For any fixed-causal-order process, an entropy difference is bounded below by the logarithm of a dimension ratio, and violations witness causally nonseparable or indefinite causal order.
Pearl, Causality (Cambridge University Press, Cambridge, England, 2009)
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Entropic limitations on fixed causal order
For any fixed-causal-order process, an entropy difference is bounded below by the logarithm of a dimension ratio, and violations witness causally nonseparable or indefinite causal order.