RIS-Assisted MIMO CV-QKD at THz Frequencies: Channel Estimation and SKR Analysis

Debasish Ghose; George C. Alexandropoulos; Soumya P. Dash; Sushil Kumar

REVIEW 4 major objections 4 minor 60 references

RIS-Assisted MIMO CV-QKD at THz Frequencies: Channel Estimation and SKR Analysis

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that the secret key rate of an RIS-assisted MIMO CV-QKD link at THz frequencies, under least-squares channel estimation and a collective Gaussian attack, is given by a closed-form per-mode sum in Eq.

desk verdict New combination of RIS, MIMO THz CV-QKD, and LS channel estimation, but Eq. (44) relies on an unproven diagonalization of the estimation-error covariance, so the closed-form claim is premature. read the letter →

arxiv 2412.18771 v1 pith:O6ECZNS5 submitted 2024-12-25 cs.IT eess.SPmath.IT

Sushil Kumar , Soumya P. Dash , Debasish Ghose , George C. Alexandropoulos This is my paper · ORCID

classification cs.ITeess.SPmath.IT

keywords continuous-variablequantumkeydistributionreconfigurableintelligentsurfaceterahertzMIMOleast-squareschannelestimationsecretratecollectiveGaussianattackhomodynedetectionheterodyne

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to answer a practical question: how much secret key can an RIS-assisted MIMO terahertz link still generate when the channel is learned from pilots rather than assumed known. Its main claim is a closed-form secret key rate, Eq. (44), that takes the least-squares channel estimate, the extra noise it injects, detector noise, and a collective Gaussian eavesdropping attack into account. The formula splits the estimated MIMO channel into $r$ parallel modes, adds each mode's estimation-noise variance to the signal-to-noise ratio, and subtracts the corresponding Holevo information. A sympathetic reader would care because this gives a parameter-level handle on RIS element count, phase configuration, pilot power and length, and homodyne versus heterodyne detection, all in one expression. The paper's numerics use that expression to argue that the RIS keeps the key rate from falling at longer distances and that heterodyne detection is better at low pilot power.

What carries the argument

The argument is carried by the SVD of the least-squares channel estimate, $H_{RIS}^{LS} = U_{RISLS} \Sigma_{RISLS} V_{RISLS}^{\dagger}$, which splits the MIMO link into $r$ parallel Gaussian channels with gains $\beta_i$. The estimation-error covariance $C_{RIS} = \mathbb{E}[n_{RIS} n_{RIS}^{\dagger}] = \frac{2 V_a N_T}{V_p L_p} U_{RISLS}^{\dagger} \tilde{C}_n U_{RISLS}$ supplies the extra noise variance $\sigma_{RISi}^2 = 0.5 C_{RIS}(i,i)$ used in each parallel channel; the paper estimates $\tilde{C}_n$ by maximum likelihood from pilot residuals. The Holevo information terms are evaluated through symplectic eigenvalues $\lambda_{i1},\dots,\lambda_{i4}$ of Eve's covariance matrices, giving closed-form entropy expressions. This machinery turns a full MIMO quantum protocol into a sum over independent scalar Gaussian channels.

What would settle it

Compute the full multi-mode secret key rate from the joint Gaussian state using the entire estimated covariance $C_{RIS}$, including its off-diagonal entries, for a scenario where $C_{RIS}$ is strongly non-diagonal, and compare it with Eq. (44). If the difference is non-negligible, the closed-form sum is not the exact secret key rate. A Monte Carlo simulation of the LS pilot phase followed by key exchange would provide the same check.

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Extended reading notes

Core claim

The central claim is that Eq. (44) is a closed-form expression for the secret key rate of the RIS-assisted MIMO CV-QKD system under reverse reconciliation: $SKR_{RR}^{MIMO} = (1 - L_p/T_c) \sum_{i=1}^{r} \left( \eta \frac{d_m}{2} \log_2\left(1 + \frac{\beta_i V_s}{\beta_i V_o + (1-\beta_i)V_e + \sigma_d^2 + \sigma_{RISi}^2}\right) - h_o(\lambda_{i1}) - h_o(\lambda_{i2}) + h_o(\lambda_{i3}) + h_o(\lambda_{i4}) \right)$, with $\sigma_{RISi}^2 = 0.5 C_{RIS}(i,i)$ and $C_{RIS} = \frac{2 V_a N_T}{V_p L_p} U_{RISLS}^{\dagger} \tilde{C}_n U_{RISLS}$, where $\tilde{C}_n$ is the maximum-likelihood estimate of the pilot-noise covariance. The $\beta_i$ are the squared singular values of the LS-estimated channel, the $\lambda$'s are symplectic eigenvalues of Eve's covariance matrices, and $h_o(\cdot)$ is the Holevo entropy function. This extends the earlier MIMO THz CV-QKD analysis to the practical case of imperfect channel state information, using the estimated channel for both beamforming and key extraction.

Load-bearing premise

The formula assumes the $r$ parallel SVD modes are independent and that each mode's channel-estimation noise is fully described by the single diagonal variance $\sigma_{RISi}^2 = 0.5 C_{RIS}(i,i)$; if the covariance $C_{RIS}$ has significant off-diagonal correlations, summing per-mode rates does not give the exact total secret key rate.

Editorial extensions

If this is right

With Eq. (44), designers can compute the secret key rate of an RIS-assisted THz CV-QKD link directly from system parameters without running full protocol simulations, provided the per-mode independence assumption holds.
The number of RIS elements $K$ and their common phase $\phi$ enter the key rate through the effective channel and through $C_{RIS}$; the numerics show the rate rises with $K$ and varies sinusoidally with $\phi$, giving a tuning knob for the RIS.
Heterodyne detection is preferable at low pilot powers, while homodyne can win at longer distances for smaller MIMO arrays, yielding a detector-selection rule.
Longer pilot lengths and higher pilot powers do not simply help: both reduce the fraction of coherence time available for key exchange and change the noise covariance, so the secret key rate can decrease with $L_p$ and $V_p$.
The RIS prevents the secret key rate from decaying with distance in the simulated range, which is exactly the regime where THz links are otherwise loss-limited.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The per-mode sum in Eq. (44) uses only the diagonal of $C_{RIS}$; if the off-diagonal entries are large, the true multi-mode secret key rate would require the full covariance, so Eq. (44) may be an approximation. A numerical comparison against the full-covariance rate would settle this.
Eve in this model learns the estimated channel from the feedback link but does not appear to influence the estimation itself. An active pilot-corruption attack, where Eve injects noise during training, is a natural stress test of the formula.
Because pilot power and length enter both the estimation-noise covariance and the overhead factor $(1 - L_p/T_c)$, the paper's degradation curves imply an optimal pilot budget per coherence block, though the paper does not solve for it.
The same derivation structure should carry to MMSE or other estimators: replace the LS error covariance with the corresponding estimator's covariance and re-derive $C_{RIS}$; the secret key rate formula would keep its shape.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper combines two strands that hadn't been put together before: RIS-assisted MIMO THz CV-QKD with LS channel estimation. Prior work had MIMO THz QKD with estimation error but no RIS ([16]), and RIS-assisted MIMO THz QKD with perfect CSI ([50]). This paper inserts the RIS channel and the estimation-error covariance into the standard CV-QKD SKR machinery, and the algebraic chain from (16) to (44) is internally consistent under its stated Gaussian assumptions. That is a real contribution: it gives a closed-form design tool, and the authors are upfront about the pilot overhead and the noise terms. The numerical study is thorough in terms of parameter sweeps.

The soft spots are in proportion. The biggest one is the step from (19) to (44). Expression (44) sums per-mode rates and uses only the diagonal entries σ²_RISi = 0.5 C_RIS(i,i). But C_RIS in (19) is generically a full matrix: U†_RISLS C_tilde_n U_RISLS is not diagonal, because C_tilde_n includes the term H_RIS H_RIS† and U_RISLS comes from the SVD of the estimated channel, not the true one. The off-diagonal entries couple the r modes, so the r parallel channels are not independent. The authors never prove that the covariance is diagonalized in this basis, and I don't see how it could be. As written, (44) is an unproven approximation, not a derived exact expression. This is load-bearing, because the whole claim of a closed-form SKR depends on it.

Second, the ML estimate in (21) is the biased residual sample covariance: its mean is ((Lp - NT)/Lp) C_tilde_n. They plug it into (22) without a degrees-of-freedom correction. For small Lp relative to NT, the numerical values will shift. This is fixable but genuine, and it means the computed SKR curves should be read with caution.

Third, the per-mode independence assumption also affects the Holevo information: Eve's attack is modeled as per-mode with a common Ve, but if the modes are coupled, the optimal collective attack should account for the joint state. The authors don't justify that choice physically.

The central argument—that RIS helps at larger distances and that heterodyne beats homodyne at low pilot power—probably survives a proper fix, but the numbers as printed are not reliable. My recommendation: send it to peer review, but reviewers should demand either a proof of the diagonalization or an explicit acknowledgment that (44) is an approximation, plus the bias correction. The paper is for researchers working on THz CV-QKD and RIS-assisted quantum links; it's a useful incremental extension, not a breakthrough.

Referee Report

4 major / 4 minor

Summary. This paper considers an RIS-assisted MIMO CV-QKD link at THz frequencies. Bob estimates the effective channel by least squares from pilot symbols, and the estimated channel state information is fed back to Alice over a public authenticated channel that is also accessible to Eve. Alice uses Gaussian modulation, Bob uses homodyne or heterodyne detection, and Eve is modeled as employing a collective Gaussian entanglement attack. The main claim is Eq. (44), a closed-form secret key rate (SKR) expression for reverse reconciliation that includes the channel estimation overhead and the noise covariance induced by LS estimation. Numerical evaluations study the SKR versus distance, pilot power, pilot length, number of RIS elements, and RIS phase configuration.

Significance. If correct, the closed-form SKR would be a useful design tool for RIS-assisted THz CV-QKD under imperfect channel knowledge, extending the perfect-CSI analyses in [48] and [50]. The paper is clearly structured and the algebraic chain leading to Eq. (44) follows the standard CV-QKD security template, which makes the claimed extension attractive. However, the exactness of Eq. (44) rests on at least two security-sensitive assumptions that are not proved—the diagonalization of the estimation-error covariance and the inclusion of the estimation noise in Eve's output state—and the numerical section uses a biased covariance estimator. Until these points are resolved, the quantitative SKR values and design conclusions should be treated as tentative. The paper does not provide independent Monte Carlo validation or reproducible code.

major comments (4)

[§IV, Eq. (19); §V, Eq. (44)] The per-mode summation in Eq. (44) is exact only if the channel-estimation noise n_RIS has independent components in the SVD basis. The covariance C_RIS computed in Eq. (19) is generically a full matrix: C_RIS = (2Va NT/(Vp Lp)) U†_RISLS C_tilde_n U_RISLS, and C_tilde_n includes 2V0 H_RIS H_RIS† plus detector noise. Because U_RISLS is the left singular-vector matrix of H_RISLS = H_RIS + Δ, not of H_RIS, there is no reason for U†_RISLS C_tilde_n U_RISLS to be diagonal; the off-diagonal entries of the cross term U†_RISLS Δ H_RIS V_RISLS in Eq. (16) are exactly cross-mode couplings. The paper defines σ²_RISi = 0.5 C_RIS(i,i) but never proves that the off-diagonal entries vanish or that the multimode Gaussian state separates into r independent single-mode channels. Thus Eq. (44) is an unproven approximation rather than a derived exact closed-form SKR. The authors should either prove the diagonalization under the specific channel/pilot model, or compute an exact multimode SKR from the full covariance matrix, or explicitly present Eq. (44) as an approximation with a justification and a bound on the error.
[§IV, Eqs. (17)–(18); §V, Eq. (28)] The same realization n_RIS appears in Eve's output mode in Eq. (18), and consequently σ²_RISi enters the Eve-side variance V_eoi = (1−β_i)V_a + β_i V_e + σ²_RISi in Eq. (28). In a collective Gaussian entangling-cloner attack, Eve's stored half of the two-mode squeezed vacuum is not affected by Bob's channel-estimation noise realization. An additive noise at Bob's measurement stage should appear in Bob's measured variance V_bi, but it should not appear in Eve's unconditional covariance matrix unless the attack model explicitly copies that noise to Eve. No physical justification is given for this common-noise assumption. This modeling choice reduces the Holevo information and therefore inflates the SKR. The authors should derive the joint state of Alice, Bob, and Eve from the actual attack and measurement model, recompute the conditional entropies, and, if a common-noise assumption is intended, state and defend it explicitly.
[§IV, Eqs. (21)–(22)] The ML estimator in Eq. (21) is a biased residual sample covariance. Because the residual vectors are (I − P)Ñ, where P is the projector onto the row space of Ψ_p, the expectation satisfies E[Ĉ_ML] = ((Lp − NT)/Lp) C_tilde_n. Substituting Ĉ_ML into Eq. (22) therefore underestimates C_RIS by the factor (Lp − NT)/Lp. In the numerical section, where NT is as large as 256 and Lp can be comparable (Fig. 4), this bias materially changes the σ²_RISi values and hence the reported SKR curves. The paper should use an unbiased estimator, or explicitly correct for the bias and show that the numerical conclusions are unchanged.
[§VI, Fig. 3] The text reports that the SKR degrades with increasing pilot power Vp and explains this by saying that higher pilot power implies a longer estimation duration. Vp is a power, not a duration, so the explanation is dimensionally inconsistent with the authors' own formula: in Eq. (22), Vp appears only in the denominator of the estimated noise covariance, so an increase in Vp should reduce σ²_RISi and, all else being equal, increase the SKR. The figure or the explanation should be reconciled with Eq. (22).

minor comments (4)

[§VI, Fig. 6 caption] The caption reads 'NT = NR = {32, 12}'; this appears to be a typo, since the surrounding text specifies NT = NR = {32, 128}.
[§III, Eqs. (10)–(11)] The LS solution and the DFT pilot design require Lp ≥ NT so that Ψ_p Ψ_p† is invertible; this constraint should be stated explicitly, and the plotted Lp ranges for large NT in Fig. 4 should be checked against it.
[§VI] The section says the numerical results validate the analytical framework, but all curves are evaluations of Eq. (44) and no independent simulation is reported; the wording should be changed to 'illustrate the derived formula' or an actual Monte Carlo validation should be added.
[§VI] The claim that this is 'the first work proposing the use of RISs for such quantum communication systems employing a practical channel estimation scheme' should be benchmarked against the existing literature, including the authors' own RIS-assisted CV-QKD study [50], to avoid an unsupported priority statement.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eq. (44) is a parameter-level extension of standard CV-QKD security formulas to an LS-estimated RIS-assisted MIMO channel, and no load-bearing step reduces the claimed result to its inputs.

full rationale

The claimed SKR formula (44) is assembled from two ingredients that are external to the target result: the per-mode collective-Gaussian-attack CV-QKD SKR expression (Eq. (23), with symplectic eigenvalues (29)-(43), attributed to [48] and [59]) and the LS channel-estimation error covariance C_RIS derived in Eq. (19) from the pilot model (7)-(10). Neither ingredient is defined in terms of the SKR, and the ML covariance estimate (21)-(22) is an input parameter estimate used to evaluate a performance expression, not a fitted quantity relabeled as a prediction. The self-citations [16], [48], and [50] supply the DFT pilot construction, the base MIMO THz CV-QKD model, and the prior RIS-assisted system context; those cited items are parameter-free, state assumptions that do not include the present target result, and are therefore real evidence that does not raise the circularity score under the review rules. The known technical concern that Eq. (44) sums per-mode rates using only diagonal variances sigma_RISi^2 = 0.5 C_RIS(i,i) while C_RIS is generically a full matrix is a validity/approximation question about whether the exact multimode SKR reduces to the summed expression; it is not a case of the derivation being equivalent to its inputs by construction. No circular step can be exhibited from the text, so the appropriate finding is no significant circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The derivation relies on standard CV-QKD security machinery plus two paper-specific modeling choices: giving Eve the same channel estimation noise as Bob, and using only the diagonal of a generally non-diagonal noise covariance. No new physical entities are introduced.

free parameters (1)

Reconciliation efficiency eta = 0.95
Set to 0.95 in Section VI; it multiplies the mutual information term in Eq. (44) but is not derived or fitted.

assumptions (4)

domain assumption Standard CV-QKD security formulas for collective Gaussian attacks and reverse reconciliation apply to the RIS-assisted channel
Eqs. (23)-(44) import the SKR structure from refs. [48] and [59] without re-deriving the security proof for the RIS scenario.
ad hoc to paper Channel estimation noise n_RIS is common to Bob and Eve
Eqs. (17)-(18) put the same n_RIS_i in Bob's and Eve's output quadratures; this strengthens Eve and is not justified from the estimation model.
ad hoc to paper C_RIS is effectively diagonal for the parallel-channel decomposition
Eq. (44) uses only C_RIS(i,i) from Eq. (19); no proof that off-diagonal terms vanish is provided.
domain assumption No finite-size corrections or error correction inefficiencies beyond eta are needed
The SKR expression is asymptotic; finite-size effects and practical reconciliation overhead are not modeled.

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Cite this review

Pith. "Pith review of RIS-Assisted MIMO CV-QKD at THz Frequencies: Channel Estimation and SKR Analysis." pith.science (2026). https://pith.science/paper/O6ECZNS5

@misc{pith2026241218771,
  author       = {Pith},
  title        = {Pith review of: RIS-Assisted MIMO CV-QKD at THz Frequencies: Channel Estimation and SKR Analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O6ECZNS5}},
  note         = {Machine review of arXiv:2412.18771}
}

read the original abstract

In this paper, a multiple-input multiple-output (MIMO) wireless system incorporating a reconfigurable intelligent surface (RIS) to efficiently operate at terahertz (THz) frequencies is considered. The transmitter, Alice, employs continuous-variable quantum key distribution (CV-QKD) to communicate secret keys to the receiver, Bob, which utilizes either homodyne or heterodyne detection. The latter node applies the least-squared approach to estimate the effective MIMO channel gain matrix prior to receiving the secret key, and this estimation is made available to Alice via an error-free feedback channel. An eavesdropper, Eve, is assumed to employ a collective Gaussian entanglement attack on the feedback channel to avail the estimated channel state information. We present a novel closed-form expression for the secret key rate (SKR) performance of the proposed RIS-assisted THz CV-QKD system. The effect of various system parameters, such as the number of RIS elements and their phase configurations, the channel estimation error, and the detector noise, on the SKR performance are studied via numerical evaluation of the derived formula. It is demonstrated that the RIS contributes to larger SKR for larger link distances, and that heterodyne detection is preferable over homodyne at lower pilot symbol powers.

Figures

Figures reproduced from arXiv: 2412.18771 by the authors.

**Figure 2.** SKR versus the transmission distance between Alice [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 5.** SKR versus transmission distance d between Alice and Bob for a 32×32 MIMO configuration, fc = 15 THz, φ = π/4 for K = {25, 225, 400, 1600} RIS elements, with homodyne detection at Bob’s side. the performance of the homodyne and heterodyne detectors for varying Lp. Additionally, the saturation effect is not as prominent with increasing Lp as it was with increasing Vp. A similar trend to Vp is also observed for differ… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: SKR versus the pilot length Lp for NT = NR = {32, 128, 256} MIMO configurations, fc = 15 THz, d = 30 m, K = 100, φ = π/4, with Bob employing homodyne and heterodyne measurements for secret key detection. estimation of the noise variance. As seen in the figure, this eff…

**Figure 6.** Figure 6: SKR versus the RIS phase configuration φ for NT = NR = {32, 12} MIMO configurations, fc = 15 THz, d = {10, 80} m, and K = 100, with Bob employing heterodyne detection for secret key detection. VII. CONCLUSIONS This paper presented an RIS-assisted point-to-point MIMO co…

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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