Heisenberg Uncertainty Relation Revisited — Universality of Robertson's Relation

Abstract

It is shown that all the known uncertainty relations are the secondary consequences of Robertson's relation. The basic idea is to use the Heisenberg picture so that the time development of quantum mechanical operators incorporate the effects of the measurement interaction. A suitable use of triangle inequalities then gives rise to various forms of uncertainty relations. The assumptions of unbiased measurement and unbiased disturbance are important to simplify the resulting uncertainty relations and to give the familiar uncertainty relations such as a naive Heisenberg error-disturbance relation. These simplified uncertainty relations are however valid only conditionally. Quite independently of uncertainty relations, it is shown that the notion of precise measurement is incompatible with the assumptions of unbiased measurement and unbiased disturbance. We can thus naturally understand the failure of the naive Heisenberg's error-disturbance relation, as was demonstrated by the recent spin-measurement by J. Erhart, et al..

Entanglement-enhanced lidars for simultaneous range and velocity measurements

“…Significant complications occur, in the independent, identical interactions setting, when multiple unknown parameters are to be estimated [12][13][14][15]. In particular, if these parameters are associated with noncommuting observables, then the uncertainty principle would seem to forbid obtaining unlimited simultaneous knowledge of them from a single returned probe [16][17][18][19][20][21][22]. In such cases quantum-enhanced accuracy can be obtained by entangling probes with locally-stored idlers [12][13][14][23][24][25][26][27], in addition to the benefit derived from entangling different probes.…”