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) ≥ 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) ≥ I (A, B), is satisfied. Our results suggest that for general regions characterized by length-scales LA ∼ LB ∼ L and separated a distance ℓ, the reflected entropy in the large-separation regime (x ≡ L/ℓ ≪ 1) behaves as R(x) ∼ −I(x) log x for general CFTs in arbitrary dimensions.

Capacity of entanglement in local operators

“…This might shed light on the universal scaling behavior of capacity in different time regimes, in a similar spirit to the scaling of entropies [72]. Another exciting direction will be to explore the evolution of capacity in pure states in equilibrium [73] and connect with other information-theoretic measures like reflected entropy [74][75][76], complexity [77][78][79][80][81][82][83][84][85], and logarithmic negativity [86][87][88] to have a better understanding of the complete picture.…”

The Markov gap for geometric reflected entropy

“…One can think of this as a rather delicate energy condition on quantum fields, where quantum field theory states violating some classical energy conditions must have a sufficiently large S R,bulk − I bulk to make up for the violation. Any research effort in this direction would likely start with the results of [71][72][73], where the authors compared reflected entropy and mutual information in simple quantum field theories.…”

Entanglement wedge cross section in holographic excited states

“…Further, the nonequilibrium evolution of EWCS for various quench protocols has been considered in [21][22][23][24][25][26]. Related investigations attempting to better understand the corresponding measures from the perspective of the boundary field theory have also appeared in [27][28][29][30][31]. This paper presents another step in this research program, in which we investigate the behavior of EWCS for different bulk geometries dual to boundary excited states.…”

“…An interesting question is if either of these behaviors is reflected in the QFT calculations of mixed state correlation measures dual to EWCS. Indeed, free field theories provide a particularly well-controlled setup for studying various aspects of the corresponding measures, e.g., EoP [4,5] and reflected entropy [29,30]. Let us emphasize that to the best of our knowledge, these studies mainly focused on vacuum (Gaussian) states without any excitations or Ising model at the critical point which has conformal symmetry.…”