# Distance between Bound Entangled States from Unextendible Product Bases and Separable States

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## Abstract

**:**

## 1. Introduction

## 2. Bound Entangled States from Unextendible Product Bases

## 3. Entanglement Witnesses

## 4. The Gilbert Algorithm

- Choose at random a pure product state ${\rho}_{2}$.
- Maximize $\mathrm{tr}({\rho}_{0}-{\rho}_{1})({\rho}_{2}-{\rho}_{1})$, or go to step 1 if $\mathrm{tr}({\rho}_{0}-{\rho}_{1})({\rho}_{2}-{\rho}_{1})\le 0$.
- Find the point ${\rho}_{1}^{\prime}$, which lies on line ${\rho}_{1}d-{\rho}_{2}$, and minimizes the Hilbert–Schmidt distance $D(\sigma ,\rho )=\sqrt{\mathrm{tr}{(\rho -\sigma )}^{2}}$.
- Update ${\rho}_{1}\leftarrow {\rho}_{1}^{\prime}$.
- Go to point 1 until a given HALT condition is met.

## 5. Numerical Results

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Visualization of the structure of bound entangled states from unextendible product bases considered in this article. The size of the central tile is $(n-l+1)(m-o+1)$.

**Figure 2.**The comparison between last distance between a UPB BE state found by the Gilbert algorithm after 25,100 corrections ($3\times 3$, top left), 4000 corrections ($4\times 4$, top right, and $5\times 5$, bottom left), and 3500 corrections ($6\times 6$, bottom right). Red points correspond to distances from Equation (3) with BGR witnesses, the black points show the distance to the hyperplanes of the witnesses found by the Gilbert algorithm. The last distance found by the algorithm is marked with blue points and the extrapolation of its decay is represented with green points. The data are segregated by the size of the central tile.

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**MDPI and ACS Style**

Wieśniak, M.; Pandya, P.; Sakarya, O.; Woloncewicz, B.
Distance between Bound Entangled States from Unextendible Product Bases and Separable States. *Quantum Rep.* **2020**, *2*, 49-56.
https://doi.org/10.3390/quantum2010004

**AMA Style**

Wieśniak M, Pandya P, Sakarya O, Woloncewicz B.
Distance between Bound Entangled States from Unextendible Product Bases and Separable States. *Quantum Reports*. 2020; 2(1):49-56.
https://doi.org/10.3390/quantum2010004

**Chicago/Turabian Style**

Wieśniak, Marcin, Palash Pandya, Omer Sakarya, and Bianka Woloncewicz.
2020. "Distance between Bound Entangled States from Unextendible Product Bases and Separable States" *Quantum Reports* 2, no. 1: 49-56.
https://doi.org/10.3390/quantum2010004