Universal entanglement and correlation measure in two-dimensional conformal field theories

TL;DR

This paper introduces a universal entanglement measure for two intervals in 1+1D conformal field theories, providing a formula dependent only on geometry, central charge, and partition function, with broad implications for quantum correlations.

Contribution

It presents a universal expression for entanglement between two intervals in CFTs based on the CCNR criterion, valid for disjoint intervals, and proves it using the replica approach with torus topology.

Findings

Universal entanglement measure depends only on geometry, central charge, and partition function.

Derived formulas for two-interval purity and N-partite information for N≤4.

Numerical verification in the spin-1/2 XXZ chain confirms theoretical results.

Abstract

We calculate the amount of entanglement shared by two intervals in the ground state of a (1+1)-dimensional conformal field theory (CFT), quantified by an entanglement measure $\mathcal{E}$ based on the computable cross norm (CCNR) criterion. Unlike negativity or mutual information, we show that $\mathcal{E}$ has a universal expression even for two disjoint intervals, which depends only on the geometry, the central charge c, and the thermal partition function of the CFT. We prove this universal expression in the replica approach, where the Riemann surface for calculating $\mathcal{E}$ at each order n is always a torus topologically. By analytic continuation, result of n=1/2 gives the value of $\mathcal{E}$. Furthermore, the results of other values of n also yield meaningful conclusions: The n=1 result gives a general formula for the two-interval purity, which enables us to calculate the Renyi-2 N-partite information for N<=4 intervals; while the $n=\infty$ result bounds the correlation function of the two intervals. We verify our findings numerically in the spin-1/2 XXZ chain, whose ground state is described by the Luttinger liquid.