Nature and measure of entanglement in quantum phase transitions

Abstract

Characterizing and quantifying quantum correlations in states of many-particle systems is at the core of a full understanding of phase transitions in matter. In this work, we continue our investigation of the notion of generalized entanglement [Barnum et al., Phys. Rev. A 68, 032308 (2003)] by focusing on a simple Lie-algebraic measure of purity of a quantum state relative to an observable set. For the algebra of local observables on multi-qubit systems, the resulting local purity measure is equivalent to a recently introduced global entanglement measure [Meyer and Wallach, J. Math. Phys. 43, 4273 (2002)]. In the condensed-matter setting, the notion of Lie-algebraic purity is exploited to identify and characterize the quantum phase transitions present in two exactly solvable models: the Lipkin-Meshkov-Glick model, and the spin-1/2 anisotropic XY model in a transverse magnetic field. For the latter, we argue that a natural fermionic observable-set arising after the Jordan-Wigner transformation, better characterizes the transition than alternative measures based on qubits. This illustrates the usefulness of going beyond the standard subsystem-based framework while providing a global disorder parameter for this model. Our results show how generalized entanglement leads to useful tools for distinguishing between the ordered and disordered phases in the case of broken symmetry quantum phase transitions. Additional implications and possible extensions of concepts to other systems of interest in condensed matter physics are also discussed.

Efficient Solvability of Hamiltonians and Limits on the Power of Some Quantum Computational Models

“…3 can be expressed in terms of generalized entanglement [1,14,21]. A pure state is generalized unentangled (GU) with respect to a preferred set O of observables if it is extremal among states considered as linear functionals on O, otherwise it is generalized entangled.…”

“…We use the term GMFH [1] for Hamiltonians belonging to √ −1h for h in a sequence of semisimple compact operator Lie algebras of dimension M ≤ polylog(d) acting on d-dimensional Hilbert spaces. A GMFH is necessarily specified in terms of a basis of h that can be efficiently transformed to a CW basis.…”

“…In this paper, we show close connections between these issues and the efficient (or exact) solvability of Hamiltonians. In particular, we show that a class of quantum models we call generalized mean-field Hamiltonians (GMFHs) [1] is efficiently solvable and furthermore does not provide a stronger-than-classical model of computation: A quantum device engineered to have dynamical gates generated by Hamiltonians from such a set cannot directly simulate universal efficient quantum computation and can be efficiently simulated by a classical computer (CC).…”

Entanglement of indistinguishable particles as a probe for quantum phase transitions in the extended Hubbard model

“…In fact, there are a multitude of distinct approaches and an ongoing debate around the entanglement in these systems [6,[17][18][19][20][21][22][23][24][25][26][27][28][29][30]. Nevertheless, despite the variety, the approaches consist essentially in the analysis of correlations under two different aspects: the correlations genuinely arising from the entanglement between the particles (entanglement of particles) [6,[17][18][19][20][21][22][23][24], and the correlations arising from the entanglement between the modes of the system (entanglement of modes) [25][26][27][28]. These two notions of entanglement are complementary, and the use of one or the other depends on the particular situation under scrutiny.…”

“…The only non-classical correlation present in such states is the exchange, due to the antissymetrization, which does not constitute entanglement. For example, in [17] the analysis follows by using a very elegant mathematical formalism, called GNS (Gelfand-Naimark-Segal) construction, for the case of two fermions, each one with Hilbert space dimension 3 or 4, and two bosons with dimension 3; in [23,24] the authors propose a "Generalized Entanglement (GE)" measure, obtaining a simple formula for the "partial trace", and the set of fermionic unentangled states for an arbitrary number of particles; or also in [6], where a general notion of quantum correlation beyond entanglement (the quantumness of correlations) is investigated by means of an "activation protocol", which yields the same set of states with no quantumness as the above unentangled one.…”