Coherent states are important information carriers in the field of communication and are widely used in star-ground coherent optical communication and star-ground quantum key distribution scenarios. Improving the discrimination accuracy for coherent states is a key method for achieving longer communication distances. The quantum-enhanced measurement technique can overcome the limitations of shoot noise and the standard quantum limit (SQL), allowing increased communication distance of star-ground optical communication systems. Researchers have conducted a series of studies on quantum-enhanced measurements and proposed various schemes including adaptive feedback strategies and optimization of displacement operators. Although the SQL is surpassed in quantum-enhanced measurement schemes that use adaptive feedback strategies, the experimental structures of these schemes are usually complex. Therefore, designing such a scheme with a simple structure and good performance for weak signals remains an unresolved research problem. We hope to construct a measurement scheme for weak coherent states with a simple experimental structure that can surpass the SQL.

To determine weak coherent states, this paper proposes a new hybrid measurement scheme based on homodyne and binary quantum measurements, as depicted in Fig.1. The scheme includes two successive measurements of the coherent states: the first involves a homodyne detector, which can preliminarily determine the received coherent states. The results of the first measurement are then provided as feedback for the second stage. Subsequently, the second measurement stage involves a binary quantum measurement with two partitions, which discriminates the signal states. The received coherent state is first divided using a beam splitter (BS) with transmittance T and reflectivity R. The transmitted and reflected parts of the received signal state are output to the quantum measurement and HD stages, respectively. T and R can be varied to adjust the signal amplitude for the different measurement stages. Accordingly, the hybrid measurement scheme was optimized via enhancement of the ratio of each measurement stage. In addition, the displacement operator was optimized to improve the performance of the quantum measurement stage.

First, the displacement operator was improved, and the results are shown in Fig.6. It is clear that for weak signals, optimization of the displacement operator significantly reduces the symbol error rate in the quantum measurement stage, particularly when the average number of photons in the signal is 0.25, and the decrease is close to 0.1. Second, we analyzed the partition ratio in the binary quantum measurement stage and calculated the optimal partition ratio using a global search algorithm, as shown in Fig.7. The experimental results indicate that the partitioning scheme reduces the symbol error rate during the quantum measurement stage by approximately 1% compared with the normal scheme. Finally, the partition ratio between the quantum and classical measurements was enhanced using the gradient descent method, as presented in Fig. 8. From the experimental results, the partition ratio gradually decreases as the signal is enhanced, implying that more energy of the signal is allocated to the classical measurement stage. Under the optimal partition ratio, the error probability of the hybrid scheme for weak QPSK signals with an average photon number of less than two is significantly lower than the 66% SQL (considering the SQL when the detection efficiency is 66%). In addition, the hybrid solution has some advantages over 100% SQL. The experimental results show that the designed hybrid measurement scheme, with simpler experimental structure, can exceed the SQL under the current experimental conditions, and thus is of great significance for future engineering applications in practical coherent optical communications.

In this study, we developed a hybrid measurement scheme including homodyne and binary quantum measurements and optimized the corresponding parameters based on the experiment. In addition, the performance of the hybrid measurement scheme under the experimental parameters was calculated using Monte Carlo and numerical simulations. The simulation results illustrate that for weak signals with a mean photon number of less than two, the hybrid measurement scheme can surpass the SQL under practical conditions. For example, at the mean photon number of two, the error probability was reduced by 10% compared with the SQL. The proposed hybrid measurement scheme has a simpler experimental structure, compared with others, and can surpass the SQL under the current experimental conditions for weak coherent states.