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arXiv:quant-ph/9710043 · Physica D Nonlinear Phenomena

The maximum speed of dynamical evolution

Norman Margolus, Lev B. Levitin

Published Sep 1, 1998• 887 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1016/s0167-2789(98)00054-2

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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.
I report a tight upper bound of the maximum speed of evolution from one quantum state $\ensuremath{\rho}$ to another ${\ensuremath{\rho}}^{\ensuremath{'}}$ with fidelity $F(\ensuremath{\rho},{\ensuremath{\rho}}^{\ensuremath{'}})$ less than or equal to an arbitrary but fixed value under the action of a time-independent Hamiltonian. Since the bound is directly proportional to the average absolute deviation from the median of the energy of the state DE\mathcal{D}EDE, one may interpret DE\mathcal{D}EDE as a meaningful measure of the maximum information-processing capability of a system.
The presence of noise or the interaction with an environment can radically change the dynamics of observables of an otherwise isolated quantum system.We derive a bound on the speed with which observables of open quantum systems evolve.This speed limit is divided into Mandelstam and Tamm's original time-energy uncertainty relation and a time-information uncertainty relation recently derived for classical systems, and both are generalized to open quantum systems.By isolating the coherent and incoherent contributions to the system dynamics, we derive both lower and upper bounds on the speed of evolution.We prove that the latter provide tighter limits on the speed of observables than previously known quantum speed limits and that a preferred basis of speed operators serves to completely characterize the observables that saturate the speed limits.We use this construction to bound the effect of incoherent dynamics on the evolution of an observable and to find the Hamiltonian that gives the maximum coherent speedup to the evolution of an observable.
The limited distinctness of physical systems is roughly expressed by uncertainty relations. Here we show distinctness is a finite resource we can exactly count to define basic physical quantities, limits to the resolution of space and time, and informational foundations for classical mechanics. Our analysis generalizes quantum speed limits: we count the distinct (orthogonal) states that can occur in a finite length of unitary change. As in Nyquist's bound on distinct signal values in classical waves, widths of superpositions bound the distinct states per unit length -- and basic conserved quantities are widths. Maximally distinct unitary evolution is effectively discrete -- and this characterizes classical systems. [see also Popular Summary in arxiv ancillary files]
Maximum counts of distinct states define important physical quantities, the most famous being entropy. Here we show that energy, momentum and other basic quantities of mechanics are also maximum counts. This generalizes a property familiar in communication with classical waves: frequency-width equals maximum number of distinct values that can be sent, per unit time. Basic mechanical quantities are frequency-widths of wavefunction evolution, and similarly bound number of distinct states. Evolutions that achieve maximum distinctness are effectively discrete. These properties refine our understanding of uncertainty, measurement, classical mechanics, and relativity.
The speed limit on entanglement is defined as the maximal rate at which entanglement can be generated or degraded in a physical process. We derive the speed limits on entanglement, using the relative entropy of entanglement and trace-distance entanglement, for unitary and arbitrary quantum dynamics, where we assume that the dynamics of the closest separable state can be approximately described by the closest separable dynamics of the actual dynamics of the system. For the unitary dynamics of isolated bipartite systems which are described by pure states, the rate of entanglement production is bounded by the product of fluctuations of the system's driving Hamiltonian and the surprisal operator, with an additional term reflecting the time-dependent nature of the closest separable state. Removing restrictions on the purity of the input and on the unitarity of the evolution, the two terms in the bound get suitably altered. Furthermore, we find a lower bound on the time required to generate or degrade a certain amount of entanglement by arbitrary quantum dynamics. We demonstrate the tightness of our speed limits on entanglement by considering quantum processes of practical interest.