Open quantum systems interact with environments that remember. The Markov approximation — memoryless baths, instantaneous relaxation — works for weakly coupled systems at long times, but biological light-harvesting complexes and molecular junctions violate those assumptions. Memory Kernel Coupling Theory captures non-Markovian dynamics through integral kernels that encode the environment's influence over time. The original formulation, though, was scalar — limited to population dynamics, unable to track coherences or cross-correlations.
The tensorial extension changes what the theory can see. By reformulating the memory kernel as a matrix-valued object acting on the full density operator, the method resolves coherences in the spin-boson model, excitonic absorption spectra in the Fenna-Matthews-Olson complex, and charge mobility in lattice models. The mathematical upgrade — from scalar to tensor — corresponds to a physical upgrade from tracking where probability sits to tracking how it interferes.
The structural lesson: a system's relationship to its past is not a single number but a structured object. Scalar memory captures history's effect on occupation; tensorial memory captures history's effect on identity — which coherences survive, which correlations form, which interference patterns the environment permits. The dimensionality of the memory kernel determines the dimensionality of what the system can remember about itself.
(arXiv:2603.01458)