Unitary causal decompositions: a combinatorial characterisation via lattice theory

If a unitary transformation has a decomposition into a quantum circuit with no directed path from input $a$ to output $b$, then $a$ does not influence $b$ through the overall unitary. Conversely, it is known that if $a$ does not influence $b$, one may always find a circuit decomposition lacking a path between these systems, thus making the no-influence condition directly apparent in the connectivity of the circuit. Causal decompositions are circuit decompositions in which, more generally, multiple such no-influence conditions are made apparent simultaneously. They bridge two fundamental concepts in quantum causality: causal structure, as expressed by influences through unitary transformations (and related to signalling through quantum channels); and compositional structure, expressed in terms of the shape of quantum circuits or networks. The general existence of causal decompositions remains unknown. This work focusses on unitary causal decompositions, i.e. decompositions in terms of unitary circuits in the traditional quantum circuit formalism that do not require the generalisation to `extended' or `routed' quantum circuits prompted by earlier research on this topic. We identify a combinatorial condition that characterises precisely those sets of causal no-influence constraints $G$ for which any unitary transformation satisfying $G$ has a unitary causal decomposition compositionally representing those constraints. Our methods are based on finite-dimensional operator algebra as well as the concept lattice construction, which was recently shown to provide a canonical shape $L_G$ for causal decompositions. The combinatorial condition we identify can be formulated in terms of $G$ as the absence of a forbidden substructure $C_3$ and in terms of $L_G$ as the existence of no more than one path between each input and output.