Adaptable Continuous Variable Quantum Network with Finite Size Security

Abstract: In recent years, continuous-variable quantum key distribution (CV-QKD) has become a promising paradigm for enabling secure communication among multiple end users sharing the same telecommunication backbone. CV-QKD with reverse reconciliation naturally enables scalability from conventional point-to-point links to quantum access networks based on passive quantum broadcasting channels. Here, we report an experimental demonstration on an active $1:4$ multi-user CV quantum network (QN) in the finite-size regime. With $1.25\cdot10<sup>9$ coherent states exchanged on each $11\text{km}$ quantum channel, the highest performance for secret key generation totaling $1.9\cdot10<sup>{-1}$ bits/channel use. Furthermore, we investigate adaptable CV-QN protocols that comprehensively allow network operation in various security and key rates requirements of individual users. The results establish the practical security of CV-QN compatible with existing telecommunication for broad deployment, and allowing additional degree of freedom for connected end users in existing infrastructures.

• The paper demonstrates simultaneous finite-size-secure key generation for four users over an 11 km broadcast network, achieving a trusted total rate of 0.19 bits per channel use from 1.25×10⁹ coherent states.
• The paper introduces untrusted, collaborative, and trusted post-processing modes, with total rates of 0.1192, 0.1688, and approximately 0.19 bits per channel use, respectively, enabling security assumptions to adapt to network conditions.
• The paper derives a chain-rule decomposition that exactly partitions the joint key rate among users without double-counting shared correlations, although individual allocations depend on user ordering.

Overview

This paper reports an experimental demonstration of an active $1{:}4$ multi-user continuous-variable quantum key distribution (CV-QKD) network with composable finite-size security against collective attacks. The work addresses two shortcomings of prior CV quantum network (CV-QN) implementations. First, previous experiments evaluated secret key rates only in the asymptotic limit, which overestimates extractable key and can undermine security; simultaneous finite-size key generation among all network users had not been demonstrated. Second, prior per-user rate constructions either conservatively discard inter-user correlations (the "untrusted broadcast" protocol) or sum independently constructed per-user rates that risk double-counting correlations shared across users, so that the summed rate may exceed what is extractable from the joint Alice–users state. The authors resolve both issues: experimentally, by generating finite-size keys for all four users simultaneously over $1.25\times10^9$ coherent states per channel; and theoretically, by a chain-rule decomposition of the joint network key rate into exactly exhaustive per-user contributions (2604.27943).

Unified trust-scenario framework

The network architecture follows the passive optical network (PON) topology: Alice broadcasts Gaussian-modulated coherent states through a fiber backbone to a $1{:}M$ beam splitter, after which each user receives an independently noise-corrupted mode over a last-mile fiber link. Security analysis proceeds within the Gaussian framework, where the covariance matrix of the joint Alice–$\mathbf B$ state fully determines security under collective attacks. Parameter estimation is performed independently by each user using standard finite-size techniques, and channel parameters ($\eta_m$, $\epsilon_m$) are minimized within their confidence intervals to yield a lower-bound key rate.

The paper formalizes three protocols that differ only in classical post-processing and trust attribution, all operating on identical physical data:

• Untrusted protocol: each user $B_k$ distills a key with Alice while all other users are treated as part of Eve's purification. Both mutual information and the Holevo bound are computed from the reduced covariance matrix $\Gamma_{AB_k}$. This is the most conservative setting and matches standard point-to-point CV-QKD performance.
• Trusted protocol: non-reference users are assumed to faithfully receive their signals without disclosing outcomes to Eve. The Holevo bound is evaluated from the global covariance matrix $\Gamma_{AB_1\cdots B_M}$, excluding trusted users from Eve's system. This yields the highest per-user rate.
• Collaborative protocol: assisting users publicly disclose their measurement outcomes, which are assumed private from Eve, but their detectors are not trusted. Both $I(A{:}B_k)$ and $1.25\times10^9$0 are evaluated from the conditional covariance matrix $1.25\times10^9$1, so disclosed outcomes directly constrain Eve's information rather than being conservatively assigned to her.

The collaborative protocol is positioned as an intermediate regime: stricter assumptions than the trusted protocol, but substantially better than the untrusted case because disclosed data tighten the bound on Eve's knowledge.

Exact decomposition of the joint key rate

The paper's central theoretical contribution is a consistent attribution of the network-wide key content. The joint finite-size key rate,

$1.25\times10^9$2

upper-bounds the total extractable key. Rather than constructing per-user rates and summing them (which the authors note risks overcounting cross-user correlations), they apply a chain-rule expansion of the mutual information and a telescopic decomposition of the Holevo term, so that

$1.25\times10^9$3

where each $1.25\times10^9$4 is the incremental contribution of user $1.25\times10^9$5 conditioned on users preceding it in a chosen ordering. Intermediate conditional entropies cancel pairwise, and the sum of individual contributions is analytically equal to the joint key rate by construction—not merely an approximation approaching it. The decomposition is optimal in the sense that it exactly exhausts the available joint key content without reusing shared correlations. Individual allocations depend on the chosen ordering, but the total is order-invariant; the ordering is an attribution rule, not a temporal constraint on reconciliation. An appendix tabulates all 24 orderings for four users, each summing to the same joint rate of $1.25\times10^9$6 bits/channel use at $1.25\times10^9$7, confirming invariance numerically. Finite-size corrections can be applied at the joint level or distributed consistently among contributions.

This construction contrasts with prior work that builds per-user end-to-end rates and sums them toward a collective upper bound [(2604.27943), citing Bian et al.], and it implies that the network key-rate limit is not only approachable but exactly attainable within the Gaussian finite-size framework.

Experimental implementation

The transmitter generates Gaussian-modulated coherent states at 125 MBaud with symbols drawn from a QRNG, single-sideband modulation with a 150 MHz offset, a 10 MHz pilot tone for carrier recovery, and RRC pulse shaping (roll-off 0.2). Signals traverse a 10 km backbone fiber, a passive $1.25\times10^9$8 splitter, and 1 km last-mile links. Each user performs RF heterodyne detection with a locally generated local oscillator (LLO), polarization matching, and balanced detection at 1 GSample/s. The DSP chain includes whitening, machine-learning-aided carrier recovery via an unscented Kalman filter, delay compensation, and matched-filter downsampling. Around $1.25\times10^9$9 symbols were exchanged with each Bob.

System calibration was performed iteratively across the four receiver stations, jointly optimizing LO clearance above electronic noise, detector gain, and pilot tone power. Estimated parameters at $1{:}M$0 (modulation variance $1{:}M$1 SNU, trusted detector efficiency $1{:}M$2):

User	Trusted noise (mSNU)	Transmittance $1{:}M$3	Excess noise (mSNU, I/Q)
1	54.00	0.13 (−8.94 dB)	4.24 / 4.10
2	49.80	0.12 (−9.23 dB)	2.94 / 2.98
3	60.22	0.11 (−9.42 dB)	5.02 / 5.00
4	51.08	0.10 (−10.02 dB)	5.14 / 5.18

The deviation of user transmittances from the intrinsic ~8.2 dB loss is attributed to beam splitter imperfection and polarization mode mismatch. Excess noise was kept low (2.9–5.2 mSNU) through parameter fine-tuning. Because each Bob's mode is coupled to independent vacua, measurement outcomes remain distinct despite the common broadcast state, and reverse reconciliation yields independent keys with each user.

Results and comparison

Finite-size key rates converge toward asymptotic values as $1{:}M$4 grows from $1{:}M$5 to $1{:}M$6; at $1{:}M$7 the system operates close to the asymptotic limit. Per-user rates at full block size ($1{:}M$8):

User	Untrusted	Collaborative	Trusted
1	0.0396	0.0529	0.0596
2	0.0451	0.0579	0.0644
3	0.0225	0.0348	0.0408
4	0.0120	0.0232	0.0286

The trusted protocol yields the highest total network rate of 0.19 bits/channel use (sum over users), with User 2 reaching 0.0644 bits/channel use individually. The fully untrusted scenario drops the total to 0.1192 bits/channel use. Notably, the collaborative protocol sacrifices little network performance—total 0.1688 bits/channel use, with a maximum individual rate of 0.0579—while requiring only that assisting users' disclosed outcomes remain private from Eve, not that their detectors be trusted. This indicates that the collaborative mode is a practical compromise when full user trust cannot be assumed.

Against other reported Gaussian-modulated 1-to-$1{:}M$9 CV networks secure against collective attacks, this work achieves the largest summed secret key rate while being the only one with finite-size security:

Work	Users	Distance (km)	Sum SKR (bit/use)	Finite-size
This work	4	10 + 1	$\mathbf B$0	Yes
Bian et al.	2	25 + 5	$\mathbf B$1	No
Hajomer et al.	8	10 + 1	$\mathbf B$2	No
Pan et al.	16	5 + 1	$\mathbf B$3	No

The roughly order-of-magnitude advantage in total key volume over prior demonstrations is attributable to both the finite-size-optimized hardware (low excess noise, high reconciliation efficiency) and the exact joint-rate decomposition, which exploits rather than discards inter-user correlations.

Limitations and open questions

The paper concedes several constraints. The trusted protocol's performance depends on the assumption that trusted users' measurement results are not disclosed to Eve and that detector efficiency and trusted noise are calibrated correctly; the collaborative protocol additionally assumes that disclosed measurement outcomes remain private from Eve while making no trust assumption on detectors. The decomposition's per-user allocations depend on the chosen ordering, so a fair allocation policy among users with heterogeneous demands remains a design question rather than something the framework prescribes. Key generation is conditional on successful information reconciliation for all Bobs, and the security proof covers collective attacks within the Gaussian framework; extension to general (coherent) attacks and to composable security proofs specific to the multi-user setting is not addressed. The demonstration uses uniform splitting and symmetric channel parameters; performance under strongly asymmetric splitting ratios or heterogeneous user distances is left unexamined.

Conclusion

This work provides both an experimental and a theoretical foundation for adaptable CV quantum access networks. Experimentally, it demonstrates simultaneous finite-size secret key generation for four users on an 11 km broadcast channel with a total rate of $\mathbf B$4 bits/channel use—the highest reported for a Gaussian-modulated CV network and the first with finite-size security. Theoretically, it establishes a chain-rule decomposition that exactly attains, rather than approximates, the joint network key rate, eliminating double-counting of shared correlations. The three-protocol framework—untrusted, collaborative, and trusted—allows the same physical infrastructure to elastically adapt to varying security assumptions and per-user bandwidth requirements, which the authors identify as essential for deployment in networks where user demands are not static.