Quantum information theory proves a universal speed limit for thermalization

For two decades, physicists have suspected that nature imposes a fundamental speed limit on how fast quantum systems can reach thermal equilibrium. The suspected bound is set by the Planckian time, τ_Pl = ℏ/(k_B T), a duration built solely from the reduced Planck constant, Boltzmann’s constant and the temperature. Now a team of theorists has delivered what the field has long lacked: a general, model-independent proof that this bound is real, derived not from the details of any particular material but from the basic structure of quantum mechanics itself.

The work, published in Nature Physics by Paolo Abiuso, Alberto Rolandi, John Calsamiglia, Pavel Sekatski and Martí Perarnau-Llobet, frames thermalization as an information-processing task. The authors imagine an abstract ‘thermalization machine’ that must cool or equilibrate a quantum system toward the Gibbs state, the canonical thermal ensemble, and they ask how quickly quantum mechanics allows such a machine to operate. Crucially, the machine must work for many different Hamiltonians, not just one. This adversarial framing is what makes the proof universal.

The motivation traces back to Jan Zaanen’s 2004 proposal of a ‘Planck scale of dissipation’ to explain the strikingly universal linear-in-temperature resistivity of strange metals and superconductors above their critical temperature. Experiments on cuprates and other materials later confirmed that scattering rates across wildly different systems cluster around ℏ/(k_B T), and the same timescale has appeared in studies of quantum chaos, where it bounds the exponential growth of out-of-time-ordered correlators. Yet all of this evidence was circumstantial. Apparent counterexamples were easy to construct: in collisional models, where a system swaps with pre-thermalized copies of itself, relaxation to the thermal state of a known Hamiltonian can happen arbitrarily fast.

The resolution lies in the second requirement of the new framework: a genuine thermalization machine must output states close to the Gibbs ensemble for a whole set of distinct Hamiltonians, within some error ε. A swap machine that instantly prepares the thermal state of one specific Hamiltonian is merely a state-preparation device, not a thermalizer. Once the machine must cope with Hamiltonians it does not know in advance, a universal lower bound emerges. The authors prove that the thermalization time τ must satisfy τ ≥ τ_Pl/2 at finite temperature whenever the thermal states involved are sufficiently mixed, and τ ≥ ℏ/Δ in the low-temperature regime, where Δ is the spectral gap between the ground state and the first excited state.

The mathematical engine behind the proof is quantum information geometry, specifically the Bures angle, a distance measure between quantum states built from the quantum fidelity. The argument proceeds in two steps. First, if the machine thermalizes two different Hamiltonians H_S^(1) and H_S^(2), the triangle inequality for the Bures angle forces the two output states to be distinguishable by at least the distance between the corresponding Gibbs states, minus twice the tolerated error. Second, the Schrödinger equation limits how much the machine’s output can vary as the local Hamiltonian is changed: after time τ, the outputs can differ by at most τ times the norm of the Hamiltonian difference, divided by 2ℏ. Combining these two constraints and optimizing over the choice of Hamiltonians yields the universal bound.

In the limit of locally exact thermalization, where the machine must prepare the exact Gibbs state for infinitesimally perturbed Hamiltonians, the bound takes an elegant metrological form involving the quantum Fisher information of thermal states. The interpretation is intuitive: a thermalization machine is implicitly a Hamiltonian-estimation device, because its output reveals which Hamiltonian it acted on. Estimating a Hamiltonian is subject to the Heisenberg limit of quantum metrology, and the Planckian bound is precisely what happens when that limit is applied to thermal state preparation. The machine simply cannot violate the Heisenberg limit, no matter how cleverly it is engineered.

The quantitative details are robust. When the thermal state is mixed enough that its population can be split roughly in half, the dimensionless prefactor χ in the bound is at least about 0.47, recovering the τ_Pl/2 result. Even allowing a 5 percent error in the output state, the bound remains within roughly 20 percent of the Planckian value, and at 20 percent error it still exceeds 0.3 τ_Pl. Remarkably, all of these lower bounds can be derived using only Hamiltonians that commute with each other, meaning the result holds even for effectively classical thermalization tasks. In the low-temperature limit, where the thermal state concentrates in the ground state, the bound smoothly crosses over to ℏ/Δ, connecting the result to the quantum adiabatic theorem and to optimal protocols for adiabatic state preparation.

The authors also show the bound is tight. Inspired by Hamiltonian discrimination protocols, they construct an abstract machine that, after a time exactly saturating their inequality, maps dynamically generated states onto purifications of the two target Gibbs states, using Uhlmann’s theorem to preserve the correct fidelity. As a physical case study, they analyze the resonant-level model, in which a single fermionic mode tunnels into a fermionic bath under a tunable coupling schedule. Solving the exact dynamics, they demonstrate that trajectories approaching the Planckian limit exist: if the system thermalizes one Hamiltonian faster than τ_Pl/2, it necessarily fails to thermalize a second Hamiltonian with a different energy gap in the same time, exactly as the bound demands.

Because the framework is model-independent, it extends well beyond idealized Gibbs-state preparation. The authors derive refined bounds for scenarios where only certain observables can be measured, where the target states are generalized Gibbs ensembles or steady states, and where additional structure such as locality is known. For many-body systems, combining their Fisher-information inequality with Lieb–Robinson bounds on entanglement growth yields scaling relations in system size, coupling the thermalization time to the light-cone velocity of the lattice. These results connect Gibbs-state sampling algorithms on quantum computers with the Hamiltonian-learning program, two fields that had developed largely in parallel.

The proof also carries thermodynamic weight. Since τ_Pl grows inversely with temperature, the bound τ ≳ τ_Pl becomes harder and harder to satisfy as T approaches zero, echoing the unattainability formulation of the third law of thermodynamics and providing it with a geometric underpinning. Open questions remain, including the role of entanglement in multipartite systems and how closely realistic baths can approach the limit, but the central message stands: the Planckian timescale is not a curiosity of particular materials but a consequence of quantum mechanics itself, written into the geometry of quantum states.

Subject of Research: An information-theoretic proof of the universal Planckian bound on quantum thermalization timescales

Article Title: An information-theoretic proof of the Planckian bound for thermalization

Article References: Abiuso, P., Rolandi, A., Calsamiglia, J., Sekatski, P., & Perarnau-Llobet, M. (2026). An information-theoretic proof of the Planckian bound for thermalization. Nature Physics. https://doi.org/10.1038/s41567-026-03397-y

Image Credits: AI Generated

DOI: 10.1038/s41567-026-03397-y

Keywords: thermalization, Planckian time, quantum information theory, quantum metrology, quantum Fisher information, Gibbs states, quantum speed limits, thermodynamics, many-body physics, strange metals, quantum chaos, Nature Physics