We describe a categorical model of MALL (Multiplicative Additive Linear Logic) inspired by the Heisenberg-Schrödinger duality of finite-dimensional quantum theory. Proofs of formulas with positive logical polarity correspond to CPTP (completely positive trace-preserving) maps in our model, i.e. the quantum operations in the Schrödinger picture, whereas proofs of formulas with negative logical polarity correspond to CPU (completely positive unital) maps, i.e. the quantum operations in the Heisenberg picture. The mathematical development is based on noncommutative geometry and finite-dimensional von Neumann (co)algebras, which can be defined as special kinds of (co)monoid objects internal to the category of finite-dimensional operator spaces.

翻译：我们提出一种受有限维量子理论中海森堡-薛定谔对偶性启发的MALL（乘性加法线性逻辑）范畴模型。在该模型中，具有正逻辑极性的公式证明对应于CPTP（完全保迹正）映射，即薛定谔绘景中的量子操作；而具有负逻辑极性的公式证明则对应于CPU（完全保单位正）映射，即海森堡绘景中的量子操作。该数学理论建立在非交换几何与有限维冯·诺依曼（余）代数的基础上，这些代数可定义为有限维算子空间范畴内特殊类型的（余）幺半群对象。