Recent years have witnessed a surge of discoveries in the studies of
thermodynamic inequalities: the thermodynamic uncertainty relation (TUR) and
the entropic bound (EB) provide a lower bound on the entropy production (EP) in
terms of nonequilibrium currents; the classical speed limit (CSL) expresses the
lower bound on the EP using the geometry of probability distributions; the
power-efficiency (PE) tradeoff dictates the maximum power achievable for a heat
engine given the level of its thermal efficiency. In this study, we show that
there exists a unified hierarchical structure encompassing all of these bounds,
with the fundamental inequality given by a novel extension of the TUR (XTUR)
that incorporates the most general range of current-like and state-dependent
observables. By selecting more specific observables, the TUR and the EB follow
from the XTUR, and the CSL and the PE tradeoff follow from the EB. Our
derivations cover both Langevin and Markov jump systems, with the first proof
of the EB for the Markov jump systems and a more generalized form of the CSL.
We also present concrete examples of the EB for the Markov jump systems and the
generalized CSL.Comment: 19 pages, 4 figure