Abstract
It is difficult to evaluate the precision of quantum measurements because it is not possible to conduct a second reference measurement on the same physical system to compare the measurement outcome with a more accurate value of the measured quantity. Here, I show that a direct evaluation of measurement uncertainties is possible when the measurement outcomes are used to compensate the small amount of decoherence induced in a probe qubit by carefully controlled interactions with the system. Since the original uncertainty of the target observable causes fluctuating phase shifts in the probe qubit, any additional information obtained about the target observable can be used to compensate a part of the decoherence by applying a conditional phase shift to the reference qubit. The magnitude of this negative feedback corresponds to an estimate of the target observable, and the uncompensated decoherence defines the uncertainty of that estimate. The results of the analysis show that the uncertainties of the estimates are given by the uncertainties introduced by Ozawa in Phys. Rev. A 67, 042105 (2003) and the optimal estimates are given by the weak values associated with the different measurement outcomes. Feedback compensation of decoherence therefore demonstrates the empirical validity of definitions of errors and estimates that combine the initial information of the input state with the additional information provided by each measurement outcome.
Highlights
It is shown that measurement uncertainties can be observed directly by evaluating the feedback compensation of the decoherence induced by the measured system on a probe qubit in a weak interaction occurring between state preparation and measurement
A mathematical definition of the error based on the representation of values by their corresponding operators was proposed by Ozawa [4], but this mathematical definition has been criticized precisely because it refers to hypothetical properties that do not appear in the observable statistics of quantum states [5,6,7,8,9]
The consistency of Ozawa’s theory with the results of weak measurements and the fact that the results of error-free measurements can be anomalous weak values have not been sufficiently recognized as convincing evidence in favor of either Ozawa’s theory or the theory of weak values, possibly because it is suspected that both theories might be misrepresentations of quantum interference effects [10,11,12,13,14,15]
Summary
It is shown that measurement uncertainties can be observed directly by evaluating the feedback compensation of the decoherence induced by the measured system on a probe qubit in a weak interaction occurring between state preparation and measurement. It will be shown that this is the case: Ozawa’s uncertainties describe the experimentally observable fluctuations of weak forces in a quantum interaction, with the optimal estimate given by the corresponding weak values. A standardized setup for feedback compensation of decoherence using a qubit as a probe can be used to implement an operational definition of measurement uncertainty.
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