LOv-Calculus: A Graphical Language for Linear Optical Quantum Circuits

We introduce the LO v -calculus, a graphical language for reasoning about linear optical quantum circuits with so-called vacuum state auxiliary inputs. We present the axiomatics of the language and prove its soundness and completeness: two LO v -circuits represent the same quantum process if and only if one can be transformed into the other with the rules of the LO v -calculus. We give a confluent and terminating rewrite system to rewrite any polarisation-preserving LO v -circuit into a unique triangular normal form, inspired by the universal decomposition of Reck et al. (1994) for linear optical quantum circuits. 2012