Open problem in composite hypothesis testing posed by Berta et al.
Given arbitrary nonempty families of finite-dimensional quantum channels and , characterize the optimal asymptotic Stein exponent for discriminating the hypotheses versus under each admissible tester--jammer information pattern and input structure, including parallel, adaptive, entangled-input, and IID-input strategies. In addition, determine for each model whether a strong converse holds: namely, whether every sequence of tests whose type-II error decays at a rate strictly larger than the optimal Stein exponent must have its type-I error converging to .
References
Primary source
Additional references
Progress summary
A new paper advances several versions of the problem, but it does not settle every composite-testing formulation.
The problem seeks a general understanding of composite quantum hypothesis testing across different information patterns between tester and jammer. Berta et al. posed the underlying challenge; the retrieved literature treats several important special cases rather than the full formulation.
Known results
- Composite channel discrimination has a characterized parallel Stein exponent and an adaptive upper bound; classically, convex hypothesis sets remove the adaptive advantage, while the corresponding quantum question remains open.
- Adversarial channel discrimination claims a strong converse and an asymptotic exponent, but identifies the best-case setting as an open problem.
- A 2025 work develops a minimax channel divergence, proves its super-additivity, and presents the result as progress toward the problem.
September 9, 2026 development
The paper Minimax games for quantum channel discrimination reports nine minimax divergences, twelve Stein exponents, and strong converses for several game models, while separating tester–jammer information patterns. This is substantial claimed progress, but it explicitly covers selected models rather than every formulation and has not been independently verified here.
Current status (as of September 2026): Several important game models now have claimed strong-converse results, but the full composite-testing problem and related best-case formulations remain open.
Solutions 0
No solutions have been posted yet.