The fundamental limit on the rate of quantum dynamics: the unified bound is tight

TL;DR

This paper investigates the fundamental quantum speed limit for state evolution, demonstrating that the unified bound based on energy spread and average energy is tight and nearly attainable, setting a fundamental limit on quantum information processing rates.

Contribution

The paper proves the tightness of the unified quantum speed limit bound and characterizes initial states approaching this bound, clarifying the relation between energy parameters and evolution speed.

Findings

The unified bound is tight but not saturated unless DeltaE equals E.

A family of states can approach the bound arbitrarily closely.

The relation between energy levels and orthogonalization time is clarified.

Abstract

The question of how fast a quantum state can evolve has attracted a considerable attention in connection with quantum measurement, metrology, and information processing. Since only orthogonal states can be unambiguously distinguished, a transition from a state to an orthogonal one can be taken as the elementary step of a computational process. Therefore, such a transition can be interpreted as the operation of "flipping a qubit", and the number of orthogonal states visited by the system per unit time can be viewed as the maximum rate of operation. A lower bound on the orthogonalization time, based on the energy spread DeltaE, was found by Mandelstam and Tamm. Another bound, based on the average energy E, was established by Margolus and Levitin. The bounds coincide, and can be exactly attained by certain initial states if DeltaE=E; however, the problem remained open of what the situation is otherwise. Here we consider the unified bound that takes into account both DeltaE and E. We prove that there exist no initial states that saturate the bound if DeltaE is not equal to E. However, the bound remains tight: for any given values of DeltaE and E, there exists a one-parameter family of initial states that can approach the bound arbitrarily close when the parameter approaches its limit value. The relation between the largest energy level, the average energy, and the orthogonalization time is also discussed. These results establish the fundamental quantum limit on the rate of operation of any information-processing system.