Complexity of Mixed States in QFT and Holography

Understand

We study the complexity of Gaussian mixed states in a free scalar field theory using the 'purification complexity'.

• The latter is defined as the lowest value of the circuit complexity, optimized over all possible purifications of a given mixed state.
• We argue that the optimal purifications only contain the essential number of ancillary degrees of freedom necessary in order to purify the mixed state.
• We also introduce the concept of 'mode-by-mode purifications' where each mode in the mixed state is purified separately and examine the extent to which such purifications are optimal.