Within these applications, TEURs can provide useful information and practical bounds for parameter estimation. Such revival has been prompted and paralleled by a steady advancement of experimental techniques dealing with single ion traps, qubits, neutron interferometry, and a countless and growing number of other quantum-information developments since then, e.g., nonlinear quantum metrology. In the present paper, we are motivated by the past two decades of important advancements in our understanding of the general structure of dynamical models for non-equilibrium thermodynamics, including non-equilibrium quantum thermodynamic models. Several reviews are available on the pioneering discussions and the subsequent developments. The time–energy uncertainty relation has remained an open and at times controversial issue throughout the history of quantum theory. For practical applications, however, such bounds are insufficient when Hamiltonian dynamics must be complemented by models of dissipation and decoherence. In the same spirit as fluctuation theorems that allow to estimate some statistical features of the dynamics from suitable state properties, also the Mandelstam–Tamm–Messiah time–energy uncertainty relations (MTM-TEURs) have been long known to provide bounds on lifetimes of quantum decaying states under Hamiltonian (non-dissipative) evolution. 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. For purely dissipative dynamics this reduces to the time–entropy uncertainty relation τ F Δ S ≥ k B τ, meaning that the nonequilibrium dissipative states with longer lifetime are those with smaller entropy uncertainty Δ S. For example, we obtain the time–energy-and–time–entropy uncertainty relation ( 2 τ F Δ H / ℏ ) 2 + ( τ F Δ S / k B τ ) 2 ≥ 1 where τ 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. 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 Δ S (square root of entropy fluctuations). A useful practical consequence is that in unitary dynamics the states with longer lifetimes are those with smaller energy uncertainty Δ H (square root of energy fluctuations). In the domain of nondissipative unitary Hamiltonian dynamics, the well-known Mandelstam–Tamm–Messiah time–energy uncertainty relation τ F Δ H ≥ ℏ / 2 provides a general lower bound to the characteristic time τ F = Δ F / | d 〈 F 〉 / d t | with which the mean value of a generic quantum observable F can change with respect to the width Δ F of its uncertainty distribution (square root of F fluctuations).
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