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arXiv:cond-mat/0603004 · Physical Review A

Measuring entanglement entropies in many-body systems

Israel Klich, Gil Refael, Alessandro Silva

Published Sep 11, 2006• 79 citations
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Abstract

We explore the relation between entanglement entropy of quantum many-body systems and the distribution of corresponding, properly selected, observables. Such a relation is necessary to actually measure the entanglement entropy. We show that, in general, the Shannon entropy of the probability distribution of certain symmetry observables gives a lower bound to the entropy. In some cases this bound is saturated and directly gives the entropy. We also show other cases in which the probability distribution contains enough information to extract the entropy: we show how this is done in several examples including BEC wave functions, the Dicke model, $XY$ spin chain, and chains with strong randomness.

DOI
10.1103/physreva.74.032306

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As two of the most important entanglement measures---the entanglement of formation and the entanglement of distillation---have so far been limited to bipartite settings, the study of other entanglement measures for multipartite systems appears necessary. Here, connections between two other entanglement measures---the relative entropy of entanglement and the geometric measure of entanglement---are investigated. It is found that for arbitrary pure states the latter gives rise to a lower bound on the former. For certain pure states, some bipartite and some multipartite, this lower bound is saturated, and thus their relative entropy of entanglement can be found analytically in terms of their known geometric measure of entanglement. For certain mixed states, upper bounds on the relative entropy of entanglement are also established. Numerical evidence strongly suggests that these upper bounds are tight, i.e., they are actually the relative entropy of entanglement.
We study the Shannon entropy of the probability distribution resulting from the ground-state wave function of a one-dimensional quantum model. This entropy is related to the entanglement entropy of a Rokhsar-Kivelson-type wave function built from the corresponding two-dimensional classical model. In both critical and massive cases, we observe that it is composed of an extensive part proportional to the length of the system and a subleading universal constant S0{S}_{0}S0​. In c=1c=1c=1 critical systems (Tomonaga-Luttinger liquids), we find that S0{S}_{0}S0​ is a simple function of the boson compactification radius. This finding is based on a field-theoretical analysis of the Dyson-Gaudin gas related to dimer and Calogero-Sutherland models. We also performed numerical demonstrations in the dimer models and the spin-1/2 XXZXXZXXZ chain. In a massive (crystal) phase, S0{S}_{0}S0​ is related to the ground-state degeneracy. We also examine this entropy in the Ising chain in a transverse field as an example showing a c=1/2c=1/2c=1/2 critical point.
The uncertainty principle is an important tenet and active field of research in quantum physics. Information-theoretic uncertainty relations, formulated using entropies, provide one approach to quantifying the extent to which two non-commuting observables can be jointly measured. Recent theoretical analysis predicts that general quantum measurements are necessary to saturate some such uncertainty relations and thereby overcome certain limitations of projective measurements. Here, we experimentally test a tight information-theoretic measurement uncertainty relation with neutron spin- 1 / 2 qubits. The noise associated to the measurement of an observable is defined via conditional Shannon entropies and a tradeoff relation between the noises for two arbitrary spin observables is demonstrated. The optimal bound of this tradeoff is experimentally obtained for various non-commuting spin observables. For some of these observables this lower bound can be reached with projective measurements, but we observe that, in other cases, the tradeoff is only saturated by general quantum measurements (i.e. positive-operator valued measures) as predicted theoretically. These results showcase experimentally the advantage obtainable by general quantum measurements over projective measurements when probing certain uncertainty relations.