Entanglement of Purification and Projective Measurement in CFT

Abstract

We investigate entanglement of purification in conformal field theory. By using Reeh-Schlieder theorem, we construct a set of the purification states for ${\rho }_{AB}$, where ${\rho }_{AB}$ is reduced density matrix for subregion $AB$ of a global state $\rho$. The set can be approximated by acting all the unitary observables,located in the complement of subregion $AB$, on the global state $\rho$, as long as the global state $\rho$ 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.

arXiv comment5+1 pages,5 figures

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Cited by in corpus (13)

• Entanglement Wedge Cross Section from CFT: Dynamics of Local Operator Quench
• Balanced Partial Entanglement and the Entanglement Wedge Cross Section
• Generalizations of Reflected Entropy and the Holographic Dual
• Entanglement of Purification in Many Body Systems and Symmetry Breaking
• Quantum and Classical Correlations Inside the Entanglement Wedge
• Mixed State Entanglement for Holographic Axion Model
• Entanglement of Purification in Holographic Systems
• A Field Theory Study of Entanglement Wedge Cross Section: Odd Entropy
• Inequalities of Holographic Entanglement of Purification from Bit Threads
• Entanglement of purification and disentanglement in CFTs
• Improved proof-by-contraction method and relative homologous entropy inequalities
• Non-Conformal Behavior of Holographic Entanglement Measures
• Explicit Pfaffian Formula for Amplitudes of Fermionic Gaussian Pure States in Arbitrary Pauli Bases