State disturbance and pointer shift in protective quantum measurements

 
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arXiv: 141 1.2642V 1 [quant-ph] 10 Nov 2014 State disturbance and pointer shift in protective quantum measurements Maximilian Schlosshauer Department of Physics, University of Portland, 5000 North Willamette Boulevard, Portland, Oregon 97203, USA We investigate the disturbance of the state of a quantum system in a protective measurement for finite measurement times and different choices of the time-dependent system apparatus coupling function. The ability to minimize this state disturbance is essential to protective measurement. We show that for a coupling strength that remains constant during the measurement interaction of duration T, the state disturbance scales as Г , while a simple smoothing of the coupling function significantly improves the scaling behavior to T~e. We also prove that the shift of the apparatus pointer in the course of a protective measurement is independent of the particular time dependence of the coupling function, suggesting that the guiding principle for choosing the coupling function should be the minimization of the state disturbance. Our results illuminate the dynamics of protective measurement under realistic circumstances arid may aid in the experimental realization of such measurements. Journal reference: Phys. Rev. A 90, 052106 (2014) FACS numbers: 03.65.Ta, 03.65.\V.j I. INTRODUCTION Protective measurement [1-5] is a quantum measure¬ ment scheme in which an apparatus is weakly coupled to a quantum system for an extended period of time. If the system starts out in a nondegenerate eigenstate of its Hamiltonian and the interaction is sufficiently weak and long, then expectation values of observables of the sys¬ tem can be measured without appreciably disturbing the state of the system. Since measurement of a sufficient number of expectation values allows one to reconstruct a quantum state, protective measurement, if suitably im¬ plemented (see Ref. [4] for a discussion of constraints and complications), may enable reconstruction of the quan¬ tum state of an individual system. This provides a per¬ spective on state reconstruction different from that as¬ sociated with conventional ensemble state tomography based on strong [6-9] or weak [10-12] measurements. Only an infinitely weak or infinitely slowly changing measurement interaction will not disturb the state of a protectively measured system; tills follows directly from perturbation theory and the quantum adiabatic theorem [13]. Outside these limiting cases, however, protective measurement, if it is to yield new information, cannot avoid disturbing the state of the system, in agreement with general results concerning the fundamental trade¬ off between quantum state disturbance and information gain [14] and the independence of the maximum possible information gain in a quantum measurement from the method of measurement [15]. From a fundamental point of view, this inevitable state disturbance disproves sug¬ gestions [1, 2, 16] that protective measurement permits state measurement akin to a classical state and bears on the meaning of the wavefunetion (see Refs. [4, 17, 18] for discussions of this important foundational point). This limitation, however, does not invalidate the po¬ tential practical usefulness of protective measurement. Implementation of protective measurement would be in¬ teresting and important both from a fundamental point of view (as the realization of a new quantum measure¬ ment scheme) and from a practical point of view (en¬ abling quantum state tomography for single systems). Just like traditional ensemble quantum state tomogra¬ phy, protective measurement provides a way of (approx¬ imately) reconstructing a quantum state. The fidelity of any such reconstruction can be measured in terms of the disturbance of the initial state of the system incurred during the measurement. At the heart of protective mea¬ surement is the idea that this state disturbance can be made arbitrarily small, such that repeated measurements on the same system permit reconstruction of its initial state with arbitrarily high fidelity [1-4]. Therefore, for practical implementations of protective measurement it is essential to gain a precise and quantitative understand¬ ing of how one may reduce the state disturbance incurred during a protective measurement while simultaneously maintaining appreciable information gain. Despite its significance, however, the problem of state disturbance in protective measurement has not yet been adequately studied. Instead, the existing literature (see, e.g., Refs. [1-3, 5]) has relied on the consideration of mathematical limits involving infinitely long, infinitely weak, and/or infinitely slowly changing (adiabatic) mea¬ surement interactions, for which the state of the system can be shown to remain unchanged during the measure¬ ment. This, however, leaves open the important question of precisely how much the initial state will be disturbed in the physically relevant case of finite measurement times and interaction strengths, and how this disturbance de¬ pends on the particular choice of the coupling function describing the time dependence of the system-apparatus interaction. This paper addresses this question. We study the state disturbance in a protective measurement for differ¬ ent coupling functions and make precise the dependence of the state disturbance on the physical parameters of the