[Paper Review] Scrambling is Necessary but Not Sufficient for Chaos

Neil Dowling, Pavel Kos+1 more|PubMed|Apr 14, 2023

Quantum many-body systems75 references4 citations

TL;DR

This paper establishes that operator scrambling, measured by out-of-time-order correlators (OTOCs), is necessary but not sufficient for quantum chaos. Using analytically solvable local-circuit models—including dual-unitary circuits—it demonstrates that rapid OTOC decay (indicating scrambling) does not guarantee linear growth of local-operator entanglement (LOE), the true dynamical signature of quantum chaos, thereby clarifying the distinction between scrambling and chaos in many-body quantum systems.

ABSTRACT

We show that out-of-time-order correlators (OTOCs) constitute a probe for local-operator entanglement (LOE). There is strong evidence that a volumetric growth of LOE is a faithful dynamical indicator of quantum chaos, while OTOC decay corresponds to operator scrambling, often conflated with chaos. We show that rapid OTOC decay is a necessary but not sufficient condition for linear (chaotic) growth of the LOE entropy. We analytically support our results through wide classes of local-circuit models of many-body dynamics, including both integrable and nonintegrable dual-unitary circuits. We show sufficient conditions under which local dynamics leads to an equivalence of scrambling and chaos.

Motivation & Objective

• To clarify the relationship between quantum operator scrambling (via OTOCs) and quantum chaos (via LOE).
• To resolve the long-standing ambiguity in defining quantum chaos, especially in systems without a semiclassical limit.
• To establish that OTOC decay—commonly equated with chaos—is insufficient for true quantum chaos, as defined by linear LOE entropy growth.
• To provide exact analytical results for OTOCs and LOE in wide classes of local quantum circuits, including integrable and non-integrable models.
• To unify the OTOC and LOE frameworks via a CP operator formalism, revealing their intrinsic connection.

Proposed method

• Analytical computation of OTOCs in dual-unitary circuits using the Pauli basis and exact time evolution under local unitary gates.
• Derivation of exact expressions for OTOC decay in the XXZ model, showing exponential decay with a rate dependent on the coupling parameter $ J $.
• Use of the Choi state formalism to define and compute local-operator entanglement (LOE) as the entanglement of the time-evolved operator.
• Introduction of a generalized operator-free framework using completely positive (CP) operators: the 'out-of-time-order tensor' (OTOT) and 'local tensor entanglement' (LTE), which unify OTOC and LOE.
• Establishment of a link between OTOT and LTE via a link product construction, showing that OTOC is a reduced version of the full CP operator structure.
• Application of the classification of dual-unitary circuits to derive exact OTOC and LOE dynamics in qubit systems, enabling analytical verification of the scrambling-chaos distinction.

Experimental results

Research questions

• RQ1 Is rapid OTOC decay sufficient to conclude quantum chaos in many-body systems?
• RQ2 Does linear growth of local-operator entanglement (LOE) entropy always accompany OTOC decay?
• RQ3 Can exact analytical models demonstrate a system that scrambles (exhibits OTOC decay) but does not exhibit chaotic dynamics (no linear LOE growth)?
• RQ4 How are OTOCs and LOE related at a fundamental level, and can this relationship be formalized beyond specific models?
• RQ5 What conditions make scrambling and chaos equivalent in local quantum circuits?

Key findings

• Rapid OTOC decay is necessary but not sufficient for quantum chaos, as defined by linear growth of LOE entropy.
• In the XXZ model with $ J \neq \pi/4 $, the OTOC decays exponentially with a rate $ \alpha = \ln(1/\sin(2J)) $, confirming scrambling.
• For $ a_z = 0 $, the OTOC approaches $ -1/(d_A^2 - 1) $ in the long-time limit, indicating complete scrambling, yet LOE does not grow linearly.
• Exact analytical results in dual-unitary circuits confirm that OTOC decay can occur without linear LOE growth, proving that scrambling ≠ chaos.
• The operator-free formalism using OTOT and LTE reveals a deep structural link between OTOCs and LOE, with the OTOC emerging as a partial trace of the full CP operator.
• The study establishes that LOE is a more faithful dynamical indicator of quantum chaos than OTOCs, especially in systems where spectral methods are inapplicable.