A Sharp Quantum Information-Combining Bound for Rank-One-Block and Qubit Side Information

Hirche and Reeb conjectured a sharp erasure-channel upper bound for the entropy of the XOR of independent uniform bits with arbitrary quantum side information. We establish this bound for a noncommuting class of binary classical–quantum ensembles: one average-state/signal pair admits an orthogonal decomposition into signed rank-one signal blocks, while the other acts on a qubit with an arbitrary traceless Hermitian signal. The principal analytic ingredient is a Schur-convexity theorem for symmetrized even-order qubit resolvent trace kernels. Combined with a positive-association principle for uniform Dirichlet vectors under majorization, this yields tensor-product resolvent-moment inequalities at every even order. A positive entropy-moment representation and Chebyshev association then establish the sharp conditional-entropy inequality. Additional corollaries cover two arbitrary-dimensional rank-one-block ensembles and gauge-balanced higher-rank signal blocks. The manuscript provides complete analytical proofs, including the simplex association and trace-kernel results, together with reproducible deterministic verification of the finite trace algebra and numerical tests on physically admissible noncommuting quantum ensembles. The results establish the sharp Hirche–Reeb information-combining inequality for the stated structural classes. The unrestricted conjecture for arbitrary quantum side information remains open.