POVMs and Naimark's Theorem Without Sums

We provide a definition of POVM in terms of abstract tensor structure only. It is justified in two distinct manners. i. At this abstract level we are still able to prove Naimark's theorem, hence establishing a bijective correspondence between abstract POVMs and abstract projective measurements (cf. [B. Coecke and D. Pavlovic (2007) Quantum measurements without sums. In: Mathematics of Quantum Computing and Technology. Chapman & Hall, pp. 559–596. E-print available from arXiv:quant-ph/0608035]) on an extended system, and this proof is moreover purely graphical. ii. Our definition coincides with the usual one for the particular case of the Hilbert space tensor product. We also provide a very useful normal form result for the classical object structure introduced in [B. Coecke and D. Pavlovic (2007) Quantum measurements without sums. In: Mathematics of Quantum Computing and Technology. Chapman & Hall, pp. 559–596. E-print available from arXiv:quant-ph/0608035].