Observational Indistinguishability Between Classical Diffusion and Quantum Thermal Dynamics on Networks

• DOI: https://doi.org/10.1098/rspa.2026.0513

The heat kernel of a graph Laplacian is widely used to describe diffusion, transport, correlations and ge-ometry in complex networks. In this work, we show that the same heat-kernel operator emerges both as the propagator of classical diffusion on a network and as the thermal Green’s function of a network of cou-pled quantum harmonic oscillators restricted to the one-particle sector. We demonstrate that the geome-try induced by this operator can be reconstructed from multiple observational experiments without direct ac-cess to the microscopic physical substrate generating the dynamics. Consequently, the same experimentally observed heat-kernel geometry may arise from fun-damentally different physical mechanisms. This im-plies that heat-kernel-based observations alone may be insufficient to discriminate between classical and quantum interpretations of hidden dynamics in black-box systems. The results are discussed in the context of complex networks and observational geometries reconstructed from indirect measurements, including potential implications for neuroscience and brain con-nectivity analysis.

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