Ancilla-assisted linear optical Bell measurements and their optimality

In the last decade Grice [Phys. Rev. A 84, 042331 (2011)] and Ewert and van Loock [Phys. Rev. Lett. 113, 140403 (2014)] found linear optical networks achieving near-unit efficiency unambiguous Bell state discrimination, when fed with increasingly complex ancillary states. However, except for the vacuum ancilla case [Appl. Phys. B 72, 67 (2001)], the optimality of these schemes is unknown. Here, the optimality of these networks is investigated through analytical and numerical means. We show an analytical upper bound to the success probability for interferometers that preserve the polarization of the input photons, saturated by both Grice's and Ewert–van Loock's strategies. Furthermore, such an upper bound links the complexity of their ancilla states with the scaling of their performance. We also show a computer-aided approach to the optimization of such measurement schemes for generic interferometers, by simulating an optical network supplied with various kinds of ancillary input states. We numerically confirm the optimality of known small schemes. We use both methods to investigate other ancilla states, some of them never studied before.

Figure 1

The first “half” Ewert–van Loock single-photon scheme. It performs a Bell measurement with ${\mathcal{P}}_{\text{succ}}=\frac{5}{8}$ on the state $|\beta \rangle$ using two unentangled extra photons $|1\rangle$, two polarization-independent beam splitters, four polarizing beam splitters, two phase shifters, and six photocounters.