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arXiv:2502.10129

Planckian Bounds From Local Uncertainty Relations

Saurish Chakrabarty, Zohar Nussinov

Published Feb 14, 2025• 0 citations
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Abstract

We introduce ``local uncertainty relations'' in thermal many-body systems, from which fundamental bounds in quantum systems can be derived. These lead to universal non-relativistic speed limits (independent of interaction range) and transport coefficient bounds (e.g., those of the diffusion constant and viscosity) that are compared against experimental data.

DOI
10.48550/arxiv.2502.10129

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We introduce ``local uncertainty relations'' in thermal many-body systems, from which fundamental bounds in quantum systems can be derived. These lead to universal non-relativistic speed limits (independent of interaction range) and transport coefficient bounds (e.g., those of the diffusion constant and viscosity) that are compared against experimental data.
We introduce “ local uncertainty relations ” in thermal quantum many-body systems from which fundamental bounds can be derived. These inequalities imply rigorous universal non-relativistic speed limits (independent of interaction range) and, as we focus on in the current work, bounds associated with transport (e.g., the diffusion constant and shear viscosity). Using the chaos bounds and other considerations, we introduce simplified non-rigorous universal transport coefficient inequalities and compare our resulting bounds against experimental data.
Using local uncertainty relations in thermal many body systems, we illustrate that there exist universal speed limit (regardless of interaction range), bounds on acceleration or force/stress, acceleration or material stress rates, diffusion constant, viscosity, other transport coefficients, electromagnetic or other gauge field strengths, correlation functions of arbitrary spatio-temporal derivatives, Lyapunov exponents, thermalization times, and transition times between orthogonal states in non-relativistic quantum thermal systems, and derive analogs of the Ioffe-Regel limit. In the ℏ→0\hbar \to 0ℏ→0 limit, all of our bounds either diverge (e.g., our speed and acceleration limit) or vanish (as in, e.g., our viscosity and diffusion constant bounds). Our inequalities hold at all temperatures and, as corollaries, imply general power law bounds on response functions at both asymptotically high and low temperatures. Our results shed light on measurements and on how apparent nearly instantaneous effective collapse to an eigenstate may arise in macroscopic interacting many body quantum systems.
We discuss our recent study of local quantum mechanical uncertainty relations in quantum many body systems. These lead to fundamental bounds for quantities such as the speed, acceleration, relaxation times, spatial gradients and the Lyapunov exponents. We additionally obtain bounds on various transport coefficients like the viscosity, the diffusion constant, and the thermal conductivity. Some of these bounds are related to earlier conjectures, such as the bound on chaos by Maldacena, Shenker and Stanford while others are new. Our approach is a direct way of obtaining exact bounds in fairly general settings. We employ uncertainty relations for local quantities from which we strip off irrelevant terms as much as possible, thereby removing non-local terms. To gauge the utility of our bounds, we briefly compare their numerical values with typical values available from experimental data. In various cases, approximate simplified variants of the bounds that we obtain can become fairly tight, i.e., comparable to experimental values. These considerations lead to a minimal time for thermal equilibrium to be achieved. Building on a conjectured relation between quantum measurements and equilibration, our bounds, far more speculatively, suggest a minimal time scale for measurements to stabilize to equilibrium values.
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.
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.
The precision of transport observables can be bounded by the entropy production and the activity of the transport process via kinetic and thermokinetic uncertainty relations. In linear bosonic systems, such uncertainty relations can provide tight bounds when the activity is replaced by a local activity of the measurement contact of interest---even in the strong-coupling regime. How much nonlinearities (or interactions) impact the validity and predictiveness of these local bounds and how much this impact depends on the concrete definition of activity are open questions that we address for two experimentally relevant model systems, a harmonic oscillator network and the intrinsically nonlinear spin-boson model. We thereby provide experimentally testable predictions on how nonlinearities impact or even break local kinetic and thermokinetic precision bounds.