# The Einstein–Podolsky–Rosen Steering and Its Certification

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries and Notations

#### 2.1. The Box Framework

#### 2.1.1. The No-Signaling Principle

#### 2.1.2. Trust and Untrust

#### 2.2. Entanglement and Nonlocality

## 3. The EPR Steering

#### 3.1. Definition

**Definition**

**1**(EPR steering).

#### 3.2. One-Sided Measurement Device Independence

**Remark**

**1.**

#### 3.3. Schrödinger’s Steering Theorem

**Theorem**

**1**(Schroödinger’s steering theorem).

- 1.
- For any quantum state ${\rho}_{AB,}$, let ${\left\{{E}_{a}^{x}\right\}}_{a}$ be a complete set of POVMs satisfying ${\sum}_{a}{E}_{a}^{x}=\mathbb{I}$, $\forall x$. Then, the conditional states ${\tilde{\rho}}_{B}^{a|x}={\mathrm{tr}}_{A}\left[{E}_{a}^{x}\otimes \mathbb{I}{\rho}_{AB}\right]$ for all x and a form an assemblage.
- 2.
- For any assemblage ${\left\{{\tilde{\rho}}_{a|x}\right\}}_{a,x}$ with ${\sum}_{a}{\tilde{\rho}}_{a|x}=\sigma $, there always exist a pure quantum state ${\left|\psi \right.\u232a}_{AB}$ satisfying ${\mathrm{tr}}_{A}\left[{\left|\psi \right.\u232a}_{AB}\left.\u2329\psi \right|\right]=\sigma $ and complete sets of POVMs $\left\{{E}_{a}^{x}\right\}$ satisfying ${\sum}_{a}{E}_{a}^{x}=\mathbb{I}$ for all x, such that ${\tilde{\rho}}_{a|x}$ can be produced, i.e., ${\tilde{\rho}}_{a|x}={\mathrm{tr}}_{A}\left[{E}_{a}^{x}\otimes \mathbb{I}{\left|\psi \right.\u232a}_{AB}\left.\u2329\psi \right|\right]$.

**Proof.**

## 4. Criteria of EPR Steering

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Theorem**

**2.**

**Proof.**

#### 4.1. Linear EPR Steering Inequality

**Theorem**

**3**(The linear EPR steering inequality).

**Proof.**

**Example**

**1.**

#### 4.1.1. Optimal Observables for Alice

**Lemma**

**3.**

**Proof.**

**Theorem**

**4.**

- 1.
- ${A}_{k}$ and ${\tilde{\rho}}_{k}={\mathrm{tr}}_{\mathrm{B}}\left[\left({\mathbb{I}}_{A}\otimes {B}_{k}\right){\rho}_{AB}\right]$ are diagonalized in the same bases $\left\{{e}_{i}^{A}\right\}$.
- 2.
- Eigenvalues of ${A}_{k}$ and eigenvalues of ${\tilde{\rho}}_{k}={\mathrm{tr}}_{\mathrm{B}}\left[\left({\mathbb{I}}_{A}\otimes {B}_{k}\right){\rho}_{AB}\right]$ have the same order.

**Proof.**

**Example**

**2.**

#### 4.1.2. A Flexible Bound on Unsteerable Correlations

**Theorem**

**5.**

**Proof.**

#### 4.2. EPR Steering Inequality Based on Local Uncertainty Relations

**Theorem**

**6**(Steering inequality based on LUR).

**Proof.**

**Remark**

**2.**

**Example**

**3.**

#### 4.3. Realignment Method

**Theorem**

**7**(Realignment for EPR steering).

**Proof.**

## 5. Summary

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

EPR | Einstein–Podolsky–Rosen |

LHV | local hidden variable |

LHS | local hidden state |

LOO | local orthogonal observable |

LUR | local uncertainty relations |

POVM | positive operator valued measure |

PPT | positive partial transpose |

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**Figure 1.**The box framework: The source distributes state W to Alice and Bob. In their own closed labs, Alice and Bob make operations on received local states. Alice’s operations are labeled by inputs x, with outputs labeled by a. Bob’s operations are labeled by inputs y, with outputs labeled by b. After the experiment, Alice and Bob publicize their results and the corresponding statistics are denoted by probability distribution $\left\{P\left(ab|xy;W\right)\right\}$. According to such a distribution, the local property of W can be inferred.

**Figure 2.**The set of quantum states: All quantum states form a convex set, with the boundary being the pure state. The region I represents the convex subset of separable states. The complement set, i.e., regions II, III, and IV, represent entangled states. Particularly, regions III and IV represent Einstein–Podolsky–Rosen (EPR) steerable states, and the region IV represents nonlocal states. Region II are entangled states which is neither EPR steerable nor nonlocal.

**Figure 3.**The box framework for nonlocality, entanglement, and EPR steering. The color black represents untrusted, gray represents unknown, and white represents trusted. (

**a**) The nonlocality scenario, where the source is unknown and measurement devices are untrusted. (

**b**) The entanglement scenario, where source is unknown and measurement devices are trusted. (

**c**) The EPR steering scenario, where source is unknown and Alice’s measurement devices are untrusted while Bob’s are trusted.

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**MDPI and ACS Style**

Zhen, Y.-Z.; Xu, X.-Y.; Li, L.; Liu, N.-L.; Chen, K.
The Einstein–Podolsky–Rosen Steering and Its Certification. *Entropy* **2019**, *21*, 422.
https://doi.org/10.3390/e21040422

**AMA Style**

Zhen Y-Z, Xu X-Y, Li L, Liu N-L, Chen K.
The Einstein–Podolsky–Rosen Steering and Its Certification. *Entropy*. 2019; 21(4):422.
https://doi.org/10.3390/e21040422

**Chicago/Turabian Style**

Zhen, Yi-Zheng, Xin-Yu Xu, Li Li, Nai-Le Liu, and Kai Chen.
2019. "The Einstein–Podolsky–Rosen Steering and Its Certification" *Entropy* 21, no. 4: 422.
https://doi.org/10.3390/e21040422