Nonasymptotic assisted distillation of quantum coherence

We characterize the operational task of environment-assisted distillation of quantum coherence under different sets of free operations when only a finite supply of copies of a given state is available. We first evaluate the one-shot assisted distillable coherence exactly, and introduce a semidefinite programming bound on it in terms of a smooth entropic quantity. We prove the bound to be tight for all systems in dimensions 2 and 3, which allows us to obtain computable expressions for the one-shot rate of distillation, establish an analytical expression for the best achievable fidelity of assisted distillation for any finite number of copies, and fully solve the problem of asymptotic zero-error assisted distillation for qubit and qutrit systems. Our characterization shows that all relevant sets of free operations in the resource theory of coherence have exactly the same power in the task of one-shot assisted coherence distillation, and furthermore resolves a conjecture regarding the additivity of coherence of assistance in dimension 3.

Figure 1

How does increasing the number of copies affect the achievable fidelity of distillation of a single bit of coherence? The figures present a comparison between the fidelity of assisted distillation $F_{A,\mathcal{O}}({\rho }^{\otimes n},2)$ as obtained in this work, as well as the fidelity of unassisted distillation $F_{\mathcal{O}}({\rho }^{\otimes n},2)$ for $\mathcal{O}\in \{\text{MIO},\text{DIO}\}$ as obtained in [18] and for $\mathcal{O}=\text{SIO}$ as obtained in [20, 21]. In each case, we consider the distillation from the state ${\rho }^{\otimes n}$ for $n\in \{1,2,3,4\}$, where (a) $\rho =p|0\rangle \langle 0|+(1-p)|1\rangle \langle 1|$, (b) $\rho =\left(\begin{array}{cc}p& p(1-p)\\ p(1-p)& 1-p\end{array}\right)$, and (c) $\rho =p{\mathrm{\Psi }}_2+(1-p)\mathbb{1}/2$. (The quantities plotted are dimensionless.) Panel (a) shows in particular the fundamental difference between assisted and unassisted distillation in that assistance allows for distillation from incoherent states. Figure (b) showcases the advantage provided by the assistance, as one can see that the state $\rho$ admits a value of $p<\frac{1}{2}$ such that perfect distillation of ${\mathrm{\Psi }}_2$ is possible already from one copy of $\rho$ with assistance, while even four copies are insufficient without it. We further recall from [21] that the SIO fidelity is bounded away from 1 for any number of copies of the state $\rho$ in (b). We finally remark the curious phenomenon in (c), where increasing the number of copies does not increase the achievable fidelity under SIO (previously noted in [21]), while the increase in fidelity of distillation for MIO/DIO occurs only for odd number of copies ${\rho }^{\otimes n}$. At the same time, perfect assisted distillation is possible already for one copy of the state, since the maximally mixed state is as useful as the maximally coherent state for assisted distillation.