Size and momentum of an infalling particle in the black hole interior

Ying Zhao,Felix M. Haehl

doi:10.1007/jhep06(2021)056

Abstract

The future interior of black holes in AdS/CFT can be described in terms of a quantum circuit. We investigate boundary quantities detecting properties of this quantum circuit. We discuss relations between operator size, quantum complexity, and the momentum of an infalling particle in the black hole interior. We argue that the trajectory of the infalling particle in the interior close to the horizon is related to the growth of operator size. The notion of size here differs slightly from the size which has previously been related to momentum of exterior particles and provides an interesting generalization. The fact that both exterior and interior momentum are related to operator size growth is a manifestation of complementarity.

Highlights

Quantum gates.1 This is shown in figure 1, which was argued to be the circuit representing the future interior of the bulk dual geometry [4, 7, 8]
We show that a particular four-point function counts the number of healthy gates in the interior circuit
If we say that a circuit like figure 1 is stored in the future interior, what boundary quantities can we use to detect the properties of the circuit? In particular, what boundary quantity gives the number of healthy gates in the circuit as a function of right time? In figure 5, F denotes the fraction of healthy gates at time step tRi

Summary

Interior circuit corresponding to thermofield double

We first look at the quantum circuit without perturbations. We represent the thermofield double by S Bell pairs and model the dynamics by a Hayden-Preskill type circuit [13]: at each time step the qubits are randomly grouped into. S 2 pairs, and on each pair a randomly chosen 2-qubit gate is applied. As the left (right) boundary time increases, the circuit grows toward left (right). Note that the orange gates can be undone from both sides. As we scan through the circuit, we can ask about the number of healthy gates per unit circuit time. In the case of the thermofield double it is a constant as the circuit is uniform in time

Interior circuit corresponding to perturbed thermofield double

Counting the number of healthy gates

Growth of an operator in the interior circuit

Operator growth in the SYK model

Operator growth in the interior circuit

Time dependence of complexity and properties of the interior circuit

Operator size and exterior momentum

Operator size and interior momentum

Interpretation in terms of complementarity

Conclusion and discussion

A Computation of two-sided four-point function

B Momentum calculations in JT gravity