Complexity of mixed Gaussian states from Fisher information geometry

Understand

We study the circuit complexity for mixed bosonic Gaussian states in harmonic lattices in any number of dimensions.

• By employing the Fisher information geometry for the covariance matrices, we consider the optimal circuit connecting two states with vanishing first moments, whose length is identified with the complexity to create a target state from a reference state through the optimal circuit.
• Explicit proposals to quantify the spectrum complexity and the basis complexity are discussed.
• The purification of the mixed states is also analysed.