Structure of the Resource Theory of Quantum Coherence

Quantum coherence is an essential feature of quantum mechanics which is responsible for the departure between the classical and quantum world. The recently established resource theory of quantum coherence studies possible quantum technological applications of quantum coherence, and limitations that arise if one is lacking the ability to establish superpositions. An important open problem in this context is a simple characterization for incoherent operations, constituted by all possible transformations allowed within the resource theory of coherence. In this Letter, we contribute to such a characterization by proving several upper bounds on the maximum number of incoherent Kraus operators in a general incoherent operation. For a single qubit, we show that the number of incoherent Kraus operators is not more than 5, and it remains an open question if this number can be reduced to 4. The presented results are also relevant for quantum thermodynamics, as we demonstrate by introducing the class of Gibbs-preserving strictly incoherent operations, and solving the corresponding mixed-state conversion problem for a single qubit.

Figure 1

Achievable region for single-qubit SIO, IO, and MIO. Colored areas show the projection of the achievable region in the $x\text{−}z$ plane for initial Bloch vectors $(0.5,0,0.5{)}^T$ [blue], $(-0.8,0,-0.6{)}^T$ [green], and $(1,0,0{)}^T$ [red]. Note that the last two states are pure. The magenta line corresponds to the achievable region of an incoherent state with Bloch vector $(0,0,0.65{)}^T$.

Figure 2

Achievable region [blue area] of Gibbs-preserving SIO on a single qubit. The initial state has the Bloch vector $\mathit{r}=(0.5,0,0.5{)}^T$ [blue dot] and the preserved Gibbs state has the Bloch vector $\mathit{t}=(0,0,-0.2{)}^T$ [green dot].