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arXiv:1706.03826

Quantum speed limits and the maximal rate of quantum learning

Thiago Vaz Acconcia, Sebastian Deffner

Published Jun 12, 2017• 2 citations
arXivHTMLPDF

Abstract

The Bremermann-Bekenstein bound is a fundamental bound on the maximal rate with which information can be transmitted. However, its derivation relies on rather weak estimates and plausibility arguments, which make the application of the bound impractical in the laboratory. In this paper, we revisit the bound and extend its scope by explicitly accounting for the back action of quantum measurements and refined expressions of the quantum speed limit. Our result can be interpreted as an upper bound on the maximal rate of quantum learning, and we show that the Bremermann-Bekenstein bound follows as a particular limit. Our results are illustrated, by first deriving a tractable expression from time-dependent perturbation theory, and then evaluating the bound for two time-dependent systems -- the harmonic oscillator and the Pöschl-Teller potential.

DOI
10.13016/m2fb3r-eswx

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The Bremermann-Bekenstein bound is a fundamental bound on the maximal rate with which information can be transmitted. However, its derivation relies on rather weak estimates and plausibility arguments, which make the application of the bound impractical in the laboratory. In this paper, we revisit the bound and extend its scope by explicitly accounting for the back action of quantum measurements and refined expressions of the quantum speed limit. Our result can be interpreted as an upper bound on the maximal rate of quantum learning, and we show that the Bremermann-Bekenstein bound follows as a particular limit. Our results are illustrated, by first deriving a tractable expression from time-dependent perturbation theory, and then evaluating the bound for two time-dependent systems -- the harmonic oscillator and the Pöschl-Teller potential.
In this paper, we give two independent and rigorous derivations for the quantum bound on the information transmission rate proposed independently by Bekenstein [Phys. Rev. Lett. 46, 623 (1981)] and Bremermann .ul2 [Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, edited by L. M. LeCam and J. Neyman (University of California Press, Berkeley, 1967], preceded by a heuristic argument showing why such a bound must hold. In both approaches, information carriers are quanta for some field. The first method resembles the microcanonical approach to statistical mechanics, where the strategy of overestimating the real number of states by relaxing the indistinguishability of quanta was adopted. The second is based entirely upon maximum-entropy methods. Amazingly enough, the results obtained by these physically unrelated premises turn out to be identical, namely, that the single (noiseless) channel capacity is I${\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}_{\mathrm{max}}$=E/2\ensuremath{\pi}\ensuremath{\Elzxh} bits ${\mathrm{s}}^{\mathrm{\ensuremath{-}}1}$. It is further shown that, in a finite time \ensuremath{\tau}, no information can ever be conveyed unless the energy threshold 2\ensuremath{\pi}\ensuremath{\Elzxh}/\ensuremath{\tau} is reached, allowing the reinterpretation of the time-energy uncertainty in informational-theoretic language.
The quantum speed limit is a fundamental upper bound on the speed of quantum evolution. However, the actual mathematical expression of this fundamental limit depends on the choice of a measure of distinguishability of quantum states. We show that quantum speed limits are qualitatively governed by the Schatten- p -norm of the generator of quantum dynamics. Since computing Schatten- p -norms can be mathematically involved, we then develop an alternative approach in Wigner phase space. We find that the quantum speed limit in Wigner space is fully equivalent to expressions in density operator space, but that the new bound is significantly easier to compute. Our results are illustrated for the parametric harmonic oscillator and for quantum Brownian motion.
Quantum asymmetry and coherence are genuinely quantum resources that are essential to realize quantum advantage in information technologies. However, all quantum processes are fundamentally constrained by quantum speed limits, which raises the question on the corresponding bounds on the rate of consumption of asymmetry and coherence. In the present work, we derive a formulation of the quantum speed limit for observables in terms of the trace-norm asymmetry of the time-dependent quantum state relative to the observable. This quantum speed limit can be directly observed in experiment through weak value measurement and provides a lower bound to the quantum Fisher information about the parameter conjugate to the observable. It can be further related to quantum coherence relative to the eigenbasis of the observable. We obtain a complementary relation for the speed of three mutually unbiased observables for a single qubit. As an application, we derive a notion of a quantum thermodynamic speed limit.
The quantum speed limit is a fundamental upper bound on the speed of quantum evolution. However, the actual mathematical expression of this fundamental limit depends on the choice of a measure of distinguishability of quantum states. We show that quantum speed limits are qualitatively governed by the Schatten-p-norm of the generator of quantum dynamics. Since computing Schatten-p-norms can be mathematically involved, we then develop an alternative approach in Wigner phase space. We find that the quantum speed limit in Wigner space is fully equivalent to expressions in density operator space, but that the new bound is significantly easier to compute. Our results are illustrated for the parametric harmonic oscillator and for quantum Brownian motion.
Quantum speed limits provide upper bounds on the rate with which a quantum system can move away from its initial state. Here, we provide a different kind of speed limit, describing the divergence of a perturbed open system from its unperturbed trajectory. In the case of weak coupling, we show that the divergence speed is bounded by the quantum Fisher information under a perturbing Hamiltonian, up to an error which can be estimated from system and bath timescales. We give three applications of our speed limit. First, it enables experimental estimation of quantum Fisher information in the presence of decoherence that is not fully characterized. Second, it implies that large quantum work fluctuations are necessary for a thermal system to be driven quickly out of equilibrium under a quench. Moreover, it can be used to bound the response to perturbations of expectation values of observables in open systems.