Polar duality is a well-known concept from convex geometry and analysis. In the present paper, we study two symplectically covariant versions of polar duality keeping in mind their applications to quantum mechanics. The first variant makes use of the symplectic form on phase space and allows a precise study of the covariance matrix of a density operator. The latter is a fundamental object in quantum information theory., The second variant is a symplectically covariant version of the usual polar duality highlighting the role played by Lagrangian planes. It allows us to define the notion of "geometric quantum states" with are in bijection with generalized Gaussians.

翻译：极对偶性（Polar duality）是凸几何与凸分析中的经典概念。本文研究了两种具有辛协变性的极对偶变体，并着重探讨其在量子力学中的应用。第一种变体利用相空间上的辛形式，能够精确分析密度算子的协方差矩阵——这是量子信息理论中的基本对象。第二种变体是经典极对偶性的辛协变扩展，凸显了拉格朗日平面的作用，由此可定义与广义高斯分布一一对应的"几何量子态"概念。