We propose an error-disturbance relation for general observables on finite dimensional Hilbert spaces based on operational notions of error and disturbance. For two-dimensional systems we derive tight inequalities expressing the trade-off between accuracy and disturbance.
In order to discuss Ozawa's uncertainty relation @cite_6 @cite_0 @cite_13 @cite_15 , we need to introduce an auxiliary Hilbert space @math and a fixed state @math on @math . The measurement apparatus is then described by two observables One usually writes @math (and similarly for @math ), where @math is an observable on @math and @math is a unitary describing some interaction between the system of interest and the auxiliary system. @math @math on @math . Define the errors (the @math is part of the name, not an index) ^2_ O, = ( -A aux )^2 aux , ^2_ O, = ( - A aux )^2 aux , and the standard deviations ^2_ = (A- A )^2 , ^2_ = ( A- A )^2 . One can then derive the following error-disturbance relation In Ref. @cite_13 Hall derives a very similar inequality, but with @math ( @math ) defined in terms of @math ( @math ). A tight variant of is given by Branciard @cite_15 . See also @cite_17 and @cite_8 for related inequalities. @cite_0 O, O, + O, + O, 1 2 | [A, A] |.
@cite_8 @cite_6 @cite_0 @cite_15 @cite_13 @cite_17
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Heisenberg's uncertainty principle is quantified by error-disturbance tradeoff relations, which have been tested experimentally in various scenarios. Here we shall report improved new versions of various error-disturbance tradeoff relations by decomposing the measurement errors into two different components, namely, operator bias and fuzziness. Our improved uncertainty relations reveal the tradeoffs between these two components of errors, and imply various conditionally valid error-tradeoff relations for the unbiased and projective measurements. We also design a quantum circuit to measure the two components of the error and disturbance. The Heisenberg uncertainty principle states that the product of the noise in a position measurement and the momentum disturbance caused by that measurement should be no less than the limit set by Planck’s constant 2 as demonstrated by Heisenberg’s thought experiment using a g-ray microscope. Here it is shown that this common assumption is not universally true: a universally valid trade-off relation between the noise and the disturbance has an additional correlation term, which is redundant when the intervention brought by the measurement is independent of the measured object, but which allows the noise-disturbance product much below Planck’s constant when the intervention is dependent. A model of measuring interaction with dependent intervention shows that Heisenberg’s lower bound for the noise-disturbance product is violated even by a nearly nondisturbing precise position measurement. An experimental implementation is also proposed to realize the above model in the context of optical quadrature measurement with currently available linear optical devices. Universally valid uncertainty relations are proven in a model independent formulation for inherent and unavoidable extra noises in arbitrary joint measurements on single systems, from which Heisenberg's original uncertainty relation is proven valid for any joint measurements with statistically independent noises. Complementarity restricts the accuracy with which incompatible quantum observables can be jointly measured. Despite popular conception, the Heisenberg uncertainty relation does not quantify this principle. We report the experimental verification of universally valid complementarity relations, including an improved relation derived here. We exploit Einstein-Poldolsky-Rosen correlations between two photonic qubits to jointly measure incompatible observables of one. The product of our measurement inaccuracies is low enough to violate the widely used, but not universally valid, Arthurs-Kelly relation.