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arXiv:1905.03614 · Physical Review Research

Measurement incompatibility and steering are necessary and sufficient for operational contextuality

Armin Tavakoli, Roope Uola

Published Jan 6, 2020• 62 citations
arXivHTMLPDF

Abstract

Two fundamental features of quantum theory are that measurements cannot be performed jointly and that entangled states can be steered through action-at-a-distance. Here it is shown that both these phenomena are in one-to-one correspondence with the failure of noncontextual models to explain the predictions of quantum theory.
DOI
10.1103/physrevresearch.2.013011

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The existence of incompatible measurements is often believed to be a feature of quantum theory which signals its inconsistency with any classical worldview. To prove the failure of classicality in the sense of Kochen-Specker noncontextuality, one does indeed require sets of incompatible measurements. However, a more broadly applicable notion of classicality is the existence of a generalized-noncontextual ontological model. In particular, this notion can imply constraints on the representation of outcomes even within a single nonprojective measurement. We leverage this fact to demonstrate that measurement incompatibility is neither necessary nor sufficient for proofs of the failure of generalized noncontextuality. Furthermore, we show that every proof of the failure of generalized noncontextuality in a quantum prepare-measure scenario can be converted into a proof of the failure of generalized noncontextuality in a corresponding scenario with no incompatible measurements.
Abstract We investigate the connection between steering and contextuality in general probabilistic theories. We show that for a class of bipartite states the steerability of the state by given set of measurements is equivalent to non-existence of preparation noncontextual hidden variable model for certain restricted theory constructed from the given state and measurements. The connection between steering and contextuality is provided by the concept of incompatibility in restricted theories, which we also investigate.
This thesis concerns how attempting to measure certain fundamental properties of a physical theory can lead us to new insights. The idea of general probabilistic models is introduced and developed, in order to provide a way to be able to consider and compare different theories. A measure of incompatibility of two observables is defined, through the amount of smearing needed to make them jointly measureable. This measure is then used to characterise the degree of incompatibility that exists in a given theory. Quantum theory is then shown to be as incompatible as any other theory, but only in a very course grained sense. A related way of measuring the strength of incompatibility of a pair of observables is then shown to put a bound on a measure of the strength of non-local correlations. The notion of steering, or remote ensemble preparation, is then shown to be a sufficient condition for the saturation of that bound. Examples are considered that demonstrate that the given sufficient notion of steering is not necessary, and it is proposed how the measures can be modified in cases where the link does not hold. The idea of formulating measures of error and disturbance associated with a measurement device is discussed. The notion of a direct test of error or disturbance is used to analyse current proposed measures, and show their shortcomings.
One of the central features of quantum theory is that there are pairs of quantum observables that cannot be measured simultaneously. This incompatibility of quantum observables is a necessary ingredient in several quantum phenomena, such as measurement uncertainty relations, violation of Bell inequalities, and steering. Two quantum observables that admit a simultaneous measurement are, in this respect, classical. A finer classification of classicality can be made by formulating four symmetric relations on the set of observables that are stronger than compatibility; they are broadcastability, one-side broadcastability, mutual nondisturbance, and nondisturbance. It is proven that the five relations form a hierarchy and their differences in terms of the required devices needed in a simultaneous measurement are explained. All four relations that are stronger than compatibility are completely characterized in the case of qubit observables.