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arXiv:2203.00556

Mixed State Entanglement For Holographic Systems With A Scalar Hair

Zheng-Chen Song, Zhe Yang, Chong-Ye Chen, Peng Liu

Published Mar 1, 2022• 0 citations
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Abstract

We study the mixed state entanglement of an asymptotic AdS black hole system with scalar hair. Through numerical calculations, we find that the holographic entanglement entropy (HEE) presents a non-monotonic behavior with the system parameter, depending on the size of the subregion. In addition, the mutual information (MI) also shows non-monotonic behavior in certain ranges of system parameters. However, the entanglement wedge minimum cross-section (EWCS), which is a mixed state entanglement measure, increases monotonically with the AdS radius; meanwhile, shows non-monotonic with the temperature. We also give analytical understandings of the phenomena above.

DOI
10.48550/arxiv.2203.00556

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We investigate the holographic mixed-state entanglement measures in the Einstein-Maxwell-Scalar (EMS) theory. Several quantities are computed, including the holographic entanglement entropy (HEE), mutual information (MI), entanglement wedge cross-section (EWCS), and butterfly velocity (vBv_BvB​). Our findings demonstrate that these measures can effectively diagnose phase transitions. Notably, EWCS and MI, as mixed-state entanglement measures, exhibit behavior opposite to that of the HEE. Additionally, we study the butterfly velocity, a dynamic quantum information measure, and observe that it behaves differently from the static quantum information measures. We analyze the butterfly velocity and find that its non-monotonic behavior arises from the competition between two contributions in its expression, which the analytic structure suggests may be correlated with distinct physical interpretations. Moreover, we examine the scaling behavior of the holographic entanglement measures and find that all the critical exponents are equal to 111, which is twice that of the scalar field. We also explore the inequality between EWCS and MI, noting that the growth rate of MI consistently exceeds that of EWCS during phase transitions. These features are expected to be universal across thermodynamic phase transitions, with the inequalities becoming more significant as one moves away from the critical point.
Abstract We investigate the holographic mixed-state entanglement measures in the Einstein–Maxwell-Scalar (EMS) theory. Several quantities are computed, including the holographic entanglement entropy (HEE), mutual information (MI), entanglement wedge cross-section (EWCS), and butterfly velocity ( vBv_BvB​ v B ). Our findings demonstrate that these measures can effectively diagnose phase transitions. Notably, EWCS and MI, as mixed-state entanglement measures, exhibit behavior opposite to that of the HEE. Additionally, we study the butterfly velocity, a dynamic quantum information measure, and observe that it behaves differently from the static quantum information measures. We analyze the butterfly velocity and find that its non-monotonic behavior arises from the competition between two contributions in its expression, which the analytic structure suggests may be correlated with distinct physical interpretations. Moreover, we examine the scaling behavior of the holographic entanglement measures and find that all the critical exponents are equal to 1, which is twice that of the scalar field. We also explore the inequality between EWCS and MI, noting that the growth rate of MI consistently exceeds that of EWCS during phase transitions. These features are expected to be universal across thermodynamic phase transitions, with the inequalities becoming more significant as one moves away from the critical point.
We study the relationship between mixed state entanglement and thermal phase transitions. As a typical example, we compute the holographic entanglement entropy (HEE), holographic mutual information (MI), and the holographic entanglement wedge minimum cross section (EWCS) over the superconductivity phase transition. We find that HEE, MI, and EWCS can all diagnose the superconducting phase transition. They are continuous at the critical point, but their first derivative with respect to temperature is discontinuous. MI decreases with increasing temperature and exhibits a convex behavior, while HEE increases with increasing temperature and exhibits a concave behavior. However, EWCS can exhibit either the same or the opposite behavior as MI, depending on the size of the specific configuration. These results show that EWCS captures more abundant information than HEE and MI. We also provide a new algorithm to compute the EWCS for general configurations.
Abstract We study the mixed state entanglement in a holographic axion model. We find that the holographic entanglement entropy (HEE), mutual information (MI) and entanglement of purification (EoP) exhibit very distinct behaviors with system parameters. The HEE exhibits universal monotonic behavior with system parameters, while the behaviors of MI and EoP relate to the specific system parameters and configurations. We find that MI and EoP can characterize mixed state entanglement better than HEE since they are less affected by thermal effects. Specifically, the MI partly cancels out the thermal entropy contribution, while the holographic EoP is not dictated by the thermal entropy in any situation. Moreover, we argue that EoP is more suitable for describing mixed state entanglement than MI. Because the MI of large configurations are still dictated by the thermal entropy, while the EoP will never be controlled only by the thermal effects.
Abstract In this paper, we investigate the mixed-state entanglement in a model of p-wave superconductivity phase transition using holographic methods. We calculate several entanglement measures, including holographic entanglement entropy (HEE), mutual information (MI), and entanglement wedge cross-section (EWCS). Our results show that these measures display critical behavior at the phase transition points, with the EWCS exhibiting opposite temperature behavior compared to the HEE. Furthermore, we explore the behavior of thermodynamics and holographic quantum information at the zeroth-order phase transition point and find that it is opposite to that observed in the first-order phase transition. Additionally, we find that the critical exponents of all entanglement measures are twice those of the condensate. Our findings also suggest that the EWCS is a more sensitive indicator of the critical behavior of phase transitions than the HEE. Lastly, we uncover a universal inequality in the growth rates of EWCS and MI near critical points in thermal phase transitions, such as p-wave and s-wave superconductivity, suggesting that MI captures more information than EWCS when a phase transition first occurs.
We study mixed state entanglement measures in a higher dimensional T T ¯ deformed field theory at finite temperature. The holographic dual is described by AdS d + 1 black brane geometry with a finite cutoff. We compute the entanglement wedge cross section (EWCS), proposed to be dual to entanglement of purification and holographic entanglement negativity (HEN) for striplike subsystems. The behavior of EWCS and HEN is studied across different regimes of temperature and deformation parameter. It is observed that the deformation and temperature exhibit similar effects on these two entanglement measures. Increasing the deformation leads to a decrease in the entanglement between the subsystems.