Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0810.2564 · Physical Review A

Entanglement under the renormalization-group transformations on quantum states and in quantum phase transitions

Tzu-Chieh Wei

Published Jun 15, 2010• 15 citations
arXivHTMLPDF

Abstract

We consider quantum states under the renormalization-group (RG) transformations introduced by Verstraete et al. [Phys. Rev. Lett. 94, 140601 (2005)] and propose a quantification of entanglement under such RGs (via the geometric measure of entanglement). We examine the resulting entanglement under RG transformations for the ground states of ``matrix-product-state'' Hamiltonians constructed by Wolf et al. [Phys. Rev. Lett. 97, 110403 (2006)] that possess quantum phase transitions. We find that near critical points, the ground-state entanglement exhibits singular behavior. The singular behavior within finite steps of the RG obeys a scaling hypothesis and reveals the correlation length exponent. However, under the infinite steps of RG transformation, the singular behavior is rendered different and is universal only when there is an underlying conformal-field-theory description of the critical point.

DOI
10.1103/physreva.81.062313

Related papers

Ground state entanglement in quantum spin chains

José Ignacio Latorre, E. Rico, Guifré Vidal

A microscopic calculation of ground state entanglement for the XY and Heisenberg models shows the emergence of universal scaling behavior at quantum phase transitions. Entanglement is thus controlled by conformal symmetry. Away from the critical point, entanglement gets saturated by a mass scale. Results borrowed from conformal field theory imply irreversibility of entanglement loss along renormalization group trajectories. Entanglement does not saturate in higher dimensions which appears to limit the success of the density matrix renormalization group technique. A possible connection between majorization and renormalization group irreversibility emerges from our numerical analysis.

•Apr 14, 2003•300 citations

Local measures of entanglement and critical exponents at quantum phase transitions

Lorenzo Campos Venuti, C. Degli Esposti Boschi, M. Roncaglia +1 more

We discuss on general grounds some local indicators of entanglement that have been proposed recently for the study and classification of quantum phase transitions. In particular, we focus on the capability of entanglement in detecting quantum critical points and related exponents. We show that the singularities observed in all local measures of entanglement are a consequence of the scaling hypothesis. In particular, as every nontrivial local observable is expected to be singular at criticality, we single out the most relevant one (in the renormalization group sense) as the best suited for finite-size scaling analysis. The proposed method is checked on a couple of one-dimensional spin systems. The present analysis shows that the singular behavior of local measures of entanglement is fully encompassed in the usual statistical mechanics framework.

arXiv:quant-ph/0508236•Physical Review A•Jan 19, 2006•55 citations

Ground state entanglement in quantum spin chains

José Ignacio Latorre, E. Rico, Guifré Vidal

A microscopic calculation of ground state entanglement for the XY and Heisenberg models shows the emergence of universal scaling behavior at quantum phase transitions. Entanglement is thus controlled by conformal symmetry. Away from the critical point, entanglement gets saturated by a mass scale. Results borrowed from conformal field theory imply irreversibility of entanglement loss along renormalization group trajectories. Entanglement does not saturate in higher dimensions which appears to limit the success of the density matrix renormalization group technique. A possible connection between majorization and renormalization group irreversibility emerges from our numerical analysis.

Quantum Information and Computation•Jan 1, 2004•414 citations

Ground state entanglement in quantum spin chains

José Ignacio Latorre, E. Rico, Guifré Vidal

A microscopic calculation of ground state entanglement for the XY and Heisenberg models shows the emergence of universal scaling behavior at quantum phase transitions. Entanglement is thus controlled by conformal symmetry. Away from the critical point, entanglement gets saturated by a mass scale. Results borrowed from conformal field theory imply irreversibility of entanglement loss along renormalization group trajectories. Entanglement does not saturate in higher dimensions which appears to limit the success of the density matrix renormalization group technique. A possible connection between majorization and renormalization group irreversibility emerges from our numerical analysis.

arXiv:quant-ph/0304098•Apr 14, 2003•2 citations

Universal Geometric Entanglement Close to Quantum Phase Transitions

Román Orús

Under successive renormalization group transformations applied to a quantum state |Psi of finite correlation length xi, there is typically a loss of entanglement after each iteration. How good it is then to replace |Psi by a product state at every step of the process? In this Letter we give a quantitative answer to this question by providing first analytical and general proofs that, for translationally invariant quantum systems in one spatial dimension, the global geometric entanglement per region of size L>>xi diverges with the correlation length as (c/12)log(xi/epsilon) close to a quantum critical point with central charge c, where is a cutoff at short distances. Moreover, the situation at criticality is also discussed and an upper bound on the critical global geometric entanglement is provided in terms of a logarithmic function of L.

arXiv:0711.2556•Physical Review Letters•Apr 4, 2008•103 citations

The universal behavior of the quantum correlation of Ising model in a transverse field

Meng Qin, Zhongzhou Ren, Xin Zhang

We study the ground state quantum correlation of Ising model in a transverse field by implementing the quantum renormalization group (QRG) theory. It is shown that various quantum correlation measures and the Clauser-Horne-Shimony-Holt inequality will highlight the critical point related with quantum phase transitions, and demonstrate nonanalytic phenomenon and scaling behavior when the size of the system becomes large. Our results also indicate a universal behavior of the critical exponent of Ising model under QRG that the critical exponent will not change with the variation of different quantities, even when the quantities vary from entanglement measures to quantum correlation measures. This means that the two kinds of quantum correlation criterion, including the entanglement/separability paradigm and the information-theoretic paradigm, have some connections between them. These remarkable behaviors may have important implications on condensed matter physics because the critical exponent directly associates with the correlation length exponent.

•Sep 16, 2015•0 citations
We consider quantum states under the renormalization-group (RG) transformations introduced by Verstraete et al. [Phys. Rev. Lett. 94, 140601 (2005)] and propose a quantification of entanglement under such RGs (via the geometric measure of entanglement). We examine the resulting entanglement under RG transformations for the ground states of ``matrix-product-state'' Hamiltonians constructed by Wolf et al. [Phys. Rev. Lett. 97, 110403 (2006)] that possess quantum phase transitions. We find that near critical points, the ground-state entanglement exhibits singular behavior. The singular behavior within finite steps of the RG obeys a scaling hypothesis and reveals the correlation length exponent. However, under the infinite steps of RG transformation, the singular behavior is rendered different and is universal only when there is an underlying conformal-field-theory description of the critical point.
A microscopic calculation of ground state entanglement for the XY and Heisenberg models shows the emergence of universal scaling behavior at quantum phase transitions. Entanglement is thus controlled by conformal symmetry. Away from the critical point, entanglement gets saturated by a mass scale. Results borrowed from conformal field theory imply irreversibility of entanglement loss along renormalization group trajectories. Entanglement does not saturate in higher dimensions which appears to limit the success of the density matrix renormalization group technique. A possible connection between majorization and renormalization group irreversibility emerges from our numerical analysis.
We discuss on general grounds some local indicators of entanglement that have been proposed recently for the study and classification of quantum phase transitions. In particular, we focus on the capability of entanglement in detecting quantum critical points and related exponents. We show that the singularities observed in all local measures of entanglement are a consequence of the scaling hypothesis. In particular, as every nontrivial local observable is expected to be singular at criticality, we single out the most relevant one (in the renormalization group sense) as the best suited for finite-size scaling analysis. The proposed method is checked on a couple of one-dimensional spin systems. The present analysis shows that the singular behavior of local measures of entanglement is fully encompassed in the usual statistical mechanics framework.
A microscopic calculation of ground state entanglement for the XY and Heisenberg models shows the emergence of universal scaling behavior at quantum phase transitions. Entanglement is thus controlled by conformal symmetry. Away from the critical point, entanglement gets saturated by a mass scale. Results borrowed from conformal field theory imply irreversibility of entanglement loss along renormalization group trajectories. Entanglement does not saturate in higher dimensions which appears to limit the success of the density matrix renormalization group technique. A possible connection between majorization and renormalization group irreversibility emerges from our numerical analysis.
A microscopic calculation of ground state entanglement for the XY and Heisenberg models shows the emergence of universal scaling behavior at quantum phase transitions. Entanglement is thus controlled by conformal symmetry. Away from the critical point, entanglement gets saturated by a mass scale. Results borrowed from conformal field theory imply irreversibility of entanglement loss along renormalization group trajectories. Entanglement does not saturate in higher dimensions which appears to limit the success of the density matrix renormalization group technique. A possible connection between majorization and renormalization group irreversibility emerges from our numerical analysis.
Under successive renormalization group transformations applied to a quantum state |Psi of finite correlation length xi, there is typically a loss of entanglement after each iteration. How good it is then to replace |Psi by a product state at every step of the process? In this Letter we give a quantitative answer to this question by providing first analytical and general proofs that, for translationally invariant quantum systems in one spatial dimension, the global geometric entanglement per region of size L>>xi diverges with the correlation length as (c/12)log(xi/epsilon) close to a quantum critical point with central charge c, where is a cutoff at short distances. Moreover, the situation at criticality is also discussed and an upper bound on the critical global geometric entanglement is provided in terms of a logarithmic function of L.
We study the ground state quantum correlation of Ising model in a transverse field by implementing the quantum renormalization group (QRG) theory. It is shown that various quantum correlation measures and the Clauser-Horne-Shimony-Holt inequality will highlight the critical point related with quantum phase transitions, and demonstrate nonanalytic phenomenon and scaling behavior when the size of the system becomes large. Our results also indicate a universal behavior of the critical exponent of Ising model under QRG that the critical exponent will not change with the variation of different quantities, even when the quantities vary from entanglement measures to quantum correlation measures. This means that the two kinds of quantum correlation criterion, including the entanglement/separability paradigm and the information-theoretic paradigm, have some connections between them. These remarkable behaviors may have important implications on condensed matter physics because the critical exponent directly associates with the correlation length exponent.