Duality versus Dual Flatness in Quantum Information Geometry
Nihat Ay and Wilderich Tuschmann
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Submission date: 14. Aug. 2002 (revised version: August 2002)
published in: Journal of mathematical physics, 44 (2003) 4, p. 1512-1518
DOI number (of the published article): 10.1063/1.1556192
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We show that the set of density operators of a given rank admits dually flat connections for which one connection is complete if and only if this rank is maximal. We prove moreover that there is never a dually flat structure on the set of pure states. Thus any general theory of quantum information geometry that involves duality concepts must inevitably be based on dual structures which are non-flat.