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AIP conference proceedings

A unified bound on the rate of quantum dynamics

Lev B. Levitin, Tommaso Toffoli, Alexander I. Lvovsky

Published Jan 1, 2009• 0 citations
Publisher

Abstract

Two different bounds on the least time required for a quantum state to evolve to an orthogonal one are reconsidered and unified. It is shown that these bounds can be attained only when they coincide. Otherwise, they still remain tight, but can be approached only asymptotically.
DOI
10.1063/1.3131291

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Some quantum states can dynamically evolve to an orthogonal state after a time interval, while some states can never evolve to their orthogonal states. In this paper, we investigate three issues on the orthogonalization properties of quantum states. First, we answer the question as to under what conditions an initial state can evolve to an orthogonal state. Second, we discuss the orthogonalization rate of quantum states, i.e. , the total number of orthogonal states that the evolution can go through in unit time. Finally, we show that the Margolus-Levitin bound for orthogonalization time remains tight for superpositions of more than two eigenstates.
How fast a quantum state can evolve has attracted considerable attention in connection with quantum measurement and information processing. 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 attained by certain initial states if DeltaE=E. Yet, the problem remained open when DeltaE not equal E. We consider the unified bound that involves both DeltaE and E. We prove that there exist no initial states that saturate the bound if DeltaE not equal E. However, the bound remains tight: for any 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. These results establish the fundamental limit of the operation rate of any information processing system.
If an initial state ∣ψI⟩|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle ∣ψI​⟩ and a final state ∣ψF⟩|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle ∣ψF​⟩ are given, then there exist many Hamiltonians under whose action ∣ψI⟩|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle ∣ψI​⟩ evolves into ∣ψF⟩|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle ∣ψF​⟩ . In this case, the problem of the transition of ∣ψI⟩|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle ∣ψI​⟩ to ∣ψF⟩|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle ∣ψF​⟩ in the least time is very interesting. It was previously shown that for a Hermitian Hamiltonian, there is an optimum evolution time if ∣ψI⟩|\psi_{ \scriptscriptstyle{\mathrm{I}} } \rangle ∣ψI​⟩ and ∣ψF⟩|\psi_{ \scriptscriptstyle{\mathrm{F}} } \rangle ∣ψF​⟩ are orthogonal. But for a PTPTPT -symmetric Hamiltonian, this time can be arbitrarily small, which seems amazing. We discuss the optimum time evolution for pseudo-Hermitian Hamiltonians and obtain a lower bound for the evolution time under the condition that the Hamiltonian is bounded. The optimum evolution time can be attained in the case where two quantum states are orthogonal with respect to some inner product. The results in the Hermitian and pseudo-Hermitian cases coincide if the evolution is unitary with some well-defined inner product. We also analyze two previously studied examples and find that they are consistent with our theory. In addition, we give some explanations of our results with two examples.