagnetic, $h_{\text{ref}}=2$), as indicated by the labels. The inset plots the crossing point of $I_{N/2}\left(\text{F},E\right)$ and $I_{N/2}\left(\text{P},E\right)$, with ${\Psi }_{\text{P}}$ evaluated for $h_{\text{ref}}=100$ versus $N^{-1}$. A straight-line fit yields a quantum critical point at $h_{\text{cr}}=0.999\left(1\right)$ as $N\to \infty$, in agreement with standard results (Ref. 23).Reuse & Permissions

Figure 3

(Color online) The quantum Chernoff bound versus magnetic field using the extrapolation of indistinguishability, ${\xi }_{\text{CB}}^{\text{lim}}$ (dotted line) and the reduced density matrices directly, ${\xi }_{\text{CB}}^{\rho }$ (solid, dashed, and dot-dashed). A suppression of ${\xi }_{\text{CB}}$ indicates success for the ferromagnetic, F, black lines (paramagnetic, P, blue lines) ansatz for $h\lesssim 1\left(h\gtrsim 1\right)$. The inset plots shows a log plot of $I_{N/2}\left(\text{F},E\right)$ versus $N$ for several $h$ to show an abrupt change in scaling near $h=1$.Reuse & Permissions

Figure 4

(Color online) Indistinguishability versus $\theta$ for the bilinear-biquadratic Heisenberg model, Eq. (13), with $N=36$ and $n=18$. Distinct correlator classes surround five different reference states for which $I_{N/2}=1/2$: the ferromagnetic (solid), quadrupolar (dot-double dashed), Haldane (dot-dashed), and dimerized (dashed) phases. The AKLT point (double dot-dashed) at $\theta ={\text{tan}}^{-1}\left(1/3\right)$ appears within the Haldane phase.Reuse & Permissions

Figure 5

(Color online) Indistinguishability susceptibility, Eq. (12), computed for the bilinear-biquadratic chain, Eq. (13). Here we use $N=36,72$, $n=N/2$, and $\epsilon =0.02$ for the relevant part of the phase diagram. The peaks indicate phase transitions. In the case of discontinuous transitions, peaks remain finite only due to the discretization of the values of $h_{\text{ref}}$.Reuse & Permissions

Figure 6

(Color online) Analysis of the dimerized phase. (a) MPS ground state for $\theta =-\pi /2$ as the reference state. For small system sizes, the reference state remains a good ansatz state far beyond $\theta /\pi =-0.25$ into the Haldane regime. (b) $I_{N/2}$ for a strongly dimerized reference state obtained for $\theta =-\pi /2$ by omitting even-bond terms from the Hamiltonian, Eq. (13). The indistinguishability remains small even around the maximally dimerized point near $\theta /\pi =-0.5$, indicating that a product of dimers only poorly characterizes the system. (c) Dimerization order parameter [Eq. (14)] for three system sizes. The dimerization remains finite far towards the AKLT point, rendering the Haldane and dimerized phases hard to distinguish on small length scales.Reuse & Permissions