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arXiv · 1902.11069

Inequalities for the Schmidt Number of Bipartite States

Daniel Cariello
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Abstract

In this short note we show two completely opposite methods of constructing entangled states. Given a bipartite state $γ\in M_k\otimes M_k$, define $γ_S=(Id+F)γ(Id+F)$, $γ_A=(Id-F)γ(Id-F)$, where $F\in M_k\otimes M_k$ is the flip operator. In the first method, entanglement is a consequence of the inequality $\text{rank}(γ_S)<\sqrt{\text{rank}(γ_A)}$. In the second method, there is no correlation between $γ_S$ and $γ_A$. These two methods show how diverse is quantum entanglement. We prove that any bipartite state $γ\in M_k\otimes M_k$ satisfies $\displaystyle SN(γ)\geq\max \left\{ \frac{\text{rank}(γ_L)}{\text{rank}(γ)}, \frac{\text{rank}(γ_R)}{\text{rank}(γ)}, \frac{SN(γ_S)}{2}, \frac{SN(γ_A)}{2} \right\},$ where $SN(γ)$ stands for the Schmidt number of $γ$ and $γ_L,γ_R$ are the marginal states of $γ$. 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.

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Daniel Cariello. 2019-03-09. Inequalities for the Schmidt Number of Bipartite States. https://doi.org/10.1007/s11005-019-01244-1

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