# Inequalities for the Schmidt number of bipartite states

Article

## Abstract

In this short note, we show two completely opposite methods of constructing bipartite entangled states. Given a bipartite state $$\gamma \in M_k\otimes M_k$$, define $$\gamma _S=(Id+F)\gamma (Id+F)$$, $$\gamma _A=(Id-F)\gamma (Id-F)$$, where $$F\in M_k\otimes M_k$$ is the flip operator. In the first method, entanglement is a consequence of the inequality $${\text {rank}}(\gamma _S)<\sqrt{{\text {rank}}(\gamma _A)}$$. In the second method, there is no correlation between $$\gamma _S$$ and $$\gamma _A$$. These two methods show how diverse is quantum entanglement. We show that any bipartite state $$\gamma \in M_k\otimes M_k$$ satisfies
\begin{aligned} \displaystyle \mathrm{SN}(\gamma )\ge \max \left\{ \frac{ {\text {rank}}(\gamma _L)}{ {\text {rank}}(\gamma )}, \frac{ {\text {rank}}(\gamma _R)}{ {\text {rank}}(\gamma )}, \frac{\mathrm{SN}(\gamma _S)}{2}, \frac{\mathrm{SN}(\gamma _A)}{2} \right\} , \end{aligned}
where $$\mathrm{SN}(\gamma )$$ stands for the Schmidt number of $$\gamma$$ and $$\gamma _L$$ and $$\gamma _R$$ are the marginal states of $$\gamma$$. These inequalities are useful to compute the Schmidt number of many bipartite states. We prove that $$\mathrm{SN}(\gamma )=\min \{ {\text {rank}}(\gamma _L), {\text {rank}}(\gamma _R)\}$$, if $$\displaystyle {\text {rank}}(\gamma )= \frac{\max \{ {\text {rank}}(\gamma _L), {\text {rank}}(\gamma _R)\}}{\min \{ {\text {rank}}(\gamma _L), {\text {rank}}(\gamma _R)\}}$$. We also present a family of PPT states in $$M_k\otimes M_k$$, whose members have Schmidt number equal to n, for any given $$\displaystyle 1\le n\le \left\lfloor \frac{k}{2}\right\rfloor$$. This is a new contribution to the open problem of finding the best possible Schmidt number for PPT states.

## Keywords

Schmidt number Entanglement Separability PPT states

15A69 81P40

## References

1. 1.
Cariello, D.: A gap for PPT entanglement. Linear Algebra Appl. 529, 89–114 (2017)
2. 2.
Chen, L., Yang, Y., Tang, W.S.: Schmidt number of bipartite and multipartite states under local projections. Quantum Inf. Process. 16(3), 75 (2017)
3. 3.
Gühne, O., Tóth, G.: Entanglement detection. Phys. Rep. 474(1–6), 1–75 (2009)
4. 4.
Gurvits, L.: Classical deterministic complexity of Edmonds’ problem and quantum entanglement. In: Proceedings of the Thirty-fifth Annual ACM Symposium on Theory of Computing, 9–11 Jun, San Diego, CA, pp. 10–19. ACM press, New York (2003)Google Scholar
5. 5.
Gurvits, L.: Classical complexity and quantum entanglement. J. Comput. Syst. Sci. 69(3), 448–484 (2004)
6. 6.
Horodecki, M., Horodecki, P., Horodecki, R.: Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A. 223, 1–8 (1996)
7. 7.
Horodecki, P., Smolin, J.A., Terhal, B.M., Thapliyal, A.V.: Rank two bipartite bound entangled states do not exist. Theor. Comput. Sci. 292(3), 589–596 (2003)
8. 8.
Huber, M., Lami, L., Lancien, C., Müller-Hermes, A.: High-dimensional entanglement in states with positive partial transposition. Phys. Rev. Lett. 121(20), 200503 (2018)
9. 9.
Li, C.K., Poon, Y.-T., Wang, X.: Ranks and eigenvalues of states with prescribed reduced states. Electron. J. Linear Algebra 27, 935–950 (2014)
10. 10.
Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77(8), 1413 (1996)
11. 11.
Sanpera, A., Bruß, D., Lewenstein, M.: Schmidt-number witnesses and bound entanglement. Phys. Rev. A 63(5), 050301 (2001)
12. 12.
Sperling, J., Vogel, W.: The Schmidt number as a universal entanglement measure. Phys. Scr. 83(4), 045002 (2011)
13. 13.
Terhal, B.M., Horodecki, P.: Schmidt number for density matrices. Phys. Rev. A 61(4), 040301 (2000)
14. 14.
Yang, Y., Leung, D.H., Tang, W.S.: All 2-positive linear maps from M3 (C) to M3 (C) are decomposable. Linear Algebra Appl. 503, 233–247 (2016)