bibkey: chaturvedi2013nullphase
title: Null Phase Curves and Manifolds in Geometric Phase Theory
authors: S. Chaturvedi, E. Ercolessi, G. Morandi, A. Ibort, G. Marmo, N. Mukunda, R. Simon
year: 2013
url: https://arxiv.org/abs/1302.0206v1
doi: 10.1063/1.4811346
claim: Positive three-state Bargmann invariants characterize null-phase curves and yield a globally in-phase lift; its real Gram data admit real-coordinate representations.
strata_touched: []
license: citation-only
triage: anchor

Null-phase positivity, anchor gauge, and real coordinates

Chaturvedi et al., arXiv:1302.0206v1 (1 February 2013), equation (2.10), define a null-phase curve using strictly positive three-state Bargmann invariants for all triples along it. Equations (2.11) and (3.6) relate this to vanishing geometric phase and globally in-phase lifts.

The discussion before equation (3.6) constructs the lift from an anchor by normalizing the vector obtained by applying each rank-one projection to that anchor. Positivity guarantees nonzero overlaps and makes all pairwise inner products in this gauge positive. Equations (4.11)-(4.13) then describe a fixed orthonormal basis in which the lifted vectors have real coefficients, using their real Gram relations.

This is a prior source for the anchor-gauge and real-Gram ingredients. Its positivity condition is stronger than simultaneous realification: real vectors may have a negative three-state cyclic product, with phase pi instead of zero. It also excludes orthogonal pairs. A curve argument allowing real products of either sign, or isolated anchor-orthogonality points, must handle those cases separately. In the quantum volume, analyticity supplies density away from such zeros and continuation from open parameter rectangles; the channel-specific support lower bound supplies the obstruction to a common real structure.

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