Subsystem quantum Rényi-Bhattacharyya distance in 2D CFTs and its holographic applications

Abstract

We develop a systematic method to compute the subsystem fidelity and Rényi fidelity in two-dimensional conformal field theories (CFTs) from the operator product expansion of twist operators, and apply it to the subsystem quantum Bhattacharyya distance (BD) and Rényi-Bhattacharyya distance (RBD). We validate the method against exact results in the free massless fermion and boson theories, and then use it to compute the BD and RBD in holographic CFTs between two heavy primary states, between two thermal states, and between a heavy primary state and a thermal state. When the two states differ in their classical, order-$c$ data, the distances are of order $c$, and the states are distinguishable already at the level of classical gravity; when the leading data coincide, the distances are of order $1/c$ at every order of the short interval expansion, so that the distinguishability becomes a perturbative quantum gravity effect. For a heavy primary state and its thermal counterpart, the leading contribution to the BD appears at order $\ell^8$ and is of order $c^0$, whereas the RBD is generically of order $c$, in line with the earlier finding that a heavy primary state and a thermal state share the same order-$c$ entanglement entropy but have different order-$c$ Rényi entropies in the short interval expansion.