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arXiv:2609.07319

Symmetry-resolved tree tensor network analysis of Bell-state discrimination with ancilla-assisted passive linear optics

Anand Kumar, Wojciech Roga, Devendra Kumar Mishra, Masahiro Takeoka

quant-phSubmitted Sep 7, 2026• Announced Sep 8, 2026• 0 citations
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Abstract

Ancillary photons allow passive linear-optical Bell-state discrimination to exceed the one-half probability of success limit achievable with vacuum auxiliary modes. For a fixed analyzer and ancillary resource, the discrimination probability is determined by the photon-counting patterns that occur uniquely for each Bell input. We develop a photon-number-resolved tree tensor network method for evaluating these detector supports in recursively structured analyzers. Photon-number conservation and mode symmetries separate most of the Bell-state outputs before the remaining detector support is examined. We establish a tree tensor network based construction to evaluate the remaining support without need of recalling all relevant photon detector patterns separately. We apply the method to recursive, product, and asymmetric ancillary states with up to $32$ optical modes and reproduce known analytical and literature benchmarks. For factorized ancillary resources with fixed photon numbers in halves of the analyzer and the symmetry property used in our analysis, we also derive an exact composition relation; for reflected asymmetric pairings the success probability is the arithmetic mean of the corresponding symmetric configurations. The complete enumeration for smaller systems, permanent-based amplitude calculations, and independent evaluations provide additional checks of the numerical results.

Comments
17 pages, 2 figures, to be submitted to journal soon. Comments will be appreciated

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Ancillary photons allow passive linear-optical Bell-state discrimination to exceed the one-half probability of success limit achievable with vacuum auxiliary modes. For a fixed analyzer and ancillary resource, the discrimination probability is determined by the photon-counting patterns that occur uniquely for each Bell input. We develop a photon-number-resolved tree tensor network method for evaluating these detector supports in recursively structured analyzers. Photon-number conservation and mode symmetries separate most of the Bell-state outputs before the remaining detector support is examined. We establish a tree tensor network based construction to evaluate the remaining support without need of recalling all relevant photon detector patterns separately. We apply the method to recursive, product, and asymmetric ancillary states with up to 323232 optical modes and reproduce known analytical and literature benchmarks. For factorized ancillary resources with fixed photon numbers in halves of the analyzer and the symmetry property used in our analysis, we also derive an exact composition relation; for reflected asymmetric pairings the success probability is the arithmetic mean of the corresponding symmetric configurations. The complete enumeration for smaller systems, permanent-based amplitude calculations, and independent evaluations provide additional checks of the numerical results.
Quantum information processing using linear optics is challenging due to the limited set of deterministic operations achievable without using complicated resource-intensive methods. While techniques such as the use of ancillary photons can enhance the information processing capabilities of linear optical systems, they are technologically demanding. Therefore, determining the constraints posed by linear optics and optimizing linear optical operations for specific tasks under those constraints, without the use of ancillae, can facilitate their potential implementation. Here, we consider the task of unambiguously discriminating between Bell-like states using linear optics and without the use of ancillary photons. This is a basic problem relevant in diverse settings, for example, in the measurement of the output of an entangling quantum circuit or for entanglement swapping at a quantum repeater station. While it is known that exact Bell states of two qubits can be discriminated with an optimal success probability of 50%, we find, surprisingly, that for Bell-like states the optimal probability can be only 25%. We analyze a set of Bell-like states in terms of their distinguishability, entanglement as measured by concurrence, and parameters of the beam-splitter network used for unambiguous discrimination. Further, we provide the linear optical configuration comprising single-photon detectors and beam splitters with input-state-dependent parameters that achieves optimal discrimination in the Bell-like case.
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We propose a framework to analyze linear optical circuits based on an analogy with stabilizer formalism in quantum circuits, which provides efficiently computable formulas related to state discriminations. Hence, we analyze a Bell-state discrimination scheme with linear optics and ancillary single photons. With an increasing number of ancilla photons, the success probability of Bell-state discrimination has a maximum of 403512≃0.787\frac{403}{512} \simeq 0.787512403​≃0.787 at 282828 ancilla photons. By contrast, the corresponding two-qubit measurement asymptotically approaches a maximally entangling measurement.
Pairwise fusion gates are linear-optical measurements that herald Bell projections onto two-rail subspaces of two ddd-rail single-photon qudits. Without ancillary input photons, these passive measurements succeed with probability 1−d−11-d^{-1}1−d−1, with all failures confined to the diagonal logical subspace. We show that identical single-mode squeezers applied to the 2d2d2d interferometer outputs before photon-number-resolving detection recover part of this structured failure sector. Photon-number parity preserves the successful off-diagonal fusion signatures, while selected all-even patterns yield POVM elements proportional to definite pairwise Bell projectors. We derive the exact logical-space POVM and prove that a diagonal pattern is accepted if and only if its photon-number-imbalance vector has exactly two nonzero components of equal magnitude. The resulting closed elliptic-integral expression increases the ideal success probability, for instance, from 75%75\%75% to 79.62%79.62\%79.62% for d=4d=4d=4, and from 83.33%83.33\%83.33% to 87.15%87.15\%87.15% for d=6d=6d=6. With a representative finite detector saturation threshold, nsat=7n_{\rm sat}=7nsat​=7, the respective certified values remain 78.84%78.84\%78.84% and 86.71%86.71\%86.71%. These results establish active Gaussian processing as a method for recycling structured measurement failures without ancillary input photons, at the cost of 2d2d2d squeezing operations and a larger photon-number range at detection.
One major challenge in utilizing linear optical setups for quantum computing, is that only a limited set of operations can be preformed. Achieving universal quantum computing using only linear optical components comes at the cost of either requiring an untenable number of ancillary photons or the acceptance of some probability of failure. Understanding the constraints and optimizing general linear optical operations under these constraints is both an important and challenging task. Here we consider the problem of perfectly discriminating between Bell-like states without the use of ancillary photons. For the discrimination between Bell states, the optimal success probability cannot exceed 50%. Here we derive the optimal approach for Bell-like states and show that the success probability without ancillary photons can only reach 25%.