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arXiv:2606.24591

Auxiliary Schmidt Rank as a Resource for Photonic Bell Measurements

Pradip Laha, Peter van Loock

quant-phSubmitted Jun 23, 2026• Announced Jun 24, 2026• 0 citations
v1 · Jun 23, 2026v2 · Jun 30, 2026
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Abstract

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 $r_Φ$. We show that a single conclusive Bell-label functional can occur for $r_Φ\geqslant\lceil d/2\rceil$, but deterministic discrimination of all $d^2$ Bell-state labels requires $r_Φ\geqslant d$. 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.

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Bell measurements, entailing the projection onto one of the Bell states, play a key role in quantum information and communication, where the outcome of a variety of protocols crucially depends on the success probability of such measurements. Although in the case of qubit systems, Bell measurements can be implemented using only linear optical components, the same result is no longer true for qudits, where at least the use of ancillary photons is required. In order to circumvent this limitation, one possibility is to introduce nonlinear effects. In this work, we adopt the latter approach and propose a scalable Bell measurement scheme for high-dimensional states, exploiting multiple squeezer devices applied to a linear optical circuit for discriminating the different Bell states. Our approach does not require ancillary photons, is not limited by the dimension of the quantum states, and is experimentally scalable, thus paving the way toward the realization of an effective high-dimensional Bell measurement.
Abstract Bell state measurements, which project bipartite states onto the maximally entangled Bell basis, are central to a wide range of quantum information processing tasks, including quantum teleportation, entanglement swapping, and fusion-gate quantum computation. In photonic platforms, where information is encoded in optical degrees of freedom, the realization of efficient Bell state measurements is particularly challenging, especially when constrained to linear optical elements. In this review, we provide a comprehensive examination of existing proposals for the implementation of Bell state measurements, highlighting their fundamental limitations and the strategies developed to overcome them. Moreover, we survey recent advances and discuss open challenges, with a particular focus on Bell state measurements for high-dimensional systems, an area of growing interest due to its relevance for quantum repeaters and scalable quantum networks.