Non-negativity of the photon-addition coefficients

Abstract

For every k at least 2, the coefficients c_n^(kk) that compare k-photon addition with single-photon addition on a two-mode squeezed vacuum are non-negative. This proves the conjecture of Z. Van Herstraeten, N. J. Cerf, S. Guha and C. N. Gagatsos (arXiv:2312.02066), who proved the cases k = 2, …, 8.

Definition 1.1 (The expansion defining the coefficients).

$$\mathrm{Expansion}\left(k,c\right)\iff (\forall n\in \mathbb{N},C{\left(n+k+1,k\right)}^2=\sum _{i=0}^{n}c\left(n-i\right)\cdot (i+1{)}^2+(n+2{)}^2)$$

Formalization. D5/S3/Quantum/Entanglement/PhotonAddedMajorizationCoefficients.Expansion (✓ std3).

Citation. Zacharie Van Herstraeten; Nicolas J. Cerf; Saikat Guha; Christos N. Gagatsos (2024). Majorization theoretical approach to entanglement enhancement via local filtration. DOI: 10.1103/PhysRevA.110.042430. URL: https://arxiv.org/abs/2312.02066v2.

Commentary.

For natural numbers k and a sequence c of real numbers indexed by n >= 0, Expansion(k, c) is the expansion of the paper, C(n+k+1, k)^2 = sum over i from 0 to n+1 of c_(n-i)^(kk) (i+1)^2 for every n >= 0, with c_n^(kk) = c(n) for n >= 0 and c_(-1)^(kk) = 1, so that the term i = n+1 is (n+2)^2. Here C(a, b) is the binomial coefficient.

Definition 1.2 (The photon-addition conjecture).

$$claim\iff (\forall k\in \mathbb{N},(k\ge 2)\Rightarrow \left((\exists c\in {\mathbb{R}}^{\mathbb{N}},\mathrm{Expansion}\left(k,c\right))\wedge (\forall c\in {\mathbb{R}}^{\mathbb{N}},(\mathrm{Expansion}\left(k,c\right))\Rightarrow \forall n\in \mathbb{N},0\le c\left(n\right))\right))$$

Formalization. D5/S3/Quantum/Entanglement/PhotonAddedMajorizationCoefficients.claim (✓ std3).

Citation. Zacharie Van Herstraeten; Nicolas J. Cerf; Saikat Guha; Christos N. Gagatsos (2024). Majorization theoretical approach to entanglement enhancement via local filtration. DOI: 10.1103/PhysRevA.110.042430. URL: https://arxiv.org/abs/2312.02066v2.

Commentary.

For every natural number k >= 2, a sequence c with Expansion(k, c) exists, and every such sequence takes only non-negative values. The expansion determines c(n) from c(0), …, c(n-1), so these are the coefficients c_n^(kk) of the paper, and their non-negativity is the column stochasticity of the paper’s matrix D.

Theorem 1.3 (Proof of the conjecture).

$$claim$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Entanglement/PhotonAddedMajorizationCoefficients.result (✓ std3). ∎

Resolves. Problems/van-herstraeten-2024-photon-addition-coefficients (proved) by D5/S3/Quantum/Entanglement/PhotonAddedMajorizationCoefficients.result.

Source. Repository-derived.

Acknowledgement. Zacharie Van Herstraeten; Nicolas J. Cerf; Saikat Guha; Christos N. Gagatsos (2024). Majorization theoretical approach to entanglement enhancement via local filtration. DOI: 10.1103/PhysRevA.110.042430. URL: https://arxiv.org/abs/2312.02066v2.

Commentary.

Let N_k(x) be the power series with coefficients C(n+k, k)^2. The expansion says (1 + sum of c(n) x^(n+1)) N_1 = N_k. First, N_1 = (1+x)(1-x)^(-3), since C(n+2, 2) + C(n+1, 2) = (n+1)^2. Second, by Vandermonde’s identity C(n+k, k) is the sum over j of C(k, j) C(n, j), and C(n+k, k) C(n, j) = C(k+j, j) C(n+k, k+j); so with a_j = C(k, j) C(k+j, j), N_k is the sum over j from 0 to k of a_j x^j (1-x)^(-(k+j+1)). Let E = (1-x^2)^(-1), the series 1 + x^2 + x^4 + …, and let B = (1 + (k^2+k-1) x)(1-x)^(-(k-2)) E + sum over j from 2 to k of a_j x^j (1-x)^(-(k+j-3)) E. Since E(1+x) = (1-x)^(-1), (1 - x)(1-x)^(-(k+2)) = (1-x)^(-(k+1)), a_0 = 1 and a_1 = k(k+1), one gets B N_1 = N_k. For k >= 2 every factor of B has non-negative coefficients and B has constant term 1, so c(n) = [x^(n+1)] B is a solution. As N_1 is not a zero divisor, every solution equals it, and its values are non-negative.

References

• Truth anchor: D5/S3/Quantum/Entanglement/PhotonAddedMajorizationCoefficients.Expansion
• Truth anchor: D5/S3/Quantum/Entanglement/PhotonAddedMajorizationCoefficients.claim
• Truth anchor: D5/S3/Quantum/Entanglement/PhotonAddedMajorizationCoefficients.result