Long-distance entanglement in Motzkin and Fredkin spin chains

Abstract

We derive some entanglement properties of the ground states of two classes of quantum spin chains described by the Fredkin model, for half-integer spins, and the Motzkin model, for integer ones. Since the ground states of the two models are known analytically, we can calculate the entanglement entropy, the negativity and the quantum mutual information exactly. We show, in particular, that these systems exhibit long-distance entanglement, namely two disjoint regions of the chains remain entangled even when the separation is sent to infinity, i.e. these systems are not affected by decoherence. This strongly entangled behavior, occurring both for colorful versions of the models (with spin larger than 1/2 or 1, respectively) and for colorless cases (spin 1/2 and 1), is consistent with the violation of the cluster decomposition property. Moreover we show that this behavior involves disjoint segments located both at the edges and in the bulk of the chains. Contents arXiv:1904.05205v3 [cond-mat.stat-mech]The study of nonlocal properties and their consequences on the dynamics in addition to the violation of the area law for the entanglement entropy are certainly, at the present date, a very challenging field of research. The concept of locality plays a crucial role in physical theories, with far reaching consequences, a fundamental one being the cluster decomposition property [1,2]. This property implies that two-point connected correlation functions go to zero when the separation of the points goes to infinity. This is the reason why two systems very far apart, separated by a large distance, behave independently.Another aspect related to correlations is the quantum entanglement. In bipartite systems the von Neumann or entanglement entropy quantifies how the two parts of the whole system in a quantum state are entangled. This quantity measures non-local quantum correlations and has universal properties, like the fact that, for gapped systems in the ground states, it scales with the area of the boundary of the two subsystems [3]. This property is called area law and is valid for systems with shortrange interactions. In other words, if the interactions are short-ranged the information among the constituents of the system propagates with a finite speed involving a surface surrounding the source of the signal, like an electromagnetic impulse propagating with the speed of light. For critical onedimensional short-range systems the area law is violated logarithmically [4]. Quantum spin chains are promising tools for universal quantum computation [5] and the efficiency may be related to the amount of quantum entanglement. Spin systems with entanglement entropy larger than that dictated by the area law can be used for quantum computing even more efficiently, and breaking down the speed of the propagation of the excitations can represent a breakthrough for quantum information processing.Recently, novel quantum spin models have been introduced, with integer [6-8] (Motzkin model) and half-integer [9, 10] spins (Fredkin model), which...

Dynamics and correlations in Motzkin and Fredkin spin chains

“…This work extends significantly the analysis done in a previous paper 12 , where violation of the cluster decomposition has been observed for the colorful versions of the models looking at the correlation functions of spins along z-directions. The results reported here are in perfect agreement with numerical results for transverse spin correlation functions 19 and with what found for the mutual information, which remains finite even when measured between infinitely distant separated regions 21 . In this paper we considered the colorless versions of the Motzkin and Fredkin spin chains, with spins 1 and 1/2.…”

“…It has been given also a continuum description for the ground-state wavefunctions of those models, which can reproduce quite well some quantities as for example the local magnetization and the entanglement entropy, and whose scaling Hamiltonian is not conformally invariant 19 . Entanglement properties in these models have also been extensively studied, particularly the Renyi entropy 20 , the negativity and the mutual information, revealing a large-distance entanglement behavior 21 , which can produce intriguing out-of equilibrium properties 22 .…”

Entanglement and separability in continuum Rokhsar-Kivelson states

“…Correlation functions can be shown to match as well. However, our field theory result (95) for the (Rényi) entropy of a bulk interval does not agree with the spin chains calculations [41,64]. Our formula does reproduce the 'geometric' part (i.e.…”

Highly Entangled Spin Chains and 2D Quantum Gravity

“…In that case, it will be nice if any technique is developed to obtain the relevant one-point functions 1 N tr M 2n and 1 N tr (M n+r XM n−r X) with M = ∑ s f =1 M f . It will also be worth doing analogous investigation for other entanglement measures like Rényi entanglement entropy [10,11] and mutual information [19], and extending to deformed Motzkin/Fredkin spin chains in which the EEs grow linearly [20][21][22].…”

“…which is an extension of Equation (19) with Equations ( 44) and (45), thereby removing the restriction to the tree diagrams. Equation ( 53) is evaluated around the critical point.…”