Krylov complexity of purification

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, reminiscent of the quantum speed limit, and 2) the Krylov state/operator complexities are subadditive in contrast to the holographic volume complexity.
Comment: 11 pages, 5 figures; Lloyd bound with the energy is added and a slight rearrangement of texts, typos are corrected, references are added