Abstract
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.
Highlights
Quantum measurement, whilst being fundamental to quantum physics, poses perhaps the most difficult problems for the understanding of quantum theory
The question of error bounds for joint measurements of incompatible observables was already raised by Werner Heisenberg in 1927, who proposed an answer with his famous uncertainty relation [5]
The incompatible observables A and B are directly measured in a single qubit, but compatible observables C and D are obtained by joint measurements on a positive operator-valued measures (POVMs)
Summary
Whilst being fundamental to quantum physics, poses perhaps the most difficult problems for the understanding of quantum theory. The standard textbook version of the uncertainty relation is the Robertson-Schrodinger inequality: ∆A∆B ≥ | [A, B] |/2, where ∆A and ∆B are the standard deviations of two non-commuting operators A and B and the lower bound is given by the expectation value of the commutator of these operators. This relation concerns separate measurements of A and B performed on two ensembles of identically prepared quantum systems. Our results are more directly relevant to the exploration of the fundamental quantum limits of high precision measurements
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