Abstract
In Chap. 11 we have seen that noncommutativity prohibits the joint measurability of two sharp observables but not necessarily that of two unsharp observables. We have also observed that adding noise to two incompatible observables can turn them into jointly measurable observables. This leads to the idea of realising an approximate joint measurement of two incompatible sharp observables by means of a joint measurement of two compatible, possibly unsharp observables. In this chapter we develop tools to quantify the approximation errors and the degrees of unsharpness necessary to achieve compatible approximations. This will prepare the ground for formulating measurement uncertainty relations, which make precise Heisenberg’s intuitive formulations of 1927 (Heisenberg, Z Physik 43:172–198, 1927).
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Notes
- 1.
This notion of measurement error based on the statistical differences between observables was proposed, for instance, by Ludwig [2, pp. 197–198].
- 2.
Again, an extension to \(\alpha =\infty \) is possible [4] but will not be considered here.
- 3.
It may happen that the measurement noise operator \(A_\mathrm{out}-A_\mathrm{in}\) is unbounded; in this case appropriate care has to be taken with the specification of the states \(\varrho \otimes \sigma \) for which the expectation values are well defined.
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Busch, P., Lahti, P., Pellonpää, JP., Ylinen, K. (2016). Measurement Uncertainty. In: Quantum Measurement. Theoretical and Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-43389-9_13
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