# Mathematical understanding of Bell nonlocality and quantum steering

November 13, 2018In quantum information theory, Bell nonlocality, EPR steering, quantum entanglement and quantum discord reveal the relationship between subsystems of a composite quantum system from different aspects, and become important resources of quantum information technology. The relationship between them is "the former implies the latter":

Bell Nonlocality?EPR steering?quantum entanglement?quantum discord;

But the inverses are not valid.

Bell nonlocality is shown by some local quantum measurements, and the statistics of the measured results cannot be explained by a local hidden variable (lhv) model. This non-classical characteristic of quantum mechanics can be used for the processing of quantum information independent of the setting.

EPR steering arises from the famous EPR paradox, which is a bridge from Bell non-locality to quantum entanglement and can be used to construct some novel communication protocols in special situations.

A recent research reveals the mathematical connotations and equivalent characterization of these two quantum properties, referring to

*Sci. China-Phys. Mech. Astron.*, 2019, 62(3): 030311.

The research paper was entitled "Characterizing Bell nonlocality and EPR steering", published in [

*Sci. China-Phys. Mech. Astron.*, 2019, 62(3)] by Professors Cao Huaixin and Guo Zhihua, Shaanxi Normal University. Using mathematical methods, the authors give the strict definitions of Bell nonlocality and EPR steering, and reveal the mathematical connotations and equivalent characterizations of these two quantum properties.

Based on their characterization theorems, it is proved that the sets of all Bell locale states and of all EPR unsteerable states are compact convex sets, which lay a foundation for defining metric functions of Bell locality and EPR steering. At the same time, some sufficient conditions for determining the quantum state to be EPR steerable are established. It is worth emphasizing that the obtained sufficient conditions generalize the results of Chen Jingling et el. [Sec. Rep. 3, 2143 (2013)] on the 2 × 2 system to arbitrary-dimensional bipartite system and improve a result of [Phys. Rev. A 95, 042117 (2017)].

The innovative ideas of the researchers are as follows: the Bell locality and EPR unsteerability of two-body quantum states for "specific measurement assemblage" are firstly defined, and then Bell locality and EPR unsteerability are introduced; Then through the negative form of the above concepts, Bell nonlocality and EPR steerability are introduced.

In this work, from the point of view of mathematics, the ideas and methods of Bell nonlocality and EPR steering are well understood, their mathematical definitions are given, and some their equivalent characterizations are then established.

The mathematical abstractions of these physical concepts not only reveal the nature of the problem, but also deduce some new physical phenomena, so it is of great scientific significance and reference value to further study these physical concepts.

-end-

The research was funded by the National Natural Science Foundation of China (Grant Nos. 11871318 and 11771009).**See the article:**

HuaiXin Cao, ZhiHua Guo, Characterizing Bell nonlocality and EPR steering,

*Sci. China-Phys. Mech. Astron.*, 2019, 62(3): 030311

http://engine.scichina.com/doi/10.1007/s11433-018-9279-4

https://link.springer.com/article/10.1007/s11433-018-9279-4

Science China Press

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