bibkey: saied2025photondistillation
authors: Jason Saied, Jeffrey Marshall, Namit Anand, Eleanor G. Rieffel
year: 2025
title: "General protocols for the efficient distillation of indistinguishable photons"
doi: 10.1103/PhysRevApplied.23.034079
url: https://arxiv.org/abs/2404.14217v4
claim: "For n-photon distillation protocols built from an n × n unitary whose first row is identically 1/√n (for example the discrete Fourier transform), the ideal heralding rate is h_n(0) = (−1/n)^{n−1}(n − 1)! Σ_{t=0}^{n−1} (n − t)(−n)^t/t! (Theorem III.5, Eq. (III.9)), with lim h_n(0) = 1/4; the authors conjecture that for n ≥ 5 the sequence h_n(0) decreases monotonically to 1/4 (Conjecture III.7)."
strata_touched:
  - D5/S3/Quantum/FockSpace/IdealHeraldingRateMonotonicity
license: citation-only
triage: anchor

Saied–Marshall–Anand–Rieffel, efficient distillation of indistinguishable photons

J. Saied, J. Marshall, N. Anand and E. G. Rieffel, General protocols for the efficient distillation of indistinguishable photons, Phys. Rev. Applied 23, 034079 (2025), DOI 10.1103/PhysRevApplied.23.034079; arXiv:2404.14217 [quant-ph] (v4 26 March 2025).

Verified locator

DOI: 10.1103/PhysRevApplied.23.034079 (as listed on the arXiv abstract page). Primary version read: https://arxiv.org/abs/2404.14217v4 (the latest of versions v1–v4; the conjecture is numbered III.6 in v1–v3). Source: https://arxiv.org/pdf/2404.14217v4 (text read from the PDF):

• §III.2, Theorems III.4 and III.5 with Eqs. (III.8)–(III.10), the heralding rate and its ideal value;
• Remark III.6 (the integer sequence n^{n−1}h_n(0), OEIS A277458), the first values, Figure 4 and Conjecture III.7.

Source statements

Theorem III.5: “Let U = F(n₁, …, n_ℓ), where n = n₁⋯n_ℓ > 2. More generally, we may take U to be any n × n unitary with entries in the first row identically equal to 1/√n. 1. The ideal heralding rate has the following equivalent expressions: h_n(0) = (−1/n)^{n−1}(n − 1)! Σ_{t=0}^{n−1} (n − t)(−n)^t/t! (III.9) = ₂F₀(−(n − 1), 2; 1/n), (III.10) … 2. lim_{n→∞} h_n(0) = 1/4.”

After Remark III.6: “The first few values are h₃(0) = 1/3, h₄(0) = 1/4, h₅(0) ≈ 0.264. In Figure 4, we plot the difference h_n(0) − 1/4 and numerically observe that h_n(0) quickly approaches 1/4, with asymptotic h_n(0) − 1/4 ∼ 1/(16n). In particular, we conjecture Conjecture III.7. For n ≥ 5, the sequence h_n(0) monotonically decreases to 1/4.”

Figure 4 evaluates h_n(0) − 1/4 for every n from 5 to 10⁵ and for selected larger n up to 10⁷.

Reading used by the repository

The conjecture is the statement about the sequence of Eq. (III.9): for every n ≥ 5, h_{n+1}(0) < h_n(0), and h_n(0) → 1/4. The repository takes Eq. (III.9) as the definition of the sequence, as the source’s Theorem III.5 establishes it for every n > 2; it does not re-derive the formula from linear optics.