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Scalable Quantum Key Distribution via GHZ Entanglement and Qubit Reuse

Tasdiqul Islam, Rasman Mubtasim Swargo, Engin Arslan, Md Arifuzzaman

arXiv:2608.21667v1quant-phcs.CRcs.NI

TL;DR

QKD networks face bandwidth constraints because conventional protocols transmit at least as many qubits as key bits. This paper proposes GHZ entanglement with QND-based reuse so one transmitted qubit carries an L-bit key, achieving 100% raw-key fidelity through L = 12 under tested ideal and depolarizing-noise conditions. The scheme also addresses security through authenticated classical communication, CHSH-style checks, and local reset verification, while extending to multi-party and server-client settings.

  • Problem

    Conventional QKD protocols require qubit transmissions equal to or exceeding secret-key length, creating a bandwidth bottleneck for developing quantum networks.

  • Method

    The scheme generates an L+1-qubit GHZ state, sends one qubit to Bob, and reuses the entanglement through custom ancillary encoding, BSM, QND discrimination, and reset cycles.

  • Results

    100% raw-key fidelity was achieved for L ≤ 12 under ideal conditions and depolarizing noise up to p = 0.005 per round, with transmitted-qubit efficiency η = L.

  • Takeaways & Limitations

    The scheme reduces quantum-channel usage to one transmitted qubit per GHZ block and extends to multi-party QKD and server-client architectures.

  • Takeaways & Limitations

    The evaluation is limited by classical simulation memory, and future work must validate the protocol on real hardware and quantify reset success under device noise.

Abstract

from arXiv · show

Conventional Quantum Key Distribution (QKD) requires the transmission of qubits proportional to or exceeding the length of the key, as protocols such as BB84 transmit more qubits than the final key size due to basis sifting and privacy amplification. Since quantum networks are still in their infancy and have limited capacity, this overhead puts significant pressure on network resources. To address this issue, we propose a Multi-Qubit Greenberger--Horne--Zeilinger (GHZ) State-based QKD scheme that reduces the number of qubits transmitted over the quantum channel. The proposed method transmits one GHZ qubit between endpoints and reuses the resulting entanglement to convey multiple classical key bits with the help of Quantum Non-Demolition (QND) measurements. Under the stated assumptions on authenticated classical communication, local reset verification, and bounded-error QND discrimination, one can transfer $L$ classical bits by generating an (L+1)-qubit GHZ state and transferring one qubit to the remote party. We verify correctness using the NetSquid quantum network simulator: the protocol achieves 100\% raw-key fidelity for keys of length up to 12 bits under both ideal conditions and depolarizing noise up to p = 0.005 per round. We further show that the proposed QKD algorithm can be extended to multi-party QKD and server-client deployment. The proposed scheme offers a transmitted-qubit-efficient, noise-tolerant alternative for bandwidth-limited quantum networks.

1 Introduction

The paper addresses quantum-channel bandwidth limits in QKD by proposing GHZ-based qubit reuse. The scheme transmits one qubit for an L-bit key while supporting security analysis, extensions, and 100% simulated fidelity through L=12 under tested noise.

  • Motivation: QKD protocols such as BB84, B92, and E91 require transferred qubits equal to or exceeding secret-key length, stressing limited quantum-network capacity.Quantum repeaters and high-capacity transmission infrastructure remain under development.
  • Proposed Scheme: The proposed scheme generates an L+1-qubit GHZ state and sends one qubit to Bob for transmitting an L-bit key.Alice retains the other entangled qubits and uses ancillary-bit encoding with classical BSM-result communication.
  • Contributions: GHZ entanglement and QND measurement reuse a single transmitted qubit across all L key bits, yielding efficiency η = L.The scheme requires multi-qubit entanglement on Alice’s side and classical communication for BSM results.
  • Contributions: The security analysis covers entanglement measurement, intercept-and-resend, QND-based eavesdropping, and reset-stage leakage, with attacks detectable through CHSH tests.The reset stage introduces no new information leakage channel.
  • Evaluation: 100% key fidelity was achieved for keys up to L = 12 under ideal conditions and depolarizing noise up to p = 0.005 per round.The authors also report extensions to multi-party QKD and server-client deployment.

2 Related Work

Related work reduces QKD transmission costs through alternative entanglement and high-dimensional approaches. This paper instead combines GHZ entanglement with QND-based qubit reuse to target transmission independent of key length.

  • QKD: Earlier QKD approaches include BB84, Bell-state-based E91, entanglement-based methods, and transmission-reduction schemes using EPR pairs or other techniques.Cabello et al. report η = 1 by transmitting two qubits for two key bits using EPR pairs and BSM.
  • QKD: The proposed work shares prior efforts’ goal of reducing qubit transmissions but uses custom-encoded ancillary-qubit teleportation with QND measurement for key-bit recovery.Its reported efficiency is η = L under the same metric used for the cited comparison.
  • Multi-party QKD: Existing multi-party QKD schemes use GHZ states, dynamic participation, asymmetric teleportation, or experimental GHZ qubits, but require at least one qubit transmission per key bit.The paper attributes the difference to its use of QND measurement for qubit reuse.
  • QND Measurement: QND measurement in this protocol performs binary amplitude discrimination between α > β and α < β rather than full state tomography.The approach relies on Bob’s quantum memory retaining the qubit for subsequent operations.
  • High-Dimensional QKD: High-dimensional QKD encodes multiple classical bits per qudit and can achieve η = log2(d), but still requires transmissions proportional to key length.The proposed scheme instead uses GHZ entanglement and QND-based reuse to reduce transmitted-qubit growth.

3 The System Model

The system model uses an L+1-qubit GHZ state, retaining L qubits at Alice and sending one to Bob. Repeated ancillary encoding, BSM, reset, and communication support all L key bits from that shared resource.

  • System Model: Alice generates L+1 GHZ-entangled qubits, retains L in quantum memory, and sends one to Bob before key encoding begins.Alice uses one ancillary qubit per round and requires L+2 simultaneous qubits including that ancillary qubit.
  • System Model: Alice and Bob repeat the protocol until all L key bits are transmitted.The first three steps are illustrated as sending a classical bit from Alice to Bob.
  • Why Multi-Qubit GHZ is Necessary: A multi-qubit GHZ state is necessary because a Bell pair supports only one BSM round before its entanglement is consumed.After each GHZ BSM-and-reset cycle, Alice’s remaining qubits and Bob’s qubit stay entangled for reuse.

Step 1: 𝐿+ 1 Qubit GHZ State Preparation

Step 1 prepares the shared entanglement resource by creating an L+1-qubit GHZ state. Alice uses Hadamard and CNOT gates, then retains L qubits while sending one to Bob.

  • Step 1: L+1 Qubit GHZ State Preparation: Alice begins by preparing L+1 qubits in the |0⟩ state and applying Hadamard and CNOT gates to create a GHZ state.The preparation is illustrated in Figure 2.
  • Step 1: L+1 Qubit GHZ State Preparation: The GHZ-state preparation is the first step before Alice retains L qubits and sends the remaining qubit to Bob.The system model describes this transmission after preparation.
  • Step 1: L+1 Qubit GHZ State Preparation: For a 5-bit key, Alice creates a 6-qubit GHZ state.This instantiates the L+1 preparation rule with L = 5.

Step 2: Qubit Transmission to Bob

Alice retains the GHZ qubits needed for reuse and sends one qubit to Bob, using direct transmission for short distances or quantum repeaters otherwise.

  • Alice keeps the GHZ qubits in quantum memory and sends the remaining qubit to Bob.The stated memory requirement is at least L+1 qubits.
  • For short distances, Alice can transmit the qubit directly; otherwise, quantum repeaters can teleport it.The passage identifies approximately 130 km as a typical short-distance scale.

Step 3: Ancillary Qubit Teleportation

Alice encodes each key bit in an ancillary qubit by choosing unequal amplitudes, then teleports that ancilla to Bob through Bell-state measurement and classical communication.

  • Alice prepares an ancillary qubit with α1 > β1 for key bit 0 and α1 < β1 for key bit 1.The amplitude choices depend on channel noise, depolarization, and QND discrimination accuracy.
  • Figure 3 illustrates the later measurement and reset steps used to send a classical bit from Alice to Bob.The displayed state expressions represent the post-measurement branches associated with the ancillary-qubit teleportation.
  • The ancillary qubit is teleported using a Bell-state measurement between it and the first GHZ qubit.Alice sends the Bell-measurement result to Bob through classical communication.

Step 4: QND Measurement

Bob uses a bounded-error QND measurement to distinguish the ancillary amplitudes and recover the key bit without destroying the transmitted qubit.

  • Bob determines whether α1 > β1 or α1 < β1, directly recovering the encoded key bit.The discrimination error can be reduced by increasing amplitude separation and using additional QND repetitions.
  • Because QND measurement preserves Bob’s qubit, the same qubit can be reused across all L rounds.Standard teleportation without QND would require L transmissions and yield η = 1 instead of η = L.

Step 5: GHZ State Reset

After teleportation creates a complex, partially entangled state, Alice performs a success-conditioned local reset to restore the standard GHZ state before the next key round.

  • The remaining GHZ qubits retain the form α1|0...0⟩ + β1|1...1⟩ before the next ancillary qubit is encoded.The protocol then encodes the second key bit in a new ancillary qubit.
  • After teleportation, X and Z gates alone do not let Bob extract the next ancillary amplitudes from the resulting complex state.A reset stage is therefore required after every teleportation.
  • The reset restores a partially entangled GHZ-like state to balanced GHZ form only on an accepted measurement branch.This operation is characterized as heralded entanglement concentration or local filtering.
  • Alice locally verifies the reset; success restores the remaining qubits for another round, while failure causes the GHZ block to be discarded and regenerated.A failed reset is not treated as a valid key round.

4 Security Analysis

The security analysis examines attacks on the transmitted GHZ qubit, QND-based eavesdropping, and reset-stage leakage under authenticated communication and private laboratories. Sampled GHZ-integrity checks detect disturbances, while local reset handling avoids a new public leakage channel.

  • Security assumptions: Authenticated classical communication and sampled CHSH-style checks define the threat model and provide sampling-dependent attack detection.Eve may observe BSM messages and attack qubits in transit, but cannot modify authenticated messages or access private laboratories.
  • Entanglement Measure Attack: Entanglement monogamy prevents Eve from becoming maximally entangled with Bob while Bob remains maximally entangled with Alice’s GHZ subsystem.Disturbance can therefore be detected through expected CHSH violations on sampled GHZ test blocks.
  • Intercept and Resend Attack: Measuring and replacing Bob’s transmitted qubit collapses its GHZ correlation, causing failed correction or integrity checks; the intercepted qubit alone carries no key bit.The attack is detectable during affected-block checks or sampled entanglement-integrity testing.
  • QND-Based Eavesdropping: QND before encoding yields no key information because the transmitted qubit is maximally mixed, while post-encoding QND is unavailable to Eve under the threat model.Nontrivial QND interaction with the transmitted qubit is treated as an entangling disturbance detectable by integrity checks.
  • QND-Based Eavesdropping: A substituted GHZ system produces correlations inconsistent with Alice’s system, exposing Eve’s QND-based binary discrimination through sampled integrity checks.The substitute qubit is not genuinely entangled with Alice’s GHZ state.
  • Security of the Reset Stage: The reset ancillary qubit and reset measurement outcome remain local to Alice, so Eve cannot access either.The ancillary state is constructed and consumed locally and is never transmitted.
  • Security of the Reset Stage: Failed local reset verification causes the block to be discarded, preventing silent contamination of later key bits.The post-reset information remains in Alice’s and Bob’s private joint state, with no new public leakage channel.

5 Performance Analysis

The performance analysis evaluates NetSquid correctness, transmitted-qubit efficiency, classical-bandwidth trade-offs, simulation limits, and protocol extensions. The scheme maintains 100% key accuracy in tested conditions while using one transmitted qubit per GHZ block, with practical limits set by memory and GHZ resources.

  • Simulation Results: 100% key accuracy was achieved for key lengths L∈{5, 8, 10, 12} under ideal conditions and tested depolarizing noise.The evaluation used five independent NetSquid runs per configuration; p denotes the depolarizing rate per BSM round.
  • Noise Tolerance: At p=0.005 over 12 rounds, the discrimination gap decreases from 0.20 to approximately 0.189, remaining above the simulation threshold.The binary design requires only reliable discrimination of α>β versus α<β.
  • Simulation Limits: At L=12, the peak reset-step density matrix requires approximately 4.3 GB, while L=15 would require approximately 275 GB.This limitation reflects classical full-state simulation overhead rather than the protocol’s transmitted-qubit cost.
  • Communication Cost: For a key of length L, the protocol transmits one quantum-channel qubit and 2L classical bits, trading quantum bandwidth for classical bandwidth.BB84 requires O(L) qubits plus O(L) classical bits for sifting and reconciliation.
  • Transmitted-Qubit Efficiency: η=L under the transmitted-qubit metric because the scheme sends one qubit for L secret bits, whereas BB84, E91, and B92 achieve η≤1.The metric excludes local memory, GHZ preparation, reset attempts, and classical communication.
  • Practical Limits: Available GHZ size forms a practical ceiling, although experiments have demonstrated GHZ states of up to 120 superconducting qubits and 2,000 atoms.Longer keys can instead use multiple smaller GHZ states sequentially, such as three 10-qubit states for a 30-bit key.
  • Practical Operating Considerations: The key-length parameter L is bounded by GHZ fidelity, memory lifetime, QND back-action, reset success probability, and integrity-check sampling rate.Shorter blocks reduce device pressure, while longer blocks amortize one quantum transmission over more key bits; failed reset blocks are discarded.
  • Extensions: For n-party QKD, an (L+n)-qubit GHZ state requires n transmitted qubits and achieves η=L/n; a trusted server can distribute the two-party case to clients.Each party receives BSM results, applies corrections, and performs QND locally.

6 Conclusion

The GHZ-based QKD scheme reduces quantum-channel usage to one transmitted qubit per GHZ block while maintaining 100% raw-key fidelity for keys up to 12 bits in tested simulations. Its security relies on authenticated communication, integrity checks, and reset verification, with real-hardware validation and overhead reduction left for future work.

  • Efficiency: One transmitted qubit per GHZ block yields transmitted-qubit efficiency η = L under the adopted metric.The scheme reuses entanglement across key bits.
  • Simulation results: 100% raw-key fidelity was achieved for L ≤ 12 under ideal conditions and depolarizing noise up to p = 0.005 per round.The reported noise resilience follows from the binary discrimination design in the tested regime.
  • Security: The security analysis covers entanglement measurement, intercept-and-resend, QND-based eavesdropping, and reset leakage.Authenticated classical communication, sampled CHSH-style integrity checks, and local reset verification mitigate these attack surfaces.
  • Limitations and future work: Future work will validate the protocol on real quantum hardware, quantify reset success probability under device noise, and explore reducing the 2L classical-bit overhead.These are stated scope boundaries beyond the current evaluation.
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