Entropy and Reversible Catalysis

I show that nondecreasing entropy provides a necessary and sufficient condition to convert the state of a physical system into a different state by a reversible transformation that acts on the system of interest and a further “catalyst,” whose state has to remain invariant exactly in the transition. This statement is proven both in the case of finite-dimensional quantum mechanics, where von Neumann entropy is the relevant entropy, and in the case of systems whose states are described by probability distributions on finite sample spaces, where Shannon entropy is the relevant entropy. The results give an affirmative resolution to the (approximate) catalytic entropy conjecture introduced by Boes et al. [Phys. Rev. Lett. 122, 210402 (2019)]. They provide a complete single-shot characterization without external randomness of von Neumann entropy and Shannon entropy. I also compare the results to the setting of phenomenological thermodynamics and show how they can be used to obtain a quantitative single-shot characterization of Gibbs states in quantum statistical mechanics.

Figure 1

A catalytic transition: A unitary operation $U$ is applied to systems $S$ and $C$ in the state $\rho \otimes \sigma$. The resulting reduced state on $S$ is (arbitrarily close to) ${\rho }^{\prime }$, while the reduced state on $C$ is preserved exactly, but correlated to $S$ (indicated by the dashed lines). The main result of this Letter shows that a matching $\sigma$ and $U$ can be found if and only if $H({\rho }^{\prime })\ge H(\rho )$.

Figure 2

The structure of the constructed catalyst $C$: It contains subsystems $S_2,\dots ,S_n$, which are copies of the target system $S$ together with an auxiliary system $A$ of dimension $n$ as well as a catalytic source of randomness $R$. The dashed lines indicate possible correlations. The source of randomness is utilized to dilate the decoherence channel ${\mathcal{D}}_{{\rho }^{\prime }}$ on $S$ in such a way that it does not become correlated to the systems $S_2,\dots ,S_n$ and $A$ in the process.

Figure 3

Illustration of the steps in the first part of the proof for $n=4$. The different columns denote the systems $S_1,\dots ,S_n$ with $S_1=S$ and the different rows indicate the state $|k⟩$ of the auxiliary system $A$. The hyphen indicates that the corresponding subsystems may be correlated. This procedure is essentially identical to the one in Ref. [34]. In the proof of the classical result (see Supplemental Material [35]), each of the subsystems with marginals called ${\chi }_i$ here is further correlated to the second part $R$ of the catalyst, which is not shown.