MIMO Terahertz Quantum Key Distribution Under Restricted Eavesdropping

Quantum key distribution (QKD) can provide unconditional security to next-generation communication networks guaranteed by the laws of quantum physics. This article studies the secret key rate (SKR) of a continuous variable QKD (CV-QKD) system using multiple-input multiple-output (MIMO) transmission and operating at terahertz (THz) frequencies. Distinct from previous works, we consider a practical “restricted” eavesdropping scenario in which Eve can collect only a fraction of photons lost in the environment. We propose a system model for the MIMO THz CV-QKD system that accounts for restricted eavesdropping via a lossy wireless channel between Alice and Eve. We derive for this system new SKR expressions for both coherent-state-based and squeezed-state-based CV-QKD protocols. Our results show that previous analysis assuming unrestricted eavesdropping leads to overly pessimistic SKRs, and that in practice, the achievable SKR can be significantly increased under restricted eavesdropping. The increase in the SKR is quantified by the simplified SKR expansions derived in this article. Our results also reveal that squeezing is beneficial for improving the SKR only for unrestricted eavesdropping. However, in a practical setting with restricted eavesdropping, increased squeezing leads to a reduction in the SKR.


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Transactions on IEEE Kundu et  Rivest-Shamir-Adleman (RSA), advanced encryption standard (AES), and Diffie-Hellman (DH), whose security relies on a computationally hard problem, QKD is a hardware solution whose security is guaranteed by the laws of quantum physics [29], [30]. The rapid advancement in quantum computing poses a threat to the security of classical encryption algorithms, such as RSA/AES/DH, since the prime factorization problem and discrete logarithm problem can be solved in polynomial time by running Shor's algorithm on a quantum processor [31], [32], [33].
QKD is a promising quantum secure technology developed for distributing secret keys between two legitimate users, say Alice and Bob, in the presence of eavesdroppers [24]. Any eavesdropping attack can be detected by Alice and Bob with provable guarantees. The eavesdropper Eve tries to steal the key information between Alice and Bob by intercepting their quantum communication, injecting entangled quantum states, and then carrying out any general operation allowed by the laws of quantum physics. Quantum secure keys distributed by QKD can be used for physical layer security in a one-time-pad-based encryption scheme, as well as for higher layer symmetric key encryption schemes. Hence, QKD is a promising technology for quantum secure data transmission in B5G/6G wireless networks [34], [35], [36], [37].
There are two main categories of QKD protocols depending on the type of quantum states used. These are discrete variable-QKD (DV-QKD) and continuous variable-QKD (CV-QKD) schemes [24], [29]. In DV-QKD, the key information is encoded in the polarization of photons [38], whereas in CV-QKD, the key information is encoded in the quadratures of the continuous-variable quantum states [39], [40], [41], [42]. The security of DV-QKD is guaranteed by the no-cloning theorem, whereas the security of CV-QKD is guaranteed by Heisenberg's uncertainty principle. Since DV-QKD requires single-photon sources or weak coherent sources for encoding the qubits and single photon detectors at the receiver, it is difficult to integrate DV-QKD with the current telecommunication infrastructure. On the other hand, CV-QKD requires coherent optical sources and homodyne/heterodyne detectors, which are compatible with the classical coherent optical communication technology. Therefore, it is expected that CV-QKD could be more easily integrated with current and upcoming B5G/6G telecommunication systems [30], [33]. The terahertz (THz) frequency spectrum offers a number of advantages over the optical frequencies, for example, mobility of users, less delicate pointing, immunity to ambient light, cloud, and fog [43], [44], [45], [46], [47], [48], [49]. Therefore, recent works have investigated the feasibility of CV-QKD systems operating at THz frequencies [50], [51], [52], [53], [54], [55]. Moreover, the THz band is also a potential candidate for B5G/6G wireless systems [43], [44], [45], [46], [47], [48], [49], [56], which further motivates the applicability of THz CV-QKD. Although in the classical communication literature THz band refers to the frequency range of 0.1-10 THz, for quantum communications higher frequencies upto 15 THz are required since the preparation thermal noise and atmospheric absorption losses are lower at these frequencies that can support CV-QKD [50], [51], [52]. THz CV-QKD can be implemented using bidirectional optical-to-THz converters [52]. In this article, we focus on THz CV-QKD systems for future B5G/6G wireless communication networks.
In order to guarantee unconditional security, a general QKD protocol assumes that Eve has unlimited computational power and can carry out any operation allowed by the laws of quantum mechanics [24]. Furthermore, it is assumed that Eve has full control over the quantum states transmitted by Alice and can carry out any joint quantum operation on them. In CV-QKD protocol where the quantum channel is modeled using a beamsplitter, the general eavesdropping model assumes that Eve has control over the entire environment; i.e., Eve can collect all photons that are lost in the environment during propagation from Alice to Bob [24]. This unrestricted eavesdropping is a very pessimistic assumption, and in practice, Eve can collect only a fraction of the lost photons due to the lossy wireless link between Alice and Eve. This is the restricted eavesdropping scenario where Eve does not have control over the entire quantum channel and can collect only a fraction of photons lost in the environment. Some recent works have studied the achievable secret key rate (SKR) under restricted eavesdropping for single-input-single-output (SISO) CV-QKD systems [57], [58]. The authors of [57] incorporated the effect of the lossy quantum channel between Eve and the main channel, and derived upper bounds on the SKR using the relative entropy of entanglement for a SISO CV-QKD system. The authors of [58] considered a different restricted eavesdropping scenario where Eve has an imperfect quantum memory, and analyzed the SKR under an optimal hybrid attack by Eve.
This article extends our previous work on multiple-inputmultiple-output (MIMO) CV-QKD [50], [51], and analyzes the SKR under restricted eavesdropping by incorporating the effect of the channel transmittance of the wireless lossy link between Alice and Eve. The analysis of practically achievable SKR under restricted eavesdropping is important for practical MIMO THz CV-QKD system deployment. The following points summarize the main contributions of this article.

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Transactions on IEEE to unrestricted eavesdropping considered previously in [50] and [51]. 4) We demonstrate that squeezing is beneficial to improve the SKR only for the unrestricted eavesdropping model.
Our results indicate that unlike the unrestricted eavesdropping model, in the restricted eavesdropping model positive SKRs are achievable for the DR scheme even when the transmittance of the main channel is less than 0.5, due to the lossy channel between Alice and Eve. This is important in practice since the transmittances of the main channel are generally expected to be less than 0.5 at THz frequencies due to the significant path loss and atmospheric-absorption loss at such frequencies. Our simulation results show that the SKR improves significantly in the restricted eavesdropping scenario where Eve can collect only a fraction of the lost photons. Furthermore, our results reveal that in a restricted eavesdropping setting, squeezing does not help to improve the SKR, indicating that coherent states are the preferred quantum states to be used.
The rest of this article is organized as follows. Section II describes the system model of the proposed MIMO CV-QKD scheme with restricted eavesdropping. Sections III and IV present the SKR analysis for coherent-state-based and squeezed-state-based CV-QKD protocols, respectively. Section V provides the numerical results. Finally, Section VI concludes this article.
Notation: Boldface upper case (A) and lower case (a) letters denote matrices and vectors, respectively. A † and A T denote the conjugate transpose and transpose of a matrix A, respectively. The inverse and determinant of a matrix A are denoted as A −1 and det(A), respectively. A real Gaussian distribution with mean μ and variance σ 2 is denoted as N (μ, σ 2 ). The expectation and variance of a random variable X are denoted as E{X} and V {X}, respectively. The annihilation and creation operators of a quantized electromagnetic field are denoted asâ,â † , respectively. Furthermore, [x] + = max(0, x) denotes the positive part of x.

II. SYSTEM MODEL
We consider a quantum communication system operating at THz frequencies, where Alice transmits Gaussian coherent states to Bob for establishing a quantum secure key. The quantum secure key is extracted by carrying out postprocessing over an additional classical authenticated channel. Similar to [50], we consider a MIMO scenario where Alice has N t transmit antennas and Bob has N r receive antennas, and the MIMO wireless channel between them is represented by H ∈ C N r ×N t . The MIMO THz channel model is given by [49], [50], [51] where f c denotes the frequency of the carrier signal, M denotes the total number of multipath components, γ m models the path-loss of the mth multipath component given by Alice and Bob use the spatial multiplexing and beamforming capability of the MIMO CV-QKD system by employing singular value decomposition (SVD)-based transmit beamforming at Alice and receive combining at Bob. Alice generates N t coherent states using Gaussian modulation, which are then transmitted from the N t transmit antennas to Bob [50], [51]. To steal the secret key information, Eve introduces Gaussian modes during the transmission. We assume that Eve uses a Gaussian entangling attack where she generates a pair of two mode squeezed vacuum states (TMSV) for each transmit mode of Alice. Eve keeps one of the modes in her quantum memory and mixes the other mode with the incoming coherent state from Alice. For a 2 × 2 MIMO CV-QKD system, the effective MIMO channel can be pictorially depicted, as shown in Fig. 1. In Fig. 1, B η is a 2 × 2 matrix that relates the annihilation operators at the input and output modes in a 2-port beamsplitter with transmissivity η, given by Consider H = U V † as the SVD of the MIMO channel. Alice uses V as the beamforming matrix and Bob uses U † as the combining matrix, as shown in Fig. 1. In order to steal the key information, Eve generates two pairs of TMSV {e 1 , E 1 } and {e 2 , E 2 }, of which the first modes (e 1 , e 2 ) are stored in Eve's quantum memory, whereas the other two modes (E 1 , E 2 ) are mixed with the modes transmitted by Alice. In contrast to the unrestricted eavesdropping model (worst case scenario), the output modes E 1 , E 2 are not accessible to Eve. Instead, E 1 , E 2 get mixed with environmental thermal vacuum modes (v 1 , v 2 ) on a beam-splitter of transmissivity κ, as depicted in Fig. 1. Note that κ models the fraction of photons (lost during the transmission from Alice to Bob) that are accessible to Eve. Therefore, only the output modes E 1 , E 2 are accessible to Eve for joint measurement with the stored ancilla modes (e 1 , e 2 ) for carrying out eavesdropping. In a practical setting, κ can represent the channel transmittance of the quantum wiretap channel from Alice to Eve. In the beamsplitter model, the first input mode is the signal mode and the second input mode is the noise introduced by Eve (E i ) or the environmental thermal mode (v i ). Therefore, the beamsplitter matrices in the top part of Fig  2 × 2 MIMO CV-QKD system depicted in Fig. 1 can be extended for an arbitrary N r × N t MIMO model, since an arbitrary M × M unitary matrix can be represented as a mesh of interconnected 2-port beamsplitters, as shown in [59].
After using SVD-based transmit beamforming and receive combining, the effective MIMO channel between Alice and Bob decomposes into parallel SISO bosonic thermal channels. The annihilation operators of the input-output modes at Alice (â A,i ) and Bob (â B,i ) for the ith parallel channel are related aŝ where the ith nonzero singular value of H is denoted by √ T i and the rank of the MIMO channel is denoted by r. The annihilation operator of the output mode accessible to Eve is given bŷ where κ models the transmissivity of the wireless link between Alice and Eve,â v,i is the annihilation operator of the vacuum thermal mode, andâ E ,i admitŝ Eve performs general quantum measurements (positive operator value measure) on the stored ancilla modes e i and the received modes E i in order to extract the maximum key information. The quadrature of the ancilla mode accessible to Eve is given by where X E ,i admits Here, X A,i is the quadrature of the coherent state transmitted by Alice, and X E,i is the quadrature of the TMSV state injected by Eve for stealing the key information. Note that X Z,i denotes either the real or imaginary quadrature of the Gaussian quantum state corresponding to mode Z. The variance of the Gaussian noise injected by Eve is shot-noise units (SNU), and the variance of the environment thermal mode, which arises due to the lossy bosonic channel between Alice and Eve, is given by

A. CHANNEL ESTIMATION
The input-output relation of the annihilation operators at Alice and Bob in (3) assumes the availability of perfect channel state information (CSI) at both ends. However, CSI is estimated in practice to realize the SVD-based beamforming and combining at Alice and Bob, respectively. Alice and Bob can carry out a channel estimation protocol prior to the key generation protocol as proposed in [51]. Alice transmits pilot symbols and Bob uses a least squares (LS) channel estimator to estimate the MIMO channel matrix H LS using [51, Eq. (8)].
The signal-to-noise ratio (SNR) during the channel estimation phase is defined as where γ 1 is the path loss of the line-of-sight component of the MIMO channel, V p is the pilot power in SNU, V 0 is the preparation thermal noise power in SNU, and σ 2 det is the detector noise in SNU. The estimated channel matrix is then fed back to Alice using a classical authenticated communication (CAC) channel. The CAC channel is also necessary for carrying out the information reconciliation and error correction steps of the QKD protocol.
Bob randomly performs homodyne measurement on the received mode and measures one of the quadratures to extract the quantum secure keys. Accounting for imperfect CSI and detector noise at Bob, the input-output relation of the quadratures between Alice and Bob obtained from homodyne measurement is given by [51] where T i denotes the ith nonzero singular value of H LS . Furthermore, n h,i denotes the additional noise term due to channel estimation error, which is distributed as where the matrix C h is given by [51,Eq. (21)], and n det,i denotes the detector noise distributed as Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.

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Transactions on IEEE a homodyne detector and σ 2 det = 2v el + 1 SNU for a heterodyne detector with v el being the electronic noise in SNU.
There can be two types of CV-QKD protocols depending on the type of Gaussian quantum states used by Alice: coherent-state-based and squeezed-state-based. We consider each of these protocols in the sections that follow.

III. COHERENT-STATE-BASED CV-QKD PROTOCOL
In this section, we describe the coherent-state-based CV-QKD protocol and analyze the SKR of the system with restricted eavesdropping by Eve. For the ith transmit antenna, Alice first generates two independent zero-mean Gaussian random variables x i , p i of variance V s , and then creates a displaced Gaussian coherent state |α i such that The coherent-state-based CV-QKD protocol can further be classified into two categories depending on the type of detection scheme used by Bob: homodyne detection to randomly measure one of the quadratures or heterodyne detection to simultaneously measure both quadratures (albeit with a higher detection noise) of the received quantum state. The homodyne detector gives one real-valued measurement outcome for each received quantum state, whereas the heterodyne detector gives two real-valued measurement outcomes, which can be used for generating the quantum secure keys. It was shown in [51] that at practical transmission distances, the SKR performance of the homodyne and heterodyne detection-based schemes are almost the same, due to the higher detector noise in the heterodyne case. Here, we focus on the homodyne detection scheme for the coherent-state-based CV-QKD protocol. After exchanging the quantum states, Alice and Bob carry out a sifting procedure where Bob declares over the CAC channel, which of the two quadratures was measured by Bob, and consequently Alice keeps either x i or p i for the ith mode depending on Bob's choice of the quadrature measurement. Therefore, Alice uses only one of the two random variables x i or p i for generating the secret key. After this, Alice and Bob carry out information reconciliation, classical error correction, and privacy amplification in order to extract the secret keys. In particular, there are two types of reconciliation schemes: (i) RR where Bob's data is used as a reference for classical error correction, and (ii) DR where Alice's data is used as a reference for classical error correction. We present the SKR of the coherent-state-based MIMO CV-QKD system with restricted eavesdropping for both RR and DR schemes in the following.

A. REVERSE RECONCILIATION
For the coherent-state-based CV-QKD protocol with RR, the SKR for the ith parallel SISO channel is given by [60] where I(X A,i : X B,i ) is the classical Shannon's mutual information between the random variables obtained from quadrature measurements by Alice and Bob, and χ (X B,i : e i E i ) is the information leaked to Eve, which can be upper bounded by the Holevo information between Bob's measurement outcome X B,i and the quantum state available to Eve ρ e i E i . Furthermore, β ∈ (0, 1) denotes the reconciliation efficiency, and T p , T c denote the pilot length and the channel coherence block length, respectively. Note that T c depends on the coherence bandwidth (B c ) and coherence time (τ c ) of the wireless channel, given by T c = τ c B c , and for the LS channel estimator the pilot length should satisfy T p ≥ N t .
Proposition 1: The total SKR of the MIMO CV-QKD system in RR for the coherent-state-based protocol with restricted eavesdropping is given by (10) shown at the bottom of the page. In (10) is the function defined in (31), V s is the variance of the signal quadrature transmitted by Alice, and V 0 , W are the variances of the quadratures of the preparation thermal noise and Eve's injected noise, respectively. Furthermore, ζ i is the quantum correlations between the mode E i accessible to Eve and Bob's measurement outcome given by Proof: See Appendix A. Now, we present a simplified expression for the SKR in order to quantify the effect of the restricted eavesdropping parameter κ on the SKR.
Proposition 2: For small κ (i.e., κ → 0), the SKR expression can be approximated as (11), shown at the bottom of the next page.
Proof: The proof follows from a first-order Taylor series expansion of (10) with respect to κ.  (11), we observe that κ introduces a linear penalty on the SKR. The information leaked to Eve, which is given by the Holevo information, increases as κ → 1. Therefore, the SKR decreases and attains the lowest value for unrestricted eavesdropping with κ = 1.

B. DIRECT RECONCILIATION
It is known that under the pessimistic assumption of unrestricted eavesdropping, positive SKR can be achieved in DR only under the constraint of T i > 0.5 [60]. However, it is important to investigate if positive SKR could potentially be achieved in DR with T i < 0.5 under a restricted eavesdropping scenario where only a fraction of the photons lost in the environment are accessible to Eve. In this section, we derive the SKR for the MIMO CV-QKD system with the DR scheme. For coherent-state-based CV-QKD protocols with DR, the SKR of the ith parallel SISO channel is given by [60] where I(X A,i : X B,i ) is given by (18) in Appendix A, and χ (X A,i : e i E i ) denotes the maximum information leaked to Eve given Alice's quadrature measurement X A,i . Proposition 3: The total SKR of the MIMO CV-QKD system in DR for the coherent-state-based protocol with restricted eavesdropping is given by (13), shown at the bottom of the page, where V E i |A = i (W, V 0 ).
Proof: See Appendix B.
As before, we now present a simplified expression for the SKR in DR in order to quantify the effect of the restricted eavesdropping parameter κ on the SKR.
Proposition 4: For small κ (κ → 0) the SKR in DR can be approximated by Proof: The proof follows from a first-order Taylor series expansion of (13) with respect to κ.
Similar to RR, it can be observed from (14) that the SKR for DR also decreases linearly as κ increases.
Remark: It is known that for the unrestricted eavesdropping model, positive SKR can be achieved only forT i > 0.5, which is generally not valid at THz frequencies due to significant path-loss and atmospheric absorption loss. However, for the restricted eavesdropping case the SKR can be positive even forT i > 0.5. This is because the information lost to Eve is reduced for κ < 1, as is evident from (14). For the restricted eavesdropping case, there exists a threshold κ max above which positive SKR cannot be achieved in DR. This threshold value for κ can be determined from the approximate SKR expression in (14) by constraining the first term in the summation to be positive. Therefore, κ max can be approximated as whereT 1 is the maximum eigenvalue of H † LS H LS .

IV. SQUEEZED-STATE-BASED CV-QKD PROTOCOL
In the squeezed-state-based CV-QKD protocol, Alice uses a displaced squeezed quantum state. More specifically, for the ith transmit antenna mode, Alice first draws a zero-mean Gaussian random variable a i of variance V s . Alice prepares a squeezed vacuum state by applying a squeezing operation on one of the randomly chosen quadratures. Alice then displaces the squeezed quadrature by a i [30]. This displaced squeezed quantum state is then transmitted from the ith transmit antenna. The variance of the squeezed quadrature in which the key information is encoded by Alice is given by where s is the squeezing parameter and V 0 is the variance of the thermal noise [61]. Note that as compared to the coherent-state-based CV-QKD protocol, in this protocol, the preparation noise is reduced for the squeezed quadrature. At the receiving   end, Bob can perform either homodyne or heterodyne detection, which gives two categories of squeezed-state-based CV-QKD protocols.

A. HOMODYNE DETECTION
With homodyne detection, Bob randomly measures one of the quadratures to obtain one real-valued measurement outcome for each received quantum state. During the sifting process, Alice declares over the CAC channel, which of the two quadratures was squeezed for each of the transmitted quantum states. Bob then stores only those measurement outcomes for which the measurement bases matches Alice's squeezing basis. On average, only 50% of the times the measurement basis of Bob and Alice's squeezing basis will match. Therefore, only 50% of the measurement outcomes at Bob can be used for extracting the secret key. Hence, the SKR of the squeezed-state-based CV-QKD protocol with homodyne detection scheme for RR and DR are given by where R r c MIMO and R d c MIMO are evaluated from (10) and (13), respectively, by replacing V a → V a and V 0 → V 0 since the preparation noise variance is reduced due to squeezing.
Remark: There are two competing factors that affect the SKR in the squeezed-state-based CV-QKD protocol with homodyne detection. On the one hand, the reduced noise variance (V 0 ) in the squeezed state increases the first term in (11) (i.e., the mutual information between Alice and Bob). For the second term, the numerator is a quadratic function of V 0 (since V a = V s + V 0 ), and the denominator is a linear function of V 0 : therefore, the second term decreases with decreasing V 0 . On the other hand, there is a factor of 0.5 in the SKR expression of the squeezed-state-based CV-QKD protocol, since only 50% of the measurement outcomes could be utilized for extracting the keys. All the measurements cannot be used for extracting the secret key since the noise variance on the orthogonal quadrature will increase due to Heisenberg's uncertainty principle.

B. HETERODYNE DETECTION
With heterodyne detection, Bob measures both quadratures of the received quantum state. As before, during the sifting process, Alice declares which quadrature was squeezed by her. For each quantum state, Bob now has two real-valued measurement outcomes; however, Bob keeps only one of the measurement outcomes which corresponds to the squeezed quadrature of Alice. Thus, in contrast to the homodyne case, the measurement outcomes from all the received quantum states are used for extracting the secret key. However, in this case the detector noise is increased to σ 2 det = 1 + 2v el . Therefore, the SKR for the squeezed-state-based CV-QKD protocol with heterodyne detection for both RR and DR can be obtained from the SKR expressions of the coherent-statebased CV-QKD protocol derived in the previous section [i.e., (10), (13)] by replacing V a → V a , V 0 → V 0 , and using σ 2 det = 1 + 2v el in the SKR expressions.
Remark: There are again two competing factors which affect the SKR. On one hand the preparation noise, V 0 , reduces due to squeezing, however, on the other hand, the variance of the detector noise, σ 2 det , increases since a heterodyne detector is used to measure both quadratures simultaneously. The effect of squeezing on the SKR of the MIMO CV-QKD scheme with restricted eavesdropping is numerically investigated in the section that follows.

A. COHERENT-STATE-BASED CV-QKD PROTOCOL
We consider a simulation scenario similar to [51] and study the SKR at a frequency of 15 THz. We assume a VOLUME 4, 2023 Engineering uantum  propagation scenario with a dominant line of sight (LoS) path such that L = 1 [51]. The general system parameters (applicable to all the figures) are: V s = 10, W = 1, W = 1, T e = 296 K [60], v el = 0.01, β = 0.95, T p = N t + 500, T c = 5 × 10 5 , = 20 dB, and the antenna gain is 30 dBi. Fig. 2 shows the plot of SKR of the MIMO CV-QKD system in RR R r c MIMO (in bits/channel use) as a function of transmission distance with N t = N r = 64 for two types of eavesdropper models, the unrestricted eavesdropping model from [51] and the restricted eavesdropping model proposed in this article with κ = 0.1. It can be observed that the SKR under restricted eavesdropping is orders of magnitude larger than the pessimistic unrestricted eavesdropping model of [51], where Eve has access to all the photons lost in the environment. This is due to the fact that the Holevo information leaked to Eve is reduced in restricted eavesdropping with κ < 1, as indicated by the simplified SKR expression in (11).
We also study how the SKR varies as a function of κ. Fig. 3 shows the plot of the SKR R r c MIMO as a function of κ with N t = N r = 64 and a fixed transmission distance of 50 m for both restricted and unrestricted eavesdropping models. SKR results for the restricted eavesdropping are plotted using the exact expression from (10) and the approximate expression from (11). It can be observed that the approximate SKR expression is accurate for small values of κ. It can also be observed that the SKR of the restricted eavesdropping model is higher than the SKR of the unrestricted eavesdropping, and the SKR with restricted eavesdropping approaches to that of the unrestricted eavesdropping as κ → 1. This observation is in agreement with the analytical SKR approximation in (11).
Next, we evaluate the SKR of the MIMO CV-QKD system obtained from the DR scheme and compare it with that of the RR scheme under the restricted eavesdropping model. Fig. 4(a) shows the plots of the SKR of DR and RR as a function of transmission distance with restricted eavesdropping (κ = 10 −5 ) for two MIMO configurations. SKR results are shown based on the exact expressions from (10), (13) and the approximate expressions from (11), (14). It can be observed that the SKR obtained from the approximate expressions are close to that of the exact SKR expressions for lower transmission distances. It can also be observed that unlike the unrestricted eavesdropping model of [50], positive SKR is achievable in DR for the restricted eavesdropping model at practical transmission distances for which T i < 0.5. This is due to the fact that the Holevo information leaked to Eve is reduced in the restricted eavesdropping model with κ < 1, as indicated by the simplified SKR expression in (14). Furthermore, it can be observed that at lower transmission distances the SKR obtained from DR and RR are almost the same, whereas at larger transmission distances the SKR obtained from RR is higher. Therefore, in practice, the RR scheme should be used for MIMO CV-QKD. We also evaluate the SKR obtained from DR and RR as a function of κ. Fig. 4(b) shows the plots of the SKR of DR and RR as a function of κ for two MIMO configurations. It can be observed that the SKR obtained from the approximate expressions are close to those of the exact SKR expressions for small κ (κ → 0). It can also be observed that the SKR decreases as κ increases for both DR and RR. However, the DR scheme is more sensitive to κ, since the SKR rapidly decreases and drops to zero as κ increases beyond a threshold value. This is due to the fact that the Holevo information leaked to Eve is higher in DR as compared to RR. The reason for this is that in DR the quantum state transmitted by Alice is accessible to Eve, whereas in RR the quantum state received by Bob is not accessible to Eve. From Fig. 4(b), it can be observed that in RR positive SKR can be achieved for all values of 0 < κ ≤ 1, however, for DR there is a threshold κ max above which positive SKR cannot be achieved. This threshold κ max depends on the channel transmittanceT i , which depends on the transmission distance and MIMO configuration. Fig. 5 shows the plot of κ max as a function of transmission distance for different MIMO configurations. Results are shown for both the approximate theoretical κ max obtained from (15), and the exact κ max obtained from simulations. It can be observed that the approximate κ max obtained from (15) is close to the exact κ max for lower values of κ max since the approximate SKR expression in (14) is valid for small values of κ (κ → 0). In general, it can be observed that both the curves follow the general trend that the threshold κ max decreases as the transmission increases. This is due to the fact that for a fixed MIMO configuration, the channel transmittance (T i ) of the ith parallel channel decreases with increasing distance due to path-loss and atmospheric-absorption loss, therefore, the threshold κ max should be lower in order to achieve positive  SKR in DR. However, a higher threshold κ max can be tolerated for larger MIMO configurations sinceT i increases with increasing N r , N t due to the beamforming gain, which can be observed from the vertical shift in the curves for different MIMO configurations.

B. SQUEEZED-STATE-BASED CV-QKD PROTOCOL
Now we compare the SKRs of the squeezed-state-based CV-QKD protocols. Fig. 6 shows the plots of SKR in RR as a function of distance for the squeezed-state-based CV-QKD protocol for unrestricted eavesdropping (κ = 1) and restricted eavesdropping with κ = 0.1, at different squeezing levels s. Note that s = 0 corresponds to the coherentstate-based CV-QKD protocol. Results are shown for the homodyne-detection-based protocol in the top row and for the heterodyne-detection-based protocol in the bottom row for a 64 × 64 MIMO configuration. It can be observed that squeezing leads to an improvement in the SKR for the unrestricted eavesdropping model (κ = 1) for both homodyne and heterodyne-detection-based protocols. However, in the case of restricted eavesdropping with κ = 0.1, squeezing leads to degradation in the SKR.
In the pessimistic scenario of unrestricted eavesdropping, the SKR is limited by the preparation thermal noise, therefore, the SKR improves as the preparation noise decreases with increased squeezing. In this case, the improvement in the SKR due to reduced preparation noise for squeezed state is overweighted by the decrease in SKR due to the factor of 0.5 (in homodyne detection) or the increased detector noise σ 2 det (in heterodyne detection), which leads to an overall improvement in the SKR for the squeezed-state-based CV-QKD protocols. However, for the restricted eavesdropping scenario, the SKR is already higher for the coherent-statebased CV-QKD protocol since Eve can access only a fraction (κ = 0.1) of the photons lost in the environment. In this case, the reduction in preparation thermal noise does not benefit in increasing the SKR since the factor of 0.5 (in homodyne detection) or the increased detector noise σ 2 det (in heterodyne detection) leads to a more deteriorating effect in the overall SKR. This can be intuitively understood as follows: as compared to the unrestricted eavesdropping model, in restricted eavesdropping the mutual information between Alice and Bob remains the same, whereas the information leaked to Eve (given by the Holevo information) is reduced. Therefore, for the squeezed-state-based CV-QKD protocol with homodyne detection the SKR reduces due to the overall factor of 0.5, whereas for the heterodyne detection the SKR reduces since the mutual information between Alice and Bob reduces due to the higher detector noise. Hence, it is beneficial to use a coherent-state-based CV-QKD protocol (which is also easier to implement) for the restricted eavesdropping scenario.
Next, we evaluate the SKR of the squeezed-state-based CV-QKD protocol as a function of κ in order to understand the range of κ values for which squeezing can improve the SKR of the CV-QKD system for the restricted eavesdropping model. Fig. 7 shows the plots of the SKR in RR as a function of κ with restricted eavesdropping at different squeezing levels s. Results are shown for both homodyne and heterodyne detection-based protocols for a (64 × 64) MIMO configuration at a fixed transmission distance of 20 m. It can be observed that there exists a threshold κ th below which squeezing leads to a deteriorating effect on the SKR and above this threshold the SKR improves with squeezing as compared to the coherent-state-based CV-QKD. It can also be observed that for the unrestricted eavesdropping model with κ = 1, squeezing always leads to an improvement in the SKR for both the homodyne and heterodyne detectionbased CV-QKD. The SKR results can be used for practical THz CV-QKD system deployment and to determine if a squeezing-based protocol should be used or not depending on what practical setting the CV-QKD protocol is designed to operate at. Our simulation results suggest that in a practical restricted eavesdropping scenario with κ much less than unity, no real gain in SKR can be achieved by using squeezed states. Moreover, it is much easier to generate coherent states as compared to squeezed states, therefore, in practice, coherent-state-based CV-QKD protocol should be used.

VI. CONCLUSION
This article analyzes the SKR of a MIMO CV-QKD system under a restricted eavesdropping scenario where Eve can collect only a fraction of photons lost in the environment. In a practical setting, Eve does not have control over the entire environment due to the presence of the lossy wireless link between Alice and Eve. We have presented a system model and derived new SKR expressions for a MIMO CV-QKD system that incorporates the effect of the channel transmittance of the Alice-Eve link. We have investigated the SKR of the system with both coherent-state-based and squeezed-state-based CV-QKD protocols. The SKR expressions reveal that the information leaked to Eve, given by the Holevo information, is reduced for the restricted eavesdropping scenario, which improves the achievable SKR of the system. Our simulation results show that under restricted eavesdropping the SKR improves by orders of magnitude as compared to the pessimistic scenario of unrestricted eavesdropping. Furthermore, our results reveal that squeezing is beneficial for improving the SKR mainly for the unrestricted eavesdropping model. In practical settings with restricted eavesdropping due to the lossy link between Alice and Eve, the SKR degrades as squeezing increases. Therefore, in practice, coherent-state-based CV-QKD protocols should be used, which are also easier to implement. Our results reveal that THz CV-QKD is a promising solution for quantum secure data transmission in future communication networks [62].
There are certain practical challenges that need to be overcome in order to implement the MIMO THz CV-QKD system investigated in this article. A reliable estimate of the MIMO channel is crucial for realizing the SVD-based transmit-receive beamforming. The LS-based channel estimation scheme considered in this article requires a high SNR during the channel estimation phase in order to reliably estimate the channel matrix for large MIMO systems. Therefore, THz sources with high signal power are necessary during the channel estimation phase. This could be realized with the technological advancement of frequency down-conversion of high-power laser sources to THz frequencies [63]. Therefore, significant advances in high-power THz sources and lownoise homodyne/heterodyne detectors are necessary for the implementation of MIMO THz CV-QKD in future communication systems. Alternatively, the transmitted pilot power could potentially be reduced by using compressive sensingbased channel estimation schemes, since the THz MIMO channel has a sparse representation in the angular domain due to the limited number of multipath components at THz frequencies. Therefore, efficient channel estimation schemes for MIMO THz channels and achievable SKR analysis are important research problems that should be investigated in future extensions of this work.

A. PROOF OF PROPOSITION 1
Using the input-output relation of the quadratures at Alice and Bob from (8), the classical Shannon's mutual information between Alice and Bob is given by is the variance of the signal quadrature transmitted by Alice, and V 0 and W are the variances of the quadratures of the preparation thermal noise and Eve's injected noise, respectively. Furthermore, note that σ 2 det = v el for the homodyne detector used in the coherentstate-based CV-QKD protocol.
The Holevo information between Eve and Bob is given by where S(ρ) is the von-Neumann entropy of the quantum state with density operator ρ given by S(ρ) = −Tr(ρ log ρ).
In (19), ρ e i E i is the density operator of Eve's state for the ith parallel channel and ρ e i E i |X B,i is the density operator of the conditional quantum state given Bob's measurement outcome X B,i . For Gaussian quantum states the von-Neumann entropy depends on the covariance matrix of the quantum states in terms of the symplectic eigenvalues [64]. Using (6), (7), and the covariance matrix of the TMSV state, the covariance matrix of the Gaussian quantum state ρ e i E i admits where with and I 2 being the 2 × 2 identity matrix. Furthermore where and Z is the Pauli-z matrix with entries Using the properties of two-mode Gaussian quantum states [33], [65], the symplectic eigenvalues of i E admit where i = det(A i ) + det(B i ) + 2det(C i ). After some algebra, we obtain (28) For large signal variance i.e., V s W , V s W , and V s V 0 , the symplectic eigenvalues can be approximated as The von-Neumann entropy S(ρ e i E i ) is given by where the function h(x) is defined as (31) Next, we find the covariance matrix of the conditional state ρ e i E i |X B,i . This is given by where i E is given by (20), V B,i is the variance of the output mode measured by Bob given by and = 1 0 0 0 is a projection matrix for homodyne measurement. Furthermore, D i is the matrix containing the quantum correlations between the modes available to Eve (e i , E i ), and Bob's measurement outcome X B,i , and is given by where and Substituting (20) and (34)- (37) in (32), we obtain where The symplectic eigenvalues of i E|B are given by where ϒ i = det(E i ) + det(G i ) + 2det(F i ). For large signal variance, the symplectic eigenvalues of the conditional covariance matrix can be approximated as The von Neumann entropy of Eve's conditional state in terms of the symplectic eigenvalues is given by Thus, the Holevo information in (19) can be evaluated by substituting (30), (42) in (19), and the SKR for the ith parallel channel R r c i can be evaluated by substituting (18), (19) in (9). Finally, the total SKR of the MIMO CV-QKD system in RR for the coherent-state-based protocol in (10) is obtained by summing the SKR of all the parallel SISO channels.

B. PROOF OF PROPOSITION 3
In DR, the classical Shannon's mutual information between the measurement outcomes at Alice and Bob I(X A,i : X B,i ) is same as that of the RR scheme given by (18). The maximum information leaked to Eve given Alice's quadrature measurement X A,i can be upper bounded by the Holevo information.
In DR, the Holevo information admits χ X A,i : e i E i = S(ρ e i E i ) − S(ρ e i E i |X A,i ) where S(ρ e i E i ) is given by (30), and S(ρ e i E i |X A,i ) denotes the conditional von Neumann entropy of Eve's state given Alice's measurement outcome X A,i . S(ρ e i E i |X A,i ) depends on the symplectic eigenvalues of the conditional covariance matrix. Using the relationship between the different quadrature modes from (6) to (7) and the covariance matrix of TMSV Gaussian state, the conditional covariance matrix admits where In (45), a i is given by (22), and b i is given by where V E i |A = (1 − T i )V 0 + T i W . Furthermore, the matrices B i and C i are given by (21) and (24) where ν i 5 , ν i 6 are the symplectic eigenvalues of i E|A given by where i = det(J i ) + det(B i ) + 2det(C i ). For large signal modulation, the symplectic eigenvalues can be simplified as In such a case, R d ci can be evaluated by substituting (18), (43) in (12). Finally, the total SKR of the MIMO CV-QKD system in DR for the coherent-state-based protocol in (13) is obtained by summing the SKR of all the parallel SISO channels.