Inequalities for the Schmidt Number of Bipartite States

Abstract

In this short note we show two completely opposite methods of constructing 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 prove that any bipartite state $\gamma \in M_k\otimes M_k$ satisfies $SN(\gamma )\ge \max \left\{\frac{\text{rank}({\gamma }_L)}{\text{rank}(\gamma )},\frac{\text{rank}({\gamma }_R)}{\text{rank}(\gamma )},\frac{SN({\gamma }_S)}{2},\frac{SN({\gamma }_A)}{2}\right\},$ where $SN(\gamma )$ stands for the Schmidt number of $\gamma$ and ${\gamma }_L,{\gamma }_R$ are the marginal states of $\gamma$. 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 $1\le n\le \lceil \frac{k-1}{2}\rceil$. This is a new contribution to the open problem of finding the best possible Schmidt number for PPT states.

In this new version, our main inequality has been improved and the construction of a PPT state in $M_k\otimes M_k$ with Schmidt number $n\left(1\le n\le \lceil \frac{k-1}{2}\rceil \right)$ has been simplified