Joint measurability of POVM measurement assemblages and applications
XIAO Shu, Cao Huai-xin, YU XueQing · Zhongguo kexue. Wulixue Lixue Tianwenxue · 2020
The POVM measurement assemblage consisting of a set of positive operator-value measurements (POVMs) is an important tool for describing and studying the non-locality, quantum steering and quantum correlation of composite quantum systems. In this paper, we first introduce the definition of joint measurability of the POVM measurement assemblage $\mathcal{M}_S$ consisting of a finite number of POVMs and then prove that the joint measurability of $\mathcal{M}_S$ is equivalent to its compatibility. We give a general form of a POVM measurement which is jointly measurable with the one-rank projection measurement. It is proved that the POVM measurement assemblage $\mathcal{M}_S$ is jointly measurable if and only if the corresponding “row stochastic operator matrix" $\textbf{M}_S$ composed of $\mathcal{M}_S$ can be expressed as an “operator convex combination" of $\{0,I_S\}$—the row stochastic operator matrices. In particular, it is proved that the POVM measurement assemblage $\mathcal{M}(\textbf{n}_{0~},\textbf{n}_{1})$ induced by two three-dimensional real unit vectors $\textbf{n}_{i}(i=0,1)$ is jointly measurable if and only if $\textbf{n}_{0}=\pm\textbf{n}_{1}$. As an application, we obtain a necessary and sufficient condition under which a two-qubit quantum state is unsteerable with $\mathcal{M}(\textbf{n}_{0},\textbf{n}_{1})$.