Momentum/Complexity duality and the black hole interior

Abstract

We establish a version of the Momentum/Complexity (PC) duality between the rate of operator complexity growth and an appropriately defined radial component of bulk momentum for a test system falling into a black hole. In systems of finite entropy, our map remains valid for arbitrarily late times after scrambling. The asymptotic regime of linear complexity growth is associated to a frozen momentum in the interior of the black hole, measured with respect to a time foliation by extremal codimension-one surfaces which saturate without reaching the singularity. The detailed analysis in this paper uses the Volume-Complexity (VC) prescription and an infalling system consisting of a thin shell of dust, but the final PC duality formula should have a much wider degree of generality.

Proof of a momentum/complexity correspondence

“…• Even though we chose a purely gravitational solution in order to maximize the differences with our previous analysis for collapsing matter solutions (cf. [9]), notice that most of the details go through even after dropping the Ricci-flatness condition (42). In such case, the ansatz (40) describes in general a mixture of a gravitational and a (null) matter pp-wave with an energy momentum tensor given by…”

On operator growth and emergent Poincaré symmetries

“…It was found in [1,2], for the case of SYK [3][4][5], that certain notion of operator size, to be reviewed in the main text, is related to out-of-time-ordered correlation (OTOC) functions [6][7][8]. Second, because of the relation between operator growth, quantum complexity and the emergence of near horizon symmetries [9][10][11][12]. Finally, due to the broader connection between complexity and operator growth, as discussed from different perspectives in [10,[13][14][15], such as using Nielsen's geometric approach to quantum circuit complexity [16,17] or the recursion method in many-body physics [18].…”

“…This proposal is inspired by the relation (3.22), which is a specific instance of the more general Tomita-Takesaki like equation (A.18). One basically defines the simplest operators annihilating the thermofield double 12 and uses them to define a number operator. For Majorana fermions this was the path chosen in [2].…”

“…In this section we first stress the existence of a natural relation between operator growth and quantum circuit complexity. Afterwards, we discuss the link between these two concepts and quantum chaos [10,12,15]. Finally, in the context of the AdS/CFT correspondence, we also briefly comment on the relation between them and the emergence of classical bulk chaos.…”

“…This might be related to the complexity variation C |ψ(t) − C |ψ O (t) considered in[12]. Variations in quantum circuit complexity have also been considered recently in[15,49].…”