Fig. 2: Representation of standard and boosted BSM schemes.

a Schematic and output statistics of the standard BSM scheme using two photons (red circles) as an input for the 4 × 4 splitter. Each bar corresponds to the probability of measuring a certain detector click pattern. For example, the bar at \({({\hat{a}}_{3H}^{\dagger })}^{2}\) represents the probability of measuring two horizontally polarized photons in output 3. The probabilities are derived by applying the matrix \({\hat{U}}_{4\times 4}\) from equation (4) to the different Bell-state inputs. Only the states \(\vert {\Psi }^{+}\rangle\) and \(\vert {\Psi }^{-}\rangle\) lead to distinct measurement outcomes whilst the states \(\vert {\Phi }^{+}\rangle\) and \(\vert {\Phi }^{-}\rangle\) cannot be distinguished. The background colors indicate whether a measurement of the corresponding state allows an unambiguous identification of one of the Bell states, with grey denoting ambiguous measurement outcomes. b Schematic and output statistics of the boosted BSM scheme using an ancillary Bell state (blue circles). The states \(\vert {\Psi }^{+}\rangle\) and \(\vert {\Psi }^{-}\rangle\) still lead to distinct measurement patterns. Furthermore, the additional interference with the ancillary Bell state gives rise to outcomes that are unique to either \(\vert {\Phi }^{+}\rangle\) or \(\vert {\Phi }^{-}\rangle\). These unambiguous outcomes allow the identification of these states with a probability of 50%, consequently leading to a BSM success probability of 75%.