The Error Principle

TL;DR

This paper extends the understanding of measurement errors and disturbances in joint position-momentum measurements, deriving new relationships that refine the traditional Uncertainty Principle for both ideal and real instruments.

Contribution

It generalizes previous results to a broader class of measurements and distinguishes between retrodiction and prediction errors, providing new error-disturbance relationships.

Findings

Derived new error-error and error-disturbance relationships.

Extended the analysis to instruments with finite measurement range.

Clarified differences between retrodictive and predictive errors.

Abstract

The problem of characterising the accuracy of, and disturbance caused by a joint measurement of position and momentum is investigated. In a previous paper the problem was discussed in the context of the unbiased measurements considered by Arthurs and Kelly. It is now shown, that suitably modified versions of these results hold for a much larger class of simultaneous measurements. The approach is a development of that adopted by Braginsky and Khalili in the case of a single measurement of position only. A distinction is made between the errors of retrodiction and the errors of prediction. Two error-error relationships and four error-disturbance relationships are derived, supplementing the Uncertainty Principle usually so-called. In the general case it is necessary to take into account the range of the measuring apparatus. Both the ideal case, of an instrument having infinite range, and the case of a real instrument, for which the range is finite, are discussed.