Logarithmic coherence: Operational interpretation of ${\ell }_1$-norm coherence

We show that the distillable coherence—which is equal to the relative entropy of coherence—is, up to a constant factor, always bounded by the ${\ell }_1$-norm measure of coherence (defined as the sum of absolute values of off diagonals). Thus the latter plays a similar role as logarithmic negativity plays in entanglement theory and this is the best operational interpretation from a resource-theoretic viewpoint. Consequently the two measures are intimately connected to another operational measure, the robustness of coherence. We find also relationships between these measures, which are tight for general states, and the tightest possible for pure and qubit states. For a given robustness, we construct a state having minimum distillable coherence.

Figure 1

$C_r\left(\right|\psi \rangle )$ vs $C_{{\ell }_1}\left(\right|\psi \rangle )$ for (normalized) $|\psi \rangle \in {\mathbb{C}}^4$: given only $C_{{\ell }_1}$ and $d$, the bounds in Eq. (7) are the tightest possible. For any point $(x,y)$ inside the pink region (including the boundary curves), there is a $|\psi \rangle$ such that $x=C_{{\ell }_1}\left(\right|\psi \rangle )$ and $y=C_r\left(\right|\psi \rangle )$.

Figure 2

$C_r\left(\rho \right)$ vs $C_{{\ell }_1}\left(\rho \right)$ for qubit $\rho$: the bounds given by Eq. (2) are the tightest possible. For any point $(x,y)$ inside the pink region (or over the boundary curves), there is a qubit state $\rho$ such that $x=C_{{\ell }_1}\left(\rho \right)$ and $y=C_r\left(\rho \right)$. Note that for a given $C_{{\ell }_1}$, there is a unique pure state, whose $C_r$ is given by the tightest upper bound (the magenta colored curve).