An alternative to purification in CFT

Abstract

In conformal field theories, in contrast to \emph{adding} some auxiliary states into the bipartite mixed state ${\rho }_{AB}$ as the usual purifications do, we show a pure entangled state ${\psi }_{AB}$ can be constructed by \emph{subtracting} the undetectable regions. In this pure state ${\psi }_{AB}$, the von Neumann entropy $S_{\text{vN}}(A)$ naturally captures quantum entanglement between $A$ and $B$. We verify that $S_{\text{vN}}(A)$ is equal to the entanglement wedge cross-section $E_W$ in AdS spacetime, which is conjectured to be the holographic dual of the entanglement of purification. We show such constructed entanglement entropy has a phase transition. The ordinary entanglement entropies of critical and non-critical QFTs are simply limits of the two phases.

4 pages, 4 figures