Complexity of Mixed States

Quantum computational complexity estimates the difficulty of constructing quantum states from simple operations. I will explain how to study mixed state complexity using the idea of 'purification complexity'. The latter is defined as the lowest value of the circuit complexity, optimized over all possible purifications of a given mixed state. I will study the complexity of Gaussian mixed states in free scalar field theory, focusing on thermal states and subregions of the vacuum state. I will highlight a surprising connection to the Cardy-Calabrese formula for the entanglement entropy of an interval in two dimensions. I will comment on the relation to holography.


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