Some realizable joint measurements of complementary observables

Paul Busch, PEKKA J. LAHTI

Sixia Yu, Changhun Oh

Heisenberg's uncertainty relations for measurement quantify how well we can jointly measure two complementary observables and have attracted much experimental and theoretical attention recently. Here we provide an exact tradeoff between the worst-case errors in measuring jointly two observables of a qubit, i.e., all the allowed and forbidden pairs of errors, especially asymmetric ones, are exactly pinpointed. For each pair of optimal errors we provide an optimal joint measurement that is realizable without introducing any ancilla and entanglement. Possible experimental implementations are discussed and Toronto experiment [Rozema et al., Phys. Rev. Lett. 109, 100404 (2012)] can be readily adapted to an optimal joint measurement of two orthogonal observables.

Tian‐Jian Xiong, L.-L. Yan, Zhihao Ma +5 more

The uncertainty relations, pioneered by Werner Heisenberg nearly 90 years ago, set a fundamental limitation on the joint measurability of complementary observables. This limitation has long been a subject of debate, which has been reignited recently due to new proposed forms of measurement uncertainty relations. The present work is associated with a new error trade-off relation for compatible observables approximating two incompatible observables, in keeping with the spirit of Heisenberg's original ideas of 1927. We report the first direct test and confirmation of the tight bounds prescribed by such an error trade-off relation, based on an experimental realisation of optimal joint measurements of complementary observables using a single ultracold ion trapped in a harmonic potential. Our work provides a prototypical determination of ultimate joint measurement error bounds with potential applications in quantum information science for high-precision measurement and information security.

Heinz Dieter Dombrowski

Kengo Matsuyama, Holger Friedrich Hofmann, M. Iinuma

To clarify the relation between a Bell’s inequality and measurement uncertainty, we have experimentally introduced joint measurements of two complementary polarizations. Joint statistics can be obtained because measurement uncertainties reduce the observed correlations.

STEPHEN D. BRIERLEY, Andrei Yu. Khrennikov