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arXiv:quant-ph/0110004 · Physical Review A

Measuring energy, estimating Hamiltonians, and the time-energy uncertainty relation

Yakir Aharonov, Serge Massar, Sandu Popescu

Published Nov 22, 2002• 58 citations
arXivHTMLPDF

Abstract

Suppose that the Hamiltonian acting on a quantum system is unknown and one wants to determine which is the Hamiltonian. We show that, in general, this requires a time $\ensuremath{\Delta}t$ that obeys the uncertainty relation $\ensuremath{\Delta}t\ensuremath{\Delta}H\ensuremath{\gtrsim}1,$ where $\ensuremath{\Delta}H$ is a measure of how accurately the unknown Hamiltonian must be estimated. We apply this result to the problem of measuring the energy of an unknown quantum state. It has been previously shown that if the Hamiltonian is known, then the energy can, in principle, be measured with arbitrarily large precision in an arbitrarily short time. On the other hand, we show that if the Hamiltonian is not known then an energy measurement necessarily takes a minimum time $\ensuremath{\Delta}t$ which obeys the uncertainty relation $\ensuremath{\Delta}t\ensuremath{\Delta}E\ensuremath{\gtrsim}1,$ where $\ensuremath{\Delta}E$ is the precision of the energy measurement. Several examples are studied to address the question of whether it is possible to saturate these uncertainty relations. Their interpretation is discussed in detail.

DOI
10.1103/physreva.66.052107

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Suppose that the Hamiltonian acting on a quantum system is unknown and one wants to determine which is the Hamiltonian. We show that, in general, this requires a time \ensuremathΔt\ensuremath{\Delta}t\ensuremathΔt that obeys the uncertainty relation \ensuremathΔt\ensuremathΔH\ensuremath≳1,\ensuremath{\Delta}t\ensuremath{\Delta}H\ensuremath{\gtrsim}1,\ensuremathΔt\ensuremathΔH\ensuremath≳1, where \ensuremathΔH\ensuremath{\Delta}H\ensuremathΔH is a measure of how accurately the unknown Hamiltonian must be estimated. We apply this result to the problem of measuring the energy of an unknown quantum state. It has been previously shown that if the Hamiltonian is known, then the energy can, in principle, be measured with arbitrarily large precision in an arbitrarily short time. On the other hand, we show that if the Hamiltonian is not known then an energy measurement necessarily takes a minimum time \ensuremathΔt\ensuremath{\Delta}t\ensuremathΔt which obeys the uncertainty relation \ensuremathΔt\ensuremathΔE\ensuremath≳1,\ensuremath{\Delta}t\ensuremath{\Delta}E\ensuremath{\gtrsim}1,\ensuremathΔt\ensuremathΔE\ensuremath≳1, where \ensuremathΔE\ensuremath{\Delta}E\ensuremathΔE is the precision of the energy measurement. Several examples are studied to address the question of whether it is possible to saturate these uncertainty relations. Their interpretation is discussed in detail.
Uncertainty relations play a crucial role in quantum mechanics. Well-defined methods exist for the derivation of such uncertainties for pairs of observables. Other approaches also allow the formulation of time-energy uncertainty relations, even though time is not an operator in standard quantum mechanics. However, in these cases, different approaches are associated with different meanings and interpretations for these relations. The one of interest here revolves around the idea of whether quantum mechanics inherently imposes a fundamental minimum duration for energy measurements with a certain precision. In our study, we investigate within the Page and Wootters timeless framework how energy measurements modify the relative "flow of time'' between internal and external clocks. This provides a unified framework for discussing the subject, allowing us to recover previous results and derive new ones. In particular, we show that the duration of an energy measurement carried out by an external system cannot be performed arbitrarily fast from the perspective of the internal clock. Moreover, we show that during any energy measurement the evolution given by the internal clock is non-unitary.
According to the axioms of quantum mechanics, there exists a quantum measurement corresponding to each observable.Energy is one of such observables, and the operator of the energy is called Hamiltonian.The Hamiltonian specifies the dynamics of closed quantum systems and it also characterizes the property of the equilibrium state in contact with a heat bath.Therefore the energy is an essential quantity in quantum mechanics, which characterizes behavior of a quantum system.On the other hand, since the Hamiltonian is not a local observable in general, a concrete construction of how to apply the quantum measurement of energy is not trivial.Thus various measurement models of energies have been proposed, and also discussions have been made on whether the time is a resource to be consumed to increase the precision of energy measurement in connection with the time-energy uncertainty relationship.Constructing a way to implement the energy measurement on an unknown Hamiltonian system is important, since it avoids constructing a new measurement scheme for each new quantum system.On the other hand, among measurements of energy, the projective measurement of energy has useful properties for applications.The projective measurement is a measurement which satisfies the condition so-called repeatable hypothesis that when we measure the state immediately after obtaining an outcome from the same measurement, we obtain the same outcome again.In this thesis, we consider two schemes to implement the projective measurement of energy for an unknown Hamiltonian system.One is a tomography based method, which is a construction suggested in the paper of Aharanov et.al. (2002).The other is the phase estimation based method, which utilizes the Hamiltonian dynamics of the system as a resource for the quantum algorithm, and realizes the projective measurement of energy on the system without identifying all parameters of the Hamiltonian.In this thesis, we assume there is a finite dimensional quantum system (quantum computer) which is able to apply any interaction between the target system and the finite dimensional system although the interaction should not depend on the target system Hamiltonian.To evaluate the performance of the measurements implemented by the two schemes, we formulate two evaluation functions called fluctuation of measurement value and non-repeatability, which can evaluate how much a measurement is different from the ideal projective measurement of an observable.We calculate a sufficient time to guarantee the fluctuation of measurement value below some small value ε, for each of the two measurement schemes.We find that the tomography based method takes the time proportional to O(d 4 ∆ max /ε), while the phase estimation based method takes O(∆ 3 max /ε 2 ), where d is the dimension of the target system and ∆ max is the difference between the largest energy eigenvalue and the smallest energy eigenvalue.We show that the phase estimation based method performs better for large particle systems.
The physical meaning of the uncertainty relation between time and energy appears to be not completely clear until now. This fact is not surprising since the interpretation of the time-energy relation is much more difficult than the interpretation of the similar relation between the coordinate and momentum. Indeed, the latter relation is easily derived from the quantum-mechanical formalism and does not require the examination of the course of the physical process in time. The relation between time and energy, however, essentially requires such an examination, and there arises a question whether Schro¨dinger’s equation may be used or not in deriving this relation, the latter possibility being not excluded since the relation may be interpreted to correspond to a measurement act, which does not obey Schro¨dinger’s equation.
Because time does not appear in Schr\"odinger's equation as an operator but only as a parameter, the time-energy uncertainty relation must be formulated in a special way. This problem has in fact been studied by many authors and we give a summary of their treatments. We then criticize the main conclusion of these treatments; viz., that in a measurement of energy carried out in a time interval, $\ensuremath{\Delta}t$, there must be a minimum uncertainty in the transfer of energy to the observed system, given by $\ensuremath{\Delta}({E}^{\ensuremath{'}}\ensuremath{-}E)>~\frac{h}{\ensuremath{\Delta}t}$. We show that this conclusion is erroneous in two respects. First, it is not consistent with the general principles of the quantum theory, which require that all uncertainty relations be expressible in terms of the mathematical formalism, i.e., by means of operators, wave functions, etc. Secondly, the examples of measurement processes that were used to derive the above uncertainty relation are not general enough. We then develop a systematic presentation of our own point of view, with regard to the role of time in the quantum theory, and give a concrete example of a measurement process not satisfying the above uncertainty relation.
Precision control of a quantum system requires accurate determination of the effective system Hamiltonian. We develop a method for estimating the Hamiltonian parameters for some unknown two-state system and providing uncertainty bounds on these parameters. This method requires only one measurement basis and the ability to initialize the system in some arbitrary state which is not an eigenstate of the Hamiltonian in question. The scaling of the uncertainty is studied for large numbers of measurements and found to be proportional to the reciprocal of the square root of the number of measurements.
Although time measurements are routinely performed in laboratories, their theoretical description is still an open problem. Correspondingly, the status of the energy-time uncertainty relation is unsettled. In the first part of this work the necessity of positive operator valued measures (POVM) as descriptions of every quantum experiment is reviewed, as well as the suggestive role played by the probability current in time measurements. Furthermore, it is shown that no POVM exists, which approximately agrees with the probability current on a very natural set of wave functions; nevertheless, the choice of the set is crucial, and on more restrictive sets the probability current does provide a good arrival time prediction. Some ideas to experimentally detect quantum effects in time measurements are discussed. In the second part of the work the energy-time uncertainty relation is considered, in particular for a model of alpha decay for which the variance of the energy can be calculated explicitly, and the variance of time can be estimated. This estimate is tight for systems with long lifetimes, in which case the uncertainty relation is shown to be satisfied. Also the linewidth-lifetime relation is shown to hold, but contrary to the common expectation, it is found that the two relations behave independently, and therefore it is not possible to interpret one as a consequence of the other. To perform the mentioned analysis quantitative scattering estimates are necessary. To this end, bounds of the form ∥1Re−iHtψ∥22≤Ct−3\|1_Re^{-iHt}ψ\|_2^2 \leq C t^{-3}∥1R​e−iHtψ∥22​≤Ct−3 have been derived, where ψψψ denotes the initial state, HHH the Hamiltonian, RRR a positive constant, and CCC is explicitly known. As intermediate step, bounds on the derivatives of the SSS-matrix in the form $\|1_K S^{(n)}\|_\infty \leq C_{n,K} $ have been established, with n=1,2,3n=1,2,3n=1,2,3, and the constants Cn,KC_{n,K}Cn,K​ explicitly known.