Binary Quantum Hypothesis Testing with a Rejection Option: The Exact Error-Exponent Trade-off

Abstract

We characterize the exact error-exponent trade-off in binary quantum hypothesis testing with a rejection option for faithful states on a finite-dimensional Hilbert space. The probabilities of rejection under the two hypotheses are required to vanish exponentially, with exponents at least $K$ and $L$. We show that a quadruple $(a,b,K,L)$ of positive error and rejection exponents is achievable if and only if it satisfies three single-letter conditions: two conditions involve Petz--Rényi quantities and coincide with the converse bound conjectured to be tight by Ji and Regula; the third involves the $α$--$z$ Rényi quantities on the triangle $\max\{α,1-α\}\le z\le1$. The proof combines a sector decomposition inside Schur--Weyl blocks, an explicit one-shot measurement construction, and an exact minimax argument.