It is well known that in Anderson localized systems, starting from a random product state the entanglement entropy remains bounded at all times. However, we show that adding a single boundary term to an otherwise Anderson localized Hamiltonian leads to unbounded growth of entanglement. Our results imply that Anderson localization is not a local property. One cannot conclude that a subsystem has Anderson localized behavior without looking at the whole system, as a term that is arbitrarily far from the subsystem can affect the dynamics of the subsystem in such a way that the features of Anderson localization are lost.
In one-dimensional quantum systems with short-range interactions, a set of leading numerical methods is based on matrix product states, whose bond dimension determines the amount of computational resources required by these methods. We prove that a thermal state at constant inverse temperature $beta$ has a matrix product representation with bond dimension $e^{tilde O(sqrt{betalog(1/epsilon)})}$ such that all local properties are approximated to accuracy $epsilon$. This justifies the common practice of using a constant bond dimension in the numerical simulation of thermal properties.
In quantum many-body systems, a Hamiltonian is called an ``extensive entropy generator if starting from a random product state the entanglement entropy obeys a volume law at long times with overwhelming probability. We prove that (i) any Hamiltonian whose spectrum has non-degenerate gaps is an extensive entropy generator; (ii) in the space of (geometrically) local Hamiltonians, the non-degenerate gap condition is satisfied almost everywhere. Specializing to many-body localized systems, these results imply the observation stated in the title of Bardarson et al. [PRL 109, 017202 (2012)].
We propose the task of narrative incoherence detection as a new arena for inter-sentential semantic understanding: Given a multi-sentence narrative, decide whether there exist any semantic discrepancies in the narrative flow. Specifically, we focus on the missing sentence and discordant sentence detection. Despite its simple setup, this task is challenging as the model needs to understand and analyze a multi-sentence narrative, and predict incoherence at the sentence level. As an initial step towards this task, we implement several baselines either directly analyzing the raw text (textit{token-level}) or analyzing learned sentence representations (textit{sentence-level}). We observe that while token-level modeling has better performance when the input contains fewer sentences, sentence-level modeling performs better on longer narratives and possesses an advantage in efficiency and flexibility. Pre-training on large-scale data and auxiliary sentence prediction training objective further boost the detection performance of the sentence-level model.
Understanding the asymptotic behavior of physical quantities in the thermodynamic limit is a fundamental problem in statistical mechanics. In this paper, we study how fast the eigenstate expectation values of a local operator converge to a smooth function of energy density as the system size diverges. In translationally invariant systems in any spatial dimension, we prove that for all but a measure zero set of local operators, the deviations of finite-size eigenstate expectation values from the aforementioned smooth function are lower bounded by $1/O(N)$, where $N$ is the system size. The lower bound holds regardless of the integrability or chaoticity of the model, and is tight in systems satisfying the eigenstate thermalization hypothesis.
My previous work [arXiv:1902.00977] studied the dynamics of Renyi entanglement entropy $R_alpha$ in local quantum circuits with charge conservation. Initializing the system in a random product state, it was proved that $R_alpha$ with Renyi index $alpha>1$ grows no faster than diffusively (up to a sublogarithmic correction) if charge transport is not faster than diffusive. The proof was given only for qubit or spin-$1/2$ systems. In this note, I extend the proof to qudit systems, i.e., spin systems with local dimension $dge2$.
A translation-invariant gapped local Hamiltonian is in the trivial phase if it can be connected to a completely decoupled Hamiltonian with a smooth path of translation-invariant gapped local Hamiltonians. For the ground state of such a Hamiltonian, we show that the expectation value of a local observable can be computed in time $text{poly}(1/delta)$ in one spatial dimension and $e^{text{poly}log(1/delta)}$ in two and higher dimensions, where $delta$ is the desired (additive) accuracy. The algorithm applies to systems of finite size and in the thermodynamic limit. It only assumes the existence but not any knowledge of the path.
In quantum lattice systems, we prove that any stationary state with power-law (or even exponential) decay of spatial correlations has vanishing macroscopic temporal order in the thermodynamic limit. Assuming translational invariance, we obtain a similar bound on the temporal order between local operators at late times. Our proofs do not require any locality of the Hamiltonian. Applications in quantum time crystals are briefly discussed.
Suppose we would like to approximate all local properties of a quantum many-body state to accuracy $delta$. In one dimension, we prove that an area law for the Renyi entanglement entropy $R_alpha$ with index $alpha<1$ implies a matrix product state representation with bond dimension $mathrm{poly}(1/delta)$. For (at most constant-fold degenerate) ground states of one-dimensional gapped Hamiltonians, it suffices that the bond dimension is almost linear in $1/delta$. In two dimensions, an area law for $R_alpha(alpha<1)$ implies a projected entangled pair state representation with bond dimension $e^{O(1/delta)}$. In the presence of logarithmic corrections to the area law, similar results are obtained in both one and two dimensions.
Chaotic dynamics in quantum many-body systems scrambles local information so that at late times it can no longer be accessed locally. This is reflected quantitatively in the out-of-time-ordered correlator of local operators, which is expected to decay to zero with time. However, for systems of finite size, out-of-time-ordered correlators do not decay exactly to zero and in this paper we show that the residual value can provide useful insights into the chaotic dynamics. When energy is conserved, the late-time saturation value of the out-of-time-ordered correlator of generic traceless local operators scales as an inverse polynomial in the system size. This is in contrast to the inverse exponential scaling expected for chaotic dynamics without energy conservation. We provide both analytical arguments and numerical simulations to support this conclusion.
