Quantifying Resources in General Resource Theory with Catalysts

A question that is commonly asked in all areas of physics is how a certain property of a physical system can be used to achieve useful tasks and how to quantify the amount of such a property in a meaningful way. We answer this question by showing that, in a general resource-theoretic framework that allows the use of free states as catalysts, the amount of “resources” contained in a given state, in the asymptotic scenario, is equal to the regularized relative entropy of a resource of that state. While we need to place a few assumptions on our resource-theoretical framework, it is still sufficiently general, and its special cases include quantum resource theories of entanglement, coherence, asymmetry, athermality, nonuniformity, and purity. As a by-product, our result also implies that the amount of noise one has to inject locally to erase all the entanglement contained in an entangled state is equal to the regularized relative entropy of entanglement.

Figure 1

A geometric illustration of the relative entropy of a resource $E({\rho }_M)$.

Figure 2

Catalytic resource framework. (a) A free state is prepared in registers $E$ and $J$. (b) Free operations $U_j$ are performed on registers $M$ and $E$ controlled by the classical register $J$. (c) Upon discarding the register $J$, the final state of registers $M$ and $E$ should be close to some free state.