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Physica Scripta

Scaling of Einstein–Podolsky–Rosen steering in spin chains

W. W. Cheng, Jyrki Piilo

Published Oct 22, 2019• 5 citations
Publisher

Abstract

Abstract Symmetric quantum properties and correlations have been often used earlier to study quantum phase transitions in many body systems and spin models. However, the use of asymmetric quantum features, such as steering, have attracted smaller amount of attention in this context, so far. We study EPR steering and quantum phase transitions in the Ising model in transverse field and in the anisotropic XY model by using steering robustness and quantum renormalization group method. The key ingredients of the quantum criticality near the critical points, such as finite-size scaling behaviour and critical exponents, are investigated in detail with two commonly used spin models. Our results show that the first derivative of steering robustness between two blocks diverges near the quantum phase transition points for both models, and exhibits a finite-size scaling effect. Moreover, we explore in detail the asymmetric character of EPR steering, by taking into accounts finite-size effects and measurement number, in the reduced block state in the anisotropic XY model. The results imply that one-way EPR steering does not exist in large system under the limit L → ∞ despite the fact that EPR steering for the reduced block state is asymmetric.

DOI
10.1088/1402-4896/ab5029

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Abstract Symmetric quantum properties and correlations have been often used earlier to study quantum phase transitions in many body systems and spin models. However, the use of asymmetric quantum features, such as steering, have attracted smaller amount of attention in this context, so far. We study EPR steering and quantum phase transitions in the Ising model in transverse field and in the anisotropic XY model by using steering robustness and quantum renormalization group method. The key ingredients of the quantum criticality near the critical points, such as finite-size scaling behaviour and critical exponents, are investigated in detail with two commonly used spin models. Our results show that the first derivative of steering robustness between two blocks diverges near the quantum phase transition points for both models, and exhibits a finite-size scaling effect. Moreover, we explore in detail the asymmetric character of EPR steering, by taking into accounts finite-size effects and measurement number, in the reduced block state in the anisotropic XY model. The results imply that one-way EPR steering does not exist in large system under the limit L → ∞ despite the fact that EPR steering for the reduced block state is asymmetric.
Abstract We explore Einstein–Podolsky–Rosen steering, measured by steering robustness, in the ground states of several typical models that exhibit a quantum phase transition. For the anisotropic XY model, steering robustness approaches zero around the critical point and vanishes in the ferromagnetic phase despite the fact that there exist other quantum nonlocalities, e.g. quantum entanglement. For the Heisenberg XXZ model, steering robustness exhibits some similar behavior as entanglement around the infinite-order quantum phase transition point Δ = 1, e.g. reaching its maximum. As a further example, we also consider steering robustness in the Lipkin–Meshkov–Glick collective spin model. It is then shown that steering robustness disappears at the transition point and remains at zero in the fully polarized symmetric phase, just like the behavior of entanglement and Bell nonlocality.
We investigate the Einstein-Podolsky-Rosen (EPR) steering and its criticality in quantum phase transition. It is found that the EPR steerability function of the ground state of XY spin chain exhibits nonanalytic feature in the vicinity of a quantum phase transition by showing that its derivative with respect to anisotropy parameter diverges at the critical point. We then verify the universality of the critical phenomena of the EPR steerability function in the system. We also use two-qubit EPR-steering inequality to explore the relation between EPR steering and quantum phase transition.
In this paper the entanglement and quantum phase transition of the anisotropic spin-1/21/21/2 XY\mathit{XY}XY model are studied by using the quantum renormalization-group method. By solving the renormalization equations, we get the trivial and nontrivial fixed points, which correspond to the phase of the system and the critical point, respectively. The concurrence between two blocks are calculated and it is found that when the number of iterations of the renormalization tends to infinity, the concurrence develops two saturated values that are associated with two different phases, i.e., Ising-like and spin-fluid phases. We also investigate the first derivative of the concurrence and find that there exists nonanalytic behaviors at the quantum critical point, which are directly associated with the divergence of the correlation length. To gain further insight, the scaling behaviors of the system are analyzed and it is shown that the maximum value of the first derivative of the concurrence reaches infinity and the critical point is approached as the size of the system increases.
Abstract Quantum steering ellipsoids (QSEs) can serve as a useful geometric tool for describing both the strength and type of quantum correlations between two subsystems of a compound system. By employing the quantum renormalization-group method, we focus on investigating the relation between QSEs and the quantum phase transition (QPT) in the anisotropic spin XY model. The results indicate that the QPT is well visualized in terms of the shape of the QSE, i.e. it is an oblate spheroid in the spin-fluid phase and a needle in the Néel phase. Meanwhile, after several iterations of renormalization, the QSE volume V undergoes a contraction mutation, and can develop two saturated values at the critical points associated with the QPT, which correspond to two different phases: the spin-fluid phase and the Néel phase. We also find that the QSE is closely associated with quantum entanglement in the model, i.e. the volume of the QSE between blocks is more than 4π/81 when the system is in the spin-fluid phase, which indicates that the system must be entangled. Furthermore, the nonanalytic and scaling behaviors of the volume of the QSE have been analyzed in detail, and the results convince us that the quantum critical properties are connected with the behavior of the QSE.
Abstract Using the quantum renormalization group approach, we have investigated the entanglement and critical behavior of an anisotropic XXZ Heisenberg spin chain. The resolution of the quantum renormalization equations yields stable and unstable fixed points, corresponding respectively to the phases and critical point of the system. It is shown that a quantum phase transition exists at a critical anisotropy parameter , separating the Néel phase from the spin-fluid phase. Then, the concurrence between two blocks of sites and the quantum coherence have been calculated and presented for various renormalization group iterations. In the thermodynamic limit, i.e. , for higher renormalization steps, the concurrence and coherence exhibit two different saturated values in the two regions separated by Δ c . The first derivative of concurrence diverges at the critical point as the size of the chain increases, showing nonanalytic behavior. Moreover, the scaling behavior has been examined near the critical point with some universal exponential scaling laws.
Abstract Symmetric quantum correlations and properties have been often investigated earlier in the process of quantum decoherence. However, the asymmetric quantum features, such as steering, have been rarely considered in this context. In this work, we study Einstein–Podolsky–Rosen steering, measured by steering robustness R , for two-qubit systems under asymmetric noise channels. For the one-sided asymmetric noise channels, we explore the influence of several typical channels on the asymmetric decay of steering. Our results show that the asymmetric steering is directly linked to the initial state as well as the type of noises. We prove that asymmetric decay can be used to characterize the properties of the channels in some certain. For the two-sided asymmetric noise channels, the results show that the behavior of steering are also both initial states and combination types depended. Moreover, we show that there always exist asymmetric steering even the initial state is high symmetry for the depolarizing-amplitude damping combination channel.