Quantum Complexity Dynamics for Disjoint Subsystems

Quantum complexity is a powerful tool for probing chaos and black hole interiors, but most studies have focused on contiguous subsystems. This paper investigates how the complexity dynamics of a quantum system change when the subsystem is noncontiguous—composed of multiple disjoint regions—using both holographic methods and random quantum circuits.

The authors analyze the complexity of disjoint subsystems in a two-dimensional conformal field theory (CFT) dual to an eternal BTZ black hole. They use the Complexity=Volume (CV) proposal to calculate the volume of the entanglement wedge for various configurations of disjoint intervals. To verify these findings in a more generic chaotic setting, they also study the thermalization timescales of disjoint subsystems in brickwork random quantum circuits, proving that the time to reach the maximally mixed state scales inversely with the number of disjoint components.

1. Complexity Inversion: At finite temperatures, a subsystem occupying less than half the total system can exhibit exponentially large complexity if it is fragmented into multiple disjoint intervals, even while its larger complement remains in a low-complexity state. This effect is intrinsically thermal and disappears in the infinite-temperature limit.
2. Accelerated Thermalization: When a subsystem's complexity equilibrates, fragmenting it into m equal-sized disjoint components reduces the thermalization timescale by a factor of m. This result is confirmed in both holographic models and random quantum circuits.
3. Geometric Dependence: The authors map the parameter space of interval sizes and temperatures, showing that the exotic complexity behavior is sensitive to the specific arrangement of intervals and the temperature of the system.

These results demonstrate that the topology of a subsystem—not just its size—is a critical factor in its complexity dynamics. By showing that disjointness can lead to qualitatively new phenomena, the paper challenges the standard intuition that larger subsystems are always more complex. It also provides a rigorous link between holographic geometric transitions and the information-theoretic behavior of random quantum circuits.