Stability of thermal equilibrium in long-range quantum systems

Abstract

Experimental realizations of spin models are irremediably prone to errors, which can propagate through the system corrupting experimental signals. We study how such errors affect the measurement of local observables in systems with long-range interactions, where perturbations can spread more rapidly. Specifically, we focus on the stability of thermal equilibrium and investigate its relation to the correlation structure of the system, both analytically and numerically. As a main result, we prove that the stability of local expectation values follows from the decay of correlations on the Gibbs state and the Lieb-Robinson bound, and hence that stability always holds at high temperature. We also provide numerical evidence that this stability extends to an even larger regime of interacting long-range systems. Our results support the robustness of analog simulation platforms for long-range models, and provide further evidence that computing physical quantities of interest can be significantly easier than performing arbitrary quantum computations.

Figures (8)

• Figure 1: Two disjoint subregions $A$ and $B$ of a square lattice $\Lambda$.
• Figure 2: For a region $A \subset \Lambda$, graphical definition of $B_\ell^A$ and $\partial C^A_\ell$ for $\ell=2$.
• Figure 3: Absolute value of the difference in the expectation value of a local observable $O_A=\sigma_i^z \sigma_{i+1}^z$ at the middle of the system between perturbed and unperturbed Gibbs states of Hamiltonian in Eq. \ref{['eq:hamiltonian']}. This difference is plotted as a function of system size fixing $\varepsilon=0.1$ (left), and then with fixed system size N=45, against the magnitude of the perturbation $\varepsilon$ (middle) and against $\alpha$ (right). All plots are at fixed inverse temperature $1/J$. We show both numerical simulations (points) and linear fits (dashed lines). See Appendix \ref{['app:sim']} for the specific values and parameters of the simulations and further discussion of the results.
• Figure 4: A square lattice $\Lambda$ split into $B$ and $C$. The region $A$, signaled with diagonal blue lines, is a subregion of $B$.
• Figure 5: Test of local indistinguishability (top) and decay of correlations (bottom) in the model of Eq. \ref{['eq::HLRising']}. For local indistinguishability, we show the difference between thermal mean values of the observable $O_A$ in the middle of the chain of $N=75$ sites in regions $B$ and $BC$, as illustrated in Fig. \ref{['fig4']}, against the distance between the frontiers of both regions. We again see that it holds for a wider range of $\alpha$ than Lem. \ref{['lem-main:lppl-local-indist']}. For the decay of correlations, we locate the observable $O_A=\sigma_i^z \sigma_{i+1}^z$ at $i=4$ and move $O_B$ to the right of the chain to test the dependence with distance.

Theorems & Definitions (24)

• Definition 2.1: Decay of correlations
• Theorem 3.1
• Definition 3.2: LPPL
• Lemma 3.3: cf. Lem. \ref{['lem:decay-cor-lppl']}
• Lemma 3.4: cf. Lem. \ref{['lem:lppl-local-indist']}
• proof
• proof : Proof of Thm. \ref{['thm:stability']}
• Definition 4.1: Local indistinguishability
• Lemma 4.2
• Lemma 4.3