Entanglement entropy after a partial projective measurement in $1+1$ dimensional conformal field theories: exact results

TL;DR

This paper provides exact analytical results for the entanglement entropy in 1+1 dimensional conformal field theories after a partial projective measurement, revealing dependence on central charge, operator content, and showing power-law decay with distance.

Contribution

It derives exact formulas for post-measurement entanglement entropy in 1+1D CFTs, including numerical validation and insights into massive quantum field theories.

Findings

Entanglement entropy depends on central charge and operator content.

Post-measurement entanglement decreases as a power-law with distance.

Numerical results support the analytical formulas.

Abstract

We calculate analytically the R\'enyi bipartite entanglement entropy $S_{\alpha}$ of the ground state of $1+1$ dimensional conformal field theories (CFT) after performing a projective measurement in a part of the system. We show that the entanglement entropy in this setup is dependent on the central charge and the operator content of the system. When the measurement region $A$ separates the two parts $B$ and $\bar{B}$, the entanglement entropy between $B$ and $\bar{B}$ decreases like a power-law with respect to the characteristic distance between the two regions with an exponent which is dependent on the rank $\alpha$ of the R\'enyi entanglement entropy and the smallest scaling dimension present in the system. We check our findings by making numerical calculations on the Klein-Gordon field theory (coupled harmonic oscillators) after fixing the position (partial measurement) of some of the oscillators. We also comment on the post-measurement entanglement entropy in the massive quantum field theories.