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.

arXiv commentIn 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

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• Separability Criterion for Density Matrices
• Separability of Mixed States: Necessary and Sufficient Conditions
• Entanglement detection
• A Schmidt number for density matrices
• Schmidt number witnesses and bound entanglement
• High-Dimensional Entanglement in States with Positive Partial Transposition
• A gap for PPT entanglement

Cited by in corpus (3)

• Bound entangled singlet-like states for quantum metrology
• Attainability and lower semi-continuity of the relative entropy of entanglement, and variations on the theme
• Schmidt rank constraints in Quantum Information Theory