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arXiv:1211.0370 · Physical Review Letters

Experimental Test of Universal Complementarity Relations

Morgan M. Weston, Michael J. W. Hall, Matthew S. Palsson, Howard M. Wiseman, Geoff J. Pryde

Published May 29, 2013• 91 citations
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Abstract

Complementarity restricts the accuracy with which incompatible quantum observables can be jointly measured. Despite popular conception, the Heisenberg uncertainty relation does not quantify this principle. We report the experimental verification of universally valid complementarity relations, including an improved relation derived here. We exploit Einstein-Poldolsky-Rosen correlations between two photonic qubits to jointly measure incompatible observables of one. The product of our measurement inaccuracies is low enough to violate the widely used, but not universally valid, Arthurs-Kelly relation.

DOI
10.1103/physrevlett.110.220402

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The Heisenberg-Robertson uncertainty relation is the hallmark of quantum physics and has been widely investigated. However, it does not capture the concept of incompatible observables because it can be trivial even for incompatible observables. Recently some stronger uncertainty relations relating the sums of variances were proposed. Here we experimentally demonstrate that these stronger multiobservable uncertainty relations are valid in a state-dependent manner and that the lower bound is guaranteed to be nontrivial for multiple observables that are incompatible on the state of the system being measured. We find that the behavior of multiple high-dimensional observables agrees with the predictions of quantum theory. Our experimental results not only foster insight into a fundamental limitation of measurements with multiple observables but also contribute to the study of the precision measurement technology in quantum information processing.
Abstract In standard formulations of the uncertainty principle, two fundamental features are typically cast as impossibility statements: two noncommuting observables cannot in general both be sharply defined (for the same state), nor can they be measured jointly. The pioneers of quantum mechanics were acutely aware and puzzled by this fact, and it motivated Heisenberg to seek a mitigation, which he formulated in his seminal paper of 1927. He provided intuitive arguments to show that the values of, say, the position and momentum of a particle can at least be unsharply defined, and they can be measured together provided some approximation errors are allowed. Only now, nine decades later, a working theory of approximate joint measurements is taking shape, leading to rigorous and experimentally testable formulations of associated error tradeoff relations. Here we briefly review this new development, explaining the concepts and steps taken in the construction of optimal joint approximations of pairs of incompatible observables. As a case study, we deduce measurement uncertainty relations for qubit observables using two distinct error measures. We provide an operational interpretation of the error bounds and discuss some of the first experimental tests of such relations.
We present a universal formulation of uncertainty relation valid for any conceivable quantum measurement of statistical nature. Owing to its simplicity and operational tangibility, our general relation is also experimentally verifiable. Our relation violates the traditional naïve bound ħ / 2 for the position-momentum measurement while respecting Heisenberg's original philosophy of the uncertainty principle. Our relation entails, among others, the Arthurs–Kelly–Goodman and Ozawa relations as corollaries to its special cases, and also seamlessly reduces itself to the standard Kennard–Robertson (Schrödinger) relation when the measurement is non-informative.