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arXiv:1809.05984 · New Journal of Physics

Determining complementary properties using weak-measurement: uncertainty, predictability, and disturbance

Guillaume Thekkadath, Felix Hufnagel, Jeff S. Lundeen

Published Oct 31, 2018• 11 citations
arXivHTMLPDF

Abstract

It is often said that measuring a system's position must disturb the complementary property, momentum, by some minimum amount due to the Heisenberg uncertainty principle. Using a 'weak-measurement', this disturbance can be reduced. One might expect this comes at the cost of also reducing the measurement's precision. However, it was recently demonstrated that a sequence consisting of a weak position measurement followed by a regular momentum measurement can probe a quantum system at a single point, with zero width, in position-momentum space. Here, we study this 'joint weak-measurement' and reconcile its compatibility with the uncertainty principle. While a single trial probes the system with a resolution that can saturate Heisenberg's limit, we show that averaging over many trials can be used to surpass this limit. The weak-measurement does not trade away precision, but rather another type of uncertainty called 'predictability' which quantifies the certainty of retrodicting the measurement's outcome.

DOI
10.1088/1367-2630/aaecdf

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It is often said that measuring a system's position must disturb the complementary property, momentum, by some minimum amount due to the Heisenberg uncertainty principle. Using a 'weak-measurement', this disturbance can be reduced. One might expect this comes at the cost of also reducing the measurement's precision. However, it was recently demonstrated that a sequence consisting of a weak position measurement followed by a regular momentum measurement can probe a quantum system at a single point, with zero width, in position-momentum space. Here, we study this 'joint weak-measurement' and reconcile its compatibility with the uncertainty principle. While a single trial probes the system with a resolution that can saturate Heisenberg's limit, we show that averaging over many trials can be used to surpass this limit. The weak-measurement does not trade away precision, but rather another type of uncertainty called 'predictability' which quantifies the certainty of retrodicting the measurement's outcome.
In quantum mechanics, measurements disturb the state of the system being measured. This disturbance is largest for complementary properties (e.g. position and momentum) and hence limits the precision with which such properties can be determined simultaneously. Often, this fact is conflated with Heisenberg's uncertainty principle, which refers to an uncertainty relation between complementary properties that is intrinsic to quantum states. In this thesis, the distinction between these two fundamental characteristics of quantum mechanics is made clear. At the intersection of the two are "joint measurements", which circumvent measurement disturbance to simultaneously determine complementary properties. They have applications in quantum metrology and enable a direct measurement of quantum states. The focus of this thesis is on the latter. The thesis is structured in the following way. The first chapter serves as an introduction to joint measurements. It surveys the seminal works in the field, doing so in a chronological manner to provide some historical context. The remainder of the thesis discusses two strategies to experimentally achieve joint measurements. The first strategy is to sequentially measure the complementary properties, making these measurements weak so that they do not disrupt each other. The second strategy is to first clone the system being measured, and then measure each complementary property on a separate clone. Both strategies are experimentally demonstrated on polarized photons, but can be readily extended to other systems.
Heisenberg's original intuition was that there should be a tradeoff between measuring a particle's position with greater precision and disturbing its momentum. Rigorous formulations of this idea have primarily focused on the question of how well two complementary observables can be jointly measured. Here, we provide an alternative approach based on how enhancing the predictability of one observable necessarily disturbs a complementary one. The tradeoff refers to a clear operational scenario capturing the effect of the measurement process on a single quantum system. Moreover, our relation is expressed by entropic quantities with clear statistical meaning evading recent criticism directed at some previous formulations. We discuss the performance of our tradeoff relation for existing experimental setups involving qubit measurements performed in Vienna and Toronto, and show that our relation is perfectly tight for all measurement strengths in the Toronto setup.
As one of the fundamental traits governing the operation of quantum world, the uncertainty relation, from the perspective of Heisenberg, rules the minimum deviation of two incompatible observations for arbitrary quantum states. Notwithstanding, the original measurements appeared in Heisenberg’s principle are strong such that they may disturb the quantum system itself. Hence an intriguing question is raised: What will happen if the mean values are replaced by weak values in Heisenberg’s uncertainty relation? In this work, we investigate the question in the case of measuring position and momentum in a simple harmonic oscillator via designating one of the eigenkets thereof to the pre-selected state. Astonishingly, the original Heisenberg limit is broken for some post-selected states, designed as a superposition of the pre-selected state and another eigenkets of harmonic oscillator. Moreover, if two distinct coherent states reside in the pre- and post-selected states respectively, the variance reaches the lower bound in common uncertainty principle all the while, which is in accord with the circumstance in Heisenberg’s primitive framework.
While there is a rigorously proven relationship about uncertainties intrinsic to any quantum system, often referred to as "Heisenberg's uncertainty principle," Heisenberg originally formulated his ideas in terms of a relationship between the precision of a measurement and the disturbance it must create. Although this latter relationship is not rigorously proven, it is commonly believed (and taught) as an aspect of the broader uncertainty principle. Here, we experimentally observe a violation of Heisenberg's "measurement-disturbance relationship", using weak measurements to characterize a quantum system before and after it interacts with a measurement apparatus. Our experiment implements a 2010 proposal of Lund and Wiseman to confirm a revised measurement-disturbance relationship derived by Ozawa in 2003. Its results have broad implications for the foundations of quantum mechanics and for practical issues in quantum measurement.
Reports on experiments recently performed in Vienna [Erhard et al., Nature Phys. 8, 185 (2012)] and Toronto [Rozema et al., Phys. Rev. Lett. 109, 100404 (2012)] include claims of a violation of Heisenberg's error-disturbance relation. In contrast, we have presented and proven a Heisenberg-type relation for joint measurements of position and momentum [Phys. Rev. Lett. 111, 160405 (2013)]. To resolve the apparent conflict, we formulate here a general trade-off relation for errors in qubit measurements, using the same concepts as we did in the position-momentum case. We show that the combined errors in an approximate joint measurement of a pair of $\ifmmode\pm\else\textpm\fi{}1$-valued observables A,BA,BA,B are tightly bounded from below by a quantity that measures the degree of incompatibility of AAA and BBB. The claim of a violation of Heisenberg is shown to fail because it is based on unsuitable measures of error and disturbance. Finally we show how the experiments mentioned may directly be used to test our error inequality.
A review is given of precise formulations of three conceptually distinct but related manifestations of Heisenberg's Uncertainty Principle. This principle appears in the form of trade-off inequalities: for the widths of the position and momentum distributions in any quantum state; for the inaccuracies of any joint measurement of these quantities; and for the inaccuracy of a measurement of one of the quantities and the ensuing disturbance in the distribution of the other quantity. The uncertainty principle is often described as expressing a limitation of operational possibilities imposed by quantum mechanics. Here we demonstrate that apart from this negative role, the full content of the uncertainty principle also includes its positive role as a condition ensuring that mutually exclusive experimental options can indeed be reconciled if an appropriate trade-off is accepted. Finally, we survey models and experimental implementations of joint measurements of position and momentum and comment briefly on the status of experimental tests of the uncertainty principle.