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Thermodynamic uncertainty principles make up one of the few rare anchors inthe largely uncharted waters of nonequilibrium systems, the fluctuationtheorems being the more familiar. In this work we aim to trace theuncertainties of thermodynamic quantities in nonequilibrium systems to theirquantum origins, namely, to the quantum uncertainty principles. Our resultsenable us to make this categorical statement: For Gaussian systems,thermodynamic functions are functionals of the Robertson-Schrodingeruncertainty function, which is always non-negative for quantum systems, but notnecessarily so for classical systems. Here, quantum refers to noncommutativityof the canonical operator pairs. From the nonequilibrium free energy[1], wesucceeded in deriving several inequalities between certain thermodynamicquantities. They assume the same forms as those in conventional thermodynamics,but these are nonequilibrium in nature and they hold for all times and atstrong coupling. In addition we show that a fluctuation-dissipation inequalityexists at all times in the nonequilibrium dynamics of the system. Fornonequilibrium systems which relax to an equilibrium state at late times, thisfluctuation-dissipation inequality leads to the Robertson-Schrodingeruncertainty principle with the help of the Cauchy-Schwarz inequality. This workprovides the microscopic quantum basis to certain important thermodynamicproperties of macroscopic nonequilibrium systems.

Quantum Thermodynamic Uncertainties in Nonequilibrium Systems from Robertson-Schr\"odinger Relations