Quantifying the imaginarity of quantum mechanics

The use of imaginary numbers in modelling quantum mechanical systems\nencompasses the wave-like nature of quantum states. Here we introduce a\nresource theoretic framework for imaginarity, where the free states are taken\nto be those with density matrices that are real with respect to a fixed basis.\nThis theory is closely related to the resource theory of coherence, as it is\nbasis dependent, and the imaginary numbers appear in the off-diagonal elements\nof the density matrix. Unlike coherence however, the set of physically\nrealizable free operations is identical to both completely resource\nnon-generating operations, and stochastically resource non-generating\noperations. Moreover, the resource theory of imaginarity does not have a\nself-adjoint resource destroying map. After introducing and characterizing the\nfree operations, we provide several measures of imaginarity, and give necessary\nand sufficient conditions for pure state transformations via physically\nconsistent free operations in the single shot regime.\n