Entanglement of Purification in Many Body Systems and Symmetry Breaking

We study the entanglement of purification (EOP), a measure of total correlation between two subsystems $A$ and $B$, for free scalar field theory on a lattice and the transverse-field Ising model by numerical methods. In both of these models, we find that the EOP becomes a nonmonotonic function of the distance between $A$ and $B$ when the total number of lattice sites is small. When it is large, the EOP becomes monotonic and shows a plateaulike behavior. Moreover, we also show that the original reflection symmetry which exchanges $A$ and $B$ can get broken in optimally purified systems. We provide an interpretation of our results in terms of the interplay between classical and quantum correlations.

Figure 1

An example of the setup for our lattice model with $N=20$ and $|A|=|B|=4$. The distance between $A$ and $B$ is $d=1$. There is a $Z_2$ reflection symmetry.

Figure 2

Half of MI (left) and LN (right) for $|A|=|B|=4$ and $N=60$ as a function of $d$, shown for mass $m={10}^{-1},{10}^{-2},{10}^{-3},{10}^{-4}$ (bottom to top).

Figure 3

$E_P$ for $m={10}^{-4}$ and $N=60$ for various $w=|A|=|B|$ (left) and $\mathcal{A}$ for the data set at $w=4$ (right).

Figure 4

EOP for small $N$ with block width $w=2$ and mass $m={10}^{-4}$ (left). EOP for large $N$ with $w=4$ and $m={10}^{-4}$ (right).

Figure 5

EOP for $w=1$, $N=10$ for the critical Ising model (left). The same for $N=4$ (right).

Figure 6

$S_{\max }=\max \{S_{\tilde{A}},S_{\tilde{B}}\}$ and $S_{\min }=\min \{S_{\tilde{A}},S_{\tilde{B}}\}$ of the optimal purifications for $w=1$ and $N=10$ (left). The nonmonotonic behavior of EOP for $w=2$ (right).

Figure 7

A toy model for EOP for 1D many body systems, assuming only short-range quantum entanglement (zigzag lines) for $w=|\tilde{A}|=|\tilde{B}|=3$. At $d=1$, we only show one of the two optimal $Z_2$ symmetry-broken purifications.