Quantum entanglement is the most surprising feature of quantum mechanics. Entanglement is simultaneously responsible for the difficulty of simulating quantum matter on a classical computer and the exponential speedups afforded by quantum computers. Ground states of quantum many-body systems typically satisfy an "area law": The amount of entanglement between a subsystem and the rest of the system is proportional to the area of the boundary. A system that obeys an area law has less entanglement and can be simulated more efficiently than a generic quantum state whose entanglement could be proportional to the total system's size. Moreover, an area law provides useful information about the low-energy physics of the system. It is widely believed that for physically reasonable quantum systems, the area law cannot be violated by more than a logarithmic factor in the system's size. We introduce a class of exactly solvable one-dimensional physical models which we can prove have exponentially more entanglement than suggested by the area law, and violate the area law by a square-root factor. This work suggests that simple quantum matter is richer and can provide much more quantum resources (i.e., entanglement) than expected. In addition to using recent advances in quantum information and condensed matter theory, we have drawn upon various branches of mathematics such as combinatorics of random walks, Brownian excursions, and fractional matching theory. We hope that the techniques developed herein may be useful for other problems in physics as well.quantum matter | local Hamiltonians entanglement entropy | area law | spin chains | Hamiltonian gap S tudy of quantum many-body systems (QMBSs) is the study of quantum properties of matter and quantum resources (e.g., entanglement) provided by matter for building revolutionary new technologies such as a quantum computer. One of the properties of the QMBS is the amount of entanglement among parts of the system (1, 2). Entanglement can be used as a resource for quantum technologies and information processing (2-5); however, at a fundamental level it provides information about the quantum state of matter, such as near-criticality (6, 7). Moreover, systems with high entanglement are usually hard to simulate on a classical computer (8). How much entanglement do natural QMBSs possess? What are the fundamental limits on simulation of physical systems?The area law says that entanglement entropy between two subsystems of a system is proportional to the area of the boundary between them. A generic state does not obey an area law (9); therefore, obeying an area law implies that a QMBS contains much less quantum correlation than generically expected. One can imagine that any given system has inherent constraints such as underlying symmetries and locality of interaction that restrict the states to reside on special submanifolds rendering their simulation efficient (10).Since the discovery that the Affleck-Kennedy-Lieb-Tasaki (AKLT) model (11) is exactly solvable, and that the density matrix re...

Frustration-free (FF) spin chains have a property that their ground state minimizes all individual terms in the chain Hamiltonian. We ask how entangled the ground state of a FF quantum spin-s chain with nearest-neighbor interactions can be for small values of s. While FF spin-1/2 chains are known to have unentangled ground states, the case s=1 remains less explored. We propose the first example of a FF translation-invariant spin-1 chain that has a unique highly entangled ground state and exhibits some signatures of a critical behavior. The ground state can be viewed as the uniform superposition of balanced strings of left and right brackets separated by empty spaces. Entanglement entropy of one half of the chain scales as 1/2 log n+O(1), where n is the number of spins. We prove that the energy gap above the ground state is polynomial in 1/n. The proof relies on a new result concerning statistics of Dyck paths which might be of independent interest.

The explicit form of the zeroth Green's function in the Hückel model, approximated by the negative of the inverse of the Hückel matrix, has direct quantum interference consequences for molecular conductance. We derive a set of rules for transmission between two electrodes attached to a polyene, when the molecule is extended by an even number of carbons at either end (transmission unchanged) or by an odd number of carbons at both ends (transmission turned on or annihilated). These prescriptions for the occurrence of quantum interference lead to an unexpected consequence for switches which realize such extension through electrocyclic reactions: for some specific attachment modes the chemically closed ring will be the ON position of the switch. Normally the signs of the entries of the Green's function matrix are assumed to have no physical significance; however, we show that the signs may have observable consequences. In particular, in the case of multiple probe attachments - if coherence in probe connections can be arranged - in some cases new destructive interference results, while in others one may have constructive interference. One such case may already exist in the literature.

An exponential falloff with separation of electron transfer and transport through molecular wires is observed and has attracted theoretical attention. In this study, the attenuation of transmission in linear and cyclic polyenes is related to bond alternation. The explicit form of the zeroth Green's function in a Hückel model for bond-alternated polyenes leads to an analytical expression of the conductance decay factor β. The β values calculated from our model (β(CN) values, per repeat unit of double and single bond) range from 0.28 to 0.37, based on carotenoid crystal structures. These theoretical β values are slightly smaller than experimental values. The difference can be assigned to the effect of anchoring groups, which are not included in our model. A local transmission analysis for cyclic polyenes, and for [14]annulene in particular, shows that bond alternation affects dramatically not only the falloff behavior but also the choice of a transmission pathway by electrons. Transmission follows a well-demarcated system of π bonds, even when there is a shorter-distance path with roughly the same kind of "electronic matter" intervening.