  We introduce two infinite sequences of entanglement monotones, which are
constructed from expectation values of polynomials in the modular Hamiltonian.
These monotones yield infinite sequences of inequalities that must be satisfied
in majorizing state transitions. We demonstrate this for information erasure,
deriving an infinite sequence of "Landauer inequalities" for the work cost,
bounded by linear combinations of expectation values of powers of the modular
Hamiltonian. These inequalities give improved lower bounds for the work cost in
finite dimensional systems, and depend on more details of the erased state than
just on its entropy and variance of modular Hamiltonian. Similarly one can
derive lower bounds for marginal entropy production for a system coupled to an
environment. These infinite sequences of entanglement monotones also give rise
to relative quantifiers that are monotonic in more general processes, namely
those involving so-called $\sigma$-majorization with respect to a fixed point
full rank state $\sigma$; such quantifiers are called resource monotones. As an
application to thermodynamics, one can use them to derive finite-dimension
corrections to the Clausius inequality. Finally, in order to gain some
intuition for what (if anything) plays the role of majorization in field
theory, we compare pairs of states in discretized theories at criticality and
study how majorization depends on the size of the bipartition with respect to
the size of the entire chain.
