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arXiv:1003.0726 · Physical Review A

Tight upper bound of the maximum speed of evolution of a quantum state

H. F. Chau

Published Jun 30, 2010• 37 citations
arXivHTMLPDF

Abstract

I report a tight upper bound of the maximum speed of evolution from one quantum state $\ensuremath{\rho}$ to another ${\ensuremath{\rho}}^{\ensuremath{'}}$ with fidelity $F(\ensuremath{\rho},{\ensuremath{\rho}}^{\ensuremath{'}})$ less than or equal to an arbitrary but fixed value under the action of a time-independent Hamiltonian. Since the bound is directly proportional to the average absolute deviation from the median of the energy of the state $\mathcal{D}E$, one may interpret $\mathcal{D}E$ as a meaningful measure of the maximum information-processing capability of a system.

DOI
10.1103/physreva.81.062133

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I report a tight upper bound of the maximum speed of evolution from one quantum state \ensuremathρ\ensuremath{\rho}\ensuremathρ to another \ensuremathρ\ensuremath′{\ensuremath{\rho}}^{\ensuremath{'}}\ensuremathρ\ensuremath′ with fidelity F(\ensuremathρ,\ensuremathρ\ensuremath′)F(\ensuremath{\rho},{\ensuremath{\rho}}^{\ensuremath{'}})F(\ensuremathρ,\ensuremathρ\ensuremath′) less than or equal to an arbitrary but fixed value under the action of a time-independent Hamiltonian. Since the bound is directly proportional to the average absolute deviation from the median of the energy of the state DE\mathcal{D}EDE, one may interpret DE\mathcal{D}EDE as a meaningful measure of the maximum information-processing capability of a system.
We discuss the problem of counting the maximum number of distinct states that an isolated physical system can pass through in a given period of time---its maximum speed of dynamical evolution. Previous analyses have given bounds in terms of the standard deviation of the energy of the system; here we give a strict bound that depends only on E-E0, the system's average energy minus its ground state energy. We also discuss bounds on information processing rates implied by our bound on the speed of dynamical evolution. For example, adding one Joule of energy to a given computer can never increase its processing rate by more than about 3x10^33 operations per second.
We discuss the problem of counting the maximum number of distinct states that an isolated physical system can pass through in a given period of time — its maximum speed of dynamical evolution. Previous analyses have given bounds in terms of ΔE, the standard deviation of the energy of the system; here we give a strict bound that depends only on E − E0, the system's average energy minus its ground state energy. We also discuss bounds on information processing rates implied by our bound on the speed of dynamical evolution. For example, adding 1 J of energy to a given computer can never increase its processing rate by more than about 3 × 1033 operations per second.
The average energy and energy spread of a quantum system are two important factors that affect a system's evolution speed, which is usually defined as the changing rate of the distance between a quantum state and the state during its evolution. Alternatively, the evolution speed of a quantum system could also be studied based on its evolution path. It is found that if we define the evolution speed of a quantum unitary system as the changing rate of the path distance between a quantum state and the state during its evolution, the instantaneous evolution speed is determined by the Hamiltonian's eigenvalues, rather than its expectation value or variance imposed on any quantum state. More specifically, the range of the Hamiltonian's eigenvalues determines the evolution speed of a quantum system through the relation ${\ensuremath{\omega}}_{H}=\frac{1}{2}({\ensuremath{\lambda}}_{max}\ensuremath{-}{\ensuremath{\lambda}}_{min})$, which is independent of the quantum state. Given an initial state ${\ensuremath{\rho}}_{0}$ in a quantum unitary system, the minimum time required to evolve it to a target state ${\ensuremath{\rho}}_{\ensuremath{\tau}}$ is ${\ensuremath{\tau}}_{min}=\frac{{\mathcal{L}}_{min}({\ensuremath{\rho}}_{0},{\ensuremath{\rho}}_{\ensuremath{\tau}})}{{\ensuremath{\omega}}_{H}}$, where ${\mathcal{L}}_{min}({\ensuremath{\rho}}_{0},{\ensuremath{\rho}}_{\ensuremath{\tau}})$ is the path distance contained in the shortest evolution from ${\ensuremath{\rho}}_{0}$ to ${\ensuremath{\rho}}_{\ensuremath{\tau}}$. This lower bound of the evolution time is tight and achievable in a finite-energy system.
Let us have a Hamiltonian H and 2 states which can time evolve into each other via this Hamiltonian. In this particular situation, say one of the states evolves into the other in time t. Now let us fix these two states ; then we may have infinite Hamiltonians (other than H, and one of them is say H') that can time evolve one of them into the other. But the time taken for this evolution varies from t and is say t' (that is, t' is the time Hamiltonian H' takes to time evolve one of the fixed states into the other). From now on, we shall consider only those Hamiltonians that can time evolve one of the states into the other. The question is given any small positive number a, will there exist a Hamiltonian H such that t' = a? Answer: Yes , the Hamiltonian Ht/a (obtained by rescaling the original Hamiltonian) will work. But, there are bounds that show that as long as the average energy and its fluctuations are bounded (thus, no rescaling allowed), you cannot go arbitrarily fast from one state to another orthogonal state. This discussion falls under the study of quantum speed limits. This document looks into this field of active research. We shall study with care how the question of a speed limit arises: from varying the Hamiltonians between two fixed states ; and hence produce an accurate speed limit between any two fixed states. The process and results shall develop as concretely and smoothly as the problem which is produced.
The Margolus-Levitin (ML) bound says that for any time-independent Hamiltonian, the time needed to evolve from one quantum state to another is at least $\ensuremath{\pi}\ensuremath{\alpha}(\ensuremath{\epsilon})/2\ensuremath{\langle}E\ensuremath{-}{E}_{0}\ensuremath{\rangle}$, where $\ensuremath{\langle}E\ensuremath{-}{E}_{0}\ensuremath{\rangle}$ is the expected energy of the system relative to the ground state of the Hamiltonian and $\ensuremath{\alpha}(\ensuremath{\epsilon})$ is a function of the fidelity $\ensuremath{\epsilon}$ between the two states. For a long time, only an upper bound ${\ensuremath{\alpha}}_{>}(\ensuremath{\epsilon})$ and a lower bound ${\ensuremath{\alpha}}_{<}(\ensuremath{\epsilon})$ are known, although they agree up to at least seven significant figures. Recently, H\"ornedal and S\"onnerborn [arXiv:2301.10063] proved an analytical expression for $\ensuremath{\alpha}(\ensuremath{\epsilon})$, a fully classified system whose evolution time saturates the ML bound, and gave this bound a symplectic-geometric interpretation. Here I solve the same problem through an elementary proof of the ML bound. By explicitly finding all the states that saturate the ML bound, I show that ${\ensuremath{\alpha}}_{>}(\ensuremath{\epsilon})$ is indeed equal to ${\ensuremath{\alpha}}_{<}(\ensuremath{\epsilon})$. More importantly, I point out a numerical stability issue in computing ${\ensuremath{\alpha}}_{>}(\ensuremath{\epsilon})$ and report a simple way to evaluate it efficiently and accurately.
Quantum mechanics imposes fundamental constraints known as quantum speed limits (QSLs) on the information processing speed of all quantum systems. Every QSL known to date comes from the restriction imposed on the evolution time between two quantum states through the value of a single system observable such as the mean energy relative to its ground state. So far these restrictions only place upper bounds on the information processing speed of a quantum system. Here I report QSLs each with permissible information processing speeds separated by forbidden speed intervals. They are found by a systematic and efficient procedure that takes the values of several compatible system observables into account simultaneously. This procedure generalizes almost all existing QSL proofs; the QSLs show a novel first-order phase transition in the minimum evolution time.