Quantum Systems Have a Speed Limit for Reaching Thermal Equilibrium, Researchers Prove

A hot cup of coffee cools over time and eventually approaches room temperature. This drift toward thermal equilibrium is one of the basic features of thermodynamics. Yet the question of how much, in principle, the time to reach equilibrium can be shortened has long lacked a clear answer. For everyday heat transfer, cooling can be sped up by adjusting the container material, the surrounding temperature, convection and so on. In microscopic quantum systems, however, it has been suspected that there is a speed limit on thermalization that cannot be beaten no matter how sophisticated the control.

A joint research team from Austria, Spain and Switzerland has now provided a mathematical proof of this conjecture, which has stood for more than 20 years. The results were published in Nature Physics on September 24, 2026, under the title "An information-theoretic proof of the Planckian bound for thermalization". The institutions involved were the Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences in Vienna (IQOQI Vienna), TU Wien, the Universitat Autònoma de Barcelona (UAB) and the University of Geneva. Paolo Abiuso and Martí Perarnau-Llobet are the corresponding authors. A preprint is also available on arXiv.

The team combined quantum information geometry with quantum metrology. Without relying on any particular material or thermalization model, they derived a lower bound on the minimum time needed for thermalization from the basic principles of quantum mechanics. For well-mixed thermal states at finite temperature, the bound is about half the Planckian dissipation time. At very low temperatures, the gap between energy levels takes over from temperature as the relevant scale. Note, however, that this result is not a new experimental observation. It is a theoretical proof for thermalization processes that satisfy certain conditions.

Two speed limits: finite temperature versus ultra-low temperature

In condensed matter physics, the "Planckian dissipation time" has long been discussed as a time that characterizes the speed of thermalization and dissipation. It is defined as follows.

τ_{Pl} = \frac{ℏ}{k _{B} T}

Here, $ℏ$ is the reduced Planck constant, $k_{B}$ is the Boltzmann constant, and $T$ is the absolute temperature. Because this time scale is inversely proportional to temperature, it is short at high temperatures and long at low temperatures. Since physicist Jan Zaanen proposed it in 2004 in connection with research on electrical conduction in the normal state of superconductors, it has been treated as an important measure of dissipation in strongly correlated electron systems and quantum many-body systems.

The team derived the following general inequality for the time $τ$ required for thermalization.

τ \geq τ_{Pl}, χ (\overset{ˉ}{H}_{S}, δ, ε)

$χ$ is a dimensionless coefficient determined by the reference Hamiltonian $\overset{ˉ}{H}_{S}$ of the system, the range $δ$ over which the Hamiltonian is allowed to vary, and the tolerated error $ε$ relative to the thermal equilibrium state. A Hamiltonian is the operator that specifies a quantum system's energy structure and time evolution. The inequality says that the time required for thermalization is not determined by temperature alone. It also depends on the system's energy structure and on how accurately one demands that the system has thermalized.

Denoting the energy difference between the ground state and the first excited state as the spectral gap $Δ$ and the inverse temperature as $β = 1/ (k_{B} T)$ , the team organized the lower bound on thermalization time into two regimes.

First, when the thermal energy $k_{B} T$ is comparable to or larger than the spectral gap $Δ$ and the thermal state is well mixed ( $β Δ ≲ 1$ ), the Planckian dissipation time is the main scale. The paper gives the representative bound in this regime as follows.

τ \geq \frac{τ _{Pl}}{2} = \frac{ℏ}{2 k _{B} T}

In other words, there is a constraint that bringing a quantum system to the corresponding thermal equilibrium state cannot take less than roughly half the Planckian dissipation time. However, this compact formula summarizes results that assume well-mixed thermal states and high thermalization accuracy, among other things. The same factor of $1/2$ does not apply unconditionally to every finite-temperature system. The more general bound is given by the inequality containing the coefficient $\chi$ above.

On the other hand, when the temperature drops substantially and the thermal energy becomes much smaller than the spectral gap ( $β Δ ≫ 1$ ), the thermal equilibrium state approaches the ground state. In this regime, the main energy scale constraining the thermalization speed switches from the temperature $T$ to the spectral gap $Δ$ . The bound in the low-temperature limit shown in the paper takes the following form.

τ \geq \frac{ℏ}{Δ}

Importantly, although $τ_{Pl}$ itself diverges to infinity as the temperature approaches absolute zero, the theory does not predict that thermalization time becomes infinite for all quantum systems. At low temperatures, the energy difference between the ground state and excited states governs how quickly the state can change. The authors explain that this behavior is closely related to earlier work on the quantum adiabatic theorem and adiabatic state preparation.

A "thermalization machine" for unknown quantum systems, and two basic conditions

Why does quantum mechanics impose a limit on the speed of thermalization? To uncover the mechanism, the team introduced a theoretical device called a "thermalization machine."

The team formulated thermalization not simply as an operation that cools a specific material, but as "a process that generates the corresponding thermal equilibrium state for each of multiple quantum systems with different Hamiltonians." The quantum system is denoted $S$, and the device that thermalizes it is denoted $M$. The machine $M$ must not only perform operations designed in advance for one specific quantum system, but must also handle systems with different energy structures using the same control procedure.

The team set the following two basic conditions.

It obeys quantum mechanics. The microscopic time evolution of the system $S$ and machine $M$ together follows the Schrödinger equation. Even for open quantum systems that interact with an external environment, the environment is included as part of the machine, so that the whole is treated as unitary time evolution.
It can thermalize multiple different systems. The machine $M$ outputs, for each Hamiltonian in an allowed set, a state sufficiently close to the correct thermal equilibrium state, without the details of the target Hamiltonian being individually specified in advance. A tolerance $ε$ is set between the output and the target state.

The thermal equilibrium state here is the "Gibbs state" of statistical mechanics, expressed as follows.

ω (β, H_{S}) = \frac{e ^{- β H_{S}}}{Tr ( e ^{- β H_{S}} )}

The team defined thermalization as a process that generates this Gibbs state to a given accuracy. The machine $M$ is allowed any control operations and interactions with an external environment that quantum mechanics permits. In principle, no upper limit is placed on the size of the machine or the strength of the interactions used for control. This is an extremely permissive setting in which, as long as no physical law is violated, even the most advanced device may shorten the thermalization time.

This approach corresponds to what quantum information theory calls an "adversarial" formulation. It assumes a designer who tries to finish thermalization as fast as possible and uses every means quantum mechanics allows. If a constraint that cannot be overcome even under those conditions can be derived, it amounts to a speed limit that does not depend on the properties of any particular material or heat bath.

However, the machine is not given a separate optimal control procedure for each target Hamiltonian in advance. It is required to apply the same basic operations to different Hamiltonians and obtain the correct Gibbs state for each. The core of the proof is the distinction between preparing one specific thermal state once and thermalizing different quantum systems with a common mechanism.

A speed limit derived from quantum Fisher information

When the thermalization machine handles two different Hamiltonians, $H_{S}^{(1)}$ and $H_{S}^{(2)}$ , their corresponding Gibbs states are generally different too. If the machine can thermalize both correctly, the two states obtained through time evolution must each come sufficiently close to their target states.

The team evaluated the difference between these states using the "Bures angle," a measure from quantum information geometry. The Bures angle $D$ between two quantum states $ρ$ and $σ$ is defined using the quantum fidelity $F$ as follows.

D (ρ, σ) = arccos F (ρ, σ)

Quantum fidelity measures how close two states are, and the Bures angle expresses their difference as a geometric distance. If the correct Gibbs states for the two Hamiltonians are sufficiently far apart, the outputs of the thermalization machine must be distinguishable accordingly. But in the time evolution of quantum systems, there is a limit on how fast a state can change due to differences in the Hamiltonian.

This limit is captured by the "quantum Fisher information (QFI)" used in quantum metrology. QFI expresses how sensitively a quantum state responds to changes in a Hamiltonian parameter. For a Hermitian operator $κ$ representing a small change in the Hamiltonian, the team used the fact that the QFI obtained through time evolution obeys the following generalized Heisenberg limit.

F_{κ}^{dyn} \leq \frac{∣ κ ∣ ^{2} τ ^{2}}{ℏ ^{2}}

Here, $∣ κ ∣$ is the spectral seminorm of the operator representing the Hamiltonian perturbation, and $τ$ is the time spent on the interaction. This expression captures the quantum-mechanical constraint that one cannot extract unlimited information about differences in the Hamiltonian while making the time arbitrarily short.

By comparing the distinguishability of output states required of the thermalization machine with the speed of state change that quantum mechanics permits, the team derived the minimum required time. Specifically, they combined the lower bound on the Bures angle obtained from the thermalization requirement with the upper bound obtained from unitary time evolution, and optimized over the choice of Hamiltonians. With this method, they showed that a lower bound on thermalization time emerges without assuming any particular heat bath or interaction model.

The team explains that both basic conditions are essential to deriving this constraint. If the quantum-mechanical condition is dropped and one considers the classical limit $ℏ \to 0$ , the Planckian dissipation time also approaches zero. And if the requirement to handle multiple Hamiltonians is dropped, then in an idealized setting with no limit on interaction strength, a thermal state can be prepared in arbitrarily short time by swapping in a pre-prepared thermal state. The present speed limit arises not from preparing a single state, but from requiring thermalization for different Hamiltonians.

It does not apply "unconditionally to every quantum system"

This study is important as a general proof that does not depend on specific physical models, but its conditions of validity need to be understood precisely. In particular, it would be inaccurate to interpret it as saying that "no quantum state can be prepared faster than a certain time" or that "the same lower bound applies without exception to the thermalization of every quantum system."

First, there are conditions on the thermalization accuracy. The dimensionless coefficient $χ$ derived by the team has the following upper bound.

χ (\overset{ˉ}{H}_{S}, δ, ε) \leq \frac{1}{2} - \frac{4 ε}{β δ}

Here, $δ$ is the range of Hamiltonian variation the machine must handle, and $ε$ is the tolerated error relative to the thermal equilibrium state. From this expression, obtaining a nontrivial, positive lower bound on time requires at least the following condition.

ε < \frac{β δ}{8}

In other words, if the thermalization accuracy is too coarse, there is no longer a need to distinguish the thermal states corresponding to different Hamiltonians sufficiently, and the proof cannot yield a meaningful speed limit. With a large tolerance, the machine can satisfy the conditions even while outputting states somewhat off from the several target states. This inequality is a necessary condition for obtaining a nontrivial bound; it is not a sufficient condition guaranteeing that the coefficient $χ$ will be positive whenever it is met.

How much information about the Hamiltonian can be used in advance also matters. When the target Hamiltonian is completely known, the operations to prepare its Gibbs state can be designed beforehand. The team clearly distinguishes this "Gibbs state sampling" from a thermalization machine that handles unknown Hamiltonians.

The main information-processing tasks compared in the paper and their time lower bounds are summarized as follows.

Task / condition	Information given to the device	Main energy scale	Lower bound on required time
Gibbs state sampling	Complete classical description of the Hamiltonian $H$	Depends on operations designed in advance	Can approach 0 under idealized conditions
Thermalization machine at finite temperature ( $β Δ ≲ 1$ )	Oracle access to $H$	Thermal energy $k_{B} T$	$τ \geq ℏ/ (2 k_{B} T)$ for well-mixed states
Thermalization machine at ultra-low temperature ( $β Δ ≫ 1$ )	Oracle access to $H$	Spectral gap $Δ$	$τ \geq ℏ/Δ$ in the low-temperature limit
Hamiltonian learning	Oracle access to $H$	Difference between Hamiltonians to be distinguished, etc.	Time needed to distinguish Hamiltonians, $τ_{AMP}$

Oracle access here means a condition in which, rather than receiving a complete classical description of the Hamiltonian in advance, one can obtain information about it through, for example, the time evolution of the target system. The finite-temperature and ultra-low-temperature bounds in the table are the thermalization machine results summarized in Eq. (2) of the paper, and the contrast with the other two tasks is based on Table 1 of the paper.

This comparison shows that generating a Gibbs state does not always require the Planckian dissipation time. State preparation with the Hamiltonian fully given and thermalization for different, unknown Hamiltonians require different information processing. The present speed limit is derived mainly for the latter, operational kind of thermalization.

When interpreting the low-temperature regime, one should also avoid confusing the "temperature difference" between different parts of a system with the "difference between energy levels" of a quantum system. What matters in the paper is the product $β Δ$ of the inverse temperature $β$ and the spectral gap $Δ$ . $β Δ ≫ 1$ means that the energy difference between the ground state and the first excited state is sufficiently large compared with the thermal energy. As the temperature falls and thermal excitations are suppressed, the system's energy structure comes to govern the lower bound on thermalization time. This does not presuppose a temperature difference among the system's components.

On the other hand, the team's formulation does not need to require thermalization for every target Hamiltonian. According to the paper's supplementary information, the basic condition for deriving the general speed limit is that the machine can generate the correct thermal state to sufficient accuracy for at least two different Hamiltonians. The argument can also be extended to looser definitions of thermalization, in which the output need not coincide exactly with the Gibbs state at a given instant, and which target time-averaged states or states after the phase information of the quantum state has been lost.

Time scales of 25 femtoseconds at room temperature and milliseconds at ultra-low temperatures

The result draws attention against the backdrop of more than 20 years of research on strongly correlated electron systems and high-temperature superconductors.

According to an explainer published by IQOQI Vienna on September 24, 2026, the idea of the Planckian dissipation time was proposed in 2004 by Jan Zaanen to explain universal behavior seen in the electrical conduction of materials above their superconducting transition temperature. Since then, similar time scales have appeared in many condensed matter models and have been linked to chaotic behavior and dissipation in quantum many-body systems. From this body of research, it has been suspected that the Planckian dissipation time represents a universal speed limit for relaxation and thermalization of quantum systems, but a general derivation independent of specific models had not been obtained.

This time scale is extremely short at everyday temperatures. According to an explainer published by the Universitat Autònoma de Barcelona on October 5, 2026, the Planckian dissipation time at room temperature is about 25 femtoseconds. One femtosecond is one quadrillionth of a second, comparable to the time scale of rapid molecular motion. As the temperature falls, this time grows longer, reaching the millisecond range in ultracold atomic gases near absolute zero.

Calculating from $τ_{Pl} = ℏ/ (k_{B} T)$ for representative temperatures gives the following.

Temperature environment	Absolute temperature	Planckian dissipation time $τ_{Pl}$
Room temperature	300 K	About 25 femtoseconds ( $2.5 \times 1 0^{- 14}$ s)
Cryogenic	1 K	About 7.6 picoseconds ( $7.6 \times 1 0^{- 12}$ s)
Temperature assumed for ultracold atomic gases	1 nK	About 7.6 milliseconds ( $7.6 \times 1 0^{- 3}$ s)

These are values comparing the Planckian dissipation time itself. They do not indicate the time at which thermalization is actually completed at each temperature, nor a strict lower bound applicable to all systems. In particular, in the ultra-low-temperature regime where $β Δ ≫ 1$ , the spectral gap $Δ$ becomes the main scale, as explained above, so the values in the table cannot be interpreted directly as lower bounds on thermalization time.

In the everyday cooling of coffee, macroscopic heat transport such as conduction, convection and evaporation sets the speed. These processes are usually far slower than the microscopic Planckian dissipation time, and the present quantum-mechanical constraint does not directly determine everyday cooling rates. On the other hand, in atomic gases cooled to the nanokelvin range by laser cooling and similar methods, the Planckian time scale stretches to about milliseconds, which may provide conditions suited to experiments that measure the progress of thermalization with time resolution.

However, the paper itself does not report a new materials experiment. The team examines how the derived bound works using a theoretical model of a single fermion interacting with a heat bath. Specifically, they compared two Hamiltonians corresponding to an energy gap of 0 and of $k_{B} T$ . They show theoretically that even if the same thermalization machine can thermalize one system in a very short time, it cannot simultaneously thermalize the other to the same accuracy, and that both become possible once the time reaches the order of the Planckian dissipation time.

In this model, the range of states forbidden by the speed limit shrinks over time, and around $t \approx τ_{Pl} /2$ the specific bound no longer imposes a constraint. This is a theoretical example illustrating how the general inequality works, and does not show that an experimental device has reached the Planckian dissipation limit.

Can the theoretical limit be reached, and what does it mean for quantum computing?

Even when a mathematical lower bound on speed is shown, it does not follow that a device that can actually approach that limit exists. In quantum information theory, it matters whether the derived inequality can be satisfied with equality, that is, whether the bound is "tight." The team also examined this point theoretically.

For any pair of two Hamiltonians, the team mathematically constructed a thermalization machine designed to reach the general lower bound on thermalization time. The construction uses "Uhlmann's theorem," which relates the purification of quantum states to fidelity. When the thermalization error is $ε = 0$ , the time this machine needs to thermalize the two Hamiltonians is given by the following expression.

τ = \frac{2ℏ D ( ω ( β , H ^{(1)} ) , ω ( β , H ^{(2)} ) )}{H _{S}^{(1)} - H _{S}^{(2)}}

The $D$ in the numerator is the Bures angle between the Gibbs states corresponding to the two Hamiltonians. The denominator $∣ H_{S}^{(1)} - H_{S}^{(2)} ∣$ is the spectral seminorm of the difference between the two Hamiltonians. This expression means that, for thermalization of two Hamiltonians, a machine that satisfies the theoretical lower bound with equality can be constructed.

In the conventional approach based on Hamiltonian discrimination, fully distinguishing two Hamiltonians requires the following time.

τ \geq \frac{π ℏ}{H _{S}^{(1)} - H _{S}^{(2)}}

With this method, if the two Hamiltonians are very close, the time needed to tell them apart becomes large. By contrast, the thermalization machine constructed here does not aim to fully distinguish the Hamiltonians, but directly generates the thermal state corresponding to each. It can therefore achieve thermalization in less time than complete discrimination, and can saturate the general lower bound on time.

This result shows the value of treating thermalization not merely as energy dissipation but as a problem of processing quantum information. The thermalization machine must generate different output states depending on the Hamiltonian. By linking the distinguishability of information this requires with the speed of time evolution that quantum mechanics permits, the team derived a constraint that does not depend on any particular material model.

This framework is also relevant to Gibbs state preparation on quantum computers and to Hamiltonian learning for probing the properties of unknown quantum systems. Gibbs states play an important role in simulating quantum many-body systems and in statistical-mechanical calculations. For quantum algorithms that try to generate a thermal equilibrium state without sufficient knowledge of the Hamiltonian, the present speed limit could act as a theoretical constraint. By contrast, the same lower bound does not apply as is to state preparation in which the Hamiltonian is fully known and dedicated operations can be designed beforehand.

However, it cannot be concluded from this result that quantum computer processing speeds or quantum sensor performance will immediately improve. What the team showed is a limit on general quantum operations, including idealized conditions, and real devices face additional constraints such as control precision, interaction strength and environmental noise. How closely the theoretically optimal thermalization machine can be reproduced in real hardware must be verified separately.

The paper also lists, as future research topics, the role of entanglement in complex quantum many-body systems, behavior as system size grows, and how closely realistic heat baths can approach the speed limit. How insights from theoretical models with few degrees of freedom will manifest in strongly correlated electron systems and large quantum many-body systems is not yet well understood.

The significance of this study lies in providing a general mathematical foundation, based on quantum information theory, for the universality of the Planckian dissipation time, which has been debated for more than 20 years. But the speed limit is derived under the operational condition of thermalization for different Hamiltonians, and it does not unconditionally explain all thermal equilibration phenomena. The important next step is to use highly controllable quantum systems such as ultracold atomic gases to test how closely real thermalization processes approach the theoretical lower bound. How the mathematically derived limit shows up in actual quantum matter and quantum information processing devices is the next research question.