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}≤ z≤1. The proof combines a sector decomposition inside Schur--Weyl blocks, an explicit one-shot measurement construction, and an exact minimax argument.
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BibTeX
@misc{li2026binary,
title = {{Binary Quantum Hypothesis Testing with a Rejection Option: The Exact Error-Exponent Trade-off}},
author = {Yonglong Li},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00931},
eprint = {2611.00931},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00931},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Binary Quantum Hypothesis Testing with a Rejection Option: The Exact Error-Exponent Trade-off
AU - Yonglong Li
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00931
UR - https://arxiv.org/abs/2611.00931
AB - 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}≤ z≤1. The proof combines a sector decomposition inside Schur--Weyl blocks, an explicit one-shot measurement construction, and an exact minimax argument.
ER -