Thermodynamic concentration inequalities and tradeoff relations

Abstract: Thermodynamic tradeoff relations quantify the fundamental concept of ``no free lunch'' in the physical world, suggesting that faster and more precise physical processes come at a higher thermodynamic cost. The key elements in these tradeoff relations are the thermodynamic uncertainty relation and speed limit, which are closely tied to information inequalities from which other tradeoff relations are derived. Concentration inequalities are relations that complement information inequalities in statistical analyses and have been widely used in various fields. However, their role in thermodynamic tradeoff relations remains unclear. This Letter develops thermodynamic concentration inequalities that provide bounds for the distribution of observables in quantum and classical Markov processes. We derive a set of tradeoff relations that generalize speed limits and thermodynamic uncertainty relations from the developed thermodynamic concentration inequalities. The derived tradeoff relations hold under minimal assumptions of the underlying physical processes. This Letter clarifies the role of concentration inequalities in thermodynamics, paving the way for deriving new tradeoff relations.