Thermodynamic advancement in the causally inseparable occurrence of thermal maps

Quantum mechanics allows the occurrence of events without having any definite causal order. Here, it is shown that the application of two different thermal channels in the causally inseparable order can enhance the potential to extract work, in contrast to any of their definite (separable) order of compositions. This enhancement is also possible even without assigning any thermodynamic resource value to the controlling qubit. Further, we provide the first non-trivial example of causal enhancement with non-unital pin maps, for which it is still not clear how to obtain a superposition of path structure (under definite causal order). Hence, it may be a potential candidate to accentuate the difference between superposition of time and superposition of path.