Universal joint-measurement uncertainty relation for error bars

Busch, Paul; Pearson, D. B.

doi:10.1063/1.2759831

Cited by 26 publications

(45 citation statements)

References 10 publications

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“…Measurement uncertainty relations for such overall errors and calibration errors were proven in (Appleby, 1998a,b;Busch and Pearson, 2007;Werner, 2004) for various state-independent measures, and more recently in (Busch et al, 2013(Busch et al, , 2014a for a general family of error measures. Some of these results will be reviewed in Section VII.C.…”

Section: B State-specific Error Vs Device Figure Of Meritmentioning

confidence: 97%

Colloquium: Quantum root-mean-square error and measurement uncertainty relations

2014

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Recent years have witnessed a controversy over Heisenberg's famous error-disturbance relation. Here we resolve the conflict by way of an analysis of the possible conceptualizations of measurement error and disturbance in quantum mechanics. We discuss two approaches to adapting the classic notion of root-mean-square error to quantum measurements. One is based on the concept of noise operator; its natural operational content is that of a mean deviation of the values of two observables measured jointly, and thus its applicability is limited to cases where such joint measurements are available. The second error measure quantifies the differences between two probability distributions obtained in separate runs of measurements and is of unrestricted applicability. We show that there are no nontrivial unconditional joint-measurement bounds for state-dependent errors in the conceptual framework discussed here, while Heisenberg-type measurement uncertainty relations for state-independent errors have been proven.

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Section: B State-specific Error Vs Device Figure Of Meritmentioning

confidence: 97%

Colloquium: Quantum root-mean-square error and measurement uncertainty relations

2014

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“…In particular, their importance for the approximate joint measurements of position and momentum observables has long been recognized (see, for instance, the monographs [1][2][3][4]) and it has recently been shown [5] that for any approximate joint measurement of position and momentum of a quantum object there is a covariant phase space observable with improved degrees of approximations. (For details of these concepts as well as for a further analysis of these results, see the above quoted work of Werner [5] as well as the subsequent developments [6][7][8].) The mathematical structure of the covariant phase space observables is also completely known: they correspond one-to-one onto the positive operators of trace one (acting on the Hilbert space of the quantum object in question), and they have an operator density defined by the Weyl operators and the positive trace-one operator in question, see Equation (4) below.…”

Section: Introductionmentioning

confidence: 95%

A note on the measurement of phase space observables with an eight-port homodyne detector

Kiukas¹,

Lahti²

2008

TMOP

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It is well known that the Husimi Q-function of the signal field can actually be measured by the eight-port homodyne detection technique, provided that the reference beam (used for homodyne detection) is a very strong coherent field so that it can be treated classically [see e.g. Leonhardt, U.; Paul, H. Phys. Rev. A 1993, 47, R2460-R2463]. Using recent rigorous results on the quantum theory of homodyne detection observables [Kiukas, J.; Lahti, P. J. Mod. Opt., in press (see arXiv:0706.4436v1 [quant-ph])], we show that any phase space observable, and not only the Q-function, can be obtained as a high amplitude limit of the signal observable actually measured by an eight-port homodyne detector. The proof of this fact does not involve any classicality assumption.

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“…Indeed, there have been several independent appeals to find alternative and operationally motivated definitions [8,[25][26][27][28][29][30][31][32][33] that produce inequalities which faithfully reflect Heisenberg's original discussion as presented in Ref. [1], and that also have a form similar to the inequality in Eq.…”

Section: Introductionmentioning

confidence: 99%

Certainty in Heisenberg's uncertainty principle: Revisiting definitions for estimation errors and disturbance

Dressel

Nori

2014

Phys. Rev. A

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We revisit the definitions of error and disturbance recently used in error-disturbance inequalities derived by Ozawa and others by expressing them in the reduced system space. The interpretation of the definitions as meansquared deviations relies on an implicit assumption that is generally incompatible with the Bell-Kochen-Specker-Spekkens contextuality theorems, and which results in averaging the deviations over a non-positive-semidefinite joint quasiprobability distribution. For unbiased measurements, the error admits a concrete interpretation as the dispersion in the estimation of the mean induced by the measurement ambiguity. We demonstrate how to directly measure not only this dispersion but also every observable moment with the same experimental data, and thus demonstrate that perfect distributional estimations can have nonzero error according to this measure. We conclude that the inequalities using these definitions do not capture the spirit of Heisenberg's eponymous inequality, but do indicate a qualitatively different relationship between dispersion and disturbance that is appropriate for ensembles being probed by all outcomes of an apparatus. To reconnect with the discussion of Heisenberg, we suggest alternative definitions of error and disturbance that are intrinsic to a single apparatus outcome. These definitions naturally involve the retrodictive and interdictive states for that outcome, and produce complementarity and error-disturbance inequalities that have the same form as the traditional Heisenberg relation.

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Universal joint-measurement uncertainty relation for error bars

Abstract: We formulate and prove a new, universally valid uncertainty relation for the necessary errors bar widths in any approximate joint measurement of position and momentum.

Cited by 26 publications

References 10 publications

Colloquium: Quantum root-mean-square error and measurement uncertainty relations

Colloquium: Quantum root-mean-square error and measurement uncertainty relations

A note on the measurement of phase space observables with an eight-port homodyne detector

Certainty in Heisenberg's uncertainty principle: Revisiting definitions for estimation errors and disturbance

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