Chaos in quantum channels

TL;DR

The paper develops an information-theoretic framework for chaos and scrambling in unitary quantum channels by mapping channels to entangled states and using OTO correlators together with the tripartite information I_3 as scrambling diagnostics. It establishes a direct link between the decay of OTOs and shrinking bipartite mutual informations, showing that chaotic dynamics delocalize input information across the output. Through numerics on spin chains and Majorana models, and analytical results from perfect tensor networks, the work demonstrates that chaotic channels approach Haar-scrambled entanglement structures and exhibit ballistic operator growth. The findings illuminate the relationship between the butterfly effect, information propagation speeds (v_B and v_E), and the capacity for quantum information processing, with implications for holography and computational complexity.

Abstract

We study chaos and scrambling in unitary channels by considering their entanglement properties as states. Using out-of-time-order correlation functions to diagnose chaos, we characterize the ability of a channel to process quantum information. We show that the generic decay of such correlators implies that any input subsystem must have near vanishing mutual information with almost all partitions of the output. Additionally, we propose the negativity of the tripartite information of the channel as a general diagnostic of scrambling. This measures the delocalization of information and is closely related to the decay of out-of-time-order correlators. We back up our results with numerics in two non-integrable models and analytic results in a perfect tensor network model of chaotic time evolution. These results show that the butterfly effect in quantum systems implies the information-theoretic definition of scrambling.

Figures (16)

• Figure 1: Interpretations of a unitary channel: (a) a unitary operator $U$ with input and output legs. (b) state interpretation $|U\rangle$ of the unitary operator $U$. By bending the input legs down, we treat input/output equally. (c) the state interpretation is equivalent to the creation of a maximally entangled state followed by acting with $U$ on half the EPR pairs, which gives $|U\rangle$.
• Figure 2: Setup to study scrambling in a unitary quantum channel $U$. Even though we draw our channels with input and output legs, when we discuss entanglement we always mean of the state $|U\rangle$ given by the mapping to the doubled Hilbert space as in \ref{['eq:U-as-state']}.
• Figure 3: (a) Permutations of qubits. The isometric pure state $|\Psi\rangle$ consists of EPR pairs between input and output qubits. (b) A swap gate and a unitary corresponding to a perfect tensor. Note the similarity to the Feynman diagrams of a $2\to2$ scattering process for a free theory and interacting theory, respectively.
• Figure 4: The Hayden-Preskill thought experiment.
• Figure 5: The eternal AdS black hole interior is a geometric representation of the unitary quantum channel given by the time evolution operator of the dual CFT. Left: Penrose diagram for the eternal AdS black hole geometry with a spacelike slice (blue) anchored on the left boundary at the middle of the diagram anchored at time $t$ on the right boundary. Right: Geometric depiction of the spacelike slice through the Einstein-Rosen bride (ERB). The spatial coordinates on the boundary CFT are represented by $\varphi$. The renormalized length of the ERB is proportional to $t$. For small $t$, the RT surface used to compute the entanglement entropy $S_{AC}$ goes across the ERB (red). After a time proportional to the size of $A$ or $C$, the disconnected RT surface (blue) is preferred and the entanglement entropy is a sum of disjoint contributions ($S_{AC} = S_A + S_C$).