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arXiv:quant-ph/9704001

Measurability in Linear and Non-Linear Quantum Mechanical Systems

Yakir Aharonov, Benni Reznik

Published Apr 1, 1997• 1 citation
arXivHTMLPDF

Abstract

The measurability by means of continuous measurements, of an observable $\A(t_0)$, at an instant, and of a time averaged observable, $\bar \A=1/T\int \A(t')dt'$, is examined for linear and in particular for non-linear quantum mechanical systems. We argue that only when the exact (non-perturbative) solution is known, an exact measurement may be possible. A perturbative approach is shown to fail in the non-linear case for measurements with accuracy $Δ\bar \A < Δ\bar \A_{min}(T)$, giving rise to a restriction on the accuracy. Thus, in order to prepare an initial pure state of a non-linear system, by means of a continuous measurement, the exact non-perturbative solution must be known.

DOI
10.48550/arxiv.quant-ph/9704001

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The measurability by means of continuous measurements, of an observable \A(t0)\A(t_0)\A(t0​), at an instant, and of a time averaged observable, \Aˉ=1/T∫\A(t′)dt′\bar \A=1/T\int \A(t')dt'\Aˉ​=1/T∫\A(t′)dt′, is examined for linear and in particular for non-linear quantum mechanical systems. We argue that only when the exact (non-perturbative) solution is known, an exact measurement may be possible. A perturbative approach is shown to fail in the non-linear case for measurements with accuracy Δ\Aˉ<Δ\Aˉmin(T)Δ\bar \A < Δ\bar \A_{min}(T)Δ\Aˉ​<Δ\Aˉ​min​(T), giving rise to a restriction on the accuracy. Thus, in order to prepare an initial pure state of a non-linear system, by means of a continuous measurement, the exact non-perturbative solution must be known.
The linear and the nonlinear non-Markovian quantum state diffusion equation (NMQSD) are well known tools for the description of certain non-Markovian open quantum systems. In this work, we systematically investigate whether the normalized linear NMQSD or the nonlinear NMQSD solutions can be generated by means of a time-continuous measurement. By considering any conceivable measurement scheme in the framework of instruments, we derive a necessary criterion for a measurement interpretation of both equations. Concrete examples show that the normalized linear NMQSD solutions are realizable only in the Markovian limit in general. The application of the presented criterion to the nonlinear NMQSD remains an open issue.
The theory of continuous quantum measurement allows to reconstruct the state ρtρ_tρt​ of a system from a continuous stochastic measurement record ItI_tIt​. However, this truly continuous-time signal ItI_tIt​ is never available in practice. In experiments, one generally has access to its digitization, i.e., to a series of time averages IkI_kIk​ over finite intervals of duration ΔtΔtΔt. In this letter, we take this digitization seriously and define ρˉn\barρ_nρˉ​n​ as the best Bayesian estimate of the quantum state given (only) a digitized record (I1,…,In)(I_1,\dots,I_n)(I1​,…,In​). We show that ρˉn+1\barρ_{n+1}ρˉ​n+1​ can be computed recursively from In+1I_{n+1}In+1​ and ρˉn\barρ_nρˉ​n​ using an exact formula. The latter can be evaluated numerically exactly, or used as the basis for a perturbative expansion into successive powers of Δt\sqrt{Δt}Δt​. This allows reconstructing quantum trajectories in regimes of coarse ΔtΔtΔt where existing methods fail, estimating parameters at fixed ΔtΔtΔt without bias, and directly sampling digitized quantum trajectories with schemes of arbitrarily high order.
The theory of measurements continuous in time in quantum mechanics (quantum continual measurements) has been formulated by using the notions of instrument and positive operator valued measure, functional integrals, quantum stochastic differential equations and classical stochastic differential equations (SDE's). Various types of SDE's are involved, and precisely linear and non linear equations for vectors in Hilbert spaces and for trace-class operators. All such equations contain either a diffusive part, or a jump one, or both. In this paper we introduce a class of linear SDE's for trace-class operators, relevant to the theory of continual measurements, and we recall how such SDE's are related to instruments and master equations and, so, to the general formulation of quantum mechanics. We do not present the Hilbert space formulation of such SDE's and we make some mathematical simplifications: no time dependence is introduced into the coefficients and only bounded operators on the Hilbert space of the quantum system are considered. Then we introduce the notion of a posteriori state and the non linear SDE satisfied by such states and we give conditions from which such equation is assured to preserve pure states and to send any mixed state into a pure one for large times. Finally we review the known results about the existence and uniqueness of invariant measures in the purely diffusive case and we give some concrete examples of physical systems.
We study the dynamics of classical and quantum systems undergoing a continuous measurement of position by schematizing the measurement apparatus with an infinite set of harmonic oscillators at finite temperature linearly coupled to the measured system. Selective and non-selective measurement processes are then introduced according to a selection of or an average over all possible initial configurations of the measurement apparatus. At quantum level, the selective processes are described by a nonlinear stochastic Schrödinger equation whose solutions evolve into properly defined coherent states in the case of linear systems. For arbitrary measured systems, classical behaviour is always recovered in the macroscopic limit.
The possibility of determining the state of a quantum system after a continuous measurement of position is discussed in the framework of quantum trajectory theory. The initial lack of knowledge of the system and external noises are accounted for by considering the evolution of conditioned density matrices under a stochastic master equation. It is shown that after a finite time the state of the system is a pure state, and can be inferred from the measurement record alone. The relation to emerging possibilities for the continuous experimental observation of single quanta, as for example in cavity quantum electrodynamics, is discussed.
We study the instantaneity of the state-reduction process in relativistic quantum mechanics. The conclusion of various authors that this instantaneity will restrict the set of relativistic observables to purely local ones (i.e., that the measurement of any nonlocal property of a system at a well-defined time would give rise to violations of relativistic causality) is found to be erroneous, and experiments (of a kind not encountered before in measurement theory) are described whereby certain nonlocal properties of some simple physical systems can be measured at a well-defined time without violating causality. The attempts of certain authors to reconcile the reduction process with the covariance of the relativistic quantum state are considered and found wanting, and it is argued that the covariance of relativistic quantum theories resides exclusively in the experimental probabilities, and not in the underlying quantum states. The problem of nonlocal measurement is considered in general: distinctions (which are not to be met with in the nonrelativistic case) arise in relativistic quantum mechanics between what can be measured for fermions and what can be measured for bosons, between what can be measured for individual systems and what can be measured for ensembles, and between what kinds of states can be verified by measurement and what kinds of states can be prepared by measurement; and these pose difficult questions about the nature of measurement itself.