bibkey: fernandes2024imaginarity
title: Unitary-invariant witnesses of quantum imaginarity
authors: Carlos Fernandes, Rafael Wagner, Leonardo Novo, Ernesto F. Galvão
year: 2024
url: https://arxiv.org/abs/2403.15066v1
claim: Non-real Bargmann invariants witness set imaginarity; three-state and four-state overlap data have different real-realizability constraints.
strata_touched: []
license: citation-only
triage: anchor
doi: null

Set imaginarity and finite-ensemble Gram data

Fernandes et al., arXiv:2403.15066v1 (22 March 2024; manuscript dated 25 March 2024), define set imaginarity by the absence of a single unitary that makes every density matrix in an ensemble real. This differs from imaginarity of one state relative to a specified basis.

The discussion preceding equation (1) explains that a non-real trace of a product of states witnesses this property, using the unitary and gauge invariance of Bargmann invariants. For three pure states with nonzero pairwise overlaps, equation (1) displays a Gram gauge in which two entries are positive and the remaining phase is the three-state cyclic phase. Theorem 1 recalls the positive-semidefinite, unit-diagonal criterion for a candidate Gram matrix.

Theorem 4 states that three squared pairwise overlaps attainable by complex pure states have a real realization in some Hilbert space; it does not require preserving the original dimension. Theorem 5 supplies four-state pairwise-overlap data without a real realization. Thus the fact that each second-order invariant is real does not preclude using joint inequalities among several such invariants to witness imaginarity.

These are finite-ensemble results. The quantum volume’s analytic-curve argument additionally uses the identity theorem and the density of the nonorthogonality set of a fixed anchor, to infer local three-state witnesses from the absence of a common real structure. Its exact channel and prescribed-jet requirements need their own continuation proof. The underlying Bargmann and real-Gram principles are prior results.

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