  Purification maps a mixed state to a pure state and a non-unitary evolution
into a unitary one by enlarging the Hilbert space. We link the operator
complexity of the density matrix to the state/operator complexity of purified
states using three purification schemes: time-independent, time-dependent, and
instantaneous purification. We propose inequalities among the operator and
state complexities of mixed states and their purifications, demonstrated with a
single qubit, two-qubit Werner states, and infinite-dimensional diagonal mixed
states. We find that the complexity of a vacuum evolving into a thermal state
equals the average number of Rindler particles created between left and right
Rindler wedges. Finally, for the thermofield double state evolving from zero to
finite temperature, we show that 1) the state complexity follows the Lloyd
bound, and 2) the Krylov state/operator complexities are subadditive in
contrast to the holographic volume complexity.
