Under the framework of classical theory, the performance of coherence detection is limited by the standard quantum limit (SQL) corresponding to the shot noise. However, the quantum-enhanced receiving technology can break the SQL and approach the Helstrom limit by introducing the displacement operation and converting the classical measurement into the measurement of photon number states. Unambiguous state discrimination (USD) is one of the commonly used discrimination strategies for quantum-enhanced reception. However, due to the limited energy of weak signals, the traditional USD quantum-enhanced receiving method has a high error rate of weak signal recognition. A hybrid measurement scheme for quadrature phase-shift-keying (QPSK) coherent states is developed. The scheme firstly converts the discrimination of QPSK coherent states into the distinction of BPSK coherent states by homodyne detector (HD), and then realizes the unambiguous discrimination of coherent states by BPSK quantum-enhanced receiving measurement. The simulation results show that the hybrid scheme is superior to the classical measurement scheme in the average photon number between 3.2 and 11.3, and has a larger signal range than the traditional QPSK quantum-enhanced receiving scheme.

  Objective  Coherent states are a critical carrier in optic communication and quantum information processing due to their intrinsic resilience to the loss of coherence. Unambiguous state discrimination (USD), which aims to realize the error-free discrimination of coherent states by outputting "no result" for the finite ambiguous results, is especially essential in quantum key distribution and quantum digital signatures. Recently, Becerra first experimentally demonstrated a generalized quantum measurement for USD of four non-orthogonal coherent states with a displacement operator and single-photon detector (SPD). As the unambiguously correct probability is still significantly low, Ref. introduces an adaptive feedback strategy and presents an adaptive generalized measurement scheme for USD of QPSK coherent states. However, the adaptive measurement scheme still needs to be improved to realize better performance. Thus, it is crucial to develop new schemes that can unambiguously discriminate coherent states with performance surpassing the ideal heterodyne strategy. This paper presents a new hybrid scheme that unites a homodyne detector (HD) and a quantum measurement scheme for USD of QPSK coherent states.

  Methods  To realize the unambiguous state discrimination, this paper presents a new hybrid measurement scheme based on these (Fig.1). The scheme consists of two successive measurements toward the coherent states. The first measurement is conducted by a homodyne detector, which can exclude half of the four possible states of the QPSK coherent states. The result of the first measurement gives a feed-forward to the second measurement. And the second measurement is conducted by a quantum measurement scheme and finally discriminates the signal states. The received QPSK coherent state \left|\alpha _m\right\rangle=\left|\alpha \right|\rme^i\left(m-1/2\right)\pi /2, m=\mathrm1,2,\mathrm3,4 is first divided by a beam splitter (BS) with transmittance T and reflectivity R . To simplify the calculation, we use t^2=T and r^2=R to denote the transmittance and reflectivity, respectively. The transmitted part \left|\alpha _Tm\right\rangle=\left|t\alpha \right|\rme^i\left(m-1/2\right)\pi /2 and reflected part \left|\alpha _Rm\right\rangle=\left|r\alpha \right|\rme^i\left(m-1/2\right)\pi /2 of the received signal state are respectively output to the quantum measurement stage and HD stage. We can change the partitional ratio R_Q,H=t^2/\left(r^2+t^2\right)\approx t^2 by selecting the appropriate beam splitter.

  Results and Discussions  The M=1+2 hybrid scheme can realize a lower error ratio than the heterodyne strategy when achieving the same correct unambiguous results probability for coherent states with mean photon number 4.2\leqslant \left|\alpha \right|^2\leqslant 12.6 (Fig.8). And M=4 quantum measurement scheme can only beat the heterodyne strategy for coherent state with mean photon number \left|\alpha \right|^2\leqslant7.41 while the M=1+3 hybrid scheme can realize it with mean photon number 3.2\leqslant \left|\alpha \right|^2\leqslant 11.3 (Fig.10). Furthermore, the hybrid scheme has a lower error ratio for coherent state with a mean photon number more than 6.7 compared with the quantum measurement scheme. This phenomenon notes that the hybrid scheme has a more excellent application range than the quantum scheme.

  Conclusions  A hybrid measurement scheme with an adaptive feedback strategy is proposed to unambiguously discriminate the QPSK coherent states, which converts the discrimination of four coherent states to two coherent states by HD and conducts the final discrimination by quantum measurement scheme. Here, we fully consider the non-ideal factors in the practical implementations, such as detection efficiency and dark count rate of detectors, visibility and transmittance of displacements, and develop the model of the hybrid measurement scheme. The simulation results clearly show that the hybrid scheme with M=1+2 can beat the heterodyne strategy for coherent states with 4.2 \leqslant \left|\alpha \right|^2 \leqslant 12.6 . Furthermore, compared with the quantum measurement scheme, the hybrid scheme with M=1+3 has a higher probability of correct unambiguous results and a lower error ratio for coherent states with 6.7\leqslant \left|\alpha \right|^2 . The hybrid scheme uses HD with less energy to convert a complex four-state discrimination problem to a simple two-state discrimination problem, which improves the probability of obtaining an unambiguous conclusion. However, this scheme is also limited by the HD and quantum measurements and can only achieve better performance within a specific range.