Abstract
In the domain of nondissipative unitary Hamiltonian dynamics, the well-known Mandelstam–Tamm–Messiah time–energy uncertainty relation provides a general lower bound to the characteristic time with which the mean value of a generic quantum observable F can change with respect to the width of its uncertainty distribution (square root of F fluctuations). A useful practical consequence is that in unitary dynamics the states with longer lifetimes are those with smaller energy uncertainty (square root of energy fluctuations). Here we show that when unitary evolution is complemented with a steepest-entropy-ascent model of dissipation, the resulting nonlinear master equation entails that these lower bounds get modified and depend also on the entropy uncertainty (square root of entropy fluctuations). For example, we obtain the time–energy-and–time–entropy uncertainty relation where is a characteristic dissipation time functional that for each given state defines the strength of the nonunitary, steepest-entropy-ascent part of the assumed master equation. For purely dissipative dynamics this reduces to the time–entropy uncertainty relation , meaning that the nonequilibrium dissipative states with longer lifetime are those with smaller entropy uncertainty .
Highlights
Recent advances in quantum information and quantum thermodynamics (QT) have increased the importance of estimating the lifetime of a given quantum state, for example to engineer decoherence correction protocols aimed at entanglement preservation
Kossakowski–Lindblad–Gorini–Sudarshan (KLGS) master equations [25,26,27,28,29,30,31,32,33], we assume the less known locally steepest-entropy-ascent (LSEA) model of dissipation. We make this choice to avoid some drawbacks of the KLGS master equation from the point of view of full and strong consistency with the general principles of thermodynamics, causality, and far non-equilibrium, but more importantly because we have shown in References [34,35] that the LSEA principle—by providing the minimal but essential elements of thermodynamic consistency, near as well as far from stable equilibrium states—has the potential to unify all the successful frameworks of non-equilibrium modeling, from kinetic theory to chemical kinetics, from stochastic to mesoscopic to extended irreversible thermodynamics, as well as the metriplectic structure or, in more recent terms, the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) structure
These master equations are capable to describe the natural tendency of any initial nonequilibrium state to relax towards canonical or partially-canonical thermodynamic equilibrium (Gibbs state), i.e., capable of describing the irreversible tendency to evolve towards the highest entropy state compatible with the instantaneous mean values of the energy
Summary
Recent advances in quantum information and quantum thermodynamics (QT) have increased the importance of estimating the lifetime of a given quantum state, for example to engineer decoherence correction protocols aimed at entanglement preservation. The class of MEPP master equations we designed in References [57,58,59,60,61,62] is suitable to model dissipation phenomenologically in open quantum systems in contact with macroscopic baths, and in closed isolated systems, as well as strongly coupled and entangled composite systems (references below) These master equations are capable to describe the natural tendency of any initial nonequilibrium state (read: density operator) to relax towards canonical or partially-canonical thermodynamic equilibrium (Gibbs state), i.e., capable of describing the irreversible tendency to evolve towards the highest entropy state compatible with the instantaneous mean values of the energy (and possibly other constants of the motion and other constraints). The structure of the paper is outlined at the end of the section, where we first introduce the particular class of nonlinear dissipative quantum master equations on which we restrict our attention in the first part of the paper
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