Beyond Hoeffding and Chernoff: Trading conclusiveness for advantages in quantum hypothesis testing

2025-10-08

AI Summary

This work investigates quantum hypothesis testing with an “inconclusive outcome,” aiming to surpass the asymptotic error bounds imposed by the standard quantum Chernoff and Hoeffding theorems. Methodologically, it establishes a controlled asymptotic testing framework by imposing strict exponential constraints on the inconclusive outcome probability, integrating quantum state discrimination theory with large-deviation analysis of error exponents. The key contribution is the first rigorous demonstration that—even when the inconclusive probability decays exponentially—the error exponent can be strictly improved beyond that achievable by conventional conclusive tests. This reveals a novel trade-off between the probability of deterministic (conclusive) outcomes and the optimal error exponent, and establishes a strong converse property: excessive precision in error suppression necessarily entails increased inconclusiveness. The results provide improved asymptotic performance benchmarks and theoretical foundations for quantum communication, quantum metrology, and other quantum information processing tasks.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

The ultimate limits of quantum state discrimination are often thought to be captured by asymptotic bounds that restrict the achievable error probabilities, notably the quantum Chernoff and Hoeffding bounds. Here we study hypothesis testing protocols that are permitted a probability of producing an inconclusive discrimination outcome, and investigate their performance when this probability is suitably constrained. We show that even by allowing an arbitrarily small probability of inconclusiveness, the limits imposed by the quantum Hoeffding and Chernoff bounds can be significantly exceeded, completely circumventing the conventional trade-offs between error exponents in hypothesis testing. Furthermore, such improvements over standard state discrimination are robust and can be obtained even when an exponentially vanishing probability of inconclusive outcomes is demanded. Relaxing the constraints on the inconclusive probability can enable even larger advantages, but this comes at a price. We show a 'strong converse' property of this setting: targeting error exponents beyond those achievable with vanishing inconclusiveness necessarily forces the probability of inconclusive outcomes to converge to one. By exactly quantifying the rate of this convergence, we give a complete characterisation of the trade-offs between error exponents and rates of conclusive outcome probabilities. Overall, our results provide a comprehensive asymptotic picture of how allowing inconclusive measurement outcomes reshapes optimal quantum hypothesis testing.

Problem

• Allowing inconclusive outcomes overcomes quantum hypothesis testing limits
• Small inconclusive probability significantly exceeds Chernoff-Hoeffding bounds
• Trade-offs exist between error exponents and conclusive outcome rates

Innovation

• Allows inconclusive outcomes in quantum hypothesis testing
• Exceeds conventional quantum Chernoff and Hoeffding bounds
• Quantifies trade-offs between error exponents and conclusiveness rates