Over the course of two years, we demonstrated a number of improvements to the problem of efficiently telling overlapping quantum states apart. Here, we review these results.

Coherent states are the best signal carriers for optical communication over a high-loss channel since they remain pure states. On the other hand, their non-orthogonality prohibits us from discriminating different coherent states without error. This error limits the amount of information transmitted from sender to receiver. The ultimate minimum error probability for a given set of coherent states is quantum mechanically obtainable by optimizing a measurement operator. However, physical implementations of the optimal measurements are often highly non-trivial and conventional measurement strategies detecting classical physical variables of optical fields are far from the optimal error probability for coherent state discrimination. Therefore, a major goal ofquantum state discrimination is to devise a receiver that can beat the standard quantum limit (SQL) given by conventional measurements and approach the ultimate bound.

We developed a feedback receiver for the discrimination of the quaternary phase shift keying (QPSK) coherent states encoding format. The receiver consists of a displacement operation, a photon detection and fast feedback control of the displacement operation dependent on outcomes from the photon detection performed on preceding parts of a signal state. By installing a photon detector with a detection efficiency over 70% at telecom wavelength and performing the feedback procedure up to 10 times, our receiver beat the SQL under practical imperfections at telecom wavelength for the first time. Last year, the work entitled “Experimental Demonstration of a Quantum Receiver Beating the Standard Quantum Limit at Telecom Wavelength” was published in Physical Review Applied.

Following the successful implementation of such a feedback receiver that can discriminate coherent states beyond the SQL, a natural question arises: is the technically involved feedback necessary to beat the SQL? It has been believed so for the discrimination of the QPSK coherent states, especially in the weak coherent amplitude range. As a collaboration work with Jasminder S. Sidhu and Cosmo Lupo from the University of Sheffield, we devised a novel receiver that can beat the SQL in the very weak signal power regime without adopting feedback and experimentally demonstrated the proposed receiver. Although this proposed receiver requires a photon detector with almost ideal detection efficiency to surpass the SQL and does so only for very weak signal powers, our strategy does not rely on the feedback technique and may as such have higher compatibility with optical communication, since the feedback can limit the communication bandwidth. This work, “Quantum Receiver for Phase-Shift Keying at the Single-Photon Level,” was published in PRX Quantum earlier this year.

As described above, there is a certain probability of making erroneous decisions in the discrimination of coherent states due to their non-orthogonality. However, if one is allowed to discard some of the measurement results, it is indeed possible to make a conclusion without ambiguity. This type of quantum state discrimination is referred to as unambiguous state discrimination and usually aims at maximizing the success probability of reaching unambiguous conclusions. While the ultimate bound of the success probability had been theoretically obtained, there was a large gap between this ultimate bound of the QPSK unambiguous state discrimination and existing protocols. In our recent paper entitled “Adaptive Generalized Measurement for Unambiguous State Discrimination of Quaternary Phase-Shift-Keying Coherent States,” published in PRX Quantum, we theoretically proposed and experimentally demonstrated a novel feedback receiver with photon detection that can unambiguously distinguish non-orthogonal QPSK coherent states with a high success probability beating all existing protocols and provide a near-optimal performance. The feedback receiver based on photon detections can therefore with different settings provide near-optimal performance for both minimum error discrimination and unambiguous state discrimination of multiple coherent states. As such, it is expected to serve as a novel receiver technique in applications associated with classical coherent communication as well as quantum communication.

The feedback receiver combined with photon detection is a powerful measurement strategy not only for discrimination of coherent states but also for discrimination of non-classical states such as qubits, as it can implement an arbitrary two-dimensional projective measurement. We characterized our feedback receiver via quantum detector tomography and experimentally showed that our receiver can perform very well in the discrimination of single-rail qubits, one of several promising encoding schemes in optical quantum information processing. The work will pave a new way for quantum information processing with the single-rail optical qubit and contains an interesting aspect of fundamental science since it is the first demonstration of quantum detector tomography of a measurement that dynamically updates its physical structure in real time. The paper “Tomography of a Feedback Measurement with Photon Detection” was published in Physical Review Letters last year.

## Comments