Rényi Entanglement of Purification and Half Rényi Reflected Entropy in Free Scalar Theory

Abstract

In the AdS/CFT context, the entanglement of purification (EoP, denoted as $E_P$) of CFT is conjectured to be dual to the entanglement wedge cross section (EWCS) in bulk. However, another quantity called reflected entropy $S_R$ is also supposed to be dual to two times the EWCS. A natural question is whether they are the same in holographic CFTs even though they are different in general. Previous studies have shown $E_P\ge \frac{1}{2}{S}_R^{(n)},n\ge 2$ for random tensor networks. In this paper, we study this inequality beyond $n\ge 2$, and we focus on the range $0<n<2$. However, the calculations of EoP are notoriously difficult in general. Thus, our calculations mainly focus on the free scalar theory which is close to the holographic CFTs. We generalized the previous strategy for EoP in \cite{Takayanagi:2018sbw} to the Rényi case. And we have also presented two methods for Rényi reflected entropy, one is using correlators, the other one is Gaussian wavefunction ansatz. Our calculations show that the inequality still holds for $0<n<2$, and it may give us some insights into the equivalence of EoP and half reflected entropy in holographic CFTs. As byproducts of our research, we have also demonstrated the positivity of the Rényi Markov gap and the monotonicity of the Rényi reflected entropy in the free scalar theory.

arXiv comment28 pages, 8 figures; More discussions added; Published version

References in corpus (49)

• Gauge Theory Correlators from Non-Critical String Theory
• Separability Criterion for Density Matrices
• A computable measure of entanglement
• Holographic Derivation of Entanglement Entropy from AdS/CFT
• Entanglement Entropy and Quantum Field Theory
• Entanglement in quantum critical phenomena
• Aspects of Holographic Entanglement Entropy
• Entropy and Area
• Geometric and Renormalized Entropy in Conformal Field Theory
• The logarithmic negativity: A full entanglement monotone that is not convex
• Entanglement entropy in free quantum field theory
• Entanglement negativity in quantum field theory
• Holographic duality from random tensor networks
• Holographic Entanglement of Purification
• Entanglement negativity in extended systems: A field theoretical approach
• The Worldsheet Dual of the Symmetric Product CFT
• The entanglement of purification
• Deriving the ${\text{AdS}}_3/{\text{CFT}}_2$ Correspondence
• Entanglement of purification: from spin chains to holography
• Entanglement negativity and minimal entanglement wedge cross sections in holographic theories
• Tensionless String Spectra on ${\mathrm{AdS}}_3$
• Entanglement Wedge Cross Section from the Dual Density Matrix
• Beyond Toy Models: Distilling Tensor Networks in Full AdS/CFT
• Entanglement Wedge Cross Sections Require Tripartite Entanglement
• Entanglement contour
• Entanglement Wedge Reconstruction and Entanglement of Purification
• Entanglement of Purification for Multipartite States and its Holographic Dual
• Reflected Entropy for an Evaporating Black Hole
• Entanglement Wedge Cross Section from CFT: Dynamics of Local Operator Quench
• Holographic Entanglement of Purification from Conformal Field Theories
• Conditional and Multipartite Entanglements of Purification and Holography
• Partition functions of the tensionless string
• Some Aspects of Entanglement Wedge Cross-Section
• Correlation measures and the entanglement wedge cross-section after quantum quenches in two-dimensional conformal field theories
• Holographic Inequalities and Entanglement of Purification
• Balanced Partial Entanglement and the Entanglement Wedge Cross Section
• Reflected Entropy and Entanglement Wedge Cross Section with the First Order Correction
• Entanglement of Purification in Free Scalar Field Theories
• Generalizations of Reflected Entropy and the Holographic Dual
• Towards Entanglement of Purification for Conformal Field Theories
• Multipartite Reflected Entropy
• Entanglement of Purification and Multiboundary Wormhole Geometries
• Reflected entropy for free scalars
• Reflected entropy is not a correlation measure
• Time dependence of reflected entropy in rational and holographic conformal field theories
• Entanglement wedge cross-section in shock wave geometries
• Entanglement Wedge Cross Section Growth During Thermalization
• An alternative to purification in CFT
• One-half reflected entropy is not a lower bound for entanglement of purification