Detection scheme using a beam splitter and on-off detectors for non-Gaussian state-based quantum illumination with attenuation

Tiancheng Wang; Tsuyoshi Sasaki Usuda

doi:10.1364/OPTCON.514321

1. Introduction

Entanglement [1] is a non-local correlation between multi-quantum systems, and is known as an important resource in quantum protocols. In recent years, quantum metrology (also called quantum sensing), which applies entanglement in metrology techniques, has attracted particular attention. By the 2010s, several protocols of quantum metrology had been proposed, and typical examples include quantum ghost imaging for edge detection [2], quantum illumination for target detection [3], and quantum reading for reading information from an optical memory [4]. Application protocols using these quantum metrology techniques are currently expected to be wide ranging, from non-destructive cell imaging techniques [5,6] to secure Internet-of-Things communication systems [7–9].

Quantum illumination [3] is a protocol that confirms the existence of a target using two entangled beams of light (hereafter referred to as an entangled state) under the constraints of the presence of strong thermal noise and significant energy attenuation. When quantum illumination was first proposed in 2008, an entangled state that is a generalization of the Bell state was used for analysis [3]. In the same year, Tan et al. examined the application of a two-mode squeezed vacuum (TMSV) state, which is a type of Gaussian state, and showed that it has an advantage over the protocols that do not utilize entanglement, such as laser radars, in weak light regions [10]. Since then, Gaussian states such as the TMSV state have been actively discussed, and experiments using these states have been actively conducted (e.g., [11,12]). Around 2020, we focused on the quasi-Bell state [13], which is a type of non-Gaussian state, and analyzed the performance of quantum illumination in terms of the minimum error probability with a one-shot optical pulse, using a new approach that differs from previous approaches [14–17]. First, as far as we investigated, we found that the performance of quantum illumination using the quasi-Bell state is always superior to that of a classical laser radar constructed using the coherent state and homodyne detector [15]. We then lifted the constraint of the presence of strong thermal noise and investigated the case where only energy attenuation existed. As a result, we found that when the quasi-Bell state is used, the superiority of quantum illumination is maintained not only in weak light regions but also in all light regions [16,17]. By contrast, the TMSV state only maintains its superiority in weak light regions. Therefore, the use of the quasi-Bell state should greatly expand the limited application range of the original quantum illumination to tasks such as cell detection, LiDAR in autonomous cars, and probes in deep space.

In the above studies, although it is assumed that the optimum quantum measurement [18] is used for performance evaluation (i.e., the Helstrom bound), how to implement a physical scheme has not yet been clarified. Recently, we focused on a detection scheme for quasi-Bell state-based quantum illumination with an attenuation environment and evaluated its performance by considering a structure using a beam splitter and two photon counters [19]. As a result, we found that even a simple detection scheme that is constructed using a beam splitter and two photon counters can overcome the Helstrom bound for the TMSV state when the average number of photons emitted from the source is large. Furthermore, it was quantitatively shown that the range of the average number of photons in which the quasi-Bell state outperforms the TMSV state can be expanded when the reflectivity of the beam splitter has been optimized. However, the photon counter [20,21] is an element that measures the number of incident photons with high accuracy and is generally quite expensive because it has currently only been developed in laboratories. To spread the benefits of quantum technology to as widely as possible in future, it will be desirable to provide this device at a lower cost while retaining its high performance. In this paper, we aim to construct a practical detection scheme for the quasi-Bell state-based quantum illumination using some elements that are less expensive than the photon counter. Therefore, we focus on a commercially available on-off detector, which detects only the presence or absence of photons, and reconstruct the detection scheme of our previous study.

To evaluate the performance of the on-off detector-based detection scheme, we first describe the effects of the attenuation environment using the Schrödinger picture, which describes the evolution over time of the quasi-Bell state. Then, we describe the interaction between the two modes of the entangled state in the beam splitter using the unitary operator [22] proposed by Prasad et al. Finally, we analyze and numerically evaluate the performance based on the maximum likelihood detection criterion after measuring the obtained quantum state using two on-off detectors that respectively operate on the two modes. Interestingly, unlike the previous detection schemes that used photon counters in the quasi-Bell state-based quantum protocols [19,23,24], our results show that almost the same performance can be obtained by replacing the photon counters with on-off detectors.

2. Structure of the on-off detector-based scheme

We first briefly explain the quantum illumination protocol with attenuation, as presented in [16,17,19].

1. A quantum state source (light source) produces an entangled state in which the two modes are labeled S (signal mode) and A (ancilla mode).
2. The light corresponding to mode A is emitted toward the detector.
3. The light corresponding to mode S is emitted toward the target: if the target exists, then the light is subsequently reflected and collected by the detector; otherwise, only the vacuum state $|0\rangle$ is collected by the detector.
4. The detector then distinguishes between the presence or absence of a target by performing a joint measurement on both modes.

Note that the a priori probabilities of Case(0) (absence of a target) and Case(1) (presence of a target) are set to be equal. This indicates that there is no prior information about the presence or absence of a target, and thus the Bayes decision criterion with equal a priori probabilities is the optimum approach under the quantum minimax criterion in quantum detection theory [25].

When the energy attenuation occurs in the channel of mode S, the received quantum states of Case(0) and Case(1) (i.e., the quantum states before entering the detector) must be considered. The quasi-Bell state [13] that is constructed using the coherent state $|\alpha \rangle$ as a light source is represented by

(1)$$|\Psi\rangle_{\rm SA} = h\left(|\alpha\rangle_{\rm S}|\alpha\rangle_{\rm A} - |-\alpha\rangle_{\rm S}|-\alpha\rangle_{\rm A}\right),$$

where $\kappa =\mathrm {e}^{-2\alpha ^2}$, $h= (\sqrt {2 - 2\kappa ^2})^{-1}$, and the complex amplitude $\alpha$ of the coherent state is considered a real number in this paper. The average number of photons in mode S is $\langle n \rangle _{\rm QBS}= \alpha ^2 \coth ( 2\alpha ^2 )$. Here, the received quantum states are written in Stinespring representation as follows:

(2)$$\Psi_{\rm SA}^{(0)}=|0\rangle_{\rm S}\langle 0|\otimes\,{\rm Tr}\,_{\rm S}|\Psi\rangle_{\rm SA}\langle \Psi|,$$

(3)$$\Psi_{\rm SA}^{(1)}=\,{\rm Tr}\,_{\rm E}\left\{ (\mathbb{U}_{\rm SE}\otimes\mathbb{I}_{\rm A}) (|\Psi\rangle_{\rm SA}\langle \Psi| \otimes |0\rangle_{\rm E}\langle 0|) (\mathbb{U}_{\rm SE}\otimes\mathbb{I}_{\rm A})^{{\dagger}} \right\},$$

where $\mathbb {I}$ is the identity operator, and the unitary operator $\mathbb {U}_{\rm SE}$ on the composite system SE describes the interaction between mode S and environment mode E:

\begin{aligned} \mathbb{U}_{\rm SE} |\alpha\rangle_{\rm S}|0\rangle_{\rm E}&=|\sqrt{\eta}\alpha\rangle_{\rm S} |\sqrt{1-\eta}\alpha\rangle_{\rm E},\\ \mathbb{U}_{\rm SE} |-\alpha\rangle_{\rm S}|0\rangle_{\rm E}&=|-\sqrt{\eta}\alpha\rangle_{\rm S} |-\sqrt{1-\eta}\alpha\rangle_{\rm E}. \end{aligned}

In addition, $\eta$ is the energy transmissivity of the channel of mode S. Smaller values of $\eta$ indicate more significant attenuation. In general, preliminary measurements are used to determine $\eta$.

In our previous study [19], we considered a detection scheme using a beam splitter (BS) and two photon counters for the quasi-Bell state-based quantum illumination with attenuation. The light corresponding to modes S and A incident in the detector is measured by two photon counters, one for each mode, after interference on an $R$:$T$ beam splitter of which the sum of the reflectivity $R$ and transmissivity $T$ is $1$. In this study, to develop a detection scheme using practical and inexpensive elements instead of photon counters, we focused on the on-off detector (OD), which detects only the presence or absence of photons (Figs. 1, 2). In recent years, thanks to advances in technology, nearly ideal on-off detectors such as superconducting nanowire single-photon detectors have been commercially available, and the realization of ideal on-off detectors is becoming feasible. Therefore, we analyzed performance using an ideal on-off detector in this paper. Here, we consider using two on-off detectors labeled OD_S and OD_A in place of the two photon counters used in the previous study. Detectors OD_S and OD_A measure $u$ and $v$, respectively, where $u, v \in \{{\rm on}, {\rm off} \}$ represent the binary information of the presence (on) or absence (off) of photons. To derive the joint probability distribution in terms of the presence or absence of photons in the two on-off detectors, it is first necessary to describe the quantum states after interference on the $R$:$T$ beam splitter, characterized by $|\alpha \rangle _{\rm S}|\beta \rangle _{\rm A} \to |\sqrt {T}\alpha -\sqrt {R} \beta \rangle _{\rm S}|\sqrt {R}\alpha +\sqrt {T} \beta \rangle _{\rm A}$ [22]. In Case(0) and Case(1), the quantum states after interference on the beam splitter can be represented by [19]

(4)$$\begin{aligned} \Psi_{\rm SA}^{(0)'}=&h^2 (|-r\alpha\rangle_{\rm S}\langle -r\alpha| \otimes|t\alpha\rangle_{\rm A}\langle t\alpha| -\kappa|r\alpha\rangle_{\rm S}\langle -r\alpha| \otimes|-t\alpha\rangle_{\rm A}\langle t\alpha|\\ &\;\;\;\;\;\;\;\;-\kappa|-r\alpha\rangle_{\rm S}\langle r\alpha| \otimes|t\alpha\rangle_{\rm A}\langle -t\alpha| +|r\alpha\rangle_{\rm S}\langle r\alpha| \otimes|-t\alpha\rangle_{\rm A}\langle -t\alpha|), \end{aligned}$$

and

(5)$$\begin{aligned} \Psi_{\rm SA}^{(1)'}=&h^2 ( |\Delta\rangle_{\rm S}\langle \Delta|\otimes|\Theta\rangle_{\rm A}\langle \Theta| -L|\Delta\rangle_{\rm S}\langle -\Delta|\otimes |\Theta\rangle_{\rm A}\langle -\Theta|\\ &\;\;\;\;\;\;\;\;-L|-\Delta\rangle_{\rm S}\langle \Delta|\otimes |-\Theta\rangle_{\rm A} \langle \Theta|+|-\Delta\rangle_{\rm S}\langle -\Delta|\otimes|-\Theta\rangle_{\rm A} \langle -\Theta|), \end{aligned}$$

respectively, where $r=\sqrt {R}$, $t=\sqrt {T}$, $\Delta =(t\sqrt {\eta }-r)\alpha$, $\Theta =(r \sqrt {\eta }+t)\alpha$, and $L=\mathrm {e}^{-2(1-\eta )\alpha ^2}$.

Fig. 1. Detection scheme using beam splitter and on-off detectors for quasi-Bell state-based quantum illumination when the target does not exist.

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Fig. 2. Detection scheme using beam splitter and on-off detectors for quasi-Bell state-based quantum illumination when the target exists.

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Then, the joint probability distributions $P_{\rm OD}(u,v|0)$ and $P_{\rm OD}(u,v|1)$ in terms of the presence or absence of a target corresponding to Case(0) and Case(1), respectively, can be derived as

(6)$$\begin{aligned}P_{\rm OD}(u,v|0)&=\,{\rm Tr}\, \Psi_{\rm SA}^{(0)'} \left(\mathbf{\Pi}_{\rm S} \otimes \mathbf{\Pi}_{\rm A}\right)\\ &=\left\{ \begin{array}{ll} 2 h^2 (1- \kappa) \mathrm{e}^{-\alpha^2} & (u:{\rm off},v:{\rm off}) \\ \sum_{i=1}^{\infty} 2 h^2 (1- ({-}1)^i\kappa ) {\rm P_o}^{(t\alpha)^2}(i) {\rm P_o}^{(r\alpha)^2}(0) & (u:{\rm off},v:{\rm on})\\ \sum_{j=1}^{\infty} 2 h^2 (1- ({-}1)^j\kappa ) {\rm P_o}^{(t\alpha)^2}(0) {\rm P_o}^{(r\alpha)^2}(j) & (u:{\rm on},v:{\rm off})\\ \sum_{i=1}^{\infty}\sum_{j=1}^{\infty} 2 h^2 (1- ({-}1)^{i+j}\kappa ) {\rm P_o}^{(t\alpha)^2}(i) {\rm P_o}^{(r\alpha)^2}(j) & (u:{\rm on},v:{\rm on})\\ \end{array} \right. , \end{aligned}$$

and

(7)$$\begin{aligned}P_{\rm OD}(u,v|1)&=\,{\rm Tr}\, \Psi_{\rm SA}^{(1)'} \left(\mathbf{\Pi}_{\rm S} \otimes \mathbf{\Pi}_{\rm A}\right)\\ &=\left\{ \begin{array}{ll} 2 h^2 (1- L) \mathrm{e}^{-\alpha^2(1+\eta)} & (u:{\rm off},v:{\rm off})\\ \sum_{i=1}^{\infty} 2 h^2 \{1- ({-}1)^i L\} {\rm P_o}^{\Theta^2}(i) {\rm P_o}^{\Delta^2}(0) & (u:{\rm off},v:{\rm on})\\ \sum_{j=1}^{\infty} 2 h^2 \{1- ({-}1)^j L\} {\rm P_o}^{\Theta^2}(0) {\rm P_o}^{\Delta^2}(j) & (u:{\rm on},v:{\rm off})\\ \sum_{i=1}^{\infty}\sum_{j=1}^{\infty} 2 h^2 \{1- ({-}1)^{i+j} L\} {\rm P_o}^{\Theta^2}(i) {\rm P_o}^{\Delta^2}(j) \;\;\;\;\;\;\;\; & (u:{\rm on},v:{\rm on}) \\ \end{array} \right. , \end{aligned}$$

where the detection operator $\mathbf {\Pi } (\in \{|0\rangle \langle 0|, \mathbb {I}-|0\rangle \langle 0| \})$ is used. In addition, the Poisson distribution with a random variable $k$ and mean $\mu$ is expressed as ${\rm P_o}^{\mu }(k)$. Note that if neither on-off detector measures a photon, the joint probability distribution does not depend on the reflectivity $R$ of the beam splitter.

Finally, for each of the four patterns of $\{u,v\}$, we introduce the maximum likelihood detection criterion that determines Case(0) when $P_{\rm OD}(u,v|0)>P_{\rm OD}(u,v|1)$ and Case(1) when $P_{\rm OD}(u,v|0) \leq P_{\rm OD}(u,v|1)$. Therefore, the (average) error probability of the on-off detector-based detection scheme is optimized for $R$ and given as follows:

(8)$$P_{\rm e}^{({\rm OD})}= \min_{R} P_{\rm OD}^{(R)},$$

where

P_{\rm OD}^{(R)}= \frac{1}{2} \left(\sum_{\{u,v| P_{\rm OD}(u,v|0) \leq P_{\rm OD}(u,v|1)\}} P_{\rm OD}(u,v|0) +\sum_{\{u,v| P_{\rm OD}(u,v|0) >P_{\rm OD}(u,v|1)\}} P_{\rm OD}(u,v|1) \right).

3. Error performance

In this section, we show the error probability (8) of the detection scheme described in the above section using a beam splitter and two on-off detectors for the quasi-Bell state-based quantum illumination with attenuation. We also show the error probabilities of a detection scheme from our previous study that uses a beam splitter and two photon counters (PC) and a simpler detection scheme that uses only two photon counters and has no interference between modes S and A [19]. Additionally, for comparison with the above physical implementation schemes, we show the error probabilities with the optimum quantum measurement, that is, the Helstrom bound, when either the quasi-Bell or TMSV state is used as the light source [7,16,17]. Note that, based on the conventional numerical calculation method (e.g., [7,26]), we perform the calculation after a suitable truncation of $i$ and $j$, which represent the number of photons in the detectors and are initially required to handle an infinite-dimensional Hilbert space. To determine the optimal $R$ in (8), we perform an exhaustive search in the range $R \in [0.5, 1]$ with increments of $10^{-4}$.

Figure 3 plots the error probabilities when the average number of transmitted photons in mode S for both quasi-Bell and TMSV states is varied from $0.5$ to $10$ and the transmissivity of the channel is fixed at $1$ (i.e., without attenuation). We consider the minimum average number of photons to be $0.5$ because that of the quasi-Bell state is $\min \langle n \rangle _{\rm QBS}= 0.5$. In addition, because the matrix size required for the density operator in the TMSV state becomes too large to calculate even after a suitable truncation, the maximum average number of photons is set to $10$. The blue dot-dash line represents the error probability with the optimum quantum measurement (labeled as the Helstrom bound) when the TMSV state is used, and the black dashed line represents that when the quasi-Bell state is used. For the quasi-Bell state, the error probabilities for the detection scheme using only two photon counters (“PC”; green dotted line), the scheme proposed in the above section (“BS+OD”; red dot-dot-dash line), and a beam splitter and two photon counters (“BS+PC”; cyan dashed line) are given. Figures 4 and 5 show the results when the transmissivity of the channel is fixed at $0.5$ and $0.1$, respectively, and the meaning of each line is the same as that in Fig. 3.

Fig. 3. Error probabilities with respect to the average number of photons where the transmissivity of the channel equals to $1$.

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Fig. 4. Error probabilities with respect to the average number of photons where the transmissivity of the channel equals to $0.5$.

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Fig. 5. Error probabilities with respect to the average number of photons where the transmissivity of the channel equals to $0.1$.

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These figures confirm that the BS+OD scheme proposed in this paper has almost the same performance as the BS+PC scheme proposed in our previous study. The features of the detection scheme become more noticeable as the attenuation increases (i.e., $\eta$ decreases) or the average number of photons increases. It is also shown that the attenuation increases, and the improvement in the performance of the proposed BS+OD scheme when compared with the PC detection scheme also increases. Therefore, even if the entanglement is completely destroyed in Case(0) and almost completely destroyed in Case(1), interference between modes S and A on a beam splitter offers an advantage over the scheme without interference, and this advantage is achieved even without the introduction of elements with photon-number resolution such as photon counters. In addition, the performance of the approach using the quasi-Bell state and the BS+OD scheme proposed in this study is almost the same as that using the TMSV state and the optimum quantum measurement when the average number of photons is small, and the former can outperform the latter as the average number of photons increases. However, the approach using the quasi-Bell state and the optimum quantum measurement, shown by the black solid line, always offers a lower error probability than that using the TMSV state and the optimum quantum measurement. For cases in which there is attenuation and the average number of photons is small, it would be meaningful to consider how to implement an optimum quantum measurement or a near-optimum quantum measurement using simple elements.

Finally, we describe the optimal $R$ of the beam splitter used in the on-off detector-based detection scheme in detail. Figure 6 shows the value of optimal $R$ when the average number of photons of the quasi-Bell state is varied from $0.5$ to $10$. As shown in the figure, the optimal $R$ is characterized by a rapid increase near the average number of photons $1$ and then asymptotically approaches $R=1$. Regarding the reason for this threshold in terms of the average number of photons $1$, we find that there are multiple local minimum values of the error probabilities depending on $R$, and the minimum one is switched at this threshold.

Fig. 6. Optimal $R$ for the detector where the transmissivity equals to $0.1$, $0.5$, and $1$.

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4. Conclusion

In this paper, we developed and evaluated a practical detection scheme using on-off detectors and a beam splitter for quasi-Bell state-based quantum illumination with attenuation. From the results, we found that our detection scheme has the same performance as a photon counter-based detection scheme, and a quantum advantage with respect to an approach that does not utilize entanglement is achieved even without the introduction of elements that have photon-number resolution. Moreover, we clarified that the approach using the quasi-Bell state and our detection scheme, while not achieving the Helstrom bound, can outperform that using the TMSV state and the optimum quantum measurement, which does achieve the Helstrom bound, especially when the average number of photons is large. In general, a detection scheme using the TMSV state in the quantum illumination protocol requires photon-number resolution; for instance, an approach that measures mode A with an on-off detector and mode S with a photon counter [27]. In contrast, for the quasi-Bell state, we found that it is not necessary to distinguish the information of more than one photon, and the performance of this approach can exceed the approach using the TMSV state and the optimum quantum measurement. We believe the advantage of the quasi-Bell state over the TMSV state comes from its property of macroscopic qubits and will revisit this issue in the future. In this study, we also simulated a hybrid quasi-Bell state-based detection scheme that uses an on-off detector and a photon counter. However, the performance was almost the same as that of our proposed detection scheme, and hence there is little merit in introducing an expensive photon counter.

Quantum protocols using the TMSV state generally require a complex detection scheme (e.g., [12,28]). In contrast, for quantum protocols using the quasi-Bell state, it is generally possible to construct a detection scheme with simple elements such as a beam splitter and photon counters [23,24]. In this paper, we show that even if the photon counters are replaced by on-off detectors, the performance of the detection scheme can overcome the Helstrom bound of the TMSV state. In fact, it is well known that generating non-Gaussian states such as quasi-Bell states (e.g., [29]) is more difficult than generating Gaussian states such as TMSV states, and thus the research progress of the former has been slower than that of the latter. The results of this paper suggest that the detection schemes for quantum protocols using non-Gaussian states such as quasi-Bell states may be constructed using less expensive devices, and this will hopefully stimulate continual research into the generation of non-Gaussian states.

Future work includes the physical implementation of detection schemes that achieve or approach the Helstrom bound of the quasi-Bell state. In addition, the performance of the detection scheme when thermal noise occurs in the channel could be evaluated.

Funding

Japan Society for the Promotion of Science (JSPS) KAKENHI (20H00581, 20K20397, 21K04064, 22K20437); The Nitto Foundation.

Acknowledgments

We thank Kimberly Moravec, PhD, from Edanz (https://jp.edanz.com/ac) for editing a draft of this manuscript.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Detection scheme using a beam splitter and on-off detectors for non-Gaussian state-based quantum illumination with attenuation

Abstract

Corrections

1. Introduction

2. Structure of the on-off detector-based scheme

3. Error performance

4. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (10)

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