Quantum illumination with three-mode Gaussian state

The quantum illumination is examined by making use of the three-mode maximally entangled Gaussian state, which involves one signal and two idler beams. It is shown that the quantum Bhattacharyya bound between ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} (state for target absence) and σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} (state for target presence) is less than the previous result derived by two-mode Gaussian state when NS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_S$$\end{document}, average photon number per signal, is less than 0.295. This indicates that the quantum illumination with three-mode Gaussian state gives less error probability compared to that with two-mode Gaussian state when NS<0.295\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_S < 0.295$$\end{document}.


I. INTRODUCTION
As IC (integrated circuit) becomes smaller and smaller in modern classical technology, the effect of quantum mechanics becomes prominent more and more.As a result, quantum technology (technology based on quantum mechanics and quantum information theories [1]) becomes important more and more recently.The representative constructed by quantum technology is a quantum computer [2], which was realized recently by making use of superconducting qubits.In quantum information processing quantum entanglement [1,3,4] plays an important role as a physical resource.It is used in various quantum information processing, such as quantum teleportation [5,6], superdense coding [7], quantum cloning [8], quantum cryptography [9,10], quantum metrology [11], and quantum computer [2,12,13].
Few years ago the entanglement-assisted target detection protocol called the quantum illumination [14,15] and its experimental realization [16][17][18][19][20] were explored.Quantum illumination is a protocol which takes advantage of quantum entanglement to detect a low reflective object embedded in a noisy thermal bath.The typical quantum illumination can be described as follows.The transmitter generates two entangled photons called signal (S) and idler (I) modes.The S-mode photon is used to interrogate the unknown object hidden in the background.After receiving photon from a target region, joint quantum measurement for returned light and I-mode photon is performed to decide absence or presence of target.
The most surprising result of the quantum illumination is the fact that the error probability for the detection is drastically lowed even if the initial entanglement between S and I modes disappears due to decoherence.To show more concretely let us briefly review Ref. [15], where the quantum illumination was explored with two-mode Gaussian state.In the paper the initial state is chosen as a maximally entangled state as a form where N S is the average photon number per signal mode.This is a zero-mean Gaussian state whose covariance matrix is where S ≡ 2N S + 1 and C q = 2 N S (1 + N S ).Even though |ψ SI is maximally entangled, it was shown that ρ (state for target absence) and σ (state for target presence) are separable due to entanglement-breaking noise.The most important result of this paper is as follows.
Let N B and κ be average photon number of background thermal state and reflectivity from a target respectively.Let us assume that background contribution is very strong (N B 1) compared to the signal (N B N S ).We also assume that the reflectivity from a target is extremely small (κ 1).Then, the quantum Bhattacharyya (QB) bound between ρ and σ is shown to be where M is a number of identical copies of ρ and σ, and γ 2 is The last equality in Eq. ( 4) is valid only when N S 1.This is important because the QB bound for the single-mode coherent-state is When N S 1, the difference of Eq. (3) from Eq. ( 5) is a missing of factor 4 in the exponent.
This implies that the quantum illumination with two-mode Gaussian state gives much less error probability compared to that with the classical coherent state.The quantum illumination with two-mode Gaussian state was extended to the asymmetric Gaussian hypothesis testing [21,22].Also, the quantum illumination with non-Gaussian initial state generated by photon subtraction and addition was also discussed [23,24].
As emphasized above, the entanglement's benefit survives even though there is no entanglement in ρ and σ due to entanglement-breaking noise.Then, it is natural to ask how entanglement's benefit is enhanced when partial entanglement of the initial state survives in ρ and σ.In order to examine this issue we consider the same physical situation in this paper by making use of the three-mode Gaussian state (one signal (S) and two idler modes (I 1 , I 2 )).Thus, there are three biparties (S, I 1 + I 2 ), (I 1 , S + I 2 ), (I 2 , S + I 1 ).As we will show in the following, the states ρ and σ derived from the three-mode maximally entangled state still have zero bipartite entanglement for first biparty while non-zero for the remaining parties.In this reason the noise discussed in our quantum illumination scheme is not com- pletely entanglement-breaking, because it survives between two idler beams.Experimental construction of the three-mode states was recently discussed in Ref. [25].

II. MAIN RESULT
The final and main result of this paper is that the QB bound for our case becomes where when N S 1.Thus, this QB bounds is slightly different from Eq. (3) in the second order of exponent.As a result, P QB is slightly less than P QB .With relaxing the criterion N S 1, one can compute the ratio γ 3 /γ 2 , which is plotted in Fig. 1.Fig. 1 indicates that γ 3 is larger than γ 2 in the region 0 ≤ N S ≤ 0.295.This indicates that the quantum illumination with three-mode Gaussian state gives less error probability compared to that with two-mode Gaussian state when N S < 0.295.
In order to derive Eq. ( 6) we start with a state of signal and two idler modes.We assume that this state is a zero-mean three-mode Gaussian state whose covariance matrix is Λ (3) where S = 2N S + 1.The subscript "SI 1 I 2 " stands for signal, idler 1, and idler 2 respectively.

Since any cubic equation can be analytically solved in principle, C
(3) q can be completely determined.Although it is hard to express C (3) q explicitly due to its lengthy expression, one can show that it has following limits: When C = C q , the composite system becomes pure and maximally entangled state between the signal and idler modes.When 0 , the composite state becomes separable.
Let ρ and σ be quantum states for target absence and target presence respectively.Both are the zero-mean Gaussian states.Since, for ρ, the annihilation operator for the return from the target region should be âR = âB , where âB is the annihilation operator for a thermal state with average photon number N B , its covariance matrix can be written in a form: where B = 2N B + 1.It can be written as Λ ρ = S ρ 3 j=1 α j 1 1 2 S T ρ , where the symplectic eigenvalues are The symplecic transformation S ρ is where q , the state ρ has non-zero logarithmic negativity E N = − log 2 S − C (3) q in the bipartition I 1 + (S, I 2 ) and I 2 + (S, I 1 ).However, it is zero in the party S + (I 1 , I 2 ).
For σ the return-mode's annihilation operator would be âR = √ κâ S + √ 1 − κâ B , where âB is an annihilation operator for a thermal state with average photon number N B /(1 − κ).
We assume κ 1. Combining all of the facts, one can deduce that the covariance matrix for σ is where A = 2κN S + B. The symplectic eigenvalues of Λ σ are where The 2 × 2 matrices X j and Y j are and In Eq. ( 16) µ 1,± and µ 2,± are given by µ These relations are frequently used in the calculation of the QB bound.The state σ has non-zero logarithmic negativity in the bipartition I 1 + (S, I 2 ) and I 2 + (S, I 1 ), whose explicit expression is too complicated to express it explicitly.Like a ρ, however, it is zero in the party S + (I 1 , I 2 ) In order to accomplish the quantum illumination processing, we should choose the one of the null hypothesis H 0 (target absent) or the alternative hypothesis H 1 (target present).
Then, the average error probability is where P (H 0 ) and P (H 1 ) are the prior probabilities associated with the two hypotheses.We assume P (H 0 ) = P (H 1 ) = 1/2 for simplicity.Therefore, the minimization of P E naturally requires the optimal discrimination of ρ and σ.If we have M identical copies of ρ and σ, the optimal discrimination scheme presented in Ref. [26,27] yields the minimal error probability P min E in a form where ||A|| 1 = Tr √ A † A denotes the trace norm of A. However, the computation of the trace norm in Eq. ( 19) seems to be highly tedious for large M .Also, it is difficult to imagine the large M behavior of the minimal error probability from Eq. (19).In order to overcome these difficulties the quantum Chernoff bound was considered [28,29].The quantum Chernoff bound P QC of ρ and σ is defined as This is a tight upper bound of P min E , i.e., P min E ≤ P QC .This bound was analytically computed in several simple quantum systems [29].However, the computation of the optimal s, which minimizes Tr [ρ s σ 1−s ], is in general highly tedious.
The Chernoff bound for general n-mode Gaussian states ρ and σ can be expressed as follows.First we define Let the mean displacement vector of ρ and σ be xρ and xσ , and the corresponding covariance matrices are Λρ = S ρ n j=1 α j 1 1 2 S T ρ and Λσ = S σ n j=1 β j 1 1 2 S T σ .Now, we define V ρ (s) = S ρ n j=1 Λ s (α j ) 1 1 2 S T ρ and V σ (s) = S σ n j=1 Λ s (β j ) 1 1 2 S T σ .Then, the Chernoff bound for ρ and σ [30] is expressed as where In Eq. ( 23) d = xρ − xσ and Now, let us return to our case.Since d = 0 and n = 3 for our case, the quantum Chernoff bound for ρ and σ can be written as In order to understand this phenomenon more clearly, it seems to be necessary to examine the N -mode Gaussian illumination.
One can examine the advantage of the three-mode Gaussian approach in the asymmetric Gaussian hypothesis testing.Also, it is of interest to examine the effect of the Gaussian operations such as squeezing operation in quantum Gaussian illumination scheme.We hope to revisit these issues in the future.