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IEEJ Transactions on Electronics Information and Systems

Error Probabilities of Quantum Illumination with Attenuation Using Maximum and Non-maximum Quasi-Bell States

Tiancheng Wang, Souichi Takahira, Tsuyoshi Sasaki Usuda

Published Jan 31, 2022• 3 citations
Publisher

Abstract

Quantum illumination proposed by Lloyd is a protocol that detects the existence of a target by using entangled light of two-mode. Quantum illumination was first proposed using the Bell state, and then the use of a two-mode Gaussian state was proposed. On the other hand, there are many impact studies using the quasi-Bell state. However, as far as we know, there are no studies that analyze quantum illumination using the quasi-Bell state. Therefore, we focused on the performance evaluation of quantum illumination using the quasi-Bell state in this paper. We compared and considered the effects using the Bell state, the two-mode Gaussian state, the maximum quasi-Bell state, the non-maximum quasi-Bell state, and the coherent state (i.e., a laser radar) on the attenuated channel. In particular, we derived the exact solutions of the error probabilities when using the maximum and non-maximum quasi-Bell states. Then we clarified that quantum illumination using the quasi-Bell state offers a clear advantage over that using the other entangled states under almost situations and always offers a clear advantage over the laser radar.

DOI
10.1541/ieejeiss.142.151

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Quantum illumination proposed by Lloyd is a protocol that detects the existence of a target by using entangled light of two-mode. Quantum illumination was first proposed using the Bell state, and then the use of a two-mode Gaussian state was proposed. On the other hand, there are many impact studies using the quasi-Bell state. However, as far as we know, there are no studies that analyze quantum illumination using the quasi-Bell state. Therefore, we focused on the performance evaluation of quantum illumination using the quasi-Bell state in this paper. We compared and considered the effects using the Bell state, the two-mode Gaussian state, the maximum quasi-Bell state, the non-maximum quasi-Bell state, and the coherent state (i.e., a laser radar) on the attenuated channel. In particular, we derived the exact solutions of the error probabilities when using the maximum and non-maximum quasi-Bell states. Then we clarified that quantum illumination using the quasi-Bell state offers a clear advantage over that using the other entangled states under almost situations and always offers a clear advantage over the laser radar.
Quantum illumination is one of the application protocols of This protocol uses entanglement for target detection. The protocol was first proposed by Lloyd, and its performance was later studied by Tan et al. In a lossy environment, they considered a Gaussian state in the presence of thermal noise. Entanglement with nonorthogonal states, such as coherent states, produces These states have been shown to be capable of perfect entanglement. In this paper, we consider quantum illumination using quasi-Bell states, and compute the minimum error probability of quantum illumination under these states.
Quantum illumination has been a hot topic in quantum sensing since first proposed by S. Lloyd in 2008. Quantum illumination is an entanglement-based protocol that is able to detect the existence of a target in an environment with low-energy transmissivity and strong thermal noise. The Bell state and the two-mode squeezed vacuum (TMSV) state as an entangled light source have been widely discussed. In our previous study, we clarified that quantum illumination using the quasi-Bell state offers good performance at error probability even though easing the above-mentioned restrictions on the environment. In this study, we considered and discussed a physical implementation of a measurement scheme for the quasi-Bell state-based quantum illumination using a beam splitter and photon counters. As the result, although the measurement scheme is not an optimum scheme, it still offers a clear advantage over that using the TMSV state, which allows optimum quantum detection, under almost situations.
Quantum illumination makes use of the strong correlation between entangled photons, making it possible to detect targets that break through physical limits. The Gaussian entangled state is a continuous variable entangled state with a high number of Hilbert spatial modes, which is currently a viable alternative to quantum lighting. In order to explore the target detection ability of Gaussian entangled state, this paper starts from the Gaussian entanglement model of Lloyd, establishes the target detection error probability boundary model under Gaussian entangled quantum illumination, and explores the optimal detection condition of Gaussian entangled state based on the detection signal-to-noise ratio and the number of measurements. The results show that when the Gaussian entangled state is optimal for the coherent state, the number of measurements is inversely proportional to the signal-to-noise ratio. Under the same error probability, the difference between the detection distances of the two states is determined by the atmospheric attenuation coefficient. The smaller the attenuation coefficient is, the smaller the attenuation coefficient is, the difference is greater.
Quantum illumination (QI) is an entanglement-based protocol for improving LiDAR/radar detection of unresolved targets beyond what a classical LiDAR/radar of the same average transmitted energy can do. Originally proposed by Seth Lloyd as a discrete-variable quantum LiDAR, it was soon shown that his proposal offered no quantum advantage over its best classical competitor. Continuous-variable, specifically Gaussian-state, QI has been shown to offer a true quantum advantage, both in theory and in table-top experiments. Moreover, despite its considerable drawbacks, the microwave version of Gaussian-state QI continues to attract research attention. A recent QI study by Armanpreet Pannu, Amr Helmy, and Hesham El Gamal (PHE), however, has: (i) combined the entangled state from Lloyd’s QI with the channel models from Gaussian-state QI; (ii) proposed a new positive operator-valued measurement for that composite setup; and (iii) claimed that, unlike Gaussian-state QI, PHE QI achieves the Nair–Gu lower bound on QI target-detection error probability at all noise brightnesses. PHE’s analysis was asymptotic, i.e., it presumed infinite-dimensional entanglement. The current paper works out the finite-dimensional performance of PHE QI. It shows that there is a threshold value for the entangled-state dimensionality below which there is no quantum advantage, and above which the Nair–Gu bound is approached asymptotically. Moreover, with both systems operating with error-probability exponents 1 dB lower than the Nair–Gu bound, PHE QI requires enormously higher entangled-state dimensionality than does Gaussian-state QI to achieve useful error probabilities in both high-brightness (100 photons/mode) and moderate-brightness (1 photon/mode) noise. Furthermore, neither system has an appreciable quantum advantage in low-brightness (much less than 1 photon/mode) noise.
Quantum illumination is an entanglement-based protocol for target detection. The use of a two-mode squeezed vacuum (TMSV) state as a type of Gaussian state has been widely discussed. In our previous study, we found that the quasi-Bell state, which is a non-Gaussian state, outperforms the TMSV state in the quantum illumination protocol with attenuation. In this paper, we construct a detection scheme using practical elements and demonstrate that it can outperform the TMSV state without using the photon-number resolution.
Quantum reading is the protocol that uses entanglement for reading a bit stored in digital memory. By using entanglement, it was shown by S. Pirandola that the data rate was improved. O. Hirota showed that in noiseless environments, error-free reading is possible by using a quasi-Bell state that has maximum entanglement. After that, some non-ideal cases were considered, such as symmetric loss and incomplete phase shift. In this study, we would like to clarify properties of the protocol in more realistic situations for experiment. In this paper, we compare and consider the effects using maximum and non-maximum quasi-Bell states on quantum reading with (1) only non-symmetric loss, (2) non-symmetric loss and incomplete phase shift, and (3) non-symmetric loss and phase noise. In the case (1), we derive the exact solutions of the error probabilities when using maximum and non-maximum quasi-Bell states. Then we clarify that the error probability using maximum quasi-Bell states is always lower than that using non-maximum quasi-Bell states. In the other two cases, we compute error probabilities numerically. In cases (2) and (3), the results are different from that of the case (1). That is, both maximum and non-maximum quasi-Bell states are not always supremer than the other.