Lower bounds on the complexity of preparing mixed states
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Abstract
We establish a relationship between the correlations in a many-qubit mixed state and the minimum circuit depth needed for its preparation. If the mutual information between two subsystems exceeds the mutual information between one of those subsystems and the environment, which purifies the mixed state of the system, then the past lightcones of the subsystems must intersect one another. This results in a lower bound on the circuit depth of any ensemble of geometrically local unitaries that prepares the state to some specified degree of approximation. As an application, we derive lower bounds on the circuit depth needed to prepare thermal states of one-dimensional quantum critical systems described by conformal field theory, showing that the depth diverges as temperature is decreased up to a cutoff set by the preparation error.