Comments on Entanglement Negativity in Holographic Field Theories

TL;DR

This work examines entanglement negativity as a computable measure of distillable entanglement in relativistic quantum field theories and holographic duals. It first clarifies negativity (and logarithmic negativity) via PPT and Renyi constructions, then derives a concise thermofield-double result ${\mathscr E}(\psi_\beta) = \log \frac{Z(\beta/2)^2}{Z(\beta)} = \beta\big(F(\beta) - F(\beta/2)\big)$, establishing a bridge between entanglement and thermodynamics. It proceeds to compute negativity for CFT vacua across dimensions by exploiting the Casini mapping to hyperbolic space, obtaining explicit results in $d=2$ and for free theories, and extends to holographic CFTs where ${\mathscr E}(\psi_p)$ scales with the entanglement entropy as ${\mathscr E}(\psi_p) = S(\rho_A) \ {\cal X}_d^{\text{hol}}$, with a dimension-dependent factor ${\cal X}_d^{\text{hol}}$. The paper then outlines broad holographic expectations for pure and mixed-state negativity, discusses possible phase-transition-like behavior in multi-interval setups, and emphasizes open questions about the geometric meaning of ${\cal X}_{\cal A}$ and the link between negativity and spacetime geometry. These insights illuminate how distillable entanglement behaves in strongly coupled quantum fields and their gravity duals, with potential implications for the emergence of spacetime from quantum information.

Abstract

We explore entanglement negativity, a measure of the distillable entanglement contained in a quantum state, in relativistic field theories in various dimensions. We first give a general overview of negativity and its properties and then explain a well known result relating (logarithmic) negativity of pure quantum states to the Renyi entropy (at index $1/2$), by exploiting the simple features of entanglement in thermal states. In particular, we show that the negativity of the thermofield double state is given by the free energy difference of the system at temperature $T$ and $2\,T$ respectively. We then use this result to compute the negativity in the vacuum state of conformal field theories in various dimensions, utilizing results that have been derived for free and holographic CFTs in the literature. We also comment upon general lessons to be learnt about negativity in holographic field theories.