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arXiv:1504.03210 · Physical Review A

Measurement incompatibility and channel steering

Manik Banik, Subhadipa Das, A. S. Majumdar

Published Jun 23, 2015• 10 citations
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Abstract

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.
DOI
10.1103/physreva.91.062124

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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 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.
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 formally link the concept of steering (a concept created by Schr\"odinger but only recently formalized by Wiseman, Jones, and Doherty [Phys. Rev. Lett. 98, 140402 (2007)]) and the criteria for demonstrations of Einstein-Podolsky-Rosen (EPR) paradox introduced by Reid [Phys. Rev. A 40, 913 (1989)]. We develop a general theory of experimental EPR-steering criteria, derive a number of criteria applicable to discrete as well as continuous-variable observables, and study their efficacy in detecting that form of nonlocality in some classes of quantum states. We show that previous versions of EPR-type criteria can be rederived within this formalism, thus unifying these efforts from a modern quantum-information perspective and clarifying their conceptual and formal origin. The theory follows in close analogy with criteria for other forms of quantum nonlocality (Bell nonlocality and entanglement), and because it is a hybrid of those two, it may lead to insights into the relationship between the different forms of nonlocality and the criteria that are able to detect them.
Abstract Quantum paradoxes are essential means to reveal the incompatibility between quantum and classical theories, among which the Einstein-Podolsky-Rosen (EPR) steering paradox offers a sharper criterion for the contradiction between local-hidden-state model and quantum mechanics than the usual inequality-based method. In this work, we present a generalized EPR steering paradox, which predicts a contradictory equality “2Q=(1+δ)C2_{Q}=\left(1+\delta\right)_{C}2Q​=(1+δ)C​” (0≤δ＜10\leq\delta＜10≤δ＜1) given by the quantum (QQQ) and classical (CCC) theories. For any NNN-qubit state in which the conditional state of the steered party is pure, we test the paradox through a two-setting steering protocol, and find that the state is steerable if some specific measurement requirements are satisfied. Moreover, our construction also enlightens the building of EPR steering inequality, which may contribute to some schemes for typical quantum teleportation and quantum key distributions.
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.