An alternative to purification in CFT

In conformal field theories, in contrast to adding some auxiliary states into the bipartite mixed state ρAB as the usual purifications do, we show a pure entangled state ψAB can be constructed by subtracting the undetectable regions. In this pure state ψAB, the von Neumann entropy SvN(A) naturally captures quantum entanglement between A and B. We verify that SvN(A) is equal to the entanglement wedge cross-section EW 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.