Temporal Entanglement from Twist Correlators in 2d Conformal Field Theory and Holography

This paper addresses the challenge of defining and calculating entanglement entropy for timelike-separated subregions, a quantity known as timelike entanglement entropy (TEE). While previous attempts relied on geometric prescriptions or piecewise geodesic constructions, this work seeks a rigorous foundation for TEE within 2D conformal field theory (CFT) and its holographic duals.

The authors propose that TEE and its Rényi extensions should be derived by analytically continuing the replica twist correlator—the standard tool for calculating entanglement in CFT—to time-ordered, timelike-separated insertions. By applying this to various states, including the vacuum, excited states, and quench protocols, they compare the resulting field-theoretic expressions with holographic calculations. In the bulk, they identify the relevant saddles as complex geodesics, which they show are selected by the boundary operator ordering.

The study establishes that the time-ordered replica twist correlator provides a consistent, unambiguous definition of TEE. In holographic settings, this approach resolves ambiguities found in earlier piecewise geometric models by selecting the complex geodesic that minimizes the real part of the length. The imaginary part of the TEE is found to be quantized in units of cπ/6 and is sensitive to the effective causal structure of the system, rather than the underlying dynamics. The authors demonstrate that this framework correctly reproduces results for diverse settings, such as global quenches in AdS-Vaidya spacetimes, where it outperforms previous piecewise geodesic approximations.

This work provides a robust, first-principles derivation of timelike entanglement, bridging the gap between boundary CFT definitions and bulk gravitational descriptions. By grounding TEE in the replica trick, the authors clarify the physical origin of the complex-valued entropy and provide a reliable tool for probing time-dependent processes in quantum many-body systems and holographic spacetimes, where standard entanglement measures are often difficult to define.