A fundamental feature of quantum mechanics is the incompatibility between quantum measurements, which leads to the Heisenberg uncertainty relations and is intrinsically related to detection of many other quantum features such as nonlocality and steering. Given the significance and ubiquity of incompatibility, it is desirable to quantify the degree of incompatibility between two quantum measurements which are mathematically described by positive operator valued measures (POVMs). The commutator (Lie product) between operators has widely been used in quantifying incompatibility (non-commutativity) and plays a crucial role in quantum mechanics. In contrast, the anti-commutator (Jordan product) between operators has been relatively less employed in the study of quantum mechanics. In this work, we explore the role of the Jordan product in characterizing measurement incompatibility. The key observation lies in that while the product of two non-negative classical observables is automatically non-negative, this is not the case for quantum observables. Indeed the Jordan product of two non-negative observables is not necessarily non-negative. This negativity of the Jordan product is a purely quantum phenomenon and is here exploited to quantify a kind of measurement incompatibility, which interpolates between non-commutativity and joint measurability. We reveal basic features of this approach, make some comparisons with existing quantifiers of incompatibility.

骆顺龙，2001年任中国科学院数学与系统科学研究院研究员，博士生导师，2013--2015年任华罗庚应用数学首席研究员， 2023年当选中国工业与应用数学学会会士。现任中国科学院数学与系统科学研究院应用数学研究所所长。主要从事概率统计，量子信息和数学物理研究，论文被引用9500余次（Google Scholar），应邀在第八届国际工业与应用数学大会作一小时报告。在Proceedings of the American Mathematical Society等数学刊物发表论文60余篇， 在Physical Review Letters、Physical Review A 等物理刊物发表论文120余篇。