One-shot coherence distillation with catalysts

The resource theory of quantum coherence is an important topic in quantum information science. Standard coherence distillation and dilution problems have been thoroughly studied. In this paper, we introduce and study the problem of one-shot coherence distillation with catalysts. In order to distill more coherence from a state of interest, a catalytic system can be involved and a jointly free operation is applied to both systems. The joint output state should be a maximally coherent state in tensor product with the unchanged catalysts, with some allowable fidelity error. We consider several different definitions of this problem. First, with a small fidelity error in both systems, we show that, even via the smallest free operation class (physically implementable incoherent operations), the distillable coherence of any state with no restriction on the catalysts is infinite, which is a “coherence embezzling phenomenon.” We then define and calculate a lower bound for the distillable coherence when the dimension of catalysts is restricted. Finally, in consideration of physical relevance, we define the “perfect catalysts” scenario where the catalysts are required to be pure and precisely unchanged. Interestingly, we show that in this setting catalysts basically provide no advantages in pure state distillation via incoherent operations and strictly incoherent operations under certain smoothing restriction. Our work enhances the understanding of catalytic effect in quantum resource theory.

Figure 1

Different schemes for one-shot coherence distillation via incoherent operation ${\mathrm{\Lambda }}_{\mathcal{O}}$. (a) Standard coherence distillation. The output state ${\rho }^{\text{out}}={\mathrm{\Lambda }}_{\mathcal{O}}\left(\rho \right)$ is close to the $N$-dimensional maximally coherent state ${\mathrm{\Psi }}_N$ with purified distance $\mathsf{P}({\rho }^{\text{out}},{\mathrm{\Psi }}_N)\le \varepsilon$. (b) Catalytic coherence distillation with global smoothing. The joint output state ${\rho }^{\mathrm{out}}={\mathrm{\Lambda }}_{\mathcal{O}}(\rho \otimes \sigma )$ is close to ${\mathrm{\Psi }}_N\otimes \sigma$ with purified distance $\mathsf{P}({\rho }^{\text{out}},{\mathrm{\Psi }}_N\otimes \sigma )\le \varepsilon$. (c) Catalytic coherence distillation with perfect catalysts. The catalyst $\phi$ is required to be pure and precisely unchanged, and the output state ${\rho }^{\text{out}}$ in the distilled system satisfies $\mathsf{P}\left({\rho }^{\text{out}},{\mathrm{\Psi }}_N\right)\le \varepsilon$.