Critical Examination of Incoherent Operations and a Physically Consistent Resource Theory of Quantum Coherence

Considerable work has recently been directed toward developing resource theories of quantum coherence. In this Letter, we establish a criterion of physical consistency for any resource theory. This criterion requires that all free operations in a given resource theory be implementable by a unitary evolution and projective measurement that are both free operations in an extended resource theory. We show that all currently proposed basis-dependent theories of coherence fail to satisfy this criterion. We further characterize the physically consistent resource theory of coherence and find its operational power to be quite limited. After relaxing the condition of physical consistency, we introduce the class of dephasing-covariant incoherent operations as a natural generalization of the physically consistent operations. Necessary and sufficient conditions are derived for the convertibility of qubit states using dephasing-covariant operations, and we show that these conditions also hold for other well-known classes of incoherent operations.

Figure 1

This figure depicts the general process of implementing an incoherent operation on the joint system $AB$ whose reduced action on $A$ is the incoherent CPTP map ${\rho }_A\mapsto \mathcal{E}({\rho }_A)$. A second system $B$ is introduced in an incoherent state ${\widehat{\rho }}^B$. Both the unitary $U_{AB}$ and projective measurement are coherence nongenerating. All measurement outcomes are stored in a classical register of system $B$ so that the joint system is in a QC state at time $t_2$. Only maps $\mathcal{E}$ implemented in this way are physically consistent within a resource-theoretic picture.

Figure 2

Heuristic comparison between the five classes of incoherent operations MIO, DIO, IO, SIO, and PIO.