# Inequalities for the Schmidt Number of Bipartite States

## TL;DR

This paper establishes new inequalities relating the Schmidt number of bipartite states to their ranks and constructs PPT states with specified Schmidt numbers, advancing understanding of quantum entanglement and PPT state properties.

## Contribution

It introduces novel bounds for the Schmidt number based on state ranks and presents a family of PPT states with controllable Schmidt numbers, addressing open problems in quantum entanglement theory.

## Key findings

- Derived lower bounds for the Schmidt number using rank inequalities.
- Constructed PPT states with any desired Schmidt number within a certain range.
- Demonstrated the diversity of quantum entanglement through two contrasting state construction methods.

## 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 $\displaystyle SN(\gamma)\geq\max \left\{ \frac{\text{rank}(\gamma_L)}{\text{rank}(\gamma)}, \frac{\text{rank}(\gamma_R)}{\text{rank}(\gamma)}, \frac{SN(\gamma_S)}{2}, \frac{SN(\gamma_A)}{2} \right\},$ 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 $1\leq n\leq \left\lceil\frac{k-1}{2}\right\rceil$. This is a new contribution to the open problem of finding the best possible Schmidt number for PPT states.