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Quantum illumination is a protocol for detecting a low-reflectivity target by using two-mode entangled states composed of signal and idler modes. In this study, we extend the two-mode qubit states to three-mode qubit states, exploring the following configurations: (i) three signals, (ii) two signals and one idler, and (iii) one signal and two idlers. Each configuration considers various three-qubit states, such as GHZ state, W state, bipartite entangled states, and product states. We derive single-shot detection limits of the three-qubit states under white noise environment, evaluating the detection error probabilities for each configuration. We obtain that the performance is enhanced by the entanglement between signal and idler qubits, whereas it is degraded by the entanglement between signal qubits. In particular, the optimal probe state is not a maximally entangled state but a bipartite entangled state. Moreover, we show that the order of detection error probabilities is consistently certified by the quantum mutual information for the three-qubit probe states. Furthermore, we devote attention to whether this tendency maintains even in multipartite qudit states., Comment: 28 pages, 15 figures, 9 tables, minor correction

Artificial neural networks (NNs) can be implemented using chemical reaction networks (CRNs), where the concentrations of species act as inputs and outputs. In such biochemical computing, noise-robust computing is crucial due to the intrinsic and extrinsic noise present in chemical reactions. Previously suggested CRNs for feed-forward networks often utilized the rectified linear unit (ReLU) or discrete activation functions. However, one concern in this case is the discontinuities of the derivatives of those non-smooth functions, which can cause significant noise disruption during backpropagation. In this study, we propose a CRN that performs both feed-forward and training processes using smooth activation functions to avoid discontinuities in the backpropagation. All reactions occur in a single pot, and the reactions for training are bimolecular. Our case studies on XOR, Iris, MNIST datasets, and a non-linear regression model demonstrate that computation via the CRN (i) maintains accuracy despite noise in the reaction rates and the concentration of species and (ii) is insensitive to the choice of the running time and the magnitude of the noise in comparison to NNs with a non-smooth activation function. This work presents a noise-robust CRN for full NN computation, including backpropagation, paving the way for more stable and efficient biochemical computing systems., Comment: 33 pages