Beyond Heisenberg: Linking Thermodynamics and Quantum Uncertainty

Author: Denis Avetisyan

New research connects the fundamental limits of quantum precision with the laws governing energy and information flow.

A system of inequalities demonstrates a relationship where a tighter upper bound is geometrically derived from the lower bound and a conventional variance bound, establishing a constrained interplay between these values and revealing inherent dependencies within the system.

This paper establishes bounds on quantum Fisher information for thermodynamically conjugate variables, revealing a link between uncertainty relations and the fluctuation-dissipation theorem.

The foundational limits imposed by uncertainty relations in quantum mechanics typically address mechanically conjugate variables, yet analogous constraints for thermodynamic systems remain largely unexplored. This work, ‘Bounds on quantum Fisher information and uncertainty relations for thermodynamically conjugate variables’, establishes a rigorous framework to derive uncertainty relations for thermodynamically conjugate variables, quantifying the precision with which a classical intensive parameter and its corresponding quantum operator can be simultaneously determined. By bounding the quantum Fisher information via thermodynamic properties, we demonstrate a fundamental limit—$\Delta\theta\,\overline{\Delta O} \ge k_\text{B}T$—connecting uncertainty to fluctuations and response functions. Could this framework unlock improved sensing precision and fundamentally reshape our understanding of quantum measurements in thermal environments?

The Intrinsic Limits of Knowing

Quantum mechanics inherently limits the precision with which certain pairs of physical properties can be simultaneously known, formalized by the Heisenberg-Robertson Uncertainty Relation. This isn’t merely a limitation of measurement, but a fundamental property of the quantum state itself. The principle is readily demonstrated through the Position-Momentum Uncertainty, where greater precision in position implies less accuracy in momentum, and vice versa. This extends beyond position and momentum; the Angle-Angular Momentum Uncertainty reveals that conjugate variables are fundamentally limited in simultaneous definition. Every deviation from expectation, then, is not error, but a glimpse into the hidden dependencies of the quantum world.

Thermodynamic Uncertainty and Hidden Connections

The concept of conjugate variables, central to thermodynamics, mirrors the non-commuting observables of quantum mechanics, establishing a link between seemingly disparate fields. Analogous to the position-momentum relationship, variables like magnetization and magnetic field exhibit comparable uncertainty. This fundamental limit on simultaneous precision isn’t a measurement issue, but an inherent property of the system. A Thermodynamic Uncertainty Relation can be derived, providing a quantifiable bound on the precision of thermodynamic conjugate pairs. Current work refines this bound, particularly within the paramagnetic phase, offering improved sensitivity for characterizing these systems.

Information’s Role in Precise Estimation

The Quantum Fisher Information quantifies the information a quantum state holds about an unknown parameter, exceeding the capabilities of classical Fisher Information. The relationship between Quantum Fisher Information and estimation precision is formalized through the Cramer-Rao Inequality, where a higher Quantum Fisher Information corresponds to tighter lower bounds on variance. Recent work establishes a rigorous lower bound on achievable precision – β²⟨(ΔO²)⟩ – derived using the Quantum Fisher Information, representing an improvement over prior limits and providing a valuable tool for quantum metrology.

Locality, Interaction, and the Boundaries of Complexity

The Gibbs Ensemble provides a framework for analyzing systems at thermal equilibrium, allowing macroscopic properties to be calculated from microscopic interactions. Characterizing sensitivity to external perturbations requires the Symmetric Logarithmic Derivative, which assesses system stability by quantifying changes in the probability distribution. The Lieb-Robinson Bound establishes a fundamental limit on information propagation within localized quantum systems. Recent work demonstrates an exponential decay of the commutator – e^{-min(a, μ/v)⋅r} – quantitatively bounding error—2C||O_loc||e^{-min(a, μ/v)⋅r}—when approximating operators, confirming a principle of locality and informing scalable quantum simulations. Ultimately, the pursuit of understanding complex systems reveals that limitations are not failures, but the very contours of the landscape awaiting exploration.

The exploration within this work centers on establishing bounds for quantum Fisher information, effectively mapping the limits of precision in estimating thermodynamic properties. This resonates with Niels Bohr’s assertion: “Every great advance in natural knowledge begins with an investigation of popular prejudice.” The authors challenge conventional understandings of uncertainty relations, particularly concerning thermodynamic conjugates, by leveraging the fluctuation-dissipation theorem and the Lieb-Robinson bound. This methodological approach mirrors Bohr’s emphasis on questioning established norms. By carefully bounding the quantum Fisher information, the study demonstrates a structured relationship between information gain and the locality of estimators, proving that if a pattern cannot be reproduced or explained, it doesn’t exist.

What’s Next?

The established bounds, while offering a formal link between fluctuation-dissipation principles and the precision of parameter estimation, reveal a subtle truth: the pursuit of tighter uncertainty relations is less about achieving absolute limits and more about mapping the structural dependencies within a system. Each derived inequality doesn’t simply define a constraint; it reveals a previously obscured relationship between seemingly disparate quantities. The current work, therefore, acts as a demonstration of principle, indicating that the true value lies in extending this approach to more complex, many-body systems.

A critical limitation remains the reliance on local estimators and the Lieb-Robinson bound. While mathematically convenient, this introduces a bias toward systems where locality is readily apparent. Future investigations should explore whether alternative bounding techniques, perhaps leveraging entanglement measures directly, can circumvent this restriction and yield more general uncertainty relations – relations that do not inherently favor spatially separated degrees of freedom.

Ultimately, the progression of this field hinges on a shift in perspective. Interpreting the models, uncovering the hidden symmetries that dictate the precision of measurements, is demonstrably more important than simply producing ever-more-complex inequalities. The next step isn’t to refine the bounds, but to understand why these bounds exist, and what those reasons tell us about the fundamental nature of thermodynamic systems.

Original article: https://arxiv.org/pdf/2511.05042.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

The Intrinsic Limits of Knowing

Thermodynamic Uncertainty and Hidden Connections

Information’s Role in Precise Estimation

Locality, Interaction, and the Boundaries of Complexity

What’s Next?

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