论文标题
在保留半阳性和消除实数的量词理论的地图上
On maps which preserve semipositivity and quantifier elimination theory for real numbers
论文作者
论文摘要
假设$φ：\ mathbb {m} _ {n}（\ mathbb {c}）\ rightarrow \ Mathbb {m} _ {n}（\ Mathbb {c}）$是一个超级驱动器，可以保留遗传性。我们给出了一种算法，以确定$φ$保存是否固定半阳性（在这种情况下，我们称$φ$为正）。我们解决问题的方法具有模型理论性质，即，我们将消除量化器理论的技术应用于实数。一种基于这些技术的方法似乎是唯一允许决定任意寄居性的$φ$的方法。在我们对问题进行详细分析之前，我们认为消除实数（以及复数）的量词可以在量子信息理论和其他领域中发挥重要作用。
Assume that $Φ:\mathbb{M}_{n}(\mathbb{C})\rightarrow\mathbb{M}_{n}(\mathbb{C})$ is a superoperator which preserves hermiticity. We give an algorithm determining whether $Φ$ preserves semipositivity (we call $Φ$ positive in this case). Our approach to the problem has a model-theoretic nature, namely, we apply techniques of quantifier elimination theory for real numbers. An approach based on these techniques seems to be the only one that allows to decide whether an arbitrary hermiticity-preserving $Φ$ is positive. Before we go to detailed analysis of the problem, we argue that quantifier elimination for real numbers (and also for complex numbers) can play a significant role in quantum information theory and other areas as well.
