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Inequalities for the Schmidt number of bipartite states

  • Daniel CarielloEmail author
Article
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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 

Mathematics Subject Classification

15A69 81P40 

Notes

Acknowledgements

The author would like to thank the referees for providing constructive comments and helping in the improvement in this manuscript.

Compliance with ethical standards

Conflict of interest

No potential conflict of interest was reported by the author.

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Faculdade de MatemáticaUniversidade Federal de UberlândiaUberlândiaBrazil

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