Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1502.05779 · Journal of Mathematical Physics

Measurement incompatibility and Schrödinger-Einstein-Podolsky-Rosen steering in a class of probabilistic theories

Manik Banik

Published May 1, 2015• 35 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/1.4919546

Related papers

Measurement incompatibility and channel steering

Manik Banik, Subhadipa Das, A. S. Majumdar

Incompatible measurements in quantum theory always lead to Einstein-Podolsky-Rosen (EPR)-Schr\"odinger steering. Channel steering, which is a generalized notion of EPR-Schr\"odinger steering, has been introduced recently. Here we establish a connection between lack of joint measurability and channel steering.

arXiv:1504.03210•Physical Review A•Jun 23, 2015•10 citations

Joint measurability and temporal steering

H. S. Karthik, J. Prabhu Tej, A. R. Usha Devi +1 more

Quintino et al. [Phys. Rev. Lett.113, 160402 (2014)10.1103/PhysRevLett.113.160402PRLTAO0031-9007] and Uola et al. [Phys. Rev. Lett.113, 160403 (2014)10.1103/PhysRevLett.113.160403PRLTAO0031-9007] have recently established an intrinsic relation between nonjoint measurability and Einstein–Podolsky–Rosen steering. They showed that a set of measurements are incompatible (i.e., not jointly measurable) if and only if they can be used for the demonstration of steering. In this paper, we prove the temporal analog of this result, viz., a set of measurements are incompatible if and only if they exhibit temporal steering in a single quantum system.

arXiv:1412.0871•Journal of the Optical Society of America B•Mar 25, 2015•17 citations

Joint Measurability, Einstein-Podolsky-Rosen Steering, and Bell Nonlocality

Marco Túlio Quintino, Tamás Vértesi, Nicolas Brunner

We investigate the relation between the incompatibility of quantum measurements and quantum nonlocality. We show that a set of measurements is not jointly measurable (i.e., incompatible) if and only if it can be used for demonstrating Einstein-Podolsky-Rosen steering, a form of quantum nonlocality. Moreover, we discuss the connection between Bell nonlocality and joint measurability, and give evidence that both notions are inequivalent. Specifically, we exhibit a set of incompatible quantum measurements and show that it does not violate a large class of Bell inequalities. This suggests the existence of incompatible quantum measurements which are Bell local, similarly to certain entangled states which admit a local hidden variable model.

arXiv:1406.6976•Physical Review Letters•Oct 14, 2014•278 citations

Incompatibility in restricted operational theories: connecting contextuality and steering

Martin Plávala

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.

arXiv:2112.10596•Journal of Physics A Mathematical and Theoretical•Mar 4, 2022•9 citations

Conditions for the compatibility of channels in general probabilistic theory and their connection to steering and Bell nonlocality

Martin Plávala

We derive general conditions for the compatibility of channels in general probabilistic theory. We introduce formalism that allows us to easily formulate steering by channels and Bell nonlocality of channels as generalizations of the well-known concepts of steering by measurements and Bell nonlocality of measurements. The generalization does not follow the standard line of thinking stemming from the Einstein-Podolsky-Rosen paradox, but introduces steering and Bell nonlocality as entanglement-assisted incompatibility tests. We show that all of the proposed definitions are, in the special case of measurements, the same as the standard definitions, but not all of the known results for measurements generalize to channels. For example, we show that for quantum channels, steering is not a necessary condition for Bell nonlocality. We further investigate the introduced conditions and concepts in the special case of quantum theory and we provide many examples to demonstrate these concepts and their implications.

arXiv:1707.08650•Physical Review A•Nov 21, 2017•17 citations

Role of Measurement Incompatibility and Uncertainty in Determining Nonlocality

Guruprasad Kar, Sibasish Ghosh A, Sujit K. Choudhary +1 more

It has been recently shown that measurement incompatibility and fine grained uncertainty—a particular form of preparation uncertainty relation—are deeply related to the nonlocal feature of quantum mechanics. In particular, the degree of measurement incompatibility in a no-signaling theory determines the bound on the violation of Bell-CHSH inequality, and a similar role is also played by (fine-grained) uncertainty along with steering, a subtle non-local phenomenon. We review these connections, along with comments on the difference in the roles played by measurement incompatibility and uncertainty. We also discuss why the toy model of Spekkens (Phys. Rev. A 75, 032110 (2007)) shows no nonlocal feature even though steering is present in this theory.

Mathematics•Aug 15, 2016•13 citations
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.
Quintino et al. [Phys. Rev. Lett.113, 160402 (2014)10.1103/PhysRevLett.113.160402PRLTAO0031-9007] and Uola et al. [Phys. Rev. Lett.113, 160403 (2014)10.1103/PhysRevLett.113.160403PRLTAO0031-9007] have recently established an intrinsic relation between nonjoint measurability and Einstein–Podolsky–Rosen steering. They showed that a set of measurements are incompatible (i.e., not jointly measurable) if and only if they can be used for the demonstration of steering. In this paper, we prove the temporal analog of this result, viz., a set of measurements are incompatible if and only if they exhibit temporal steering in a single quantum system.
We investigate the relation between the incompatibility of quantum measurements and quantum nonlocality. We show that a set of measurements is not jointly measurable (i.e., incompatible) if and only if it can be used for demonstrating Einstein-Podolsky-Rosen steering, a form of quantum nonlocality. Moreover, we discuss the connection between Bell nonlocality and joint measurability, and give evidence that both notions are inequivalent. Specifically, we exhibit a set of incompatible quantum measurements and show that it does not violate a large class of Bell inequalities. This suggests the existence of incompatible quantum measurements which are Bell local, similarly to certain entangled states which admit a local hidden variable model.
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.
We derive general conditions for the compatibility of channels in general probabilistic theory. We introduce formalism that allows us to easily formulate steering by channels and Bell nonlocality of channels as generalizations of the well-known concepts of steering by measurements and Bell nonlocality of measurements. The generalization does not follow the standard line of thinking stemming from the Einstein-Podolsky-Rosen paradox, but introduces steering and Bell nonlocality as entanglement-assisted incompatibility tests. We show that all of the proposed definitions are, in the special case of measurements, the same as the standard definitions, but not all of the known results for measurements generalize to channels. For example, we show that for quantum channels, steering is not a necessary condition for Bell nonlocality. We further investigate the introduced conditions and concepts in the special case of quantum theory and we provide many examples to demonstrate these concepts and their implications.
It has been recently shown that measurement incompatibility and fine grained uncertainty—a particular form of preparation uncertainty relation—are deeply related to the nonlocal feature of quantum mechanics. In particular, the degree of measurement incompatibility in a no-signaling theory determines the bound on the violation of Bell-CHSH inequality, and a similar role is also played by (fine-grained) uncertainty along with steering, a subtle non-local phenomenon. We review these connections, along with comments on the difference in the roles played by measurement incompatibility and uncertainty. We also discuss why the toy model of Spekkens (Phys. Rev. A 75, 032110 (2007)) shows no nonlocal feature even though steering is present in this theory.