Entanglement Hamiltonians: from field theory, to lattice models and experiments

Abstract

We review results about entanglement (or modular) Hamiltonians of quantum many-body systems in field theory and statistical mechanics models, as well as recent applications in the context of quantum information and quantum simulation.

Local and non-local properties of the entanglement Hamiltonian for two disjoint intervals

“…The entanglement Hamiltonian of an interval A = (a, b) on the half-line x 0 has been found in [22] and it takes the form (5). In particular, the local term is still given by ( 1) with the weight function (12) and the energy density T 00 defined in (3).…”

“…Its form depends on the quantum state in question as well as on the type of partition but also shows certain universal features and it has been the topic of various studies in recent years. For a review, see [5].…”

Quantum Simulation for High Energy Physics

“…Furthermore, elastic and inelastic processes are shown to represent different rates of entanglement spreading in the final state [55], as do the evolution of confined vs. deconfined phases of matter after a quench [328]. While it is only recently that thermalization and associated questions in gauge theories have started to be explored [31,[329][330][331][332] using entanglement measures in classical and quantum Hamiltonian-simulation studies, it is conceivable that such explorations will gain a considerable boost as large-scale programmable simulators become available [333][334][335][336]. Quench experiments may reveal interesting out-of-equilibrium features of the physical system [337,338] while being relatively straightforward to set up, as demonstrated in quantum-simulation experiments of spin systems that start to push the limits of classical computing [339,340].…”

Boiling quantum vacuum- Thermal subsystems from ground state entanglement

“…The entanglement temperature (10) is of the order of the bandwidth (or the hopping amplitude) in the traced-out dimensions and, as such, very high for isotropic models. In strong contrast to generic area-law subsystems which exhibit strongly inhomogeneous spatial temperature profile [22,23], the entanglement entropy here scales as the subsystem volume, and the effective temperature for lower-dimensional systems can be typically regarded as a constant as seen below. The analogy to a true thermal equilibrium state with uniform temperature makes the phenomenon feasible to experimental studies.…”

“…For example, the emergence of effective temperature from ground-state entanglement has lately been identified in systems obeying the entanglement area law, where the entanglement entropy scales as the subsystem boundary [17][18][19]. The area law is known to give rise to subsystem density matrices which are character- * Email: teemu.ojanen@tuni.fi J d Z J j 1 l X S 1 w h Z 5 N e q e x V v R n w M v F z U o Y c 9 V 7 p q 9 u P a S q Z s l Q Q Y z q + l 9 g g I 9 p y K t i k 2 E 0 N S w g d k Q H r O K q I Z C b I Z g d P 8 K l T + j i K t S t l 8 U z 9 P Z E R a c x Y h q 5 T E j s 0 i 9 5 U / M / r p D a 6 D j K u k t Q y R e e L o l R g G + P p 9 7 j P N a N W j B 0 h V H N 3 K 6 Z D o g m 1 L q O i C 8 F f f H m Z N M + r / m X V v 7 ized by a spatially-varying effective temperature that decreases rapidly away from the boundaries [20][21][22][23][24]. Unfortunately, the strongly inhomogeneous entanglement temperature profile is mostly of theoretical interest since its experimental verification poses so far unresolved practical and conceptual issues.…”