# Binary Discrimination in Quantum Systems via Hypothesis Testing

## TL;DR

This paper explores quantum hypothesis testing for binary discrimination of Hamiltonians using continuous weak measurement, emphasizing Bayesian methods for improved efficiency and speed in decision-making.

## Contribution

It introduces a Bayesian-based approach for quantum binary discrimination that outperforms maximum a posteriori methods in efficiency and speed.

## Key findings

- Bayesian criterion provides more efficient error probability calculations.
- The method enables quick discrimination between two Hamiltonian candidates.
- Comparison shows Bayesian approach is faster than MAP in practical scenarios.

## Abstract

We investigate the discrimination of two candidates of an unknown parameter in quantum systems with continuous weak measurement, inspired by the application of hypothesis testing in distinguish-ing two Hamiltonians [Kiilerich and M{\o}lmer, Phys. Rev. A, 98, 022103 (2018)]. Based on the measurement output and stochastic master equation, temporal evolutions of posterior probabilities of two hypotheses are given by Bayes' formula. The Bayes criterion is presented by the likelihood ratio conditioned on the outcome of measurements. Different from the calculation method based on maximum a posteriori criterion, the Bayes criterion based method for calculating the average prob-ability of making errors is more suitable and efficient in general situation of binary discrimination. Finally, an example of distinguishing two candidate Hamiltonians is given and the running times of calculating the average probability of error under the Bayes criterion and the maximum a posteriori criterion are compared to illustrate the feasibility of the hypothesis testing in quickly distinguishing two candidates of the parameter to be estimated.