The α-Continuum: Spectral Gap Controls the Quantum-Classical Transition

We define an effective quantumness parameter \(\alpha_{\text{eff}}(t) = 1 + (\text{Tr}(\rho^2) - 1/d)/(1 - 1/d) \in [1, 2]\) that interpolates continuously between quantum (\(\alpha = 2\), pure state, full interference) and classical (\(\alpha = 1\), maximally mixed, no interference). The time evolution of \(\alpha_{\text{eff}}\) is controlled by a single number: the spectral gap \(|\lambda_1|\) of the Lindblad generator, giving \(\alpha_{\text{eff}}(t) \approx 1 + e^{-2|\lambda_1|t}\). This connects the quantum-classical transition to the same spectral gap that governs mixing in stochastic processes, transfer between Lagrange points in celestial mechanics, and convergence in the Universal Spectral Representation Theorem. For a qubit coupled to a thermal bath at temperature \(T\), we show: (i) \(|\lambda_1|\) increases with \(T\), giving faster decoherence at higher temperatures; (ii) the stationary \(\alpha_{\text{eff}}(\infty)\) decreases from 2 (cold, quantum) to 1 (hot, classical); (iii) the decoherence time \(T_2 = 1/(2|\lambda_1|)\) is recovered exactly from the spectral gap. The α-parameter provides a single, measurable, basis-independent quantifier of "how quantum" a system is at any moment, and the spectral gap tells you how fast that quantumness is being lost.