  We investigate and ascertain the ideal inputs to any finite-time
thermodynamic process. We demonstrate that the expectation values of entropy
flow, heat, and work can all be determined via Hermitian observables of the
initial state. These Hermitian operators encapsulate the breadth of behavior
and the ideal inputs for common thermodynamic objectives. We show how to
construct these Hermitian operators from measurements of thermodynamic output
from a finite number of effectively arbitrary inputs. Behavior of a small
number of test inputs thus determines the full range of thermodynamic behavior
from all inputs. For any process, entropy flow, heat, and work can all be
extremized by pure input states -- eigenstates of the respective operators. In
contrast, the input states that minimize entropy production or maximize the
change in free energy are non-pure mixed states obtained from the operators as
the solution of a convex optimization problem. To attain these, we provide an
easily implementable gradient descent method on the manifold of density
matrices, where an analytic solution yields a valid direction of descent at
each iterative step. Ideal inputs within a limited domain, and their associated
thermodynamic operators, are obtained with less effort. This allows analysis of
ideal thermodynamic inputs within quantum subspaces of infinite-dimensional
quantum systems; it also allows analysis of ideal inputs in the classical
limit. Our examples illustrate the diversity of 'ideal' inputs: Distinct
initial states minimize entropy production, extremize the change in free
energy, and maximize work extraction.
