Performance of various correlation measures in quantum phase transitions using the quantum renormalization-group method

Abstract

We have investigated the quantum phase transition employing the quantum renormalization-group method while, in most of the previous literature, entanglement (concurrence) has been barely demonstrated. However, it is now well known that entanglement is not the only signature of quantum correlations and a variety of computable measures have been developed to characterize quantum correlations in the composite systems. As an illustration, two cases are elaborated: a one-dimensional anisotropic (i) XXZ model and (ii) an XY model, with various measures of quantum correlations, including quantum discord, geometric discord, measurement-induced disturbance, measurement-induced nonlocality, and violation of Bell inequalities [e.g., Clauser-Horne-ShimonyHolt (CHSH) inequality]. We have proved that all of these correlation measures can effectively detect the quantum critical points associated with quantum phase transitions after several iterations of the renormalization in both cases. Nonetheless, it is shown that some of their dynamical behaviors are not totally similar with entanglement and, even when concurrence vanishes, there still exists some kind of quantum correlation which is not captured by entanglement. Intriguingly, CHSH inequality can never be violated in the whole iteration procedure, which indicates that block-block entanglement cannot revealed by the CHSH inequality. Moreover, the nonanalytic and scaling behaviors of Bell violation have also been discussed in detail. As a by-product, we verify that measurement-induced disturbance is exactly equal to the quantum discord measured by σ z for general X-structured states.

Universal quantum correlation close to quantum critical phenomena

“…Measurement-induced disturbance ( MID ) is defined as the difference of two quantum mutual information respectively of a given state ρ AB shared by two parties ( A and B ) and the corresponding post-measurement state Π ( ρ AB ) 25 29 …”

“…Inspired by the meaningful results about quantum discord, many similar quantum correlation measures based on information-theoretic have been proposed, such as measurement-induced disturbance, geometric discord, measurement-induced nonlocality, quantum deficit and so on 19 22 23 24 25 . The investigations on these methods also have obtained much attention 26 27 28 29 30 31 32 33 34 35 36 37 . Among them one of the topics deserves more attention that is the relations between these quantum correlation measures and QPTs.…”

Quantum discord and its allies: a review of recent progress

“…In addition to this, the Authors of Ref. [532] employed the quantum renormalization group method in the XXZ and the anisotropic XY models, and showed with several iterations of the renormalization that QD as well as GQD can faithfully detect the phase transition points present in the systems. Ground state properties of a 1D Heisenberg system with next-nearest-neighbor interaction has been characterized by using GQD in Ref.…”

Renormalization-group approach to quantum Fisher information in an XY model with staggered Dzyaloshinskii-Moriya interaction

“…Nevertheless, it should be noted that the non-analytic behavior of order parameters in the vicinity of critical points for a quantum spin system is hardly detected unless sophisticated formulation scheme is applied. In this aspect, quantum renormalization-group (RG) method 40 41 has been demonstrated to be valuable and convenient tool to rigorously treat the quantum phase transitions, and highly appreciated examples 42 43 44 45 dealing with the non-analytic behaviors in the vicinity of critical points for quite a few systems are available. These progresses allow a possibility to utilize the quantum RG method for the QFI description of quantum phase transitions.…”

“…These progresses allow a possibility to utilize the quantum RG method for the QFI description of quantum phase transitions. In refs 42 43 44 45 , the physical quantities which have been investigated involve concurrence, entanglement, and discord. They are all quantum correlation measures from the quantum mutual information while QFI from the quantum estimation theory is a more intrinsic and ubiquitous quantity by comparison.…”