bibkey: vanherstraeten2024majorization authors: Zacharie Van Herstraeten; Nicolas J. Cerf; Saikat Guha; Christos N. Gagatsos year: 2024 title: “Majorization theoretical approach to entanglement enhancement via local filtration” doi: 10.1103/PhysRevA.110.042430 url: https://arxiv.org/abs/2312.02066v2 claim: “With N_kk = sum_n lambda^n C(n+k,k)^2 and coefficients c_n^(kk) defined by C(n+k+1,k)^2 = sum_{i=0}^{n+1} c_{n-i}^(kk) (i+1)^2, c_{-1}^(kk) = 1, the authors conjecture that all c_n^(kk) are non-negative for every k >= 2, i.e. that the matrix D with q^(kk) = D q^(11) is column stochastic, implying the majorization relation sigma^(k,k) < sigma^(1,1) (proved for k = 2, …, 8, with closed forms printed for k = 2, 3, 4).” strata_touched:

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Majorization theoretical approach to entanglement enhancement via local filtration

Z. Van Herstraeten, N. J. Cerf, S. Guha and C. N. Gagatsos, arXiv:2312.02066 (v1 2023-12-04, v2 2026-01-21); Phys. Rev. A 110 (2024) 042430. Subject: quant-ph.

The paper compares two-mode squeezed vacuum states after local photon addition and subtraction through majorization of their Schmidt coefficients. For k photons added to each mode the Schmidt coefficients are q_n^{(kk)} = λⁿ C(n+k,k)² / N_kk with N_kk = Σ_n λⁿ C(n+k,k)². The appendix “Majorization relations among different photon-added TMSV” writes q^{(kk)} = D q^{(11)} with a lower-triangular Toeplitz matrix D whose entries are λN_11/N_kk · c_m^{(kk)} λ^m, where

\binom{n+k+1}{k}^2 = \sum_{i=0}^{n+1} c_{n-i}^{(kk)} (i+1)^2

and c_{-1}^{(kk)} = 1. The columns of D sum to 1. The paper proves c_n^{(kk)} ≥ 0 for k = 2, …, 8, printing the closed forms for k = 2, 3, 4, and then states:

At this point, we conjecture the non-negativity of Eq. \eqref{eqapp:Coeffcients}, \textit{i.e.}, we conjecture that the matrix $\mathbf{D}$ of Eq. \eqref{eqapp:Dvector} is column stochastic for all $k\ge 2$, implying the majorization relation $\hat{\sigma}^{k,k} \prec \hat{\sigma}^{1,1}$.

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• DOI: https://doi.org/10.1103/PhysRevA.110.042430 (the journal text was not read).
• URL: https://arxiv.org/abs/2312.02066v2 (source main.tex retrieved 2026-10-01): the appendix “Majorization relations among different photon-added TMSV” (the definition of c_n^{(kk)}, the closed forms for k = 2, 3, 4 and the conjecture) and the main-text paragraph stating that the cases k = 2, …, 8 were proved.