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arXiv:1205.0073

Measurement uncertainties in the quantum formalism: quasi-realities of individual systems

Holger Friedrich Hofmann

Published May 1, 2012• 2 citations
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Abstract

The evaluation of uncertainties in quantum measurements is problematic since the correct value of an observable between state preparation and measurement is experimentally inaccessible. In Ozawa's formulation of uncertainty relations for quantum measurements, the correct value of an observable is represented by the operator of that observable. Here, I consider the implications of this operator-based assignment of values to individual systems and discuss the relation with weak values and weak measurement statistics.

DOI
10.48550/arxiv.1205.0073

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The evaluation of uncertainties in quantum measurements is problematic since the correct value of an observable between state preparation and measurement is experimentally inaccessible. In Ozawa's formulation of uncertainty relations for quantum measurements, the correct value of an observable is represented by the operator of that observable. Here, I consider the implications of this operator-based assignment of values to individual systems and discuss the relation with weak values and weak measurement statistics.
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A universal formulation of uncertainty relations for quantum measurements is presented with additional focus on the representability of quantum observables by classical observables over a given state. Owing to the simplicity and operational tangibility of the framework, the resultant general relations admit natural operational interpretations and characterisations, and are thus also experimentally verifiable. In view of the universal formulation, Heisenberg's philosophy of the uncertainty principle is also revisited; it is reformulated and restated as a refined no-go theorem, albeit perhaps in a weaker form than was originally intended. In fact, the relations entail, in essence as corollaries to their special cases, several previously known relations, including most notably the Arthurs-Kelly-Goodman, Ozawa, and Watanabe-Sagawa-Ueda relations for quantum measurements. The Schr{ö}dinger relation (hence the standard Kennard-Robertson relation as its trivial corollary as well) is also shown to be a special case when the measurement is non-informative.
We present a universal formulation of uncertainty relation valid for any conceivable quantum measurement of statistical nature. Owing to its simplicity and operational tangibility, our general relation is also experimentally verifiable. Our relation violates the traditional naïve bound ħ / 2 for the position-momentum measurement while respecting Heisenberg's original philosophy of the uncertainty principle. Our relation entails, among others, the Arthurs–Kelly–Goodman and Ozawa relations as corollaries to its special cases, and also seamlessly reduces itself to the standard Kennard–Robertson (Schrödinger) relation when the measurement is non-informative.
Quantum metrology uses small changes in the output probabilities of a quantum measurement to estimate the magnitude of a weak interaction with the system. The sensitivity of this procedure depends on the relation between the input state, the measurement results, and the generator observable describing the effect of the weak interaction on the system. This is similar to the situation in weak measurements, where the weak value of an observable exhibits a symmetric dependence on initial and final conditions. In this paper, it is shown that the phase sensitivity of a quantum measurement is in fact given by the variance of the imaginary parts of the weak values of the generator over the different measurement outcomes. It is then possible to include the limitations of a specific quantum measurement in the uncertainty bound for phase estimates by subtracting the variance of the real parts of the weak values from the initial generator uncertainty. This uncertainty relation can be interpreted as the time-symmetric formulation of the uncertainty limit of quantum metrology, where the real parts of the weak values represent the information about the generator observable in the final measurement result.
Since weak measurements are known to produce measurement values that can be much larger than the maximal eigenvalues of the measured observable, it is an interesting question how this enhancement of the measurement signal relates to the sensitivity of quantum measurements as investigated in the field of quantum metrology. In this presentation, it is pointed out that the estimation of a small interaction parameter using weak measurements actually corresponds to standard quantum metrology, where the logarithmic derivatives of the final measurement probabilities are proportional to the corresponding weak values. The analysis of the general weak measurement formalism then shows that extreme weak values do not improve the sensitivity. Instead, all final measurements with real weak values have the same sensitivity as a final measurement of the eigenvalues. This result supports the view that real weak values can be interpreted as precise, zero uncertainty estimates of a quantum observable, despite their deviation from the eigenvalues of the corresponding operator.