POVMs and Naimark's theorem without sums
We provide a definition of POVM in terms of abstract <em>tensor structure</em> only. It is justified in two distinct manners. <strong>i</strong>. At this abstract level we are still able to prove Naimark's theorem, hence establishing a bijective correspondence between ab...
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| Format: | Journal article |
| Language: | English |
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Elsevier
2008
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| Summary: | We provide a definition of POVM in terms of abstract <em>tensor structure</em> only. It is justified in two distinct manners. <strong>i</strong>. 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) <em>Quantum measurements without sums. </em>In: Mathematics of Quantum Computing and Technology. Chapman & Hall, pp. 559–596. E-print available from http://www.arxiv.org/abs//quant-ph/0608035]) on an extended system, and this proof is moreover purely graphical. <strong>ii</strong>. Our definition coincides with the usual one for the particular case of the Hilbert space tensor product. We also provide a very useful <em>normal form</em> result for the classical object structure introduced in [B. Coecke and D. Pavlovic (2007) <em>Quantum measurements without sums.</em> In: Mathematics of Quantum Computing and Technology. Chapman & Hall, pp. 559–596. E-print available from http://www.arxiv.org/abs//quant-ph/0608035]. |
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