Entanglement of Purification and Projective Measurement in CFT

TL;DR

The work tackles entanglement of purification $E_P(\rho_{AB})$ in conformal field theories and constructs purifications via unitary operations in the complement region using the Reeh-Schlieder theorem. It then connects this purification framework to the surface/state correspondence to derive and validate the holographic cross-section formula $E_W(\rho_{AB})=\min\{\text{area}(\Sigma_{AB})\}/(4G)$ and analyzes when purification can be minimized. The paper also investigates conformal-basis projective measurements as a non-unitary route to approximate purification, finding a universal constant gap $\frac{c}{3}\log 2$ between holographic EoP and post-measurement entropy in certain limits, suggesting post-measurement states closely approximate minimal purifications. Overall, it bridges field-theoretic purification, holography, and conformal measurements, clarifying when geometric duals realize $E_P$ and how measurements relate to minimal purification strategies.

Abstract

We investigate entanglement of purification in conformal field theory. By using Reeh-Schlieder theorem, we construct a set of the purification states for $ρ_{AB}$, where $ρ_{AB}$ is reduced density matrix for subregion $AB$ of a global state $ρ$. The set can be approximated by acting all the unitary observables,located in the complement of subregion $AB$, on the global state $ρ$, as long as the global state $ρ$ is \text{cyclic} for every local algebra, e.g., the vacuum state. Combining with the gravity explanation of unitary operations in the context of the so-called surface/state correspondence, we prove the holographic EoP formula. We also explore the projective measurement with the conformal basis in conformal field theory and its relation to the minimization procedure of EoP. Interestingly, though the projective measurement is not a unitary operator, the difference in some limits between holographic EoP and the entanglement entropy after a suitable projective measurement is a constant $\frac{c}{3}\log 2$ up to some contributions from boundary. This suggests the states after projective measurements may approximately be taken as the purification state corresponding to the minimal value of the procedure.

Figures (5)

• Figure 1: A possible division of $\overline{AB}$ by $\tilde{A}$ (red) and $\tilde{B}$ (blue).
• Figure 2: A series of deformation of $\tilde{A}$ (blue) and $\tilde{B}$ (red) for the disconnected entanglement wedge.
• Figure 3: A series of deformation of $\tilde{A}$ (blue) and $\tilde{B}$ (red) for the connected entanglement wedge.
• Figure 4: The state with projective measurement in region $\tilde{B}:=\tilde{B}_1 \tilde{B}_2$ (blue line) can be represented by path integral on the lower half plane with two slits.
• Figure 5: Calculations of holographic EoP for two interval $A$ and $B$ with $l_A=l$ and the distances between $A$, $B$ are $s_1$ and $s_2$.