  Quantitative measure of disorder or randomness based on the entropy
production characterizes thermodynamical irreversibility, which is relevant to
the conventional second law of thermodynamics. Here we report, in a quantum
mechanical fashion, the first theoretical prediction and experimental
exploration of an information-theoretical bound on the entropy production. Our
theoretical model consists of a simplest two-level dissipative system driven by
a purely classical field, and under the Markovian dissipation, we find that
such an information-theoretical bound, not fully validating quantum relaxation
processes, strongly depends on the drive-to-decay ratio and the initial state.
Furthermore, we carry out experimental verification of this
information-theoretical bound by means of a single spin embedded in an
ultracold trapped $^{40}$Ca$^{+}$ ion. Our finding, based on a two-level model,
is fundamental to any quantum thermodynamical process and indicates much
difference and complexity in quantum thermodynamics with respect to the
conventionally classical counterpart.
