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A Fundamental Theorem on Einstein-Podolsky-Rosen Steering

Yu-Xuan Zhang, Jing-Ling Chen

Abstract

Quantum nonlocality is an essential nonlocality resource in quantum information. It has been classified into three distinct types: quantum entanglement, Einstein-Podolsky-Rosen (EPR) steering, and Bell's nonlocality. In 1991, Gisin presented a fundamental theorem on Bell's nonlocality, pointing out all pure entangled states possess Bell's nonloclaity. Many of the core protocols of quantum information science (such as quantum teleportation, quantum key distribution, and certain algorithms in quantum computing) rely on entanglement. Gisin's theorem tells us that as long as we successfully prepare a pure entangled state, we then have a Bell-nonlocality resource that can show the non-classical correlations. Such a resource is not ``virtual'' and can be tested and used through Bell-experiments. Similarly, in this work, we present a Gisin-like fundamental theorem on EPR steering, which indicates all rank-2 (and rank-1) entangled states possess EPR steerability. Thus all rank-2 entangled states can be applicable as EPR-steering resources in quantum information.

A Fundamental Theorem on Einstein-Podolsky-Rosen Steering

Abstract

Quantum nonlocality is an essential nonlocality resource in quantum information. It has been classified into three distinct types: quantum entanglement, Einstein-Podolsky-Rosen (EPR) steering, and Bell's nonlocality. In 1991, Gisin presented a fundamental theorem on Bell's nonlocality, pointing out all pure entangled states possess Bell's nonloclaity. Many of the core protocols of quantum information science (such as quantum teleportation, quantum key distribution, and certain algorithms in quantum computing) rely on entanglement. Gisin's theorem tells us that as long as we successfully prepare a pure entangled state, we then have a Bell-nonlocality resource that can show the non-classical correlations. Such a resource is not ``virtual'' and can be tested and used through Bell-experiments. Similarly, in this work, we present a Gisin-like fundamental theorem on EPR steering, which indicates all rank-2 (and rank-1) entangled states possess EPR steerability. Thus all rank-2 entangled states can be applicable as EPR-steering resources in quantum information.
Paper Structure (1 theorem, 20 equations, 1 figure)

This paper contains 1 theorem, 20 equations, 1 figure.

Key Result

Theorem 1

Under the local unitary transformations, for any two-qubit quantum state $\rho$ with rank 2 (including rank 1 as a special case), if $\rho$ is entangled, then $\rho$ must be steerable.

Figures (1)

  • Figure 1: Hierarchical structure of quantum nonlocality. Bell's nonlocality is the strongest-type nonlocality in nature, quantum entanglement is the weakest-type nonlocality, while EPR steering is a form of quantum nonlocality intermediating between them. For the innermost layer, Gisin's theorem shows that all pure entangled states (i.e., rank-1 entangled states) possess Bell's nonlocality. For the middle layer, there is also a Gisin-like fundamental theorem indicating all rank-2 (and rank-1) entangled states possess EPR steerability.

Theorems & Definitions (1)

  • Theorem 1