Disturbance trade-off principle for quantum measurements

Abstract

We demonstrate a fundamental principle of disturbance tradeoff for quantum measurements, along the lines of the celebrated uncertainty principle: The disturbances associated with measurements performed on distinct yet identically prepared ensembles of systems in a pure state cannot all be made arbitrarily small. Indeed, we show that the average of the disturbances associated with a set of projective measurements is strictly greater than zero whenever the associated observables do not have a common eigenvector. For such measurements, we show an equivalence between disturbance tradeoff measured in terms of fidelity and the entropic uncertainty tradeoff formulated in terms of the Tsallis entropy (T2). We also investigate the disturbances associated with the class of nonprojective measurements, where the difference between the disturbance tradeoff and the uncertainty tradeoff manifests quite clearly.The uncertainty principle, which is one of the cornerstones of quantum theory, has had a long history. In its original formulation by Heisenberg for canonically conjugate variables [1], the uncertainty principle was stated as an effect of the disturbance caused due to a measurement of one observable on a succeeding measurement of another. However, the subsequent mathematical formulation due to Robertson [2] and Schrödinger [3] in terms of variances departed from this original interpretation. Rather, they obtained a non-trivial lower bound on the product of the variances associated with the measurement of a pair of incompatible observables, performed on distinct yet identically prepared copies of a given system. The Robertson-Schrödinger inequality thus expresses a fundamental limitation as regards the preparation of an ensemble of systems in identical states, and is therefore a manifestation of the so-called preparation uncertainty [4].Along the same lines, the more recent entropic formulations of the uncertainty principle [5] also demonstrate the existence of a fundamental tradeoff for the uncertainties associated with independent measurements of incompatible observables on identically prepared ensemble of systems. Entropic uncertainty relations (EURs) have been obtained for specific classes of observables for both the Shannon and Rényi entropies [6][7][8][9][10][11][12][13][14], as well as for the Tsallis entropies [15][16][17].Here, we prove the existence of a similar principle of tradeoff for the disturbances associated with the measurements of a set of observables. It is a fundamental feature of quantum theory that when an observable is measured on an ensemble of systems, the density operator of the resulting ensemble is in general different from that prior to the measurement. The distance between these two density operators is indeed a measure of the disturbance due to measurement. Different measures of distance between density operators [18] give rise to different measures of disturbance. We are thus lead to a class of disturbance measures which have been used recently in the context of quantifying incompatibilit...

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“…In this article, we review our recent results on the existence of a similar principle of trade-off for the disturbances associated with the measurements of a set of observables 5 . It is a fundamental feature of quantum theory that when an observable is measured on an ensemble of systems, the density operator of the resulting ensemble is in general different from that prior to the measurement.…”