  Linear optical circuits can be used to manipulate the quantum states of
photons as they pass through components including beam splitters and phase
shifters. Those photonic states possess a particularly high level of
expressiveness, as they reside within the bosonic Fock space, an
infinite-dimensional Hilbert space. However, in the domain of linear optical
quantum computation, these basic components may not be sufficient to
efficiently perform all computations of interest, such as universal quantum
computation. To address this limitation it is common to add auxiliary sources
and detectors, which enable projections onto auxiliary photonic states and thus
increase the versatility of the processes. In this paper, we introduce the
$\textbf{LO}_{fi}$-calculus, a graphical language to reason on the
infinite-dimensional bosonic Fock space with circuits composed of four core
elements of linear optics: the phase shifter, the beam splitter, and auxiliary
sources and detectors with bounded photon number. We present an equational
theory that we prove to be complete: two $\textbf{LO}_{fi}$-circuits represent
the same quantum process if and only if one can be transformed into the other
with the rules of the $\textbf{LO}_{fi}$-calculus. We give a unique and compact
universal form for such circuits.
