A Fractional Calculus Framework for Open Quantum Dynamics: From Liouville to Lindblad to Memory Kernels

Abstract: Open quantum systems exhibit dynamics ranging from purely unitary evolution to irreversible dissipative relaxation. The Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) equation uniquely characterizes Markovian dynamics that are completely positive and trace-preserving (CPTP), yet many physical systems display non-Markovian features such as algebraic relaxation and coherence backflow beyond the reach of semigroup evolution. Fractional calculus provides a natural framework for describing such long-memory behavior through power-law temporal kernels introduced by fractional time derivatives. Here we establish a unified hierarchy that embeds fractional quantum master equations within the broader landscape of open system dynamics. The fractional master equation forms a structured subclass of memory-kernel models, reducing to the GKSL form at unit fractional order. Through Bochner--Phillips subordination, fractional evolution is expressed as an average over Lindblad semigroups weighted by a power-law waiting-time distribution. This construction ensures physical consistency, explains the algebraic origin of long-time decay, and bridges unitary, Markovian, and structured non-Markovian regimes. The resulting framework positions fractional calculus as a rigorous and unifying language for quantum dynamics with intrinsic memory, enabling new directions for theoretical analysis and quantum simulation.