Coherence number as a discrete quantum resource

We introduce a discrete coherence monotone named the coherence number, which is a generalization of the coherence rank to mixed states. After defining the coherence number in a manner similar to that of the Schmidt number in entanglement theory, we present a necessary and sufficient condition of the coherence number for a coherent state to be converted to an entangled state of nonzero $k$ concurrence (a member of the generalized concurrence family with $2\le k\le d$). As an application of the coherence number to a practical quantum system, Grover's search algorithm of $N$ items is considered. We show that the coherence number remains $N$ and falls abruptly when the success probability of a searching process becomes maximal. This phenomenon motivates us to analyze the depletion pattern of $C_c^{\left(N\right)}$ (the last member of the generalized coherence concurrence, nonzero when the coherence number is $N$), which turns out to be an optimal resource for the process since it is completely consumed to finish the searching task. The generalization of the original Grover algorithm with arbitrary (mixed) initial states is also discussed, which reveals the boundary condition for the coherence to be monotonically decreasing under the process.

Figure 1

The change of $C_c^{\left(2^{10}\right)}$ (black triangles) and $P\left(r\right)$ (red dots) with $N=2^{10}$ and $m=5$ from $r=0$ to $r=10$.

Figure 2

$C_c^{\left(2^{10}\right)}\left({\rho }_r\right)$ of ${\rho }_r$ with $N=2^{10}, m=5, ({\alpha }_i,{\alpha }_I)=(0,\frac{1}{\sqrt{2}})$ from $r=0$ to $r=15$.