bibkey: garciavelo2026schwarz authors: A. García-Velo, A. Ibort year: 2026 title: “Schwarz maps with symmetry” doi: 10.48550/arXiv.2601.02282 url: https://arxiv.org/abs/2601.02282v1 claim: “Remark V.2: the eigenvalues of the Choi matrix of a unital, Hermiticity-preserving U(n₁)⊗U(n₂)-equivariant map are the four values of Lemma V.8 with multiplicities 1, n₁²−1, n₂²−1 and (n₁²−1)(n₂²−1); Lemma V.8 states them only for n₁, n₂ ∈ {2, 3}, and the general case is stated as a conjecture.” strata_touched:

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García-Velo and Ibort, Schwarz maps with symmetry

A. García-Velo and A. Ibort, Schwarz maps with symmetry, arXiv:2601.02282v1 (5 January 2026; quant-ph).

Verified locator

DOI: 10.48550/arXiv.2601.02282. Primary version: https://arxiv.org/abs/2601.02282v1 (the only arXiv version). The TeX source of v1 supplies Eq. (eq:abcd) (the parameters a, b, c, d), the four matrix-unit cases, the displayed Choi matrix, Lemma V.8 (lem:eigChoi23, PDF page 29) and Remark V.2 (PDF page 32); the numbering is the printed one.

Source statements

Eq. (eq:abcd): “$a\equiv \frac{1-{\lambda }_{01}-{\lambda }_{10}+{\lambda }_{11}}{n_1{n}_2},b\equiv \frac{{\lambda }_{01}-{\lambda }_{11}}{n_1},c\equiv \frac{{\lambda }_{10}-{\lambda }_{11}}{n_2},d\equiv {\lambda }_{11}$”.

Choi matrix: “$C_{\Phi }=a{\mathbb{I}}_{(n_1{n}_2{)}^2}+b{\sum }_{k,l}({\mathbb{I}}_{n_1}\otimes F_{kl}{)}^{\otimes 2}+c{\sum }_{i,j}(E_{ij}\otimes {\mathbb{I}}_{n_2}{)}^{\otimes 2}+d{\sum }_{i,j,k,l}(E_{ij}\otimes F_{kl}{)}^{\otimes 2}$. We have no exact expression for its eigenvalues, but it is possible to compute their exact value for small dimension cases $(n_1,n_2)=(2,2),(2,3),(3,2),(3,3)$.”

Lemma V.8: “For $n_1=2,3$; $n_2=2,3$, the eigenvalues of the Choi matrix $C_{\Phi }$ of a unital, hermiticity-preserving $U(n_1)\otimes U(n_2)$-equivariant map are given by: $\frac{1}{n_1{n}_2}[1+(n_2^2-1){\lambda }_{01}+(n_1^2-1){\lambda }_{10}+(n_1^2-1)(n_2^2-1){\lambda }_{11}]$, with multiplicity 1. $\frac{1}{n_1{n}_2}[1+(n_2^2-1){\lambda }_{01}-{\lambda }_{10}-(n_2^2-1){\lambda }_{11}]$, with multiplicity $n_1^2-1$. $\frac{1}{n_1{n}_2}[1-{\lambda }_{01}+(n_1^2-1){\lambda }_{10}-(n_1^2-1){\lambda }_{11}]$, with multiplicity $n_2^2-1$. $\frac{1}{n_1{n}_2}[1-{\lambda }_{01}-{\lambda }_{10}+{\lambda }_{11}]$, with multiplicity $(n_1^2-1)(n_2^2-1)$.”

Remark V.2: “Our results suggest, and thus we conjecture, that the eigenvalues of the Choi matrix $C_{\Phi }$ are given by Eqs. (\ref{eq:eigChoi23a})-(\ref{eq:eigChoi23d}).”

Scope

The equivariant maps are the combinations ${\sum }_{\alpha ,\beta }{\lambda }_{\alpha \beta }{P}_{\alpha }\otimes P_{\beta }$ of the isotypic projections $P_0=D_i$ (trace to the scaled identity) and $P_1=\mathrm{id}-D_i$ of each factor, with ${\lambda }_{00}=1$ for unital maps and real ${\lambda }_{\alpha \beta }$ for Hermiticity-preserving ones. The paper uses the spectrum to describe complete positivity for $(n_1,n_2)=(2,2),(2,3)$ and to verify the PPT² property there.