Operator-entropy correspondence for out-of-time-order correlators beyond infinite temperature, four points, and the lattice

Hosur, Qi, Roberts, and Yoshida (JHEP 2016) proved that at infinite temperature the average of out-of-time-order four-point correlation functions over complete operator bases equals a simple function of the second Renyi entropy of the quantum time-evolution channel, regarded as a state. This operator-entropy correspondence makes information-theoretic scrambling diagnostics operational, but it is established only at $\beta=0$, for four-point functions, and on finite-dimensional lattices: the finite-temperature version is an uncontrolled high-temperature approximation, averaged six-point out-of-time-order correlators are argued not to reduce to Renyi entropies, and in the continuum the naive relation degenerates because of UV divergences. The problem asks for the exact finite-temperature correspondence, an information-theoretic interpretation of higher-point out-of-time-order averages, and a UV-regularized continuum formulation of the correspondence - or a rigorous demonstration that no such correspondence exists.

Background

The out-of-time-order correlator (OTOC) $\langle W(t) V W(t) V\rangle$ is a four-point function in which the Heisenberg-evolved operator $W(t)$ appears in a time-mismatched ('out of order') arrangement; its early-time decay, often exponential with a rate called the quantum Lyapunov exponent and spreading within a light cone of velocity $v_B$ (the butterfly velocity), diagnoses sensitive dependence on initial conditions in quantum many-body systems (quantum chaos) and the delocalization of locally encoded quantum information, a process called scrambling. Hosur, Qi, Roberts, and Yoshida (2016) characterized scrambling by mapping the time-evolution channel $U(t)=e^{-iHt}$ to a quantum state $|U(t)\rangle$ via the channel-state (Choi-Jamiolkowski) isomorphism, under which a unitary acting on $n$ qubits becomes a pure state on two copies of the system. In their convention the input system is partitioned into subsystems $A$ ($a$ qubits) and $B$ ($b$ qubits) and the output system into $C$ ($c$ qubits) and $D$ ($d$ qubits), with $a+b=c+d=n$; the channel state is $|U(t)\rangle=(I\otimes U(t))|I\rangle$, with $|I\rangle$ a product of EPR pairs, living on the doubled $2n$-qubit Hilbert space, and $\rho_{AC}=\mathrm{tr}_{BD}\{|U(t)\rangle\langle U(t)|\}$. Entanglement properties of this channel state then quantify the channel's information-processing behavior; for example, the tripartite information $I_3=I(A:C)+I(A:D)$, a combination of mutual informations that is negative when information is delocalized, was proposed as a diagnostic of scrambling. Their central technical result is an exact operator-entropy correspondence at infinite temperature: averaging the OTO four-point function over complete orthonormal operator bases $\{D_i\}$, $\{A_j\}$ on subsystems $A$ (input side) and $D$ (output side) gives

$$|\langle\mathcal{O}_D(t)\,\mathcal{O}_A\,\mathcal{O}_D(t)\,\mathcal{O}_A\rangle_{\beta=0}| = 2^{n-a-d-S^{(2)}_{AC}},$$

where $2^a$ and $2^d$ are the Hilbert-space dimensions of $A$ and $D$, and $S^{(2)}_{AC}=-\log_2 \mathrm{tr}\,\rho_{AC}^2$ is the second Renyi entropy of the channel state reduced to $A\cup C$ (the Renyi entropies are a one-parameter generalization of the von Neumann entropy, and order 2 is their most computable member). The basis-averaged OTOC thus measures exactly an entanglement entropy of the channel state, which is what makes entropic chaos diagnostics operational. The identity, however, is proven only at infinite temperature ($\beta=0$), for four-point functions, and for finite-dimensional lattice systems. At finite temperature the authors derived only an approximate relation, expected to hold 'as long as the temperature is high enough', by redistributing operators around the thermal circle of the thermofield double state (the purified thermal state $|TFD(\beta)\rangle=\sum_i e^{-\beta E_i/2}|i\rangle|i\rangle$) and Euclidean-evolving the operators, yielding $Z(\beta/2)^2 Z(\beta)^{-1} 2^{-a-d-S^{(2)}_{AC}(\beta/2)}$ with $Z$ the partition function. For six-point OTO correlators $\langle W(t_1) V(t_2) Q V(t_2) W(t_1) Q\rangle$, which arise when a thermal system is perturbed by two operators, the source considers the case $\beta=0$ and defines the basis average by analogy with the four-point case, as a sum over complete orthonormal Hermitian operator bases in the three regions associated with $W$, $V$, and $Q$; it argues, without a rigorous no-go proof, that the average is a contraction of three channel density matrices $\rho(t_1)$, $\rho(t_2)$, $\rho(t_1-t_2)$ with 'a complicated permutation', 'beyond Renyi entropies', and no entropic interpretation of this basis-averaged quantity is given; a 2025 construction from free probability (the theory of large random matrices treated as noncommuting variables) interprets the collection of all higher-point OTOCs of a fixed operator pair as quantifying operator ergodicity, but it does not identify the functional of the channel state that the six-point basis average measures and is itself restricted to infinite temperature. In the continuum the identity degenerates: the Renyi entropy is UV divergent and the normalized operator average vanishes, so the relation is trivially true; the authors suggested that recasting it in terms of mutual information might yield a sensible continuum limit, but no such formulation was constructed. The same paper poses sibling open directions that this problem deliberately does not subsume: building tensor-network models of scrambling whose recurrence time is doubly exponential in system size (perfect-tensor networks built from stabilizer tensors are classically simulable and recur at best exponentially), an information-theoretic channel-level formulation of the Eigenstate Thermalization Hypothesis (ETH, the statement that individual high-energy eigenstates of nonintegrable systems behave like thermal ensembles for simple observables), and a quantitative relation between chaos strength and computational power.

Problem Statement

Establish, or rigorously refute, the operator-entropy correspondence between operator-basis-averaged out-of-time-order correlators and entropic quantities of the quantum channel state $|U(t)\rangle$ in the regimes beyond its proven infinite-temperature, four-point, lattice setting. For a system with Hamiltonian $H$, channel $U(t)=e^{-iHt}$, partition function $Z(\beta)=\mathrm{tr}\,e^{-\beta H}$, and the basis average $\langle\mathcal{O}_D(t)\mathcal{O}_A\mathcal{O}_D(t)\mathcal{O}_A\rangle_\beta = \frac{1}{4^{a+d}} \sum_{ij} \mathrm{tr}\{ e^{-\beta H} D_i(t) A_j D_i(t) A_j \}/Z(\beta)$ over complete orthonormal operator bases on subsystems $A$ (input) and $D$ (output), normalized as in the source ($\mathrm{tr}\{D_i D_j\}=2^d\delta_{ij}$ and $\mathrm{tr}\{A_i A_j\}=2^a\delta_{ij}$, i.e. unnormalized generalized Pauli operators, under which the benchmark identity $2^{n-a-d-S^{(2)}_{AC}}$ holds), where $2^a$ and $2^d$ are the Hilbert-space dimensions of $A$ and $D$ so that the sums run over $4^a$ and $4^d$ operators (subsystem convention as in the Background: $A,B$ input, $C,D$ output, $a+b=c+d=n$, $\rho_{AC}=\mathrm{tr}_{BD}\{|U(t)\rangle\langle U(t)|\}$): (i) Finite temperature: determine the exact relation between the thermal basis-averaged OTOC at general $\beta>0$ and entropic quantities of the channel or thermofield-double state - clarifying the role of operator placement on the thermal circle and going beyond the source's high-temperature approximation $Z(\beta/2)^2 Z(\beta)^{-1} 2^{-a-d-S^{(2)}_{AC}(\beta/2)}$, whose derivation requires the temperature to be high enough that Euclidean redistribution of the operators is faithful. (ii) Higher point: at $\beta=0$, following the source's Appendix C, for the six-point OTO correlator $\langle W(t_1) V(t_2) Q V(t_2) W(t_1) Q\rangle$ with $W$, $V$, $Q$ simple Hermitian operators, averaged by summing over complete orthonormal Hermitian operator bases in the three regions associated with $W$, $V$, $Q$ (normalized as in the four-point average above): identify the information-theoretic functional of the channel density matrices $\rho(t_1)$, $\rho(t_2)$, $\rho(t_1-t_2)$ (with $\rho(t)=|U(t)\rangle\langle U(t)|$) that the average measures - the source argues, without a rigorous no-go proof, that it lies beyond Renyi entropies - and prove the identity it satisfies. (iii) Continuum: construct a UV-regularized continuum or field-theory formulation of the correspondence (for example along the suggested recasting in terms of mutual information) in which both sides are finite and nontrivial, or rigorously demonstrate that no such formulation exists. These three directions correspond to the three separate regimes - finite temperature, higher-point out-of-time-order averages, and the continuum - in which the source establishes the identity only with the limitations described above, and the source raises each of them as a separate open remark. Each of (i)-(iii) is resolved independently: establishing or rigorously refuting the correspondence within any single regime settles that regime and counts as a partial resolution of the problem, while a full resolution requires all three regimes to be settled, each either established or rigorously refuted.

References

1. Pavan Hosur, Xiao-Liang Qi, Daniel A. Roberts, Beni Yoshida, 'Chaos in quantum channels', JHEP 02 (2016) 004, DOI 10.1007/JHEP02(2016)004, arXiv:1511.04021, https://arxiv.org/abs/1511.04021
2. Namit Anand, Paolo Zanardi, 'BROTOCs and Quantum Information Scrambling at Finite Temperature', Quantum 6, 746 (2022), DOI 10.22331/q-2022-06-27-746, arXiv:2111.07086, https://arxiv.org/abs/2111.07086
3. Tianci Zhou, David J. Luitz, 'Operator entanglement entropy of the time evolution operator in chaotic systems', Phys. Rev. B 95, 094206 (2017), DOI 10.1103/PhysRevB.95.094206, arXiv:1612.07327, https://arxiv.org/abs/1612.07327
4. Cheryne Jonay, David A. Huse, Adam Nahum, 'Coarse-grained dynamics of operator and state entanglement', arXiv:1803.00089 (2018), https://arxiv.org/abs/1803.00089
5. Tarek Anous, Julian Sonner, 'Phases of scrambling in eigenstates', SciPost Phys. 7, 003 (2019), DOI 10.21468/SciPostPhys.7.1.003, arXiv:1903.03143, https://arxiv.org/abs/1903.03143
6. Bikram Pain, Ratul Thakur, Sthitadhi Roy, 'Imprints of information scrambling on eigenstates of a quantum chaotic system', arXiv:2507.02853 (2025), https://arxiv.org/abs/2507.02853
7. Ewan McCulloch, 'Haar averaged moments of correlation functions and OTOCs in Floquet systems', arXiv:2110.15151 (2021), https://arxiv.org/abs/2110.15151
8. Zi-Wen Liu, Andreas Winter, 'Many-Body Quantum Magic', PRX Quantum 3, 020333 (2022), DOI 10.1103/PRXQuantum.3.020333, arXiv:2010.13817, https://arxiv.org/abs/2010.13817
9. Arash Ahmadi, Eliska Greplova, 'Quantifying non-stabilizerness via information scrambling', SciPost Phys. 16, 043 (2024), DOI 10.21468/SciPostPhys.16.2.043, arXiv:2204.11236, https://arxiv.org/abs/2204.11236
10. Kaito Kobayashi, 'Irreducible Geometry of Higher-Order Correlator Families', arXiv:2607.08761 (2026), https://arxiv.org/abs/2607.08761
11. Shreya Vardhan, Jinzhao Wang, 'Free mutual information and higher-point OTOCs', arXiv:2509.13406 (2025), https://arxiv.org/abs/2509.13406
12. Jonah Kudler-Flam, 'Rényi Mutual Information in Quantum Field Theory', Phys. Rev. Lett. 130, 021603 (2023), DOI 10.1103/PhysRevLett.130.021603, arXiv:2211.01392, https://arxiv.org/abs/2211.01392