无求和的量子测量
Quantum measurements without sums
摘要：
和在量子力学的形式主义中扮演着重要的角色，无论是混合和叠加态，还是组成状态空间。令人惊讶的是，对量子测量的概念分析似乎表明，量子力学可以完全用张量积来表达，而不需要直接求和。相应的公理将经典空间定义为允许复制和删除数据的对象。事实上，量子世界和经典世界之间的信息交换基本上是由它们复制和删除数据的独特能力决定的。结果证明，这些总和是这种能力的隐式实现。通过显式公理实现它不仅省去了不必要的结构负担，而且还允许简单直观的图形演算。在范畴论中，经典数据类型是匕首紧致的Frobenius代数，而量子测量所依赖的量子谱是由这些Frobenius代数诱导的Eilenberg-Moore余代数。
英文摘要：
Sums play a prominent role in the formalisms of quantum mechanics, be it for mixing and superposing states, or for composing state spaces. Surprisingly, a conceptual analysis of quantum measurement seems to suggest that quantum mechanics can be done without direct sums, expressed entirely in terms of the tensor product. The corresponding axioms define classical spaces as objects that allow copying and deleting data. Indeed, the information exchange between the quantum and the classical worlds is essentially determined by their distinct capabilities to copy and delete data. The sums turn out to be an implicit implementation of this capabilities. Realizing it through explicit axioms not only dispenses with the unnecessary structural baggage, but also allows a simple and intuitive graphical calculus. In category-theoretic terms, classical data types are dagger-compact Frobenius algebras, and quantum spectra underlying quantum measurements are Eilenberg-Moore coalgebras induced by these Frobenius algebras.
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