A process matrix can be unitarily transformed to a fixed-causal-order process exactly when its eigenvalue multiplicities are divisible by the final output dimension; generic high-dimensional spectra are close to such spectra.
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Causality from the spectrum: Emergence of causal order from process-matrix mereology
A process matrix can be unitarily transformed to a fixed-causal-order process exactly when its eigenvalue multiplicities are divisible by the final output dimension; generic high-dimensional spectra are close to such spectra.