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Uncertainty equalities and uncertainty relation in weak measurement

Qiu-Cheng Song, Cong‐Feng Qiao

Published May 9, 2015• 4 citations
Publisher

Abstract

Uncertainty principle is one of the fundamental principles in quantum mechanics. In this work, we derive two uncertainty equalities, which hold for all pairs of incompatible observables. We also obtain an uncertainty relation in weak measurement which captures the limitations on the preparation of pre- and post-selected ensemble and holds for two non-Hermitian operators corresponding to two non-commuting observables.

DOI
10.7523/j.issn.2095-6134.2016.01.006

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Uncertainty principle is one of the fundamental principles in quantum mechanics. In this work, we derive two uncertainty equalities, which hold for all pairs of incompatible observables. We also obtain an uncertainty relation in weak measurement which captures the limitations on the preparation of pre- and post-selected ensemble and holds for two non-Hermitian operators corresponding to two non-commuting observables.
We prove uncertainty relations that quantitatively express the impossibility of jointly sharp preparation of pre- and post-selected quantum states for measuring incompatible observables during the weak measurement. By defining a suitable operator whose average in the pre-selected quantum state gives the weak value, we show that one can have new uncertainty relations for variances of two such operators corresponding to two non-commuting observables. These generalize the recent stronger uncertainty relations that give non-trivial lower bounds for the sum of variances of two observables which fully capture the concept of incompatible observables. Furthermore, we show that weak values for two non-commuting projection operators obey a complementarity relation. Specifically, we show that for a pre-selected state if we measure a projector corresponding to an observable AAA weakly followed by the strong measurement of another observable BBB (for the post-selection) and, for the same pre-selected state we measure a projector corresponding to an observable BBB weakly followed by the strong measurement of the observable AAA (for the post-selection), then the product of these two weak values is always less than one. This shows that even though individually they are complex and can be large, their product is always bounded.
Given two or more non-commuting observables, it is generally not possible to simultaneously assign precise values to each. This quantum mechanical uncertainty principle is widely understood to be encapsulated by some form of uncertainty relation, expressing a trade-off between the standard deviations or other measures of uncertainty of two (or more) observables, resulting from their non-commutativity. Typically, such relations are coarse, and miss important features. It was not until very recently that a broader perspective on quantum uncertainty was envisaged and explored, one that utilises the notion of an uncertainty region. Here we review this new approach, illustrating it with pairs or triples of observables in the case of qubit and qutrit systems. We recall some of the shortcomings of traditional uncertainty relations, and highlight their inability to identify the full uncertainty region. These shortcomings suggest a precautionary note that, we surmise, ought to accompany the presentation of the uncertainty principle in introductory quantum mechanics courses.
The uncertainty principle bounds the uncertainties about incompatible measurements, clearly setting quantum theory apart from the classical world. Its mathematical formulation via uncertainty relations, plays an irreplaceable role in quantum technologies. However, neither the uncertainty principle nor uncertainty relations can fully describe the complementarity between quantum measurements. As an attempt to advance the efforts of complementarity in quantum theories, we formally propose a complementary information principle, significantly extending the one introduced by Heisenberg. First, we build a framework of black box testing consisting of pre- and post-testing with two incompatible measurements, introducing a rigorous mathematical expression of complementarity with definite information causality. Second, we provide majorization lower and upper bounds for the complementary information by utilizing the tool of semidefinite programming. In particular, we prove that our bounds are optimal under majorization due to the completeness of the majorization lattice. Finally, as applications to our framework, we present a general method to outer-approximating all uncertainty regions and also establish fundamental limits for all qualified joint uncertainties.
Uncertainty principle is one of the fundamental principles of quantum mechanics. Exploring such uncertainty relations in pre- and postselected (PPS) systems, where weak measurements on post-selected states have been used as a powerful tool for exploring the foundation of quantum mechanics, has so far eluded experimental effort. In this work, we experimentally investigate the Robertson-Heisenberg-type uncertainty relation for two incompatible observables in a PPS system. Through conducting a von Neumann-type measurement, uncertainty relations between two non-commuting Pauli observables are experimentally unveiled. Such uncertainty relations in the PPS system impose limitations on the joint sharp preparation of pre- and postselected states for two incompatible observables. Our experiments would be useful for the development of amplification techniques for precision measurements.