论文标题
格里森的复合系统定理
Gleason's theorem for composite systems
论文作者
论文摘要
格里森的定理[A.格里森，J。Math。 Mech。，\ textbf {6}，885（1957）]是量子力学基础的重要结果，在量子力学的基础上，它作为量子形式主义的数学结果证明了天生的规则。正式地，它对希尔伯特空间的投影几何形状提供了一个关键的见解，表明在界限运算符$ \ bh $的代数上，投影晶格$ \ ph $上有限的添加度度量扩展到正线性功能。多年来，通过各种作者的努力，该定理的范围从I型到任意的von Neumann代数（无类型$ \ text {i} _2 _2 $因素）。在这里，我们证明了格里森定理对复合系统的概括。为此，我们通过两种方式加强了最初的结果：首先，我们将其范围扩展到Naimark的膨胀[M. A. Naimark，C。R.（Dokl。）学院。科学。 Urss，n。 ser。，\ textbf {41}，359（1943）]和stinespring [W. F. Stinespring，Proc。是。数学。 soc。，\ textbf {6}，211（1955）]，其次，我们需要在组成中相应（局部）代数的动态对应方面的一致性[E. E. M. Alfsen和F. W. Shultz，Commun。数学。 Phys。，\ textbf {194}，87（1998）]。我们表明，这两种情况都没有改变单个系统案例的结果，但是两者都必须获得对两分系统的概括。
Gleason's theorem [A. Gleason, J. Math. Mech., \textbf{6}, 885 (1957)] is an important result in the foundations of quantum mechanics, where it justifies the Born rule as a mathematical consequence of the quantum formalism. Formally, it presents a key insight into the projective geometry of Hilbert spaces, showing that finitely additive measures on the projection lattice $\PH$ extend to positive linear functionals on the algebra of bounded operators $\BH$. Over many years, and by the effort of various authors, the theorem has been broadened in its scope from type I to arbitrary von Neumann algebras (without type $\text{I}_2$ factors). Here, we prove a generalisation of Gleason's theorem to composite systems. To this end, we strengthen the original result in two ways: first, we extend its scope to dilations in the sense of Naimark [M. A. Naimark, C. R. (Dokl.) Acad. Sci. URSS, n. Ser., \textbf{41}, 359 (1943)] and Stinespring [W. F. Stinespring, Proc. Am. Math. Soc., \textbf{6}, 211 (1955)] and second, we require consistency with respect to dynamical correspondences on the respective (local) algebras in the composition [E. M. Alfsen and F. W. Shultz, Commun. Math. Phys., \textbf{194}, 87 (1998)]. We show that neither of these conditions changes the result in the single system case, yet both are necessary to obtain a generalisation to bipartite systems.
