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Position measuring interactions and the Heisenberg uncertainty principle

Masanao Ozawa

Published Jun 30, 2001• 36 citations
OpenAlex

Abstract

A prevailing formulation of the Heisenberg uncertainty principle is re-examined that measuring position with noise $\\epsilon$ brings the momentum disturbance no less than $\\hbar/2\\epsilon$. With rigorous formulations of noise and disturbance, a model of a measuring interaction demonstrates that this formulation of the Heisenberg uncertainty principle is by no means universally valid. An experimental realization of the model is also suggested.

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A prevailing formulation of the Heisenberg uncertainty principle is re-examined that measuring position with noise epsilon\\epsilonepsilon brings the momentum disturbance no less than hbar/2epsilon\\hbar/2\\epsilonhbar/2epsilon. With rigorous formulations of noise and disturbance, a model of a measuring interaction demonstrates that this formulation of the Heisenberg uncertainty principle is by no means universally valid. An experimental realization of the model is also suggested.
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 $\ensuremath{\Elzxh}/2$ as demonstrated by Heisenberg's thought experiment using a \ensuremath{\gamma}-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.
Heisenberg's uncertainty principle has been understood to set a limitation on measurements, whereas the long standing mathematical formulation does not allow such an interpretation. Recently, a new relation was found by the present author to give a universally valid relation between noise and disturbance in general quantum measurements, and it has become clear that the new relation plays a role of the first principle to derive various quantum limits on measurement and information processing. Here, we discuss the state-dependent notion of precise measurements of a given observable, and consider a perfect distance such that zero distance implies the perfect correlation. Then, we shall show that even in the perfect error notions both the position measuring noise and the momentum disturbance can be arbitrarily small. We also show that it is possible to generalize the new noise-disturbance uncertainty relation to perfect error distances.
The Heisenberg uncertainty relation for measurement noise and disturbance states that any position measurement with noise /spl epsiv/ brings the momentum disturbance not less than h/2/spl epsiv/. However, this relation holds only for restricted class of measurements. Here, a generalized uncertainty relation for measurement noise and disturbance is formalized and proven, which holds for all the possible quantum measurements. For this purpose, all the possible quantum measurements are characterized by naturally acceptable axioms. The measurement noise and disturbance are defined rigorously for any such general quantum measurements.
The uncertainty relation formulated by Heisenberg in 1927 describes a trade-off between the error of a measurement of one observable and the disturbance caused on another complementary observable so that their product should be no less than a limit set by Planck's constant. In 1980, Braginsky, Vorontsov, and Thorne claimed that this relation leads to a sensitivity limit for gravitational wave detectors. However, in 1988 a model of position measurement was constructed that breaks both this limit and Heisenberg's relation. Here, we discuss the problems as to how we reformulate Heisenberg's relation to be universally valid and how we experimentally quantify the error and the disturbance to refute the old relation and to confirm the new relation.