Post measurement bipartite entanglement entropy in conformal field theories

TL;DR

Addresses the problem of post-measurement bipartite entanglement entropy in 1+1D critical systems using conformal field theory. It derives exact formulas for S_B under periodic and open boundary conditions by mapping the geometry to a cylinder (or strip) with a slit and evaluating twist-operator correlators, yielding expressions such as $S_B = $ $\frac{c}{6} \ln\left( \frac{L}{\pi} \frac{\sin(\frac{\pi}{L}(l+s))\sin(\frac{\pi}{L}l)}{a\sin(\frac{\pi}{L}s)}\right) + \gamma_1$ for periodic BC and a corresponding open-BC formula. The results are numerically validated in Klein-Gordon (harmonic oscillators) and XX models in a magnetic field, and a universal lower bound for localizable entanglement is derived in the harmonic-oscillator case: $S_{loc}(B,\bar B) > \tfrac{1}{6} \ln\left( \frac{L}{\pi} \frac{\sin(\frac{\pi}{L}(l+s)) \sin(\frac{\pi}{L} l)}{a \sin(\frac{\pi}{L} s)} \right) + \gamma_1$. The work connects post-measurement entanglement to boundary CFT and suggests extensions to other bases, noncritical systems, and holography.

Abstract

We derive exact formulas for bipartite von Neumann entanglement entropy after partial projective local measurement in $1+1$ dimensional conformal field theories with periodic and open boundary conditions. After defining the set up we will check numerically the validity of our results in the case of Klein-Gordon field theory (coupled harmonic oscillators) and spin-$1/2$ XX chain in a magnetic field. The agreement between analytical results and the numerical calculations is very good. We also find a lower bound for localizable entanglement in coupled harmonic oscillators.

Figures (5)

• Figure 1: Euclidean version of the quantum chain with total length $L$. The removed slit domain $A$ has length $s$ and we are interested in calculating entanglement entropy of the region $B$ with length $s$ with the complement in the quantum spin chain. The twist operator can be put at point $P$.
• Figure 2: Entanglement entropy of subregion $B$ for short-range coupled harmonic oscillators with total length $L=50$ and the measurement region sizes $s=10,16$ and $20$ with respect to $\ln f (L,s,l)$, where $f (L,s,l)=\frac{L}{\pi}\frac{\sin\frac{\pi}{L}(l+s)\sin\frac{\pi}{L}l}{a\sin\frac{\pi}{L}s}$. The full line is the function (\ref{['SB for PBC']}) with $c=1$ and $\gamma_1=0.21$.
• Figure 3: Entanglement entropy of subregion $B$ with length $l$ after measurement on subsystem with length $s$ with ferromagnetic outcome for XX model in a magnetic field . From top to bottom we have $n_f=\frac{2}{3},\frac{1}{2}$ and $\frac{1}{3}$. In our numerics $l+s=40$ is fixed. The full line is the function (\ref{['SB for PBC']}) with $L\to \infty$ and $c=1$ and $\gamma_1=0.53$ and the arrows are at $l^c_{n_f}=(1-n_f)(l+s)$ from left to right for $n_f=\frac{2}{3},\frac{1}{2}$ and $\frac{1}{3}$ .
• Figure 4: Entanglement entropy of subregion $B$ with length $l$ after measurement on subsystem with length $s$ with antiferromagnetic outcome for XX model in half filling . In our numerics $l+s=40$ is fixed. The full line is the function (\ref{['SB for PBC']}) with $L\to \infty$ and $c=1$ and $\gamma_1=0.61$.
• Figure 5: Entanglement entropy of subregion $B$ for a system (short-range coupled harmonic oscillators) with total length $L=50$ and the measurement region sizes $s=0,10$ and $20$ with respect to $\ln f (L,s,l)$, where $f (L,s,l)=\frac{2L}{\pi}\frac{\cos\frac{\pi s}{L}-\cos\pi\frac{l+s}{L}}{a\cos^2\frac{\pi s}{2L}}\cot\frac{\pi(l+s)}{2L}$. The full line is the function (\ref{['SB for OBC']}) with $c=1$ and $\gamma_2=-0.02$.