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Thermodynamic entropic uncertainty relation
Yoshihiko Hasegawa1, Tomohiro Nishiyama2
1The University of Tokyo, Department of Information and Communication Engineering, Graduate School of Information Science and Technology, Tokyo 113-8656, Japan.
Physical Review. E
|December 23, 2025
Summary
This study establishes a quantitative link between Shannon entropy and entropy production in stochastic thermodynamics. It reveals a fundamental trade-off between decision accuracy and entropy production in decision-making models.
Area of Science:
- Stochastic Thermodynamics
- Information Theory
- Statistical Mechanics
Background:
- Thermodynamic uncertainty relations link observable precision and entropy production.
- Shannon entropy quantifies uncertainty in information theory.
- A direct quantitative link between Shannon entropy and entropy production is lacking.
Purpose of the Study:
- To establish a quantitative relationship between Shannon entropy of an observable and entropy production.
- To introduce and utilize symmetry entropy to quantify observable distribution asymmetry.
- To demonstrate a fundamental trade-off in stochastic decision-making.
Main Methods:
- Formulation of an uncertainty relation using Shannon entropy and entropy production.
- Introduction of symmetry entropy to measure distribution symmetry.
- Application of the derived relation to the diffusion decision model.
Main Results:
- Established a lower bound of ln2 for the sum of entropy production and symmetry entropy.
- Proved that the sum of entropy production and Shannon entropy is no less than ln2.
- Demonstrated a trade-off between decision accuracy and entropy production in the diffusion model.
Conclusions:
- A fundamental uncertainty relation exists between Shannon entropy and entropy production.
- Symmetry entropy provides a measure linked to entropy production.
- The findings have implications for understanding stochastic decision-making processes.
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