  Particle distinguishability is a significant challenge for quantum
technologies, in particular photonics where the Hong-Ou-Mandel (HOM) effect
clearly demonstrates it is detrimental to quantum interference. We take a
representation theoretic approach in first quantisation, separating particles'
Hilbert spaces into degrees of freedom that we control and those we do not,
yielding a quantum information inspired bipartite model where
distinguishability can arise as correlation with an environment carried by the
particles themselves. This makes clear that the HOM experiment is an instance
of a (mixed) state discrimination protocol, which can be generalised to
interferometers that discriminate unambiguously between ideal indistinguishable
states and interesting distinguishable states, leading to bounds on the success
probability of an arbitrary HOM generalisation for multiple particles and
modes. After setting out the first quantised formalism in detail, we consider
several scenarios and provide a combination of analytical and numerical results
for up to nine photons in nine modes. Although the Quantum Fourier Transform
features prominently, we see that it is suboptimal for discriminating
completely distinguishable states.
