Imaginarity of quantum superpositions
Phys. Rev. A 114, 032435 – Published 15 September, 2026
DOI: https://doi.org/10.1103/7xk4-19hp
Abstract
Quantum imaginarity is an important resource in quantum information theory, associated with the operational role of complex numbers in quantum mechanics. Here we investigate how imaginarity behaves under coherent superposition of orthogonal pure states. For orthonormal components, we express the geometric imaginarity in terms of a conjugation-overlap matrix. We characterize these matrices as complex symmetric contractions and determine the minimum dimension required for their realization. In the two-component setting, we derive the exact feasible region of the off-diagonal conjugation overlap and a sharp bound on its modulus. Moreover, the conjugation-overlap matrix of an orthonormal qubit pair is unitary, yielding the full range attainable by varying the relative phase. For multicomponent superpositions, we identify the exact condition for attaining maximal imaginarity in superpositions of real components and show that pairwise compatibility is insufficient for global realizability, although pairwise constraints still yield lower bounds. When only the diagonal conjugation overlaps are specified, the attainable imaginarity values form a closed interval whose end points can be determined by convex optimization. Our results show that the imaginarity of a superposition depends not only on the imaginarities of its components but also on their relative phases and the global compatibility of their conjugation overlaps.
