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Physical Review Letters

Bounds on Observables in Elementary Excitation Models

Duane P. Johnson, Allen Jay Kassman, William M. Visscher

Published Jan 4, 1971• 8 citations
Publisher

Abstract

A general technique is presented for finding upper and lower bounds on the value of any physical quantity consistent with measured values of other physical quantities and the assumption that all these quantities can be described by the same elementary excitation states. These bounds are independent of the details of the distribution of the states and in some instances can be quite restrictive.

DOI
10.1103/physrevlett.26.10

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A general technique is presented for finding upper and lower bounds on the value of any physical quantity consistent with measured values of other physical quantities and the assumption that all these quantities can be described by the same elementary excitation states. These bounds are independent of the details of the distribution of the states and in some instances can be quite restrictive.
Bounds are found for the values of an observable which are consistent with the measured value of a second observable and satisfy the conditions that f0(e){f}_{0}(e)f0​(e), f1(e){f}_{1}(e)f1​(e), and $Y(e)=\frac{{f}_{0}^{\ensuremath{'}}(e)}{{f}_{1}^{\ensuremath{'}}(e)}$ are well behaved and have nonzero slope on the open interval eLtoeU{e}_{L} \mathrm{to} {e}_{U}eL​toeU​. f0(e){f}_{0}(e)f0​(e) and f1(e){f}_{1}(e)f1​(e) are, respectively, the contributions to the undetermined and determined observable from an elementary excitation state of energy eee.
Putting a generally valid upper bound on equilibration times of physically relevant observables is a much pursued endeavor. Recently, such a bound has been suggested by Garcia-Pintos et al. While the mathematical correctness of the bound as such is undisputed, its concrete calculation requires the knowledge of certain quantities, which Garcia-Pintos et al. assess by means of assumptions. We show that, e.g., in standard cases of slow, exponential equilibration, (at least) one of these assumptions is not valid. This demonstration highlights the difficulty to judge the validity of the above assumptions without further information. Such information is, in general, very hard to obtain.
Traditional thermodynamic trade-off relations usually apply to quantities that depend linearly on probability distributions. In contrast, many important information-theoretic measures, such as entropies, are nonlinear and therefore difficult to analyze with existing frameworks. Motivated by replica methods in quantum information and spin-glass theory, we introduce replica Markov processes, i.e., Markovian dynamics of KKK independent, identical copies, and derive trade-off relations that bound relative moments of replica observables in terms of the dynamical activity. By choosing appropriate replica observables, these inequalities translate into bounds on nonlinear quantities of the original single process. Focusing on entropic measures of uncertainty, we obtain upper bounds on both the time derivative and the value of the Tsallis entropy for general trajectory-observable distributions. Moreover, we derive upper bounds on the Rényi and Tsallis entropies, where the bounds involve the initial dynamical activity. Analogous bounds can be derived for the entropies of the state distribution. In particular, we provide upper bounds on both the time derivative and the value of the Tsallis entropy of state distributions. Moreover, we show that the Rényi and Tsallis entropies of the state distribution are bounded from above by terms involving the local escape rate from the initial node. We also illustrate how replicas can be used to study other nonlinear functionals such as extreme-value observables. Finally, we extend the construction to continuously monitored open quantum systems, where the bounds are expressed in terms of the quantum dynamical activity. These results provide entropic counterparts to activity-based uncertainty relations and establish a general method for constraining nonlinear information-theoretic quantities in stochastic and quantum thermodynamics.