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The maximum average rate of state change

Norman Margolus

Published Sep 23, 2011• 2 citations
OpenAlex

Abstract

For an isolated quantum system, average energy above the ground state and average momentum determine the minimum time and distance needed to transition through $N$ distinct states. Other moments of energy and momentum provide similar bounds. For $N\gg1$, we can equate average energy and momentum with the maximum average rates of distinct state change in time and space. Thus at least vestiges of finite-state dynamics persist in the classical world, and in its interpretation.

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For an isolated quantum system, average energy above the ground state and average momentum determine the minimum time and distance needed to transition through NNN distinct states. Other moments of energy and momentum provide similar bounds. For N≫1N\gg1N≫1, we can equate average energy and momentum with the maximum average rates of distinct state change in time and space. Thus at least vestiges of finite-state dynamics persist in the classical world, and in its interpretation.
Finite physical systems have only a finite amount of distinct state. This finiteness is fundamental in statistical mechanics, where the maximum number of distinct states compatible with macroscopic constraints defines entropy. Here we show that finiteness of distinct state is similarly fundamental in ordinary mechanics: energy and momentum are defined by the maximum number of distinct states possible in a given time or distance. More generally, any moment of energy or momentum bounds distinct states in time or space. These results generalize both the Nyquist bandwidth-bound on distinct values in classical signals, and quantum uncertainty bounds. The new certainty bounds are achieved by finite-bandwidth evolutions in which time and space are effectively discrete, including quantum evolutions that are effectively classical. Since energy and momentum count distinct states, they are defined in finite-state dynamics, and they relate classical mechanics to finite-state evolution.
We discuss the problem of counting the maximum number of distinct states that an isolated physical system can pass through in a given period of time — its maximum speed of dynamical evolution. Previous analyses have given bounds in terms of ΔE, the standard deviation of the energy of the system; here we give a strict bound that depends only on E − E0, the system's average energy minus its ground state energy. We also discuss bounds on information processing rates implied by our bound on the speed of dynamical evolution. For example, adding 1 J of energy to a given computer can never increase its processing rate by more than about 3 × 1033 operations per second.
We discuss the problem of counting the maximum number of distinct states that an isolated physical system can pass through in a given period of time---its maximum speed of dynamical evolution. Previous analyses have given bounds in terms of the standard deviation of the energy of the system; here we give a strict bound that depends only on E-E0, the system's average energy minus its ground state energy. We also discuss bounds on information processing rates implied by our bound on the speed of dynamical evolution. For example, adding one Joule of energy to a given computer can never increase its processing rate by more than about 3x10^33 operations per second.