High-quality photonic Bell state measurements (BSMs) enable scalable universal quantum computing and long distance quantum communication. However, when implemented with linear optics, BSMs are fundamentally probabilistic, introducing substantial hardware overheads and limiting noise tolerance in photonic quantum computing architectures. Nonlinear interactions at the single-photon level can overcome these limitations by enabling near-deterministic photon-photon gates. Here, we demonstrate a passive photon-sorting circuit based on the induced nonlinearity arising from photon scattering in a solid-state quantum emitter. The scattering is implemented in a directional waveguide-emitter coupling interface and embedded on-chip into a linear optical circuit, through which we demonstrate sorting of one- and two-photon components with a success probability of 62%. We find that the current system can enable BSMs with a 57% post-selected success probability without ancillary photons, exceeding the linear-optical limit of 50%, and can be readily improved to >65% with design optimisations.
In quantum communication and fusion-based quantum computation, photonic Bell measurements are fundamentally limited when only passive linear optics is employed. While for qubits, some Bell states can be unambiguously identified with static beam splitters and no extra photons or entanglement, additional auxiliary photons or at least additional auxiliary degrees of freedom with a certain level of additional entanglement are needed to approach or attain a complete, deterministic Bell measurement. Here, we prove an exact resource threshold when the same two photons carry system qudits of dimension
d and a fixed auxiliary entangled state
Φ, possibly distributed over several additional degrees of freedom, with total Schmidt rank
rΦ. We show that a single conclusive Bell-label functional can occur for
rΦ⩾⌈d/2⌉, but deterministic discrimination of all
d2 Bell-state labels requires
rΦ⩾d. A maximally entangled rank-
d auxiliary state achieves the bound by local Bell-basis sorting between each photon's system and auxiliary degrees of freedom. Thus, the auxiliary Schmidt rank is a certified resource for ancilla-photon-free, embedded photonic Bell measurements.
Photonic linear optical networks provide a versatile platform for quantum information processing and quantum state engineering. However, the set of states that can be generated using passive linear optics alone is fundamentally constrained by bosonic symmetries. Heralding, based on conditional measurements on auxiliary modes, is a widely used technique to overcome these limitations and effectively enlarge the set of accessible states. Despite the widespread use of heralding, it is often unclear how specific ancillary resources impact the overall reachability of the target space. In this work, we investigate the local controllability of photonic states in linear optical networks by analyzing the rank of the Jacobian of the output state with respect to the underlying unitary circuit, which provides a quantitative measure of the dimension of the accessible tangent space at a given configuration. Our analysis ranges from passive linear optics to heralded linear optics, where auxiliary resources and conditional measurements are included. Within this framework, we quantify how different resources enlarge the locally accessible state space beyond that of passive linear optics and determine the resources required for the Jacobian rank to reach its maximal value, thereby achieving full local controllability. As maximal local rank is a necessary condition for global reachability, our framework offers a systematic tool to assess and compare the accessible state space of measurement-based photonic architectures, and to establish practical criteria for the resources needed in high-dimensional quantum state engineering.