Complexity of mixed Gaussian states from Fisher information geometry

TL;DR

This work develops a geometric framework to quantify circuit complexity for mixed bosonic Gaussian states in harmonic lattices using the Fisher-Rao distance on covariance matrices. By exploiting Williamson's decomposition, it defines and analyzes spectrum and basis complexities, purification paths, and bounds, all without introducing ancillae. The formalism yields closed expressions for pure, thermal, coherent, and one-mode cases, and it is instantiated in harmonic chains where analytic and numerical results illustrate the behavior of complexity, mutual complexity, and subregion variants. The approach connects to entanglement structure through entanglement-hamiltonian paths and provides a bridge to related concepts in holography and Gaussian channels, highlighting both strengths and limitations of purification-based methods.

Abstract

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. In the special case of harmonic chains on the circle or on the infinite line, we report numerical results for thermal states and reduced density matrices.

Figures (12)

• Figure 1: Pictorial representation of the optimal circuit (\ref{['optimal circuit']}) connecting $\gamma_{\textrm{\tiny R}}$ to $\gamma_{\textrm{\tiny T}}$ (solid black curve). Coloured solid curves represent the sets made by symmetric Gaussian matrices having the same symplectic spectrum. The red curve corresponds to $\mathcal{D}_{\textrm{\tiny R}}$ and the blue curve to $\mathcal{D}_{\textrm{\tiny T}}$.
• Figure 2: The solid black curve and the solid coloured curves have been defined in Fig \ref{['fig:different_D']}. Eq. (\ref{['W-path-williamson']}) tells us that the dashed black curves represent the $W_{\textrm{\tiny R}}$ path and the $W_{\textrm{\tiny T}}$ path that pass through $\gamma_{\textrm{\tiny R}}$ and $\gamma_{\textrm{\tiny T}}$ respectively (the auxiliary covariance matrices $\tilde{\gamma}_{\textrm{\tiny R}}$ and $\tilde{\gamma}_{\textrm{\tiny T}}$ have been defined in (\ref{['tilded gammas']})). The arcs of the dashed curves that connect the blue curve to the red curve have the same length given by (\ref{['compl_same_W']}).
• Figure 3: The optimal purification paths for $\gamma_{\textrm{\tiny R}}$ and $\gamma_{\textrm{\tiny T}}$ correspond respectively to the $W_{\textrm{\tiny R}}$ path and to the $W_{\textrm{\tiny T}}$ path, that are represented through dashed lines. The straight black solid line represent the set of the pure states, whose symplectic spectrum is given by $\mathcal{D} = \tfrac{1}{2} \boldsymbol{1}$.
• Figure 4: The complexity $\mathcal{C}_2$ in terms of the size $L$ of the periodic harmonic chain. The reference and the target states are the ground states with $\omega = \omega_\textrm{\tiny R}$ and $\omega = \omega_\textrm{\tiny T}$ respectively (here $\kappa=m=1$). The data reported correspond to $\omega_\textrm{\tiny R}=1$ and different $\omega_\textrm{\tiny T}$. The solid lines in the top panels represent (\ref{['dFR pure thermo-c2']}), while the horizontal solid lines in the bottom panels correspond to the constant values of $a(\tilde{\omega}_{\textrm{\tiny T}}, \tilde{\omega}_{\textrm{\tiny R}})$ obtained from (\ref{['dFR pure thermo']}).
• Figure 5: The complexity $\mathcal{C}_2$ between the ground states of harmonic chains with $\kappa=m=1$ and different frequencies, for a given $\omega_\textrm{\tiny R}$ and as function of $\omega_\textrm{\tiny T}$. The data points come from (\ref{['PureStateComplexityHC']}). The dashed lines correspond to the first order approximation (\ref{['complexitypure_smalldeltaomega']}) and the solid lines to the second order approximation (\ref{['complexitypure_smalldeltaomega_2ndorder']}), in the thermodynamic limit and in the expansion where $\tilde{\omega}_{\textrm{\tiny T}} = \tilde{\omega}_{\textrm{\tiny R}} + \delta \tilde{\omega}$.