An information-theoretic proof of the Planckian bound for thermalization

MainUnderstanding equilibration and thermalization from the underlying reversible quantum dynamics is one of the most fundamental, long-lasting open questions in physics1,2,3,4. One central aspect of thermalization in quantum many-body systems is the emergence of the Planckian dissipation time τPl (refs. 5,6), as given by:$${\tau }_{{\rm{Pl}}}=\frac{\hslash }{{k}_{{\rm{B}}}T}.$$ (1) This universal timescale grows inversely with temperature T and is determined by two fundamental constants in nature, namely the reduced Planck constant ℏ and the Boltzmann constant kB. Analogously to the ‘Planck time’ in quantum gravity, it is conjectured to represent the shortest timescale for thermalization. That is, the minimum time required to reach a Gibbs state starting from an arbitrary non-equilibrium distribution5,7,8,9,10,11.Originally, the concept of a ‘Planck scale of dissipation’, as given by equation (1), was coined by Zaanen to describe the universal behaviour of the conductance of superconductors above their critical temperature12,13. This behaviour was comprehensively empirically corroborated in ref. 14, which ignited the quest for an underlying microscopic explanation, tentatively put forwards for metals in refs. 15,16 and more recently in ref. 17, where a bound is provided for the emergence of hydrodynamic behaviour in many-body systems. In parallel, the concept of Planckian time has been linked to the rate of growth of chaos in many-body systems and field theories18,19,20,21,22.Despite the omnipresence of the Planckian time as a fundamental limit on the speed of dissipative or chaotic processes5,10, it is not difficult to come up with apparent violations of this conjecture. For instance, in collisional models, thermalization is described by letting the system interact with a bath made up of identical copies of the system in thermal equilibrium23. By letting these collisions take place arbitrarily fast, one realizes there is no fundamental restriction on the time required to relax to the thermal state of a previously known Hamiltonian. The best one can hope for, via quantum speed limits, are bounds assuming a limited interaction strength24,25,26.Motivated by this tension, in this work we propose an approach to rigorously prove the appearance of the Planckian bound for thermalization based on quantum information geometry and metrology. Unlike previous approaches, our assumptions on the structure of the quantum system and the allowed dynamics are minimal and model-independent. Our starting point is to introduce the concept of a thermalization machine M, namely a device that thermalizes a quantum system S. We make the following basic assumptions on the action of the machine: Requirement 1: validity of quantum mechanics. The microscopic evolution of S and M (including the environment for open systems) is well described by the Schrödinger equation. Requirement 2: thermalization. M should output states that are close to the thermal ensemble, independently of the specific structure of S. The independence on the exact specifics of S in Requirement 2 is crucial and physically motivated, as it makes a clear distinction between state preparation and (chaotic) thermalization processes. In this framework, standard thermal baths satisfying detailed balance27 and complex many-body system satisfying the eigenstate thermalization hypothesis28,29,30 would be considered thermalizing machines. Yet our framework is more general, and also covers highly engineered processes, such as algorithms for many-body quantum cooling.Under these two minimal assumptions, we provide a universal bound (equation (6)) on the time τ required to reach thermalization at inverse temperature β = 1/kBT. When considering thermalization in a system with a spectral gap \(\varDelta\)—defined as the energy difference between the ground state and the first excited state—our bound can be summarized as follows:$$\tau \ge \left\{\begin{array}{ll}{\tau }_{{\rm{Pl}}}/2,&\beta \varDelta \lesssim 1\\ \hslash /\varDelta ,&\beta \varDelta \gg 1.\end{array}\right.$$ (2) This is in good agreement with previous results in the literature, as Planckian dissipation behaviour is known to break down close to zero temperature5.An information-theoretic notion of thermalizationInspired by quantum information theory, our framework is adversarial: it assumes the existence of an entity (the machine M) that has full control to realize a desired task (thermalization) in the shortest possible time τ under minimal physical constraints. To minimize τ, we allow any control on M and its interaction with S, although they should be independent of the specifics of S alone. This is formalized through two main requirements on the thermalizing machine.Requirement 1 (validity of quantum mechanics)The evolution of S and M is described by quantum mechanics:$$\begin{array}{l} {\dot{\rho }}_{SM}(t)=-\frac{i}{\hslash }[{H}_{S}+{V}_{SM}(t)+{H}_{M}(t),{\rho }_{SM}(t)]\\ {\rho }_{S}(t,{H}_{S}):={{\rm{tr}}}_{M}{\rho }_{SM}(t)\end{array}$$ (3) where HS is the system Hamiltonian and HM(t) (VSM(t)) is the machine (interaction) Hamiltonian that specifies the machine and where \({\rho}_{SM}\) is the state of the system and machine combined, t the time and \({{\rm{tr}}}_{M}\)the partial trace on M.Note that for open quantum systems we include all of the environment in the machine; in other words, we consider any time-dependent unitary dilation of the dynamics. We also emphasize that there is no restriction on the size of M and the strengths of HM and VSM; that is, the machine is free to do anything allowed by quantum mechanics. To represent a non-trivial thermalization process, such a machine should thermalize S for multiple choices of HS via a fixed routine {HM(t), VSM(t)}. This is illustrated in Fig. 1 and formalized as follows:Fig. 1: Representation of thermalization process for a given machine M.Full size imageDepending on HS (in the figure, HS = ○,△,□,⨯), the dynamical evolution induced by M on the initial state of the system ρ(0) is required to bring S close to the corresponding thermal state ω, after time τ. This should happen up to an error ε and for multiple choices of HS (here forming a ball of size δ for simplicity).Requirement 2 (thermalization)For some initial state ρSM(0) and after time τ, the machine outputs states ρS(τ, HS) that are close to the corresponding thermal ensemble at temperature T = (kBβ)−1, for all choices of HS from some set. Specifically, for a ball \({{\mathcal{B}}}_{\delta }\) around \({\bar{H}}_{S}\) we require:$$(i)\quad D\left({\rho }_{S}(\tau ,{H}_{S}),\omega (\beta ,{H}_{S})\right)\le \varepsilon$$ (4) $$(ii)\quad \forall \,{H}_{S}\in {{\mathcal{B}}}_{\delta}\,\text{with}\,{{\mathcal{B}}}_{\delta }:=\{H\ \vert \parallel\! H-{\bar{H}}_{S}\!\parallel \le \delta \}\ ,$$ (5) where \(\omega (\beta ,{H}_{S}):={\mathrm{e}}^{-\beta {H}_{S}}/{{\mathrm{tr}}}{\mathrm{e}}^{-\beta {H}_{S}}\) are Gibbs states, \(D(\rho ,\sigma ):=\arccos\)\((\sqrt{F(\rho ,\sigma )})\) is chosen to be the Bures angle (F is the quantum fidelity) and \(\parallel A\!\parallel :={\lambda }_{\max }(A)-{\lambda }_{\min }(A)\) is the spectral semi-norm (see the Supplementary Informationfor details). The parameters δ and ε control the range and accuracy of the thermalization machine, respectively.We claim that requirements 1 and 2 are minimal criteria that should be satisfied to prove the existence of a minimal thermalization timescale. (1) Abandoning Requirement 1 and assuming the validity of classical mechanics at all scales (ℏ → 0), would take the Planckian time to zero. (2) Requirement 2 is essential to distinguish a thermalization machine from a single-state preparation machine. Indeed, a collisional ‘swap machine’ can prepare a fixed \(\omega (\beta ,{\bar{H}}_{S})\) instantaneously, as discussed above. Finally, it is worth mentioning that both equations (4) and (5) in Requirement 2 are chosen for simplicity and can be relaxed. In particular, for the derivation of our results it is in general sufficient to require an even weaker notion of thermalization, in which equation (4) is satisfied for at least two different HS. Furthermore, it is also sufficient to require equation (4) to be valid for some weighted-average of ρ(t, HS) with 0 ≤ t ≤ τ (this includes the possibility of a finite time resolution of the devices) or only after dephasing; for example, in the local energy eigenbasis of S. For a detailed discussion see the Supplementary Information.The Planckian bound on thermalizationWe demonstrate the following lower bound on the thermalization time.ResultFor any thermalization machine satisfying requirements 1 and 2 the thermalization time must satisfy:$$\begin{array}{rcl}&&\tau \ge {\tau }_{{\rm{Pl}}}\space \chi ({\bar{H}}_{S},\delta ,\varepsilon )\ ,\quad \quad \,\text{with}\,\\ &&\chi ({\bar{H}}_{S},\delta ,\varepsilon ):={\mathop{\max }\limits_{{H}_{S}^{(1,2)}\in {{\mathcal{B}}}_{\delta }}}\left[\frac{2D\left(\omega (\beta ,{H}_{S}^{(1)}),\omega (\beta ,{H}_{S}^{(2)})\right)-4\varepsilon }{\beta \parallel {H}_{S}^{(1)}-{H}_{S}^{(2)}\parallel }\right]\end{array}.$$ (6) This result can be understood as a universal bound on the thermalization time. As we show below, the adimensional factor \(\chi ({\bar{H}}_{S},\delta ,\varepsilon )\) in equation (6) is finite and remains bounded from zero in all regimes where thermalization (Requirement 2) remains sufficiently different from single-state preparation. Also note the upper bound \(\chi (\bar{H},\delta ,\varepsilon )\le \frac{1}{2}-\frac{4\varepsilon }{\beta \delta }\), which highlights that the accuracy and range parameters must satisfy \(\varepsilon \le \frac{\beta \delta }{8}\) to yield a non-trivial bound (in other words, the tolerated error should be sufficiently small to distinguish the different thermal states required).Proof sketchThe core idea of the proof is that the distinguishability between the outputs of the thermalization machine generated by different HS is fundamentally constrained by the sensitivity of the global unitary evolution to changes in HS (equation (3)). This can be concretized in information-geometrical arguments (see the Supplementary Information for full details): first, when Requirement 2 holds for two Hamiltonians \({H}_{S}^{(1)}\) and \({H}_{S}^{(2)}\), the triangle inequality for the Bures angle implies that \(D({\rho }_{S}(\tau ,{H}_{S}^{(1)}),{\rho }_{S}(\tau ,{H}_{S}^{(2)}))\ge D(\omega (\beta ,{H}_{S}^{(1)}),\omega (\beta ,{H}_{S}^{(2)}))-2\varepsilon\). In turn, Requirement 1 sets a limit on variations of S under variations of the local HS—after time τ the possible states of the system must satisfy \(D({\rho }_{S}(\tau ,{H}_{S}^{(1)}),{\rho }_{S}(\tau ,{H}_{S}^{(2)}))\le \frac{\tau }{2\hslash }\parallel\! {H}_{S}^{(1)}-{H}_{S}^{(2)}\!\parallel\). The result (equation (6)) is then obtained by optimizing the choice of the Hamiltonians.In what follows, we will characterize \(\chi ({\bar{H}}_{S},\delta ,\varepsilon )\) in different physically relevant limits to obtain universal bounds on thermalization, and later contrast such bounds with explicit dynamics/machines.Locally exact thermalization and quantum Fisher informationLet us now consider the case of locally exact thermalization, by making the ball of Hamiltonians δ → 0 in Requirement 2 infinitesimal, while keeping the error on the thermal state ε ≪ βδ negligible. In this limit, the machine must prepare the exact Gibbs state ρS(τ, HS) ≡ ω(β, HS), for all perturbations of the Hamiltonian \({H}_{S}(\delta ,\kappa )={\bar{H}}_{S}+\delta \kappa\) with κ any hermitian operator satisfying ∥ κ ∥ ≤ 1 and infinitesimal δ. The bound (equation (6)) then becomes:$$\tilde{\chi }({\bar{H}}_{S}):={\mathop{\lim }\limits_{\frac{\varepsilon }{\beta }\ll \delta \to 0}}\chi ({\bar{H}}_{S},\delta ,\varepsilon )={\mathop{\max }\limits_{\kappa ={\kappa }^{\dagger }}}\frac{\sqrt{{{\mathcal{F}}}_{\kappa }^{{\rm{th}}}(\beta ,{\bar{H}}_{S})}}{\beta \parallel \kappa \parallel },$$ (7) where \({{\mathcal{F}}}_{\kappa }^{{\rm{th}}}(\beta ,{\bar{H}}_{S})\equiv {\mathcal{F}}\left(\omega (\beta ,{H}_{S}(0,\kappa ))\right)\) is the quantum Fisher information (QFI) of a thermal state31,32,33. Here we used the fact that the QFI of a parametric state ρδ is related to its susceptiblity with respect to the Bures angle via \({\mathcal{F}}({\rho }_{0})=4{({\lim }_{\delta \to 0}\frac{D({\rho }_{0},{\rho }_{\delta })}{\delta })}^{2}\) (Supplementary Information).This result admits a natural interpretation in terms of quantum metrology. For locally exact thermalization, \({{\mathcal{F}}}_{\kappa }^{{\rm{th}}}(\beta ,{\bar{H}}_{S})\) must coincide with the ‘dynamical’ QFI \({{\mathcal{F}}}_{\kappa }^{{\rm{dyn}}}(\tau ,{\bar{H}}_{S})\equiv {\mathcal{F}}\left({\rho }_{S}(\tau ,{H}_{S}(0,\kappa ))\right)\), which is bounded by the generalized Heisenberg limit \({{\mathcal{F}}}_{\kappa }^{{\rm{dyn}}}(\tau ,{\bar{H}}_{S})\le\parallel\kappa \parallel ^{2}{\tau }^{2}/{\hslash }^{2}\) (ref. 34). By maximizing over the Hamiltonian perturbations κ, we then immediately recover equation (7). In other words, this bound follows from the observation that the thermalization machine cannot violate the Heisenberg limit.The maximization in equation (7) is detailed in the Supplementary Information, where we show that \(\tilde{\chi }\ge \sqrt{p(1-p)}\) for all possible bipartitions of the population of \(\omega (\beta ,{\bar{H}}_{S})\) in two sets with probabilities {p, 1 − p}. When \(\omega (\beta ,{\bar{H}}_{S})\) is sufficiently mixed that p ≈ 1/2 can be chosen, we find \(\tilde{\chi }\approx 1/2\), recovering the first line of equation (2). In particular, \(\tilde{\chi }\ge \sqrt{2}/3 \approx 0.47\) whenever the ground-state probability p0 is below 2/3. For the case p → 1, namely as \(\omega (\beta ,{\bar{H}}_{S})\) approaches the ground state, we derive a different bound that is tighter in such a regime: \(\tilde{\chi }\ge (2{p}_{0}-1)/(\beta \varDelta )\) for β\(\varDelta\) ≫ 1. This leads to the second line of equation (2).Approximate thermalizationThe limit of locally exact thermalization yields simple bounds and an intuitive understanding in terms of the Fisher information. The more general inequality (6) follows from a similar geometrical argument using finite variations of HS while admitting the possibility of a finite error ε > 0 in reaching the exact thermal state. Such a possibility is crucial to ensure the continuity and, more importantly, the wide validity of our main results: thermalization in nature is not, in general, exact.On a formal level, moving away from ε = 0 makes the function \(\chi ({\bar{H}}_{S},\delta ,\varepsilon )\) (equation (6)) more challenging to compute. However, we can compute again different lower bounds that hold for any \({\bar{H}}_{S}\) in any Hilbert space dimension and depend only on the possible bipartite coarse-graining of the thermal state populations. These bounds are shown in Fig. 2 and we observe once again how they yield χ ≳ 0.5 for states that are sufficiently mixed (that is, when \(\omega (\beta ,{\bar{H}}_{S})\) is not concentrated only in the ground state) and sufficiently small errors. As ε increases, the machine might in principle become faster; however, note that at ~5% error one still has χ ≳ 0.4 and at around 20% error, χ ≳ 0.3. Remarkably, all lower bounds in Fig. 2 can be obtained by considering a simple subset of Hamiltonians that are diagonal in the basis of \({\bar{H}}_{S}\). That is, they hold even when M is required to thermalize only classical (commuting) Hamiltonians.Fig. 2: Bounds for approximate thermalization.Full size imageFor different finite values of ε, lower bounds on \(\chi ({\bar{H}}_{S},\delta ,\varepsilon )\) are shown as a function of any bipartition {p, 1 − p} of the thermal state populations at \({\bar{H}}_{S}\). Specifically, equation (6) is partially optimized over a simple set of Hamiltonians \({H}_{S}^{(i)}\), namely those that commute with \({\bar{H}}_{S}\) and for which \({H}_{S}^{(i)}-{\bar{H}}_{S}\) has only two distinct eigenvalues. In units of Bures angle, the maximum tolerated error corresponds to εmax ≡ π/4. For ε ≲ 5% ⋅ εmax, χ is in general at least greater than ~0.4. In the limit p → 1 (that is, close to the ground state), we find that χ tends to zero slower than 1/(βΔ) (see the Supplementary Information for details).Applications to model-informed scenariosAfter deriving model-independent limits on the preparation of thermal states, we now outline how the techniques that we introduced can be applied well beyond this scenario and how refined bounds can be derived when: (1) only specific observables can be measured; (2) the required outputs are not necessarily thermal; or (3) further details of the physical systems involved are given.For simplicity, we take the limit of locally exact functioning of M (that is, negligible ε and infinitesimal δ), in that the machine is only required to operate close to \({H}_{S}={\bar{H}}_{S}+\delta \kappa\). Suppose that the user of the machine does not want to retrieve thermal states ω(β, HS) exactly, but rather a generic state \(\tilde{\omega }({H}_{S})\) that has a dependence on its Hamiltonian (this includes reduced Gibbs states, generalized Gibbs ensembles, dephased states, steady states of noisy systems and so on). Second, the user is not able to obtain full tomography of the output: rather, they can only measure some observable A = ∑aaΠa, Πa being the projector corresponding to output a. Then, the accessible statistics of the user are locally limited to \({p}_{a}={\rm{tr}}{\varPi }_{a}\tilde{\omega }({\bar{H}}_{S}+\delta \kappa )\), with δ-derivative ∂pa at \({\bar{H}}_{S}\). By definition, the corresponding accessible Fisher information is upper-bounded by the QFI of \(\tilde{\omega }\): \({\sum }_{a}\frac{{(\partial {p}_{a})}^{2}}{{p}_{a}}\le {{\mathcal{F}}}_{\kappa }^{\tilde{\omega }}\). After forcing the latter to be equal to the dynamical QFI \({{\mathcal{F}}}_{\kappa }^{{\rm{dyn}}}{| }_{S}\) on S, and noticing that this is smaller than the global QFI of the unitarily evolving S + M, one can derive a refined bound via convexity (Supplementary Information):$$\sum _{a}\frac{{(\partial {p}_{a})}^{2}}{{p}_{a}}\le {{\mathcal{F}}}_{\kappa }^{\tilde{\omega }}\le {{\mathcal{F}}}_{\kappa }^{{\rm{dyn}}}{| }_{SM}\le \frac{{\tau }^{2}}{{\hslash }^{2}}\int_{0}^{1}{\rm{d}}s\ {{\mathcal{F}}}_{\kappa }^{{\rm{U}}}({\rho }_{SM}(s\tau ))\ ,$$ (8) where \({{\mathcal{F}}}_{\kappa }^{{\rm{U}}}(\rho )\) is the Fisher information obtained by a unitary rotation of state ρ with generator κ. In the absence of further knowledge on M the latter is upper-bounded by \({{\mathcal{F}}}_{\kappa }^{{\rm{U}}}\le\parallel\kappa \parallel ^{2}\), hence when the user can measure any observable A and the required output \(\tilde{\omega }\) is thermal, this directly leads to the general bound (7). However, we stress here that equation (8) can be applied to any (model-specific) scenario of interest. Moreover, energy measurements are typically sufficient to saturate the first inequality in equation (8) (Supplementary Information).As an example, consider the system S to be N-partite and \(\kappa =\mathop{\sum }\nolimits_{i = 1}^{N}{\kappa }^{(i)}\) to be a uniform perturbation—for example, when δ is an intensive order parameter of a phase-transition. One can then immediately turn the above inequality in$$\tau \gtrsim \beta \hslash {N}^{\frac{\alpha -\phi }{2}},$$ (9) where Nα represents the scaling of the thermal QFI of S (possibly at criticality), and Nϕ that of the global dynamical QFI of S + M. Standard uncorrelated thermal systems satisfy α = 1, while it has been proved in ref. 32 that local classical observables can achieve up to α = 2 on strongly correlated thermal systems. Moreover, ϕ = 2 needs the machine dynamics to generate consistent N-partite entanglement in ρSM, whereas ϕ = 1 when a separable partition can be found at all times (Supplementary Information).One can also apply equation (8) when further structure of the model is known. Consider a many-body closed system SM thermalizing on S, under the additional assumptions that the entire SM is initially uncorrelated among its constituents and the overall Hamiltonian is local; one can then apply Lieb–Robinson-type bounds to the growth of entanglement in time35. In particular, for short-range interactions, these are known to bound \({{\mathcal{F}}}_{\kappa }^{{\rm{U}}}\lesssim N{(\nu t)}^{d}\), where ν is the light-cone speed on the many-body lattice and d its spatial dimension. It follows then from equation (8) that \({{\mathcal{F}}}_{\kappa }^{{\rm{dyn}}}\lesssim \frac{{\tau }^{2}}{{\hslash }^{2}}N\int_{0}^{1}{(\nu s\tau )}^{d}=\frac{{\tau }^{2}}{{\hslash }^{2}}N\frac{{(\nu \tau )}^{d}}{d+1}\) and therefore:$$\begin{array}{r}{\tau }^{2+d}\gtrsim {\beta }^{2}{\hslash }^{2}{N}^{\alpha -1}\frac{d+1}{{\nu }^{d}}\ .\end{array}$$ (10) As expected, equation (10) is tighter than equation (9) in the range for which the Lieb–Robinson bound is informative 1 0 in detail. We observe in Fig. 3 that these trajectories approach the Planckian limit, while satisfying our constraints.DiscussionWe presented a fundamental time bound (equation (6)) valid for any thermalization process in quantum mechanics. To formalize our results in the most general framework, we introduced the concept of the thermalization machine, M, which only has to comply with quantum mechanics and output states close to thermal equilibrium. As such, it applies to any device (including quantum computers) attempting to prepare thermal states for a set of Hamiltonians. This therefore connects two tasks that have been intensively studied in recent years, namely Gibbs state sampling39,40,41,42,43,44 and Hamiltonian learning45,46,47 (Table 1). Our results can be summarized as equation (2): if M is required to thermalize all Hamiltonians HS, or at least those leading to thermal states that are sufficiently mixed (βΔ ≲ 1), τPl/2 = ℏ/(2kBT) sets the universal minimum operational time of M. When the the machine is only required to be a ground-state cooling machine, its minimum operational time is bounded by (less stringent) ℏ/Δ, consistent with optimal protocols for adiabatic state preparation33,48. The core idea behind these results lies in comparing the distinguishability of thermal states with the global variation of unitary dynamics induced by their Hamiltonians.Table 1 Information protocols based on Hamiltonian knowledgeFull size tableWhile we focused on the ultimate bounds imposed by quantum mechanics alone, our results raise further questions: in multipartite systems, what is the role of entanglement in the Planckian bound? What happens in the large-size limit? How close to the Planckian bound can realistic baths approach? A comprehensive answer to these questions is not within the scope of this work and motivates further investigation, but we showed how our techniques can be used to refine our bound to specific scenarios, once additional details are known about the physical systems involved. Finally, we note that as τPl ∝ 1/T, the bound τ ≳ τPl is, in its range of validity, reminiscent of the third law of thermodynamics, thus providing geometric insight into it.