A Semiclassical Entanglement Wedge in Double-Scaled SYK

Abstract

We study the entanglement structure of partially entangled thermal states in double-scaled SYK , allowing flavor degrees of freedom carried by matter insertions to be in an arbitrary mixed state. In the semiclassical limit, followed by the heavy-probe limit, our replica calculation gives a Faulkner--Lewkowycz--Maldacena formula: the boundary entropy decomposes into an `area' term (independent of the flavor state) and the bulk matter entropy in a corresponding entanglement wedge. Varying the Euclidean locations of the operator insertions produces sharp transitions in the boundary entropy, which we interpret as individual matter excitations crossing the boundary of the wedge and hence as evidence for a sharply defined entanglement wedge. To derive this result, we analyze the semiclassical behavior of general Euclidean $2n$-point functions and give a probabilistic proof of their factorization into sums over products of two-point functions. We also give a complementary derivation from exact spectral representations, using the semiclassical behavior of the quantum $6j$-symbol.