HLV-Enhanced Detection of Strong-to-Weak Symmetry Breaking in Open Quantum Systems: A Geometric–Mathematical Protocol

Recent work has shown that detecting strong-to-weak spontaneous symmetry breaking (SW–SSB) in generic open quantum systems is information-theoretically hard. In the fully mixed-state, black-box setting no efficient protocol can distinguish weak-symmetry phases without exponentially many measurements. Here we demonstrate that this no-go result can be circumvented once geometric and temporal structure is imposed through the Helix–Light–Vortex (HLV) framework. Using a triadic spiral-time drive, we derive an explicit Floquet–Magnus expansion of the HLV-driven Liouvillian and obtain closed analytical expressions for the induced gap, including the first-order correction L(1) ∝ [H2,[H3,·]] and the resulting frequency splitting δΩ ∼ ϵ2 HLV∥[H2,H3]∥2/∆. This 1/∆ scaling shows that the protocol operates robustly in the high-frequency regime, simultaneously suppressing incoherent noise while amplifying coherent symmetry information. We construct the optimal observable for SW–SSB detection, provide an explicit singlequbit realisation illustrating the geometric weak symmetry generated by the HLV drive, and propose concrete implementations on NV centers, transmon qubits, and non-Hermitian photonic lattices. The resulting HLV-enhanced protocol yields a realistic, experimentally accessible method for detecting SW–SSB and offers a new geometric pathway for engineered Liouvillian dynamics.