State Diagnostics of Complexity in Open Quantum Systems

Traditional quantum complexity measures, such as operator-based out-of-time-ordered correlators (OTOCs) or entanglement entropy, often fail to capture the geometric reorganization of state space that occurs when a system transitions from closed, unitary evolution to open, interacting dynamics. This paper addresses this gap by asking: how can we define a geometric analogue for classical dynamical complexity in open quantum systems?

To answer this, the authors treat a subsystem's reduced state not merely as a density matrix, but as a probability measure over pure states on the complex projective Hilbert space (a Geometric Quantum State, or GQS). By conditioning the subsystem state on the environment's basis states, they track how interactions redistribute probability mass across the projective manifold. They then apply optimal transport theory—specifically the Wasserstein distance—to compare these measures. This allows them to define two key diagnostics: a distinguishability measure (quantifying sensitivity to initial conditions) and a state-space coverage index (measuring the extent of long-time exploration).

Applying these tools to the quantum kicked top, the authors demonstrate that both distinguishability and state-space coverage generally increase with interaction strength, mirroring the breakdown of periodic structure seen in classical chaotic systems. Furthermore, they find that the system's sensitivity to environment size is structured by parity symmetry, with integer-spin systems typically exhibiting higher complexity than half-integer-spin systems. This framework provides a direct, intuitive bridge between classical dynamical intuition and the behavior of open quantum systems.

This work provides a new, visually and mathematically intuitive way to study quantum complexity. By shifting the focus from abstract operator growth to the movement of probability mass on a geometric manifold, researchers can better understand how decoherence and entanglement reshape the state space of a quantum system, offering a clearer picture of the transition from coherent to complex dynamics.