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.
Highlights
Entanglement plays a central role in several phenomena in many-body quantum systems
This volume law is closely related to the eigenstate thermalization hypothesis [8,9,10,11], namely, the fact that generic eigenstates are able to capture the local properties of systems in thermal equilibrium, when the number of lattice sites N → ∞: The entropy of any finite region in the thermodynamic limit must be extensive, and entanglement has to obey a volume law [12,13,14]
To achieve a fixed variance δ2, both constructions presented in the previous section would require a number of states in the sum Eq (13) proportional to N/δ
Summary
Entanglement plays a central role in several phenomena in many-body quantum systems. Ground and low-excitation states of local lattice Hamiltonians typically have very little entanglement as they fulfill an area law [1,2]: the entanglement of a connected region with the rest scales with the number of particles (area) at its boundary. Rally implies that convex combinations thereof fulfill the same property Those mixed states may have an energy variance δ2 that scales according to δ ∼ Nα and still give rise to thermal averages. Product pure states typically possess a variance that scales as δ ∼ N1/2 and do not have any entanglement at all, so they cannot describe thermal properties of a system. TPQ are random states for which expectation values of local observables probabilistically converge, in the thermodynamic limit, to their values in a given statistical equilibrium ensemble The variance of their energy-density distribution vanishes as 1/N. Our numerical results confirm that we can decrease the variance as δ ∼ 1/ log(N ), keeping a polynomially scaling bond dimension and, for fixed value of L, all local observables in the region of size L converge to their thermal values in the thermodynamical limit.
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