For a finite tuple of square complex matrices, let F_infinity be the supremum of the normalized determinant over all matrix amplifications and Hilbert-Schmidt unit coefficient tuples, and let C be the maximum operator capacity over coefficient covariances of trace one. The previously established comparison C/e <= F_infinity <= C has the optimal universal constant: we construct finite tuples with C/F_infinity arbitrarily close to e. The tuples may have exactly identity Hilbert-Schmidt Gram matrix; their capacities tend to one and their amplified determinant values tend to e^(-1). The construction uses independent complex Gaussian coefficients with a fixed number of coefficients before the matrix dimension tends to infinity. A balance property of determinant maximizers makes polynomial moment comparison uniform over every amplification. A separate cut criterion turns Gaussian expansion estimates into a dimension-independent capacity lower bound. For each fixed number of Gaussian coefficients, scalar and arbitrarily amplified optima both converge in probability to e^(-1).
Working paper. Citable, but not yet screened for the AIRR collection.
Research paperWorking paperARR-2026-7WNWKBJVQ28KZAKA · v1
We transfer a quantitative unitary-relation construction to a quadratic optimization problem for quantum channels. The input is a real rational orthogonal projection on the Choi space, together with an explicit rational bistochastic Choi state supported exactly on its kernel. The minimum cost over all bistochastic channels is therefore zero. Nevertheless, minimizing the same cost over channels implemented by a unitary interaction with an unrestricted finite tracial ancilla is undecidable at one fixed additive precision. The objective is a projective measurement probability, not a determinant. Its equivalent matrix formulation is the squared distance between a unitary and an amplified linear subspace. This distance at fixed dimension is the Kraus-space quantity studied by Yu, Duan and Xu; we examine its stabilization under a completely depolarizing ancillary channel. We prove the transfer with explicit constants, derive the absence of a recursive uniform ancilla bound, and give the exact answer when the subspace projection is completely positive. The source hardness and the relation compiler are inherited results; no general novelty claim about noncomputable tracial optimization is made.
Working paper. Citable, but not yet screened for the AIRR collection.
Research paperWorking paperARR-2026-6FWR12DH3F8QBA5C · v1
We study the largest normalized determinant in a Kraus matrix space after amplification by a completely depolarizing channel. For the three-dimensional antisymmetric channel, we prove a bound uniform in the auxiliary dimension, identify its exact value 2^(2/9)/4, and show that a finite amplification attains it if and only if its dimension is divisible by three. The proof balances a determinant maximizer, fixes its second and fourth singular-value moments, and uses a quadratic majorant of the logarithm. Combined with known unitary constructions, this determines the regularized value for every Holevo-Werner channel: dimension three is the unique exceptional case. We also prove that adding depolarizing noise of strength epsilon produces a power law whose exponent is the normalized noncommutative-rank defect of the original Kraus space. The latter estimate is uniform over all auxiliary dimensions and has a matching lower bound at one fixed finite dimension. These results sharpen finite-dimensional examples and an exact-rank criterion into, respectively, an infinite-amplification optimum and a quantitative noise law. We state the operational interpretation for tests of an unknown global unitary separately from free ancillary assistance.
Working paper. Citable, but not yet screened for the AIRR collection.
