Abstract
The Heisenberg uncertainty principle, which underlies many quantum key features, is under close scrutiny regarding its applicability to new scenarios. Using both the Bell-Kochen-Specker theorem establishing that observables do not have predetermined values before measurements and the measurement postulate of quantum mechanics, we propose that in order to describe the disturbance produced by the measurement process, it is convenient to define disturbance by the changes produced on quantum states. Hence, we propose to quantify disturbance in terms of the square root of the Jensen-Shannon entropy distance between the probability distributions before and after the measurement process. Additionally, disturbance and statistical distinguishability of states are fundamental concepts of quantum mechanics that have thus far been unrelated; however, we show that they are intermingled thereupon we enquire into whether the statistical distinguishability of states, caused by statistical fluctuations in the measurement outcomes, is responsible for the disturbance’s magnitude.
Highlights
The Heisenberg uncertainty principle (HUP) is related in a complex form to other fundamental quantum phenomena and difficult concepts of quantum mechanics
“uncertainty relations” are associated with measurements on an ensemble, whereas the “uncertainty principle” is associated with a sequence of measurements on the same system. In another convention the term “uncertainty principle” is often referred to the information gained and the state change induced by the measurement process, whereas the term “uncertainty relations” relates the statistics of the measured observable to the statistics of a non-commuting one
In order to capture the disturbance caused by the measurement process, we will proceed to compare the distance between the statistical distribution of the observable Abefore the single measurement and the statistical distribution of the same observable after that single measurement
Summary
The Heisenberg uncertainty principle (HUP) is related in a complex form to other fundamental quantum phenomena and difficult concepts of quantum mechanics. According to the convention adopted in this paper, we should stress that we use the term uncertainty relation to mean the mathematical expression of the uncertainty principle, as done for example by Uffink and Hilgevoord[12] This convention takes into account the distinction between preparation and measurement[13,14,15]. In the noise-disturbance relation[16] the effort was focused on precisely define both noise and disturbance and to differentiate them from the standard deviation In this approach, Ozawa[16] initially defined disturbance in terms of what he called the disturbance operator, i.e. D(B) = Bout − Bin see reference[16] for details. Busch et al.[23] gave a proof of an uncertainty relation for position and momentum based on what they called calibrated error, in this case the disturbance is defined as the root mean square deviation from a sharp value of the observable
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