A theoretical quantum-network study identifies a pairing of passive optical measurements that jointly rejects both single-source two-photon false-herald classes while retaining useful Bell events involving photons from different sources. A false herald is an accepted detection pattern from the wrong kind of multipair event, rather than an inter-source Bell event. The result comes from calculations of cascaded source networks and their modeled performance, not a measurement.
The analysis asks which pairs of passive linear-optical analyzers can jointly reject multipair false-herald classes in cascaded entanglement swapping while retaining useful inter-source Bell events. It models Sagnac-configured SPDC singlet-pair sources joined by alternating BSM and GM stages. A herald requires every stage to succeed in the same spectral channel and pump pulse, with no intermediate memories.
The complementary pair
After nonentangling branches were excluded, the classification found one physical structural family: a direct-basis polarization-parity analyzer paired with an equatorial polarization analyzer. These are the BSM and GM stages used in the Pure Bell Pair, or PBP, network.
The two analyzers divide the false-herald problem between them. The BSM rejects same-polarization double emissions. The GM rejects mixed-polarization double emissions through two-photon interference and recovers same-polarization inter-source events as resolved phi-type Bell heralds. Together, their rejection sets cover both single-source two-photon classes.
What the calculations compared
For nonideal conditions, the performance analysis uses an all-orders Gaussian-state treatment of multipair emission. It covers three-source BSM-GM and four-source BSM-GM-BSM PBP networks under coupling, detector and channel loss. Each is compared with a source-count-matched all-BSM chain using the same brightness, multiplexing, detector model and loss. The reported outcomes include photon-photon and spin-spin fidelity, success probability and entanglement-delivery rate.
In the lossless three-source comparison, PBP has the higher modeled Bell-subspace fraction before final postselection. The expressions are 2G/(3G - 1) for PBP and 2P1/(P0 + 2P1 + P2) for matched all-BSM. The all-BSM denominator contains a vacuum term that is absent from the PBP expression.
When loss is included in the three-source model, PBP photon-photon fidelity remains well above the all-BSM result throughout the plotted transmission range. At internal transmission of 0.5, receiver transmission of 0.01 and a target spin-spin fidelity after memory loading of 0.99, the reported G - 1 operating point was 1.80 × 10^-3 for PBP, compared with 1.69 × 10^-3 for all-BSM.
The rate comparison also favored PBP at the reported settings. With internal and receiver transmission both at 0.7 and a target spin-spin fidelity of 0.99, its single-channel delivery probability at N_I = 1 was approximately 1.12 times the all-BSM value. At G - 1 = 0.005 and N_I = 10^7, PBP success was 0.366, compared with 0.00975 for all-BSM. At N_I = 10^7 and 10^8, the PBP delivery rates were 5.3 × 10^7 and 2.5 × 10^8 ebit/s, approximately 19 and 90 times the all-BSM rates.
Four sources under ideal conditions
The four-source calculation extends the lossless result. With lossless propagation, ideal photon-number-resolving detection and the stated two-photon acceptance rule, every alternating BSM-GM chain with at least four sources heralded a pure Bell state for the outer photons across all SPDC emission orders. Exhaustive checks for four, five and six sources found unit photonic fidelity for 64, 256 and 1,024 accepted record sequences, respectively.
In the lossless four-source comparison, PBP had unit photon-photon fidelity and approached unit success as multiplexing became large. The matched all-BSM network remained at a one-half ceiling for both photon-photon fidelity and Bell-subspace fraction because same-side events were false heralds.
The result depends on demanding conditions
The four-source advantage remained in the modeled loss comparison at the reported operating points. With a target spin-spin fidelity of 0.99 and internal and receiver transmission both at 0.7, the PBP operating point was G - 1 = 2.71 × 10^-3, versus 2.16 × 10^-3 for all-BSM, and its conditional-fidelity curve stayed above the comparator's. At internal transmission of 0.7, G - 1 = 0.005 and N_I = 5 × 10^9, success was 0.75 for PBP versus 0.47 for all-BSM. Near saturation, PBP's delivery rate was approximately 2 to 2.5 times the all-BSM delivery rate, reaching 1.22 × 10^9 ebit/s with N_M = 20 and N_I = 2 × 10^11.
The exact all-orders Bell-state result has narrow conditions. It applies only to lossless propagation, ideal photon-number-resolving detection and the stated two-photon acceptance rule. With loss included, the study instead uses an all-orders Gaussian-state calculation, so the lossless result does not establish exact fidelity under loss. Threshold detectors do not implement the analyzed photon-number-resolving heralding rule.
The architecture also carries a substantial synchronization and multiplexing burden. Because it has no intermediate memories, every stage must herald in the same spectral channel and pump pulse, creating many required trials. Adding the fourth source brings extra loss and mode-count burden. The analysis suggests distributing the required modes across frequency, time and space.
The document is identified as arXiv:2608.25211v1 [quant-ph], dated 25 Aug 2026. The work was partially supported by the National Science Foundation under Grant No. 1941583.
Paper data and sources
Original title: Multipair-resilient entanglement swapping with complementary linear-optical measurements
Authors: Siavash Mirzaei Ghormish, Ryan M. Camacho
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
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