Quantum Memory Effects in the Measurement of Observables with a Continuous Spectrum

In the measurement of a continuous observable Q, the pure components of the reduced state do, in general, depend on the initial state. For measurements which attempt to localize the measured system in a certain region R, the localized wave functions are proportional to the original wave function outside of R. This “quantum memory” effect shows that it is not possible to perfectly localize a quantum particle. In this paper we address the question of how to measure a quantum-mechanical observable with a continuous spectrum. Ever since the pioneering work of von Neumann [1], it has been known that for the measurement of an observable Q, the interaction energy between the observed system S and the measuring apparatus A has to be a function of Q [2,3]. Von Neumann considered an interaction term which is linear in a position-like variable Q [1]. More recently Zurek studied interactions between S and A in terms of a preferred “pointer basis” of A [2]. Haake and Walls studied the case of an oscillator coupled to an infinite reservoir [3], following the earlier work of Ullersma [4]. Unruh and Zurek recently studied state reduction in a similar model, including the case of a free particle, and presented explicit calculations of reduced density matrices for states with Gaussian characteristic functions [5]. These studies show that the variance of Q in the pure components of the reduced density matrix ρS is much less than in the initial state of S, in accord with the idea that an approximate measurement of Q has been performed by A. However, the explicit calculations of ρS also show that in general the set of eigenfunctions of ρS depends quite strongly on the initial state. For a measurement, on the other hand, we require that the set of eigenstates of ρS be independent of the initial state, and be identical to the set of eigenstates of the observable to be measured. In this sense, the linear models studied so far do not provide models for a measurement of Q. In the following, we show that measurements which determine the value of a continuous observable Q can be approximately realized by couplings of the form λS(Q) · VA, where λS is a strongly localized function which vanishes rapidly outside a finite interval I, and VA is an apparatus operator. However, in this case the localized wave functions do not vanish, but are proportional to the original wave function outside of I. Following von Neumann, we assume that the Hamiltonian of system S plus apparatus A takes the form H = λS(Q) · VA +HA (1) and that the initial state ρ(0) of system S plus apparatus A at time t = 0 is the product of a pure state |ψS〉〈ψS| of S with a state ρA of the apparatus. Due to the S-A interaction term, at later times t > 0 the state ρ(t) is no longer a product, and the reduced state ρS(t) = traceA ρ(t) of S is no longer a pure state. This state reduction models the measuring process. The Hamiltonian (1) does not contain a part for the free motion of S. This corresponds to the assumption that the system S does not change significantly over a time period t required for the measurement. We further assume that [VA, HA] = 0, and that the initial state of A is invariant under the free motion of A. The matrix elements of ρS(t) in the basis {|q〉} of eigenstates of Q are then (we choose units such that h̄ = 1) 〈q |ρS(t)| q〉 = traceA 〈q|e(|ψS〉〈ψS| ⊗ ρA)e |q〉 = 〈q|ψS〉〈ψS |q〉 traceA ( eS ′AρAe itλS(q ′′)VA ) = ψS(q ′)ψS(q ′′)∗fA ( t ( λS(q )− λS(q′′) ) )