Fundamental Limits on Correlated Catalytic State Transformations

Determining whether a given state can be transformed into a target state using free operations is one of the fundamental questions in the study of resource theories. Free operations in resource theories can be enhanced by allowing for a catalyst system that assists the transformation and is returned unchanged, but potentially correlated, with the target state. While this has been an active area of recent research, very little is known about the necessary properties of such catalysts. Here, we prove fundamental limits applicable to a large class of correlated catalytic transformations by showing that a small residual correlation between a catalyst and target state implies that the catalyst needs to be highly resourceful. In fact, the resource required diverge in the limit of vanishing residual correlation. In addition, we establish that in imperfect catalysis a small error generally implies a highly resourceful embezzling catalyst. We develop our results in a general resource theory framework and discuss its implications for the resource theory of athermality, the resource theory of coherence, and entanglement theory.

Figure 1

Example of the sets $\mathcal{FO}$, $\mathcal{CO}$ and $\mathcal{CCO}$ for classical resource theory of athermality with rational Gibbs states where we fixed the input state $\overrightarrow{p}=\{2/3,1/12,3/12\}$ and $\gamma =\{7/10,2/10,1/10\}$. We show one corner of the probability simplex (which is a triangle in this case). Each point in the triangle corresponds to a (classical) state of a three-dimensional system. The points in the red region satisfy the conditions of Theorem 2. The table contains the conditions characterizing each set, where $D_{\alpha }$ is the Rényi divergence of order $\alpha$ and $D$ is the Kullbach-Leibler divergence.