Abstract
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.
Highlights
It has recently been argued that already the first generation of real-life quantum computers will be able to perform certain tasks exponentially faster than classical computers [1]
In the present work we highlighted that the maximal quantum speed can be fully characterized by the Schatten-p-norms of the generator of quantum dynamics
We further showed that equivalent expressions can be found in Wigner phase space, where the computationally expensive operator norm is replaced by the absolute value of an imaginary valued function
Summary
It has recently been argued that already the first generation of real-life quantum computers will be able to perform certain tasks exponentially faster than classical computers [1]. Whereas in the theory of quantum computation one is after characterizing quantum speed-ups—the quest for faster and faster computations with less and less single operations—the quantum speed limit sets the ultimate, maximal speed with which any quantum system can evolve This means, in particular, that every single quantum operation takes a finite, minimal time to be accomplished—and even quantum computers will not be able to achieve any arbitrary speed-ups. This quantum speed limit originates in the Heisenberg indeterminacy principle [4, 5], DEDt 2. As an illustrative example we will discuss the semi-classical, high temperature limit, and we will confirm that the quantum speed limit is a pure quantum feature, i.e., that classical systems do not experience a fundamental bound on their rates of change
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