Uncertainty relation and probability: Numerical illustration

TL;DR

This paper numerically demonstrates that small uncertainty products in quantum measurements can be associated with very small probabilities, providing insights into the uncertainty relation and its implications for quantum detection limits.

Contribution

It introduces a numerical illustration linking small uncertainty products with low-probability events, supporting the Landau-Pollak uncertainty relation and addressing quantum measurement limits.

Findings

Small uncertainty products correspond to low-probability events.

Analysis aligns with Landau-Pollak uncertainty relation.

Potential implications for gravitational wave detection limits.

Abstract

The uncertainty relation and the probability interpretation of quantum mechanics are intrinsically connected, as is evidenced by the evaluation of standard deviations. It is thus natural to ask if one can associate a very small uncertainty product of suitably sampled events with a very small probability. We have shown elsewhere that some examples of the evasion of the uncertainty relation noted in the past are in fact understood in this way. We here numerically illustrate that a very small uncertainty product is realized if one performs a suitable sampling of measured data which occur with a very small probability. It is also shown that our analysis is consistent with the Landau-Pollak type uncertainty relation. It is suggested that the present analysis may help reconcile the contradicting views about the "standard quantum limit" in the detection of gravitational waves.