We give an exact, polynomial-size semidefinite formulation of the collective positive-error boundary for a mixture of two known pure states, for every finite copy count. Label and permutation symmetries reduce the response to small matrix blocks and a univariate polynomial; interval nonnegativity enforces the full composite null and alternative promises. The contribution is a certifiable solution of this testing problem using established symmetry and sum-of-squares tools. At overlap 0.6 and endpoint contamination 10−3, the nominal 1-percent false-alarm and 95-percent detection specification requires 55 collective copies, compared with 63 in a specified adaptive local threshold family. Rational witnesses certify feasibility and the excluding bounds. An explicit measurement resolves the Young diagram and selected directions within its spin blocks; retaining only the Young diagram instead requires 67 copies. Additional calibration margins are treated separately. The local comparison does not establish a separation from every sequential local protocol.
Polynomial-size calibration of finite-horizon quantum mixture alarms
