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 show that this limitation can be operationally avoided once additional geometric and temporal structure is imposed on the dynamics through a structured, triadic time-modulated drive inspired by the Helix–Light–Vortex (HLV) framework. Using a triadic spiral-time modulation, we derive an explicit Floquet–Magnus expansion of the driven Liouvillian and obtain closed analytical expressions for the induced Liouvillian gap, including the leading correction L(1) ∝ [H2,[H3,·]] and the resulting frequency splitting δΩ ∼ ϵ2 HLV∥[H2,H3]∥2/∆. This inverse-∆ scaling demonstrates that the protocol operates robustly in the high-frequency regime, where incoherent noise is suppressed while coherent symmetry information is selectively amplified. We construct an optimal observable for SW–SSB detection, present an explicit singlequbit realisation illustrating geometrically induced weak symmetry, and propose concrete implementations on NV centers, transmon qubits, and non-Hermitian photonic lattices. The resulting HLV-enhanced protocol provides a realistic and experimentally accessible route for detecting SW–SSB and highlights the role of geometric time-structuring as a resource for engineered Liouvillian dynamics.