Reflected entropy for free scalars

Abstract

We continue our study of reflected entropy, $R(A,B)$, for Gaussian systems. In this paper we provide general formulas valid for free scalar fields in arbitrary dimensions. Similarly to the fermionic case, the resulting expressions are fully determined in terms of correlators of the fields, making them amenable to lattice calculations. We apply this to the case of a $(1+1)$-dimensional chiral scalar, whose reflected entropy we compute for two intervals as a function of the cross-ratio, comparing it with previous holographic and free-fermion results. For both types of free theories we find that reflected entropy satisfies the conjectural monotonicity property $R(A,BC)\ge R(A,B)$. Then, we move to $(2+1)$ dimensions and evaluate it for square regions for free scalars, fermions and holography, determining the very-far and very-close regimes and comparing them with their mutual information counterparts. In all cases considered, both for $(1+1)$- and $(2+1)$-dimensional theories, we verify that the general inequality relating both quantities, $R(A,B)\ge I(A,B)$, is satisfied. Our results suggest that for general regions characterized by length-scales $L_A\sim L_B\sim L$ and separated a distance $\ell$, the reflected entropy in the large-separation regime ($x\equiv L/\ell \ll 1$) behaves as $R(x)\sim -I(x)\log x$ for general CFTs in arbitrary dimensions.

27 pages, 6 figures

References in corpus (8)

• Islands for Reflected Entropy
• Some Aspects of Entanglement Wedge Cross-Section
• Multipartite Reflected Entropy
• $T\bar{T}$, the entanglement wedge cross section, and the breakdown of the split property
• Mutual information challenges entropy bounds
• Evolution of Entanglement Wedge Cross Section Following a Global Quench
• A Field Theory Study of Entanglement Wedge Cross Section: Odd Entropy
• Entanglement wedge cross-section in shock wave geometries

Cited by in corpus (22)

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• Reflected entropy is not a correlation measure
• Holographic Study of Reflected Entropy in Anisotropic Theories
• Capacity of Entanglement in Local Operators
• Entanglement Wedge Cross Section in Holographic Excited States
• Reflected Entropy in AdS${}_3$/WCFT
• Reflected Entropy for Communicating Black Holes I: Karch-Randall Braneworlds
• Multipartite entanglement in two-dimensional chiral topological liquids
• Entanglement Wedge Cross Section Growth During Thermalization
• Universal entanglement signatures of interface conformal field theories
• The Reflected Entanglement Spectrum for Free Fermions
• Symmetry resolution of the computable cross-norm negativity of two disjoint intervals in the massless Dirac field theory
• Reflected multi-entropy and its holographic dual
• A quantum information perspective on meson melting
• Entanglement Negativity and Defect Extremal Surface
• The Markov gap for geometric reflected entropy
• $2$-Rényi CCNR Negativity of Compact Boson for multiple disjoint intervals
• Rényi Entanglement of Purification and Half Rényi Reflected Entropy in Free Scalar Theory
• Coarse-Grained Fixed-Point Tensor Networks and Holographic Reflected Entropy in 3D Gravity
• Additivity of disjoint interval entanglement in quasiparticle excited states