Thermal OTOC: Probing Quantum Chaos & Scrambling

• Thermal OTOCs are multi-operator quantum correlators with nontrivial time ordering that quantify chaos, scrambling, and phase transitions in finite-temperature systems.
• They are defined via commutator squares and Boltzmann averaging, obeying strict KMS relations and fluctuation-dissipation theorems that constrain their analytic structure.
• Experimental and numerical protocols, including TFD-based and forward-only methods, validate thermal OTOCs as practical probes of quantum dynamics in complex systems.

A thermal out-of-time-order correlator (OTOC) is a multi-operator quantum correlation function with nontrivial time ordering, designed to probe quantum information dynamics, operator scrambling, and chaotic behavior at finite temperature. The standard form for Hermitian operators $A$ and $B$ in a system with Hamiltonian $H$ is

$$C(t) = -\langle [A(t), B]^2 \rangle_\beta = -\frac{1}{Z}\operatorname{Tr}\left( e^{-\beta H} [A(t), B]^2 \right), \quad A(t) = e^{iHt}A e^{-iHt},$$

with canonical ensemble average at inverse temperature $\beta$. In many-body, semi-classical, and gravity-dual contexts, thermal OTOCs provide precise diagnostics for quantum chaos, scrambling, and phase transitions, and their formal properties are tightly constrained by equilibrium KMS relations, spectral representations, and fluctuation-dissipation theorems.

1. Definitions, Thermal Ensembles, and Operator Orderings

Thermal OTOCs generalize the zero-temperature four-point correlator to finite temperatures by averaging over the Boltzmann distribution. The prototypical OTOC involves two pairs of operators, often denoted $W$ and $V$ or $A$ and $B$. The canonical thermal OTOC is

$$F_{\mathrm{OTO}}(t) = \operatorname{Tr}\left[ e^{-\beta H} W(t) V(0) W(t) V(0) \right]/Z,$$

or its squared-commutator form

$C(t) = -\langle [W(t), V(0)]^{2} \rangle_\beta.$

Alternative regularizations include symmetrized forms (e.g., insertion of $e^{-\beta H/2}$ between operator pairs) and bipartite/Kubo-regularized variants, which admit improved analytic properties and connect to nonlinear response (Tsuji et al., 2016, Hunt, 29 Nov 2025, Chaudhuri et al., 2018, Sundar et al., 2021). The thermofield double (TFD) construction realizes the finite-temperature density matrix as a purified state in $H \otimes H^*$, facilitating explicit contour ordering and making TFD-based protocols central to both theoretical and experimental efforts (Green et al., 2021, Sundar et al., 2021, Gu et al., 2021).

2. Spectral Representation, KMS Relations, and Fluctuation-Dissipation

Thermal OTOCs satisfy strict constraints from equilibrium Kubo-Martin-Schwinger (KMS) periodicity, which relate various possible time orderings by analytic continuation in imaginary time. For any $n$-point Wightman function, under cyclic permutations and $-i\beta$ shifts,

$$G_\sigma(\omega_1,\dots,\omega_n) = e^{-\beta(\omega_{\sigma(k+1)}+\cdots+\omega_{\sigma(n)})} G_{\text{cyclic}_k\circ\sigma}(\omega_1,\dots,\omega_n),$$

reducing the $n!$ possible arrangements to $(n-1)!$ independent correlators (Haehl et al., 2017, Chaudhuri et al., 2018). Spectral decomposition further expresses thermal OTOCs in terms of generalized nested and double commutators: $$\rho_{[12][34]}(\omega_1,\dots,\omega_4) = \int \prod_{i=1}^4 dt_i\, e^{-i\sum \omega_i t_i} \langle [A(t_1),A(t_2)][B(t_3),B(t_4)] \rangle_\beta,$$ with contour ordering encoded in tensor products of column vectors that generalize the retarded-advanced basis (Chaudhuri et al., 2018). These analytic structures underpin generalized fluctuation-dissipation theorems for OTOCs, e.g.,

$$C_{\{A,B\}^2}(\omega) + C_{[A,B]^2}(\omega) = 2 \coth\left(\frac{\beta\hbar\omega}{4}\right) C_{\{A,B\}[A,B]}(\omega),$$

linking quantum fluctuation, dissipative response, and the growth of operator noncommutativity (Tsuji et al., 2016).

3. Physical Significance: Scrambling, Chaos, and Bounds

Thermal OTOCs quantify the evolution of initially commuting local operators, with the decay of $|F(t)|$ or growth of $C(t)$ identifying the spread of information and onset of quantum chaos (Haehl et al., 2017). In chaotic regimes, the OTOC typically exhibits transient exponential growth,

$$C(t) \sim \varepsilon e^{\lambda_L t}, \quad (\varepsilon \ll 1),$$

where $\lambda_L$ is the quantum Lyapunov exponent. The Maldacena-Shenker-Stanford (MSS) bound constrains this rate in holographic or large-$N$ systems: $\lambda_L \leq \frac{2\pi k_B T}{\hbar},$ ensured by analyticity, commutator positivity, and instanton suppression in the path integral (Hunt, 29 Nov 2025, Haehl et al., 2017). In single- and few-body systems, exponential OTOC growth does not always signal genuine chaos, but rather reflects local instability, with bounds still obeyed due to quantum fluctuation regularizations and instanton physics (Akutagawa et al., 2020, Hashimoto et al., 2020, Hunt, 29 Nov 2025).

4. Computation, Models, and Temperature Dependence

Explicit thermal OTOCs have been calculated for a range of models:

• Few-body systems and phase transitions: In the Rabi and Dicke models, infinite-temperature OTOCs sharply identify the normal-to-superradiant quantum phase transition via long-time averages, with scaling laws tying the position and sharpness of OTOC minima to critical parameters, and universality confirmed across finite-size and frequency-ratio scaling (Sun et al., 2018).
• Quantum chaos diagnostics: In coupled harmonic oscillators, the OTOC's exponential regime matches the classical Lyapunov exponent and scales with temperature as $T^{0.26\text{--}0.31}$ (Akutagawa et al., 2020). In classically chaotic stadium billiards, semiclassical expansions reveal that the leading quantum OTOC growth rate $\Lambda$ scales as $\sqrt{T}$ (Jalabert et al., 2018).
• Non-chaotic potentials: Inverted harmonic oscillator potentials can generate exponential OTOC growth without chaos; at high temperature, OTOC Lyapunov exponents saturate linear bounds reminiscent of MSS, but their origin is local instability, not global chaos (Hashimoto et al., 2020).

Experimentally, finite-temperature OTOCs have been measured on digital quantum computers via TFD preparation and controlled evolution, revealing a monotonic increase of scrambling rates with temperature, albeit far below the theoretical chaos bound for black-hole dynamics (Green et al., 2021).

5. Measurement Protocols and Practical Implementations

Thermal OTOC measurement protocols balance the complexity of time ordering with experimental feasibility. Techniques include:

• TFD-based digital and analog protocols: Quantum simulators and ion-trap circuits prepare the thermofield double, apply local perturbations, evolve under the appropriate Hamiltonian (and its conjugate), and measure two-copy correlators to extract temperature-dependent OTOCs (Green et al., 2021, Sundar et al., 2021).
• Forward-only protocols: Construction of thermal OTOCs via pure-state preparations and polarization identities obviate the need for time-reversal and ancillae, significantly simplifying experimental requirements. All necessary matrix elements can be extracted via forward-only evolution and projective measurement (Blocher et al., 2020).
• Error mitigation: Protocols employing normalization corrections, postselection, and symmetry projection (e.g., global parity) robustly mitigate dephasing, depolarization, and readout errors, with quantitative bounds on fidelity deviations (Sundar et al., 2021, Green et al., 2021). Numerical simulations validate protocol accuracy for modest system sizes.

6. Connections to Eigenstate Thermalization and Operator Statistics

Thermal OTOCs are sensitive probes of fine structure beyond the standard Eigenstate Thermalization Hypothesis (ETH). In chaotic many-body systems, the short-time OTOC regime reveals operator matrix element correlations not accessible to two-point functions. Saturation times reflect the crossover to effective Gaussian random matrix theory, with operator-dependent energy scales $\omega_{\mathrm{GOE}} \sim 1/L$ and OTOC plateaus scaling as system size (Brenes et al., 2021).

7. Open Problems, Generalizations, and Extensions

Key directions include:

• Extension of OTOC fluctuation-dissipation relations to $n$-partite and generalized covariant forms, enabling a hierarchy of nonlinear response relationships (Tsuji et al., 2016, Haehl et al., 2017).
• Analytical characterization and numerical validation of OTOC bounds in systems with mixed instability and coherent tunneling (e.g., via instanton and Matsubara dynamics) (Hunt, 29 Nov 2025).
• Exploration of model-specific, temperature-dependent operator growth, diffractive corrections, and saturation phenomena in low- and high-dimensional chaotic systems (Jalabert et al., 2018, Akutagawa et al., 2020).
• Implementation of OTOC measurement protocols in scalable quantum platforms, with emphasis on error mitigation at increasing system size and entanglement depth (Green et al., 2021, Sundar et al., 2021).

Thermal OTOCs thus sit at the nexus of quantum chaos, many-body dynamics, quantum thermodynamics, and experimental quantum information, providing uniquely sensitive diagnostics at the intersection of theory and laboratory realizations.