Observable bound for Gaussian illumination

Entanglement is a key element of quantum technologies, such as quantum teleportation, quantum communication, and quantum sensing. It takes advantage of the quantum correlation that cannot be revealed in classical systems. Quantum illumination (QI), which belongs to quantum sensing, takes quantum advantage over classical illumination (CI), with no output entanglementLloyd ; Tan . QI is used to discriminate the presence and absence of a low-reflectivity target using entangled states that consist of signal and idler modes. To detect the target, we send the signal mode towards the target while maintaining the idler mode. Then the reflected signal mode is measured together with the idler mode in a receiver. In continuous variable systems, a typical entangled state is a two-mode squeezed vacuum (TMSV) state which can be represented in terms of a number basis, $|\text{TMSV}\rangle=\sum^{\infty}_{n=0}\sqrt{\frac{N_{S}^{n}}{(1+N_{S})^{n+1}}}|n\rangle_{S}|n\rangle_{I}$, where $N_{S}$ is the mean photon number of the signal (or idler) mode. The TMSV state is nearly optimal in QIPalma ; Ranjith ; Bradshaw . CI is used to detect the target using unentangled states, e.g., coherent or thermal states. QI was compared with CI under a few measurement setups proposed in Refs.Guha ; Sanz ; Quntao ; Karsa ; Blakely ; Lee ; Jo ; Jeffers and implemented in Refs.Genovese ; Zheshen ; Sandbo ; England ; Aguilar ; Luong ; Shabir19 ; Sussman . To detect a long-distance target, it was studied on microwave QIShabir15 associated with the preparation of microwave signal and optical idler mode pairs. Even if optical entangled states are prepared, they can be converted to micro-optical entangled states by frequency conversionLauk20 ; Ihn ; Jo21 .

The performance of Gaussian illumination is quantified with the error probability that is a sum of the miss probability $P(\text{off}|\text{on})$ and false alarm probability $P(\text{on}|\text{off})$. Given a positive operator-valued measure, the error probability is lower bounded by the Helstrom bound (HB) and upper bounded by the quantum Chernoff bound (QCB)Aud07 ; Calsa08 ; Stefano08 which is also upper bounded by the Bhattacharyya bound. It is not known how to achieve the HB with implementable setups, but the QCB can be achieved with feasible ones. A single-mode coherent state attains its QCB by homodyne detection, and a TMSV state approaches its QCB asymptotically by sum frequency generation with feedforwardQuntao . Based on the QCB, QI using the TMSV state improves the error probability exponent by a factor of $4$ over CI using the coherent stateTan .

Here, we consider a signal-to-noise ratio (SNR) under mode-by-mode measurements. Initially we define that $\langle\hat{O}\rangle_{i}\equiv R_{i}$ is the mean value of an observable, $\Delta^{2}O_{i}\equiv\langle\hat{O}^{2}\rangle_{i}-\langle\hat{O}\rangle^{2}_{i}$ is its variance, $i=\text{on}(1),\text{off}(0)$, and $M$ is the number of modes. By repeated measurements on many copies $M\gg 1$, the sum of the measurements approaches a Gaussian distribution and subsequently it is applied to a decision threshold $R_{\text{th}}$. Given two Gaussian distributions with independent signal-idler mode pairs ($M\gg 1$), the total error probability is minimized according to the decision threshold. Each error probability is derived as follows: $P(\text{off}|\text{on})=\frac{1}{2}\text{erfc}[\frac{R_{\text{th}}-MR_{1}}{\sqrt{2M\Delta R_{1}}}]$, and $P(\text{on}|\text{off})=\frac{1}{2}\text{erfc}[\frac{MR_{0}-R_{\text{th}}}{\sqrt{2M\Delta R_{0}}}]$. At $R_{\text{th}}=\frac{M(R_{0}\sqrt{\Delta R_{1}}+R_{1}\sqrt{\Delta R_{0}})}{\sqrt{\Delta R_{0}}+\sqrt{\Delta R_{1}}}$, the total error probability is minimized as $P^{(M)}_{\text{err}}=\frac{1}{2}[P(\text{off}|\text{on})+P(\text{on}|\text{off})]=\frac{1}{2}\text{erfc}[\sqrt{\text{SNR}^{(M)}}]$, where the complementary error function $\text{erfc}[z]=\frac{e^{-z^{2}}}{\sqrt{\pi}z}[1-\frac{1}{2(z^{2}+1)}+\frac{1}{4(z^{2}+1)(z^{2}+2)}-...]$. At $z\gg 1$, $\text{erfc}[z]\leq\frac{e^{-z^{2}}}{\sqrt{\pi}z}$ such that the minimum error probability is upper bounded by $e^{-\text{SNR}^{(M)}}$, where $\sqrt{\pi\text{SNR}^{(M)}}$ is ignorable compared to $e^{\text{SNR}^{(M)}}$. Thus, minimizing the error probability corresponds to maximizing the SNR which is explicitly given by

There are four known receivers, such as the phase conjugate (PC) receiver, the optical parametric amplifier (OPA) receiverGuha , the double homodyne receiverJo , and the heterodyne receiver on each modeSandbo . We exclude the sum frequency generation with feedfowardQuntao , which remains hard to implement due to its complicated structure requiring a sequence of nonlinear processes.

In this paper we propose an observable bound for Gaussian illumination, which maximizes the SNR with respect to mode-by-mode measurements. In the quantum regime, we consider a TMSV state which is described with a $4\times 4$ covariance matrix $V_{\text{SI}}=\langle[\hat{a}_{S}~{}\hat{a}_{I}~{}\hat{a}_{S}^{{\dagger}}~{}\hat{a}_{I}^{{\dagger}}]^{T}[\hat{a}_{S}^{{\dagger}}~{}\hat{a}_{I}^{{\dagger}}~{}\hat{a}_{S}~{}\hat{a}_{I}]\rangle$, where $S$ ($I$) represents the signal (idler) mode. In the classical regime, an input two-mode state is prepared by impinging a coherent or thermal state into a beam splitter, which is described with the first moment and the covariance matrix. The input states interact with a target which is represented by a low-reflectivity beam splitter, where the thermal noise effect is simulated by impinging a thermal state into the beam splitter. Since both input states and interaction processes are in the Gaussian regime, we describe the output state with the covariance matrix and first moment.

In QI, the signal mode is reflected from a target with reflectance $\kappa$ while the idler mode is kept ideally. The output covariance matrixGuha that represents target-on is given by

where $A=\kappa N_{S}+N_{B}$, $C=\sqrt{\kappa N_{S}(N_{S}+1)}$, and $N_{B}$ is the mean photon number of thermal noise. When the target is off, the covariance matrix becomes $V_{\text{SI}}(0)$. The covariance matrix $V_{\text{SI}}(\kappa)$ differs from the covariance matrix $V_{\text{SI}}(0)$ by the correlation elements $(\langle\hat{a}^{{\dagger}}_{S}\hat{a}^{{\dagger}}_{I}+\hat{a}_{S}\hat{a}_{I}\rangle$) and the mean photon number of the signal mode ($\langle\hat{a}^{{\dagger}}_{S}\hat{a}_{S}\rangle$). Thus, we propose an observable with a combination of the three elements and the mean photon number of the idler mode $\langle\hat{a}^{{\dagger}}_{I}\hat{a}_{I}\rangle$,

where $\alpha$ and $\beta$ are real values. The idler mode element is not known for any effect on the performance, and the other elements of the covariance matrix are useless due to only increasing the variance in the SNR. Previously, the PC receiverGuha measures the observable $\hat{O}_{\text{PC}}=\nu(\hat{a}^{{\dagger}}_{S}\hat{a}^{{\dagger}}_{I}+\hat{a}_{S}\hat{a}_{I})+\mu(\hat{a}_{I}\hat{a}^{{\dagger}}_{V}+\hat{a}_{V}\hat{a}^{{\dagger}}_{I})$, where $|\mu|^{2}-|\nu|^{2}=1$ and $\hat{a}_{V}$ is a vacuum state operator. The OPA receiverGuha measures the observable $\hat{O}_{\text{OPA}}=\sqrt{G(G-1)}(\hat{a}^{{\dagger}}_{S}\hat{a}^{{\dagger}}_{I}+\hat{a}_{S}\hat{a}_{I})+(G-1)\hat{a}_{S}\hat{a}^{{\dagger}}_{S}+G\hat{a}^{{\dagger}}_{I}\hat{a}_{I}$, where $G>1$ is a gain of the OPA. The double homodyne (DH) receiverJo measures the observable $\hat{O}_{\text{DH}}=(\hat{X}^{2}_{S}+\hat{P}^{2}_{I})=-(\hat{a}^{{\dagger}}_{S}\hat{a}^{{\dagger}}_{I}+\hat{a}_{S}\hat{a}_{I})+\hat{a}_{S}\hat{a}^{{\dagger}}_{S}+\hat{a}^{{\dagger}}_{I}\hat{a}_{I}$. Since the correlation elements had a fixed relation with the other elements in the previous receivers, it is worthwhile to investigate a general relation between the correlation elements and the other ones. Note that the OPA and double homodyne receivers include the vacuum noise as $\langle\hat{a}_{S}\hat{a}^{{\dagger}}_{S}\rangle=1+\langle\hat{a}^{{\dagger}}_{S}\hat{a}_{S}\rangle$ but Eq. (3) does not.

Here, we consider QI under not only constant thermal noise but also non constant thermal noise that corresponds to the passive signature case Stefano18 . In a table-top experiment, it is natural to think about the non constant thermal noise since a thermal noise is injected into a beam splitter that represents a target. When we put the initial mean value of the thermal noise being constant without any relation with the beam-splitting parameter, the output thermal noise is related to the beam splitting parameter after the interaction with the beam-splitter, resulting in nonconstant thermal noise.

In CI, a single-mode coherent state asymptotically attains its QCB by performing homodyne detectionTan , but it is not known if a classically correlated thermal state can approach its QCB under mode-by-mode measurements and how far it is from the coherent-state QCB. Instead of a single-mode input state, we consider a two-mode input state which is produced by impinging a thermal state into a beam splitter, resulting in a classically correlated thermal (CCT) state. The output covariance matrix that represents target-on is given by

where $B=\kappa N_{S}+N_{B},~{}D=\sqrt{\kappa N_{S}N_{I}}$. $N_{S}~{}(N_{I})$ is the mean photon number of the signal (idler) mode that is controlled by the beam-splitting ratio. The off-diagonal element $D$ produces a classical correlation. When the target is off, the covariance matrix becomes $C_{\text{SI}}(0)$. By comparing the covariance matrices of the target-on and target-off, we propose an observable $\hat{O}_{\text{off}}=\hat{a}^{{\dagger}}_{S}\hat{a}_{I}+\hat{a}^{{\dagger}}_{I}\hat{a}_{S}$, consisting of off-diagonal elements. The corresponding SNR is given by

where $y\equiv N_{I}+N_{B}(1+2N_{I})$. The SNR with the observable attains the QCB, as shown in Fig.6. Under a fixed $N_{S}$, the amount of the classical correlation is proportional to $N_{I}$, resulting in an enhanced SNR. However it cannot beat the coherent-state QCB. Replacing the thermal state by a coherent state, we also obtain $\text{SNR}^{(M)}_{\text{coh}}$ by removing the term $4\kappa N_{S}N_{I}$ in the denominator of Eq. (9). But the coherent state with the observable cannot attain its QCB. In Fig. 6, the coherent state with HD shows a higher SNR than the thermal state. The tendencies are maintained regardless of constant or nonconstant thermal noise. In the limit of $N_{S},\kappa\ll 1$ and $N_{B},N_{I}\gg 1$, the $\text{SNR}^{(M)}_{\text{th}}$ asymptotically approaches $\frac{M\kappa N_{S}}{4N_{B}}$, and its error probability becomes $P^{(M)}_{\text{err}}=\frac{1}{2}\text{erfc}[\sqrt{\frac{M\kappa N_{S}}{4N_{B}}}]\leq e^{-\frac{M\kappa N_{S}}{4N_{B}}}$, resulting in a coherent-state QCB.

In Gaussian illumination, we proposed bound observables with feasible measurement setups to maximize the SNR under constant and nonconstant thermal noises. Using the bound observables under the condition of $N_{S},\kappa\ll 1$ and $N_{B}\gg 1$, we showed that QI using entangled states cannot attain its QCB but CI using classically correlated states can attain its QCB. In QI, the SNR using the bound observable outperforms the SNRs using other observables for any number of $N_{S},~{}N_{B},~{}\kappa$. However, the bound observable measurement cannot be achieved by using linear optics with heterodyne detection due to the additional vacuum noise. In CI, the measurement setup consists of PNDM after combining the reflected and idler modes by a $50:50$ beam splitter. The SNR using the bound observable can asymptotically approach the coherent-state QCB, while the SNR with a classically correlated thermal state cannot beat the SNR with a coherent state.

Since the observables we considered do not include collective measurements, the receivers with the observables do not always approach the QCBs. Note that, in the limit of $N_{S}\ll 1$, QI using our receiver asymptotically improves the error probability exponent by a factor of $2$ over the classical state QCB. It guarantees a half exponent of the TMSV state QCB by measuring the elements of the output covariance matrix. Since our bound observables belong to a class of local operations assisted with classical communicationCalsa10 ; Bandy11 , it cannot approach the QCB that requires collective protocolsQuntao . However, CI using our receiver asymptotically approaches the classical bound in the limit of $N_{S}\ll 1$ and $N_{I}\gg 1$.

In QI, we could not measure the bound observable with linear optics, HD, and HTD. As a further work, it remains an open question of how the bound observable can be experimentally measured with nonlinear systems.

For the SNR of Eq. (5) using the bound observable, we derive the corresponding optimal relation $|\beta|=\frac{(1+2N_{S})}{\sqrt{\kappa N_{S}(N_{S}+1)^{3}}}[f-\sqrt{f(f-\kappa(N_{S}+1))}]$, where $f=1+N_{S}+N_{B}+2N_{S}N_{B}$. It is shown in Fig. 7, which corresponds to the values of $|\beta|$ in Fig.1. With increasing $N_{S}$, the value of $|\beta|$ converges to $\sqrt{\kappa}$.

From the observables considered in Sec. II, we analytically derive the SNRs of three different receivers under constant thermal noise after the target interaction. For the PC receiver, the SNR is given by

which is always smaller than the SNR of Eq. (6). The parameters are designated as $\mu=\sqrt{2}$ and $\nu=1$Guha .

For the OPA receiver, the SNR is given by

where $q(\kappa)=\frac{G-1}{G}A(A+1)+\frac{G}{G-1}N_{S}(N_{S}+1)+\frac{C}{\sqrt{G(G-1)}}[(G-1)(4A+2)+G(4N_{S}+1)]+2C^{2}$. The gain is implementable as $G-1=7.4\times 10^{-5}$Zheshen . Since the additional denominator terms $q(\kappa)$ and $q(0)$ are much larger than the additional numerator term $\sqrt{\frac{G-1}{G}}\frac{\kappa N_{S}}{2}$, the SNR with the OPA is always smaller than the SNR of Eq. (6).

For the double homodyne (DH) receiver, the SNR is given by

where $p(\kappa)=A(A+1)+N_{S}(N_{S}+1)+2C^{2}-4C(A+N_{S}+1)$. Since all the additional terms diminish the SNR, the SNR with the DH is always smaller than the SNR of Eq. (6).

Initially, we assign the thermal mean photon number as $N_{B}$ which is natural in experiments. After the target interaction, the transmitted thermal noise becomes $(1-\kappa)N_{B}$, and then the matrix element $A$ is transformed as $\kappa N_{S}+N_{B}\rightarrow\kappa N_{S}+(1-\kappa)N_{B}$. For the SNR of Eq. (7) using the bound observable, we obtain the values of $\alpha$ and $\beta$ numerically, as shown in Fig. 8. At $N_{S}=0.01$, the $\alpha$ and $\beta$ correspond to $-0.54$ and $-9.08$, respectively. In Eqs. (13) and (14), due to the observable shapes, a component of the numerator is transformed as $\kappa N_{S}/2\rightarrow\kappa(N_{S}-N_{B})/2$.

After combining the reflected and idler modes by a $50:50$ beam splitter, $\hat{a}^{{\dagger}}_{S}\rightarrow\frac{1}{\sqrt{2}}(\hat{c}^{{\dagger}}+\hat{d}^{{\dagger}})$ and $\hat{a}^{{\dagger}}_{I}\rightarrow\frac{1}{\sqrt{2}}(\hat{d}^{{\dagger}}-\hat{c}^{{\dagger}})$, we obtain the output observable

which demands coincidence measurements on the position and momentum operators, together with the square of each operator. The mean value of a square quadrature operator is obtained by calculating the squared outcomes of HD, $\langle\hat{X}^{2}(\phi)\rangle=\int^{\infty}_{-\infty}dxx^{2}P(x,\phi)$, where the marginal distribution $P(x,\phi)$ is obtained by repeated measurements.

However it is unavoidable to produce additional vacuum noise after the HTD. Let us see the observable $\hat{M}_{prs}(0,0)=\frac{1}{2}[(\hat{X}^{2}_{d}-\hat{P}^{2}_{d})-(\hat{X}^{2}_{c}-\hat{P}^{2}_{c})]$ that corresponds to the nearly bound observable. As shown in Fig. 4 (a), an additional vacuum noise is included into the observable by the transformation of $\hat{X}_{d(c)}\rightarrow\frac{1}{\sqrt{2}}(\hat{X}_{d(c)}+\hat{X}_{d(c),v})$ and $\hat{P}_{d(c)}\rightarrow\frac{1}{\sqrt{2}}(\hat{P}_{d(c)}-\hat{P}_{d(c),v})$, where $\hat{X}_{d(c),v}$ and $\hat{P}_{d(c),v}$ belong to the vacuum noise. Although we measure $\hat{X}^{2}_{d(c)}$ and $\hat{P}^{2}_{d(c)}$ by heterodyne detection on the output signal mode, due to the vacuum noise, the observable is transformed into

which is applied to the output $c$ and $d$ modes, as shown in Fig.4 (b). Then, the mean and variance of Eq. (16) are given by

The output variance of Eq. (6) is enlarged as $\Delta^{2}O_{\kappa}+4[1+N_{B}+(1+\kappa)N_{S}]$, which is close to the output variance of the bound observable including the vacuum noise.

On the other hand, we directly measure the observable $\hat{O}_{\text{SI}}=\hat{a}^{{\dagger}}_{S}\hat{a}^{{\dagger}}_{I}+\hat{a}_{S}\hat{a}_{I}$ by performing HTD on the reflected and idler modes separately. After including the vacuum noise, the corresponding mean and variance are given by

The corresponding output variance is $\Delta^{2}O_{\kappa}+[1+N_{B}+(1+\kappa)N_{S}]$.