The Eigenstate Thermalization Hypothesis (ETH) posits that the reduced density matrix for a subsystem corresponding to an excited eigenstate is {\textquoteleft}{\textquoteleft}thermal.\&$\#$39;\&$\#$39; Here we expound on this hypothesis by asking: for which class of operators, local or non-local, is ETH satisfied? We show that this question is directly related to a seemingly unrelated question: is the Hamiltonian of a system encoded within a single eigenstate? We formulate a strong form of ETH where in the thermodynamic limit, the reduced density matrix of a subsystem corresponding to a pure, finite energy density eigenstate asymptotically becomes equal to the thermal reduced density matrix, as long as the subsystem size is much less than the total system size, irrespective of how large the subsystem is compared to any intrinsic length scale of the system. This allows one to access the properties of the underlying Hamiltonian at arbitrary energy densities/temperatures using just a {single} eigenstate. We provide support for our conjecture by performing an exact diagonalization study of a non-integrable 1D quantum lattice model with only energy conservation. In addition, we examine the case in which the subsystem size is a finite fraction of the total system size, and find that even in this case, many operators continue to match their canonical expectation values, at least approximately. In particular, the von Neumann entanglement entropy equals the thermal entropy as long as the subsystem is less than half the total system. Our results are consistent with the possibility that a single eigenstate correctly predicts the expectation values of \emph{all} operators with support on less than half the total system, as long as one uses a microcanonical ensemble with vanishing energy width for comparison. We also study, both analytically and numerically, a particle number conserving model at infinite temperature which substantiates our conjectures.

}, url = {https://journals.aps.org/prx/accepted/57078K7bAcf16207402997123fd00afaa074a0b75}, author = {James R. Garrison and Tarun Grover} } @article {2195, title = {Locality and digital quantum simulation of power-law interactions}, year = {2018}, abstract = {The propagation of information in non-relativistic quantum systems obeys a speed limit known as a Lieb-Robinson bound. We derive a new Lieb-Robinson bound for systems with interactions that decay with distance r as a power law, 1/rα. The bound implies an effective light cone tighter than all previous bounds. Our approach is based on a technique for approximating the time evolution of a system, which was first introduced as part of a quantum simulation algorithm by Haah et al. [arXiv:1801.03922]. To bound the error of the approximation, we use a known Lieb-Robinson bound that is weaker than the bound we establish. This result brings the analysis full circle, suggesting a deep connection between Lieb-Robinson bounds and digital quantum simulation. In addition to the new Lieb-Robinson bound, our analysis also gives an error bound for the Haah et al. quantum simulation algorithm when used to simulate power-law decaying interactions. In particular, we show that the gate count of the algorithm scales with the system size better than existing algorithms when α\>3D (where D is the number of dimensions).

}, url = {https://arxiv.org/abs/1808.05225}, author = {Minh Cong Tran and Andrew Y. Guo and Yuan Su and James R. Garrison and Zachary Eldredge and Michael Foss-Feig and Andrew M. Childs and Alexey V. Gorshkov} } @article {1986, title = {Extracting entanglement geometry from quantum states}, journal = {Physical Review Letters}, volume = {119}, year = {2017}, month = {2017/10/06}, abstract = {Tensor networks impose a notion of geometry on the entanglement of a quantum system. In some cases, this geometry is found to reproduce key properties of holographic dualities, and subsequently much work has focused on using tensor networks as tractable models for holographic dualities. Conventionally, the structure of the network - and hence the geometry - is largely fixed a priori by the choice of tensor network ansatz. Here, we evade this restriction and describe an unbiased approach that allows us to extract the appropriate geometry from a given quantum state. We develop an algorithm that iteratively finds a unitary circuit that transforms a given quantum state into an unentangled product state. We then analyze the structure of the resulting unitary circuits. In the case of non-interacting, critical systems in one dimension, we recover signatures of scale invariance in the unitary network, and we show that appropriately defined geodesic paths between physical degrees of freedom exhibit known properties of a hyperbolic geometry.

}, doi = {10.1103/PhysRevLett.119.140502}, url = {https://arxiv.org/abs/1704.01974}, author = {Katharine Hyatt and James R. Garrison and Bela Bauer} } @article {1987, title = {Lieb-Robinson bounds on n-partite connected correlations}, journal = {Physical Review A}, volume = {96}, year = {2017}, month = {2017/11/27}, abstract = {Lieb and Robinson provided bounds on how fast bipartite connected correlations can arise in systems with only short-range interactions. We generalize Lieb-Robinson bounds on bipartite connected correlators to multipartite connected correlators. The bounds imply that an\

We study the possible breakdown of quantum thermalization in a model of itinerant electrons on a one-dimensional chain without disorder, with both spin and charge degrees of freedom. The eigenstates of this model exhibit peculiar properties in the entanglement entropy, the apparent scaling of which is modified from a \“volume law\” to an \“area law\” after performing a partial, site-wise measurement on the system. These properties and others suggest that this model realizes a new, nonthermal phase of matter, known as a quantum disentangled liquid (QDL). The putative existence of this phase has striking implications for the foundations of quantum statistical mechanics.

}, doi = {10.1103/PhysRevB.95.054204}, url = {http://link.aps.org/doi/10.1103/PhysRevB.95.054204}, author = {James R. Garrison and Ryan V. Mishmash and Matthew P. A. Fisher} }