  We study quantum conditional entropy production, which quantifies the
irreversibility of system-environment evolution from the perspective of a third
system, called the reference. The reference is initially correlated with the
system. We show that the quantum unconditional entropy production with respect
to the system is less than the conditional entropy production with respect to
the reference, where the latter includes a reference-induced dissipative
information. The dissipative information pinpoints the distributive correlation
established between the environment and the reference, even though they do not
interact directly. When reaching the thermal equilibrium, the
system-environment evolution has a zero unconditional entropy production.
However, one can still have a nonzero conditional entropy production with
respect to the reference, which characterizes the informational nonequilibrium
of the system-environment evolution in the view point of the reference. The
additional contribution to the conditional entropy production, the dissipative
information, characterizes a minimal thermodynamic cost that the system pays
for maintaining the correlation with the reference. Positive dissipative
information also characterizes potential work waste. We prove that both types
of entropy production and the dissipative information follow quantum
fluctuation theorems when a two-point measurement is applied. We verify the
quantum fluctuation theorem for the dissipative information experimentally on
IBM quantum computers. We also present examples based on the qubit collisional
model and demonstrate universal nonzero dissipative information in the qubit
Maxwell's demon protocol.
