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arXiv:1412.0871 · Journal of the Optical Society of America B

Joint measurability and temporal steering

H. S. Karthik, J. Prabhu Tej, A. R. Usha Devi, A. K. Rajagopal

Published Mar 25, 2015• 17 citations
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Abstract

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.

DOI
10.1364/josab.32.000a34

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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.
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.
In contrast with classical physics, in quantum physics some sets of measurements are incompatible in the sense that they cannot be performed simultaneously. Among other applications, incompatibility allows for contextuality and Bell nonlocality. This makes it of crucial importance to develop tools for certifying whether a set of measurements respects a certain structure of incompatibility. Here we show that, for quantum or nonsignaling models, if the measurements employed in a Bell test satisfy a given type of compatibility, then the amount of violation of some specific Bell inequalities becomes limited. Then, we show that correlations arising from local measurements on two-qubit states violate these limits, which rules out in a device-independent way such structures of incompatibility. In particular, we prove that quantum correlations allow for a device-independent demonstration of genuine triplewise incompatibility. Finally, we translate these results into a semidevice-independent Einstein-Podolsky-Rosen-steering scenario.
Occupying a position between entanglement and Bell nonlocality, Einstein-Podolsky-Rosen (EPR) steering has attracted increasing attention in recent years. Many criteria have been proposed and experimentally implemented to characterize EPR-steering. Nevertheless, only a few results are available to quantify steerability using analytical results. In this work, we propose a method for quantifying the steerability in two-qubit quantum states in the two-setting EPR-steering scenario, using the connection between joint measurability and steerability. We derive an analytical formula for the steerability of a class of X-states. The sufficient and necessary conditions for two-setting EPR-steering are presented. Based on these results, a class of asymmetric states, namely, one-way steerable states, are obtained.