# Error Performance of Amplitude Shift Keying-Type Asymmetric Quantum Communication Systems
## 1 Introduction
In this channel model, a quantum state source (i.e., a light source) produces an entangled state, and these two modes are labeled S (signal mode: mode S) and A (ancilla mode: mode A). Light corresponding to mode S is directed toward one beam splitter with a reflectivity of either \(R_{0}\) or \(R_{1}\); the subsequently reflected light is then collected using a detector. The other beam, which corresponds to mode A, is sent to the detector directly. The detector then distinguishes the two channels that correspond to \(R_{0}\) and \(R_{1}\) by performing optimum quantum measurements (i.e., joint measurements) of the two light beams. In reference [9], the Einstein-Podolsky-Rosen (EPR) state, which consists of the \(m\)-fold tensor product of the two-mode squeezed vacuum state (TSVS) [10], was used as the light source, and the performance with regard to the error probability when using the EPR state was evaluated using its upper bound, which is defined by the Chernoff bound [11], and the lower bound, which is defined by the fidelity. As a result, it was found that the lower bound on the error probability when using the EPR state may almost reach the universal lower bound. In fact, if we consider that the communication system is used in such a manner that Alice operates the two beam splitters to transmit binary information to Bob based on differences in reflectivity (i.e., the differences in the amplitudes of the reflected light beams when subjected to different energy attenuation levels), the model in reference [9] would be an amplitude shift keying (ASK)-type AQC system. Therefore, it can be said that the work in this paper uses the same model for a different purpose to that of reference [9]. When considering aspects of the communication performance, it is necessary to perform instantaneous performance evaluations, but not using the Chernoff bound that corresponds to the case in which \(m\)-shot optical pulses are applied; instead, the error probability when a one-shot optical pulse is used here. In this case, the TSVS, i.e., the EPR state when \(m=1\), is considered, rather than the EPR state. In addition, there is a quasi-Bell state [12] that is constructed using nonorthogonal quantum states such as coherent states, but it becomes the maximum entangled state (maximum quasi-Bell state). It has been shown that the attenuation resistance of this state is strong, depending on application protocols, and it has been studied actively (e.g., [13; 14]) since the publication of reference [12]. In addition, study of the application of a quasi-Bell state that is not the maximum entangled state (i.e., a non-maximum quasi-Bell state) has advanced, and it has been reported that the non-maximum quasi-Bell state is superior to the maximum quasi-Bell state for use in certain protocols, such as quantum teleportation [15]. In particular, it was recently clarifiedthat the quasi-Bell state is superior to both the TSVS and conventional laser radar in terms of its performance for quantum illumination with attenuation (i.e., the model used in this paper with \(R_{0}=0\)) [16; 17; 18; 19]. Based on the discussion above, we aim to clarify Bob's communication performance based on the error probability criterion as a first step toward development of an ASK-type AQC system (hereinafter referred to simply as the AQC system). Specifically, by using the Schrodinger picture to describe the time evolution of both the quasi-Bell state and the TSVS, we evaluate and compare the error probabilities that occur when using these states, and thus consider the basic characteristics, i.e., the error performance, of the AQC system. We also compare these results with the error probability characteristics of an ACC system which was constructed using a coherent state source and an optimum classical measurement approach, along with the universal lower bound on the error probability when the use of any multimode quantum state is allowed.