Unitary-Invariant Witnesses of Quantum Imaginarity

Quantum theory is traditionally formulated using complex numbers. This imaginarity of quantum theory has been quantified as a resource with applications in discrimination tasks, pseudorandomness generation, and quantum metrology. In the standard formulation, a quantum state is said to have "imaginarity" if the associated density matrix is not real-valued in a given, fixed basis. If instead we consider a set of states, it is possible to devise tests that guarantee imaginarity of some state in the set, independently of the basis chosen. Here we propose such basis-independent witnesses for imaginarity that rely on measurements of unitary-invariant properties of sets of states. For three pure states, we completely characterize the invariant values attainable by quantum theory, and give a partial characterization for four pure states. We show that simple pairwise overlap measurements suffice to witness imaginarity of sets of four states, but not for sets of three. Our witnesses are experimentally friendly, opening up a new path for measuring and using imaginarity as a resource.