Abstract
Reports on experiments recently performed in Vienna [Erhard et al, Nature Phys. 8 , 185 (2012)] and Toronto [Rozema et al, Phys. Rev. Lett. 109 , 100404 (2012)] include claims of a violation of Heisenberg’s error-disturbance relation. In contrast, a Heisenberg-type tradeoff relation for joint measurements of position and momentum has been formulated and proven in [Phys. Rev. Lett. 111 , 160405 (2013)]. Here I show how the apparent conflict is resolved by a careful consideration of the quantum generalization of the notion of root-mean-square error. The claim of a violation of Heisenberg’s principle is untenable as it is based on a historically wrong attribution of an incorrect relation to Heisenberg, which is in fact trivially violated. We review a new general trade-off relation for the necessary errors in approximate joint measurements of incompatible qubit observables that is in the spirit of Heisenberg’s intuitions. The experiments mentioned may directly be used to test this new error inequality.
Highlights
Heisenberg’s uncertainty principle, as it was conceived in his classic work of 1927 [1], may be understood as comprising three distinct statements concerning pairs of canonically conjugate quantities such as the position and momentum observables of a quantum particle: Preparation Uncertainty: position and momentum distributions cannot both be arbitrarily sharply concentrated in the same state
Heisenberg’s focus was on joint measurement uncertainty and the error-disturbance tradeoffs but he deduced these by taking recourse to preparation uncertainty
If A and C commute in the state ρ, they have a joint probability distribution given by tr ρEA(dx)C(dx ) ; in this case the formula (7) is operationally well defined, giving the root of the mean squared deviation between the simultaneously obtained values of A and C in the joint probability distribution tr ρEA(dx)C(dx )
Summary
Heisenberg’s uncertainty principle, as it was conceived in his classic work of 1927 [1], may be understood as comprising three distinct statements concerning pairs of canonically conjugate quantities such as the position and momentum observables of a quantum particle: Preparation Uncertainty: position and momentum distributions cannot both be arbitrarily sharply concentrated in the same state. Heisenberg’s focus was on joint measurement uncertainty and the error-disturbance tradeoffs but he deduced these by taking recourse to preparation uncertainty. He expressed the error-disturbance relation by the symbolic relation p1 q1 ∼ h,. Where q1 stands for, say, the error in a position measurement and p1 the ensuing disturbance of momentum; the necessary order of magnitude for the product is given by Planck’s constant h EPJ Web of Conferences constrained by the degree of their incompatibility It has taken many decades until serious attempts started at giving rigorous formulations and derivations of such measurement uncertainty relations as consequences of quantum mechanics. Werner [10,11,12,13]
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