We characterize the von Neumann entropy as a certain concave functor from finite-dimensional non-commutative probability spaces and state-preserving $*$-homomorphisms to real numbers. This is made precise by first showing that the category of non-commutative probability spaces has the structure of a Grothendieck fibration with a fibrewise convex structure. The entropy difference associated to a $*$-homomorphism between probability spaces is shown to be a functor from this fibration to another one involving the real numbers. Furthermore, the von Neumann entropy difference is identified by a set of axioms similar to those of Baez, Fritz, and Leinster characterizing the Shannon entropy difference. The existence of disintegrations for classical probability spaces plays a crucial role in our characterization.

中文翻译：

---

冯诺依曼熵的函子表征

我们将冯诺依曼熵表征为从有限维非交换概率空间和状态保持 $*$-同态到实数的某个凹函子。这是通过首先证明非交换概率空间的范畴具有格罗腾迪克纤维化结构和纤维凸结构而变得精确的。与概率空间之间的 $*$-同态相关的熵差被证明是从这个纤维化到另一个涉及实数的函子。此外，冯诺依曼熵差由一组类似于 Baez、Fritz 和 Leinster 的公理来确定，这些公理表征香农熵差。经典概率空间解体的存在在我们的表征中起着至关重要的作用。