On the supremum and infimum of bounded quantum observables

Abstract

Let \documentclass[12pt]{minimal}\begin{document}$S(\mathcal {H})$\end{document}S(H) be the set of all bounded self-adjoint linear operators on a complex Hilbert space \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}.$\end{document}H. In 2006, Gudder [Math. Slovaca 56, 573 (2006)] introduced a new order ≼ on \documentclass[12pt]{minimal}\begin{document}$S(\mathcal {H}).$\end{document}S(H). Since then, the existence conditions and representations of the supremum and infimum of two elements in \documentclass[12pt]{minimal}\begin{document}$S(\mathcal {H})$\end{document}S(H) with respect to the order ≼ have been intensively studied. Specifically, Li and Sun [J. Math. Phys. 50, 122107 (2009)]10.1063/1.3272542 obtained simpler representations of A ∧ P and A ∨ P, where \documentclass[12pt]{minimal}\begin{document}$A\in S(\mathcal {H})$\end{document}A∈S(H) and P is an orthogonal projection on \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}.$\end{document}H. In this note, we present more intuitive and concise results on A ∨ P and extend the results of Li and Sun to more general cases. Moreover, some applications of our results are given to show that our results are easier to deal with.