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arXiv:1705.08008 · Physical Review A

Incompatible measurements in a class of general probabilistic theories

Anna Jenčová

Published Jul 25, 2018• 33 citations
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Abstract

We study incompatibility of measurements and its relation to steering and nonlocality in a class of finite-dimensional general probabilistic theories (GPTs). The basic idea is to represent finite collections of measurements as affine maps of a state space into a polysimplex and show that incompatibility is characterized by properties of these maps. We introduce the notion of an incompatibility witness and show its relation to incompatibility degree. We find the largest incompatibility degree attainable by pairs of two-outcome quantum measurements and characterize state spaces for which incompatibility degree attains maximal values possible in GPTs. As examples, we study the spaces of classical and quantum channels and show their close relation to polysimplices. This relation explains the superquantum nonclassical effects that were observed on these spaces.

DOI
10.1103/physreva.98.012133

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We study incompatibility of measurements and its relation to steering and nonlocality in a class of finite-dimensional general probabilistic theories (GPTs). The basic idea is to represent finite collections of measurements as affine maps of a state space into a polysimplex and show that incompatibility is characterized by properties of these maps. We introduce the notion of an incompatibility witness and show its relation to incompatibility degree. We find the largest incompatibility degree attainable by pairs of two-outcome quantum measurements and characterize state spaces for which incompatibility degree attains maximal values possible in GPTs. As examples, we study the spaces of classical and quantum channels and show their close relation to polysimplices. This relation explains the superquantum nonclassical effects that were observed on these spaces.
We study incompatibility of measurements and its relation to steering and nonlocality in finite dimensional general probabilistic theories (GPT). The basic idea is to represent finite collections of measurements as affine maps of a state space into a polysimplex and show that incompatibility is characterized by properties of these maps. We introduce the notion of an incompatibility witness and show its relation to incompatibility degree. We find the maximum incompatibility degree attainable by pairs of two-outcome quantum measurements and characterize state spaces for which incompatibility degree attains maximal values possible in GPT. As examples, we study the spaces of classical and quantum channels and show their close relation to polysimplices. This relation explains the super-quantum non-classical effects that were observed on these spaces.
We formulate the necessary and sufficient conditions for the existence of a pair of maximally incompatible two-outcome measurements in a finite-dimensional general probabilistic theory. The conditions are on the geometry of the state space; they require the existence of two pairs of parallel exposed faces with an additional condition on their intersections. We introduce the notion of discrimination measurement and show that the conditions for a pair of two-outcome measurements to be maximally incompatible are equivalent to requiring that a (potential, yet nonexisting) joint measurement of the maximally incompatible measurements would have to discriminate affinely dependent points. We present several examples to demonstrate our results.
Abstract In a bipartite Bell scenario involving two local measurements per party and two outcomes per measurement, the measurement incompatibility in one wing is both necessary and sufficient to reveal the nonlocality. However, such a one-to-one correspondence fails when one of the observers performs more than two measurements. In such a scenario, the measurement incompatibility is necessary but not sufficient to reveal the nonlocality. In this work, within the formalism of general probabilistic theory (GPT), we demonstrate that unlike the nonlocality, the incompatibility of N arbitrary measurements in one wing is both necessary and sufficient for revealing the generalised contextuality for the sub-system in the other wing. Further, we formulate an elegant form of inequality for any GPT that is necessary for N-wise compatibility of N arbitrary observables. Moreover, we argue that any theory that violates the proposed inequality possess a degree of incompatibility that can be quantified through the amount of violation. We claim that it is the generalised contextuality that provides a restriction to the allowed degree of measurement incompatibility of any viable theory of nature and thereby super-select the quantum theory. Finally, we discuss the geometrical implications of our results.
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.
Steering is one of the most counter intuitive non-classical features of bipartite quantum system, first noticed by Schrödinger at the early days of quantum theory. On the other hand, measurement incompatibility is another non-classical feature of quantum theory, initially pointed out by Bohr. Recently, Quintino et al. [Phys. Rev. Lett. 113, 160402 (2014)] and Uola et al. [Phys. Rev. Lett. 113, 160403 (2014)] have investigated the relation between these two distinct non-classical features. They have shown that a set of measurements is not jointly measurable (i.e., incompatible) if and only if they can be used for demonstrating Schrödinger-Einstein-Podolsky-Rosen steering. The concept of steering has been generalized for more general abstract tensor product theories rather than just Hilbert space quantum mechanics. In this article, we discuss that the notion of measurement incompatibility can be extended for general probability theories. Further, we show that the connection between steering and measurement incompatibility holds in a border class of tensor product theories rather than just quantum theory.
We introduce the notion of incompatibility witness for quantum channels, defined as an affine functional that is non-negative on all pairs of compatible channels and strictly negative on some incompatible pair. This notion extends the recent definition of incompatibility witnesses for quantum measurements. We utilize the general framework of channels acting on arbitrary finite-dimensional von Neumann algebras, thus allowing us to investigate incompatibility witnesses on measurement-measurement, measurement-channel, and channel-channel pairs. We prove that any incompatibility witness can be implemented as a state discrimination task in which some intermediate classical information is obtained before completing the task. This implies that any incompatible pair of channels gives an advantage over compatible pairs in some such state discrimination task.