# Entropic Steering Criteria: Applications to Bipartite and Tripartite Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Steering

## 3. Entropies and Entropic Uncertainty Relations

#### 3.1. Entropies

- The entropies $S,{S}_{q}$ and ${\tilde{S}}_{r}$ are positive and they are zero if and only if the probability distribution is concentrated at one value (k), i.e., ${p}_{i}={\delta}_{ik}$.
- In the limit of $q\to 1$ and $r\to 1$, the Tsallis and Rényi entropies converge to the Shannon entropy, and both decrease monotonically in q and r.
- The Rényi entropy is a monotonous function of the Tsallis entropy:$${\tilde{S}}_{r}\left(\mathcal{P}\right)=\frac{\mathrm{ln}[1+(1-r){S}_{q=r}\left(\mathcal{P}\right)]}{1-r}.$$
- Shannon and Tsallis entropy are concave functions in $\mathcal{P}$, i.e., they obey the relation$$f(\lambda {\mathcal{P}}_{1}+(1-\lambda ){\mathcal{P}}_{2})\ge \lambda f\left({\mathcal{P}}_{1}\right)+(1-\lambda )f\left({\mathcal{P}}_{2}\right),$$
- In the limit of $r\to \infty $, the Rényi entropy is known as min-entropy$$\underset{r\to \infty}{\mathrm{lim}}{\tilde{S}}_{r}\left(\mathcal{P}\right)=-\mathrm{ln}\underset{i}{\mathrm{max}}\left({p}_{i}\right).$$
- For two independent distributions, $\mathcal{P}$ and $\mathcal{Q}$, Shannon and Rényi entropies are additive, i.e.,$$\begin{array}{c}\hfill S(\mathcal{P},\mathcal{Q})=S\left(\mathcal{P}\right)+S\left(\mathcal{Q}\right),\end{array}$$$$\begin{array}{c}\hfill {\tilde{S}}_{r}(\mathcal{P},\mathcal{Q})={\tilde{S}}_{r}\left(\mathcal{P}\right)+{\tilde{S}}_{r}\left(\mathcal{Q}\right),\end{array}$$$${S}_{q}(\mathcal{P},\mathcal{Q})={S}_{q}\left(\mathcal{P}\right)+{S}_{q}\left(\mathcal{Q}\right)+(1-q){S}_{q}\left(\mathcal{P}\right){S}_{q}\left(\mathcal{Q}\right).$$

#### 3.2. Relative Entropies

#### 3.3. Entropic Uncertainty Relations

## 4. Entropic Steering Criteria

#### 4.1. Entropic Steering Criteria for Shannon Entropy

#### 4.2. Entropic Steering Criteria for Generalized Entropies

#### 4.2.1. Tsallis Entropy

#### 4.2.2. Rényi Entropy

## 5. Connection to Existing Entanglement Criteria

## 6. Applications

#### 6.1. Optimal Values of q and r for Steering Detection

#### 6.2. Isotropic States

#### 6.3. General Two-Qubit States

#### 6.4. One-Way Steerable States

#### 6.5. Bound Entangled States

## 7. Multipartite Scenario

#### 7.1. Steering from Alice to Bob and Charlie

#### 7.2. Steering from Alice and Bob to Charlie

#### 7.3. Applications

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**The critical value w for noisy two-qubit entangled states ${\varrho}_{{e}_{x}}^{\left(2\right)}\left(w\right)$ for the detection of steering. Solid black line corresponds to Werner states (${\varrho}_{{e}_{1}}^{\left(2\right)}\left(w\right)$), and the dashed blue and dotted red lines correspond to ${\varrho}_{{e}_{2}}^{\left(2\right)}\left(w\right)$ and ${\varrho}_{{e}_{3}}^{\left(2\right)}\left(w\right)$, respectively, with (

**a**) the criteria based on Rényi entropy [Equation (45)] and (

**b**) on Tsallis entropy [Equation (48)].

**Figure 3.**Zoom-in of Figure 2, for the interval $q\in [2;3]$. Solid black line corresponds to Werner states, and the dashed blue and dotted red lines correspond to two random entangled states.

**Figure 4.**The critical value w for noisy two-qutrit entangled states ${\varrho}_{{e}_{x}}^{\left(3\right)}\left(w\right)$ for the detection of steering. The solid black line corresponds to the state with $x=1$, the dotted red line with $x=0.5$, and the dashed blue line with $x=0.2$, with (

**a**) the criteria based on Rényi entropy (45) and (

**b**) on Tsallis entropy (48).

**Figure 5.**The critical value of white noise $\alpha $ of states in Equation (62) as function of the Tsallis parameter q, considering a complete set of MUBs. Here, the solid black line corresponds to $d=3$, the dotted red line to $d=4$, the dashed blue line to $d=5$, and the dot-dashed green line to $d=7$. The optimal value for the detection of steerability is given by $q=2$.

**Figure 6.**The critical value of white noise $\alpha $ for different dimensions d, considering a complete set of MUBs. In this plot, blue circles correspond to our criterion in Equation (63) for $q=2$. The yellow squares correspond to the results for the inequality presented in Ref. [62] and the green diamonds in Ref. [63], where ${\alpha}_{\mathrm{crit}}$ was calculated via SDP (numerical method). Below the red triangles the existence of an LHS model for all projective measurements (i.e., infinite amount of measurements instead of $d+1$ MUBs) is known [2]. Please note that Ref. [2] is given for comparison, this is not a steering criterion, but a bound on any criterion.

**Figure 7.**One-way steerability of states (67) for (

**a**) two and (

**b**) three measurement settings. The shaded area is the region where our criterion detects these weakly steerable states.

**Figure 8.**Plot of Equation (45) in terms of ${m}_{1}$ and ${m}_{2}$ with $q=2$ (blue curve) for ${\varrho}_{BES}$ in the region ${m}_{1}^{2}+{m}_{2}^{2}+{m}_{1}{m}_{2}\le 1$. The opaque flat plot is the entropic uncertainty bound for $q=2$ and the measurements given by Equations (73) and (74). From the plot one can see that there is no violation of Equation (45) for any state in this family.

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Costa, A.C.S.; Uola, R.; Gühne, O.
Entropic Steering Criteria: Applications to Bipartite and Tripartite Systems. *Entropy* **2018**, *20*, 763.
https://doi.org/10.3390/e20100763

**AMA Style**

Costa ACS, Uola R, Gühne O.
Entropic Steering Criteria: Applications to Bipartite and Tripartite Systems. *Entropy*. 2018; 20(10):763.
https://doi.org/10.3390/e20100763

**Chicago/Turabian Style**

Costa, Ana C. S., Roope Uola, and Otfried Gühne.
2018. "Entropic Steering Criteria: Applications to Bipartite and Tripartite Systems" *Entropy* 20, no. 10: 763.
https://doi.org/10.3390/e20100763