Majorization requires infinitely many second laws

TL;DR

This paper demonstrates that in the context of majorization and related theories, an infinite family of entropy-like functions is necessary to fully describe second laws, with implications for thermodynamics and quantum physics.

Contribution

It proves that a finite set of entropy functions cannot fully characterize second laws in majorization and thermo-majorization scenarios, extending to molecular diffusion and catalytic majorization.

Findings

Any second law family must be countably infinite for large state spaces.

Thermo-majorization does not require state space constraints if the equilibrium distribution is non-uniform.

No finite family of entropy functions suffices for molecular diffusion and catalytic majorization.

Abstract

Majorization is a fundamental model of uncertainty with several applications in areas ranging from thermodynamics to entanglement theory, and constitutes one of the pillars of the resource-theoretic approach to physics. Here, we improve on its relation to measurement apparatuses. In particular, after discussing what the proper notion of second law in this scenario is, we show that, for a sufficiently large state space, any family of entropy-like functions constituting a second law must be countably infinite. Moreover, we provide an analogous result for a variation of majorization known as thermo-majorization which, in fact, does not require any constraint on the state space provided the equilibrium distribution is not uniform. Lastly, we discuss the applicability of our results to molecular diffusion and catalytic majorization. In this regard, we consider a variation of majorization used in plasma physics as a model of molecular diffusion and show that no finite family of entropy-like functions constituting a second law of molecular diffusion exists. Moreover, we show how our results are useful when dealing with a conjecture regarding catalytic majorization (i.e. trumping). In particular, we show that the sort of characterizations of trumping that have been considered before require an infinite family of real-valued functions.