Complexity of mixed Gaussian states from Fisher information geometry

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.

Quantum extremal islands made easy. Part III. Complexity on the brane

“…That is, one identifies the purification of the mixed state in question with the smallest complexity, e.g., see [53][54][55][56] for recent studies. Let us add, however, that there are other possible definitions of circuit complexity of mixed states which do not require the concept of purifications [53,54,57]. Recently, information theoretic ideas have also produced exciting new insights for the resolution of the black hole information paradox [58][59][60].…”

Subsystem complexity after a global quantum quench

“…In order to understand the relation between entanglement and complexity, it is useful to study the optimal circuits and the corresponding circuit complexity when both the reference and the target states are mixed states [75][76][77][78][79]. The approach to the complexity of mixed states based on the purification complexity [75,76,79] is general, but evaluating JHEP05(2021)022 this quantity for large systems is technically complicated.…”

“…Some explicit results for large systems can be found by restricting to the simple case of bosonic Gaussian states and by employing the methods of the information geometry [80][81][82]. In our analysis we adopt the approach to the complexity of mixed states based on the Fisher information geometry [77], which allows to study large systems numerically. The crucial assumption underlying this approach is that all the states involved in the construction of the circuit are Gaussian.…”

Purification complexity without purifications

“…which is nothing but Uhlmann's fidelity (2.7). We also note the maximization condition also implies the two purifications are connected by the geometric mean (see [54] for more discussion about its application to the complexity of Gaussian states), i.e.,…”

“…Different from the purification complexity defined in(2.20), the authors in[54] develop a direct way to calculate the complexity for arbitrary Gaussian states with taking the Fisher-Rao metric as the complexity measure.…”