Discriminating distinguishability

Particle distinguishability is a significant challenge for quantum technologies, in particular for photonics, where the Hong-Ou-Mandel (HOM) effect clearly demonstrates it is detrimental to quantum interference. We take a representation theoretic approach in first quantization, 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 it clear that the HOM experiment is an instance of a (mixed) state discrimination protocol, which can be generalized to interferometers that discriminate unambiguously between ideal indistinguishable states and interesting distinguishable states, leading to bounds on the success probability of an arbitrary HOM generalization for multiple particles and modes. After setting out the first quantized 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.

Figure 1

Example of a Reck scheme parametrizing an arbitrary unitary transformation on four modes ($d_{\mathrm{S}}=4$), grouped into layers $T_k$. Each one-mode (phase shifter) and two-mode (beam splitter) subtransformation contributes one real parameter. Only the phase shifters situated between beam splitters (${\omega }_{1,3},{\omega }_{2,2},{\omega }_{2,3}$) contribute to our problem.

Figure 2

The best-known interferometer for discriminating completely indistinguishable from distinguishable states of three photons in three modes is $QF{T}_3$, with a success probability of $2/3$. Up to phases, it consists of two balanced beam splitters, one $2:1$ beamsplitter, and one $\pi /2$ phase shifter.