Fundamental Limit on the Rate of Quantum Dynamics: The Unified Bound Is Tight

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.1 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 ∆E, was found by Mandelstam and Tamm. 2 Another bound, based on the average energy E, was established by Margolus and Levitin.3 The bounds coincide, and can be exactly attained by certain initial states if ∆E = E. However, the problem remained open of what the situation is when ∆E = E.Here we consider the unified bound that takes into account both ∆E and E. We prove that there exist no initial states that saturate the bound if ∆E = E. However, the bound remains tight: for any given values of ∆E and E, there exists a oneparameter 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. Starting with the classical result of Mandelstam and Tamm,2 it was later shown by Fleming, 4 Anandan and Aharonov, 5 and Vaidman 6 that the minimum time τ required for arriving to an orthogonal state is bounded bywhere (∆E) 2 = ψ|H 2 |ψ − ( ψ|H|ψ ) 2 , H is the Hamiltonian, and |ψ the wavefunction of the system. A different bound was obtained in,Here, E = ψ|H|ψ is the quantum-mechanical average energy of the system (the energy of the ground state is taken to be zero). Both bounds (1) and (2) are tight, and achieved for a quantum state such that ∆E = E.Since then, a vast literature has been devoted to various aspects of this problem. In particular, inequality (2) has been proved for mixed states and for composite systems both in separable and in entangled states (e.g., Giovannetti et al.,7, 8 Zander et al. 9 ). Bound (2) obtained for an isolated system has been generalized to a system driven by an external Hamiltonian (a "quantum gate") in.10, 11 Various derivations of (1) and (2) However, what remained unnoticed is the paradoxical situation of the existence of two bounds based on two different characteristics of the quantum state, seemingly independent of one another. Since the average energy E and the energy uncertainty ∆E play the most determinative role in quantum evolution, it is important to have a unified bound that would take into account both of these characteristics.In all known cases where bounds (1) and (2) can be exactly attained, the ratio α = ∆E E equals 1. A question arises: what happens if α = 1? Some authors ...

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