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arXiv:2506.19188 · Nature Physics

An information-theoretic proof of the Planckian bound for thermalization

Paolo Abiuso, Alberto Rolandi, John Calsamiglia, Pavel Sekatski, Martí Perarnau-Llobet

Published Sep 24, 2026• 0 citations
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Abstract

Abstract It has been conjectured that the speed of thermalization is universally constrained by a Planckian timescale, τ Pl = ℏ /( k B T ), determined solely by the reduced Planck constant ℏ, Boltzmann’s constant ( k B ) and the temperature ( T ). However, a general and model-independent derivation of this limit has remained elusive. Here we provide a general proof of the emergence of Planckian thermalization based on quantum information geometry and quantum metrology. We formulate thermalization as a process that prepares states close to the corresponding thermal ensemble for a set of distinct Hamiltonians, and show that quantum mechanics imposes a universal lower bound τ ≥ τ Pl /2 on the thermalization time τ at finite temperature. In the low-temperature regime, the spectral gap Δ between the ground and first excited states replaces the temperature as the relevant energy scale and determines the lower bound on the thermalization time, in close connection with the quantum adiabatic theorem. These bounds, rooted in Hamiltonian estimation, establish operational limits on thermalization governed exclusively by fundamental constants and the intrinsic energy scale.

DOI
10.1038/s41567-026-03397-y

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Abstract It has been conjectured that the speed of thermalization is universally constrained by a Planckian timescale, τ Pl = ℏ /( k B T ), determined solely by the reduced Planck constant ℏ, Boltzmann’s constant ( k B ) and the temperature ( T ). However, a general and model-independent derivation of this limit has remained elusive. Here we provide a general proof of the emergence of Planckian thermalization based on quantum information geometry and quantum metrology. We formulate thermalization as a process that prepares states close to the corresponding thermal ensemble for a set of distinct Hamiltonians, and show that quantum mechanics imposes a universal lower bound τ ≥ τ Pl /2 on the thermalization time τ at finite temperature. In the low-temperature regime, the spectral gap Δ between the ground and first excited states replaces the temperature as the relevant energy scale and determines the lower bound on the thermalization time, in close connection with the quantum adiabatic theorem. These bounds, rooted in Hamiltonian estimation, establish operational limits on thermalization governed exclusively by fundamental constants and the intrinsic energy scale.
It has long been believed that dissipative timescales τ obey a "Planckian" bound τ≳(ℏ/k_{B}T) in strongly coupled quantum systems. Despite much circumstantial evidence, however, there is no known τ for which this bound is universal. Here we define operator size at a finite temperature, and conjecture such a τ: the timescale over which small operators become large. All known many-body theories are consistent with this conjecture. This proposed bound explains why previously conjectured Planckian bounds do not always apply to weakly coupled theories, and how Planckian timescales can be relevant to both transport and chaos.
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We present rigorous bounds on the thermalization time of the family of quantum mechanical spin systems known as stabilizer Hamiltonians. The thermalizing dynamics are modeled by a Davies master equation that arises from a weak local coupling of the system to a large thermal bath. Two temperature regimes are considered. First we clarify how in the low temperature regime, the thermalization time is governed by a generalization of the energy barrier between orthogonal ground states. When no energy barrier is present the Hamiltonian thermalizes in a time that is at most quadratic in the system size. Secondly, we show that above a universal critical temperature, every stabilizer Hamiltonian relaxes to its unique thermal state in a time which scales at most linearly in the size of the system. We provide an explicit lower bound on the critical temperature. Finally, we discuss the implications of these result for the problem of self-correcting quantum memories with stabilizer Hamiltonians.
The local equilibration time τeqτ_{\rm eq}τeq​ of quantum many-body systems is conjectured to be bounded below by the Planckian time ℏ/T\hbar /Tℏ/T. We formalize this conjecture by defining τeqτ_{\rm eq}τeq​ as the time scale at which a hydrodynamic description emerges for conserved densities. Drawing on analytic properties of real time thermal correlators, we establish a rigorous lower bound τeq≥αℏ/Tτ_{\rm eq} \geq α\hbar /Tτeq​≥αℏ/T on the onset of hydrodynamic behavior in a `regulated' thermal two-point function. The dimensionless coefficient ααα depends only on dimensionality and the type of hydrodynamic or diffusive behavior that emerges, and is independent of the thermalization mechanism or other microscopic details. This bound applies universally to local quantum many-body systems, with or without a quasiparticle description, including in the presence of inelastic scattering.
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Quantum thermalization explains how an isolated many-body system naturally evolves towards a thermal state, rendering information about the initial conditions inaccessible to local measurements. This is precisely the mechanism utilized in quantum error correction, where information is protected by design through a nonlocal encoding. In this work, we leverage this connection to port the rigorous framework of (approximate) quantum error correction to the study of quantum thermalization. Treating typical late-time states as codewords, we characterize the error-correcting properties of generic thermalizing dynamics. We numerically uncover a universal relationship between the encoding rate, distance, and thermal entropy density of the emergent code. At infinite temperature, this universal curve saturates the quantum Singleton bound, achieving the same optimal limit as Haar-random codes. At finite temperature, we introduce a code family based on the Scrooge ensemble, the natural thermal analogue of the Haar ensemble, and prove it saturates the entropic quantum Singleton bound, establishing this family as optimal within entropic constraints. Our extracted universal curve independently saturates this same bound, revealing that finite-temperature thermalization is itself optimal. Finally, we show how conserved quantities limit the error-correcting behavior of thermalization: codewords with differing energies, or other conserved charges, leak only classical information, and correctability persists until the difference reaches the scale of thermal fluctuations. Our results reveal a universal optimal coding structure in thermalizing dynamics, while introducing new optimal codes that achieve fundamental limits of approximate quantum error correction.