SIMS without sums

POVMs and Naimark's Theorem Without Sums

We introduce an abstract notion of POVM within the categorical quantum mechanical semantics in terms of †-compact categories. Our definition is justified by two facts: i. we provide a purely graphical abstract counterpart to Naimark’s theorem, which establishes a bijective correspondence between POVMs and abstract projective measurements on an extended system; ii. in the category of Hilbert spa...

A Polish group without Lie sums

We construct a Polish group with an invariant metric in which Lie sums and Lie brackets do not exist. The construction of the group and the proof of the main theorem use some facts of combinatorial nature about the free group with two generators equipped with a Graev metric.

Chapter 1 Quantum measurements without sums

POVMs and Naimark ’ s theorem without sums Extended abstract

We introduce POVMs within the categorical quantum mechanical formalism in terms of †-compact categories [2,7,8,17]. Our definition is justified by two facts: i. we provide a purely graphical abstract counterpart to Naimark’s theorem, which establishes a bijective correspondence between POVMs and abstract projective measurements on an extended system; ii. in the category of Hilbert spaces and li...

Simonides to the Sims