How much entanglement is needed to reduce the energy variance?

TL;DR

This paper investigates the relationship between entanglement and energy variance in quantum states, demonstrating that states with decreasing variance can be efficiently constructed and tend to thermal equilibrium as system size grows.

Contribution

It introduces a construction for low-variance matrix product states with polynomial bond dimension scaling, linking entanglement to energy fluctuations in many-body systems.

Findings

States with logarithmically decreasing variance approach thermal equilibrium.

States with constant variance do not converge to thermal equilibrium.

Efficient construction of low-variance states is possible with polynomial resources.

Abstract

We explore the relation between the entanglement of a pure state and its energy variance for a local one dimensional Hamiltonian, as the system size increases. In particular, we introduce a construction which creates a matrix product state of arbitrarily small energy variance $\delta^2$ for $N$ spins, with bond dimension scaling as $\sqrt{N} D_0^{1/\delta}$, where $D_0>1$ is a constant. This implies that a polynomially increasing bond dimension is enough to construct states with energy variance that vanishes with the inverse of the logarithm of the system size. We run numerical simulations to probe the construction on two different models, and compare the local reduced density matrices of the resulting states to the corresponding thermal equilibrium. Our results suggest that the spatially homogeneous states with logarithmically decreasing variance, which can be constructed efficiently, do converge to the thermal equilibrium in the thermodynamic limit, while the same is not true if the variance remains constant.