  We discuss a class of quantum speed limits (QSLs) based on unified quantum
($\alpha,\mu$)-entropy for nonunitary physical processes. The bounds depend on
both the Schatten speed and the smallest eigenvalue of the evolved state, and
the two-parametric unified entropy. We specialize these results to quantum
channels and non-Hermitian evolutions. In the first case, the QSL depends on
the Kraus operators related to the quantum channel, while in the second case
the speed limit is recast in terms of the non-Hermitian Hamiltonian. To
illustrate these findings, we consider a single-qubit state evolving under the
(i) amplitude damping channel, and (ii) the nonunitary dynamics generated by a
parity-time-reversal symmetric non-Hermitian Hamiltonian. The QSL is nonzero at
earlier times, while it becomes loose as the smallest eigenvalue of the evolved
state approaches zero. Furthermore, we investigate the interplay between
unified entropies and speed limits for the reduced dynamics of quantum
many-body systems. The unified entropy is upper bounded by the quantum
fluctuations of the Hamiltonian of the system, while the QSL is nonzero when
entanglement is created by the nonunitary evolution. Finally, we apply these
results to the XXZ model and verify that the QSL asymptotically decreases as
the system size increases. Our results find applications to nonequilibrium
thermodynamics, quantum metrology, and equilibration of many-body systems.
