Density kernel and temporal quantum information probes in double-scaled SYK

Abstract

Entanglement is conventionally defined through a Hilbert space bipartition on a fixed time slice. Recent work has generalized this notion to multiple times through a non-Hermitian density kernel constructed from temporal correlation functions, encoding causal response, temporal non-classicality and quantum dynamics. We construct this kernel in the double-scaled Sachdev--Ye--Kitaev (DSSYK) model using its exact chord Hilbert space formulation and study the resulting temporal information diagnostics. The direct sum structure of chord space naturally introduces a coarse-graining under which coherence between different chord sectors is removed. We show that the corresponding kernel measures quantify how information is dynamically transferred across subregions in the chord Hilbert space. We find that decreasing $λ$ enhances the propagation of temporal correlation across chord sectors, as well as the loss of memory within a fixed sector. This accelerated temporal dynamics in the small $λ$ limit coincides with the emergence of a semiclassical gravity description. Depending on the chord space partition, the probes exhibit either saturation, a growth followed by a decay, or temporary memory loss. The characteristic propagation time is nicely captured by a wavefront picture in Krylov space, with position-dependent hopping parameters that are determined by the DSSYK Lanczos coefficients. Finally, given the quantum sine-dilaton gravity description of DSSYK and its connections to de Sitter holography, our results provide a controlled setting for probing temporal quantum correlations across quantum and semiclassical gravitational regimes, and motivate a geometric interpretation of the density kernel.