A Convex‐Roof Affinity‐Square Measure of Quantum Imaginarity

ABSTRACT

Quantum imaginarity characterizes the part of a quantum state that cannot be represented by real density matrices in a fixed reference basis. We study an affinity‐square measure of imaginarity for pure states and its convex‐roof extension to mixed states. The pure state quantity is defined by minimizing the affinity‐square measure over the convex set of real states. This optimization admits a closed expression determined by the modulus of the conjugation overlap between a state and its complex conjugate. The measure vanishes precisely when the state is real up to a global phase. We prove that the convex‐roof extension is faithful, convex, monotone under real operations, and nonincreasing on average under selective real operations. For arbitrary single‐qubit mixed states, we evaluate the convex roof exactly and express it in terms of the imaginary component of the off‐diagonal density‐matrix element. We also compare pure‐state imaginarity with an auxiliary affinity‐square coherence functional and give the corresponding phase weights.