Macroscopic observables

Abstract

We study macroscopic observables defined as the total value of a physical quantity over a collection of quantum systems. We show that previous results obtained for infinite ensemble of identically prepared systems lead to incorrect conclusions for finite ensembles. In particular, exact measurement of a macroscopic observable significantly disturbs the state of any finite ensemble. However, we show how this disturbance can be made arbitrarily small when the measurement are of finite accuracy. We demonstrate a tradeoff between state disturbance and measurement coarseness as a function of the size of the ensemble. Using this tradeoff, we show that the histories generated by any sequence of finite accuracy macroscopic measurements always generate a consistent family in the absence of large scale entanglement, for sufficiently large ensembles. Hence, macroscopic observables behave "classically" provided that their accuracy is coarser than the quantum correlation length-scale of the system. The role of these observable is also discussed in the context of NMR quantum information processing and bulk ensemble quantum state tomography. I. OVERVIEWMacroscopic observables correspond to physical quantities which are accessible to our senses. Since the physical scale of individual quanta is generally tiny, macroscopic observables arise when a collection of quantum systems are measured jointly. Formally, they can be described by type projectors, which reveal information about the average population of single-particle states. For example, the total magnetization of an ensemble of spin-1 2 particles provides some information about the relative occupation number of the up and down states. We will derive several general properties of these measurement and discuss how they lead to the emergence of a quasiclassical domain in the absence of large scale entanglement.The effect of macroscopic observations on infinite ensemble of identically prepared quantum systems has been studied in various contexts [1,2,3,4]. The main conclusion of these studies is that the state |ψ ⊗N describing such an ensemble is an eigenstate of type projectors when N = ∞. However, for finite ensembles, things change dramatically. The measurement of a macroscopic observable induces a disturbance which increases as the size of the ensemble grows, in apparent contradiction with the infinite-copy result. This discrepancy follows from the ambiguous extension of finite-copy considerations to the nonseparable Hilbert space of an infinite-copy ensemble [5]. In this article, we show how the essence of the infinite-copy result can be recovered by "smoothing" the type projectors into coarse-grained positive operator valued measurement (POVM) (essentially going from the * Electronic address: dpoulin@iqc.ca strong to the weak law of large numbers).The paper is organized as follows. The central mathematical objects of the present study are defined in Section II. We first summarize the method of type and define type projectors. These are projectors on the degenerated eigensu...

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“…The deep reason is that one works with values of observables. Second, the theories based on coarse-grained operators [6,7,8]: the problem is the same as with the decoherence. For example, the Legget-Garg inequality [8] is a condition for the validity of the principle of macroscopic realism that works with values of observables.…”

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“…[15][16][17]. A conceptually different approach focuses on the limits of observability of quantum effects in macroscopic objects [18][19][20][21].…”