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Optical and Quantum Electronics

Effective channel representations and composable security for passive-user Loop-Back quantum key distribution

Luis Adrián Lizama-Pérez

Published Oct 21, 2026• 0 citations
Publisher

Abstract

Abstract This work develops effective channel representations and a composable-security framework for distributed passive-user Loop-Back quantum key distribution (QKD). Alice provides trusted preparation-and-measurement infrastructure at local service points, while two source-free and detector-free users, $$B_1$$ and $$B_2$$ , apply independently selected polarization rotations $$R(\pm \pi /8)$$ to the same optical carrier. Two complementary derivations describe the honest operation. Direct averaging over the composed rotations gives the observable Loop-Back map, while an operationally equivalent representation on the Bennett–Brassard 1984 (BB84) preparation-and-measurement data is a balanced mixture of conjugate-basis dephasings, or an anisotropic Pauli channel with identity, $$X$$ , and $$Z$$ components and no $$Y$$ component. Both recover the ideal conclusive-event probability $$P_{\textrm{conc}}=1/4$$ . Security is established without restricting the adversarial action to this effective channel. Eve may implement an arbitrary coherent attack with quantum memory on the single untrusted interface between the two local rotations. In the single-photon sector, a Choi-operator identity yields the exact ideal relation $$e_{1,\mathrm p}=3e_{1,\mathrm b}/2$$ , together with a low-error robust angular bound $$e_{1,\mathrm p}\le \kappa _{\textrm{rot}}e_{1,\mathrm b}$$ . Phase-randomized weak coherent pulses, decoy-state estimation and generalized entropy accumulation then give a finite-key composable bound. A common optical model compares asymptotic and finite-block rates and quantifies their convergence with block size. The analysis also identifies the trusted-interface condition required to exclude loss-assisted operation-discrimination attacks and specifies the hardware assumptions needed for optical implementation.

DOI
10.1007/s11082-026-09189-1

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Abstract This work develops effective channel representations and a composable-security framework for distributed passive-user Loop-Back quantum key distribution (QKD). Alice provides trusted preparation-and-measurement infrastructure at local service points, while two source-free and detector-free users, B1B_1B1​ and B2B_2B2​ , apply independently selected polarization rotations R(±π/8)R(\pm \pi /8)R(±π/8) to the same optical carrier. Two complementary derivations describe the honest operation. Direct averaging over the composed rotations gives the observable Loop-Back map, while an operationally equivalent representation on the Bennett–Brassard 1984 (BB84) preparation-and-measurement data is a balanced mixture of conjugate-basis dephasings, or an anisotropic Pauli channel with identity, XXX , and ZZZ components and no YYY component. Both recover the ideal conclusive-event probability Pconc=1/4P_{\textrm{conc}}=1/4Pconc​=1/4 . Security is established without restricting the adversarial action to this effective channel. Eve may implement an arbitrary coherent attack with quantum memory on the single untrusted interface between the two local rotations. In the single-photon sector, a Choi-operator identity yields the exact ideal relation e1,p=3e1,b/2e_{1,\mathrm p}=3e_{1,\mathrm b}/2e1,p​=3e1,b​/2 , together with a low-error robust angular bound e1,p≤κrote1,be_{1,\mathrm p}\le \kappa _{\textrm{rot}}e_{1,\mathrm b}e1,p​≤κrot​e1,b​ . Phase-randomized weak coherent pulses, decoy-state estimation and generalized entropy accumulation then give a finite-key composable bound. A common optical model compares asymptotic and finite-block rates and quantifies their convergence with block size. The analysis also identifies the trusted-interface condition required to exclude loss-assisted operation-discrimination attacks and specifies the hardware assumptions needed for optical implementation.
This work develops an effective channel model for distributed passive-user Loop-Back quantum key distribution. In the intended key-establishment setting, the two passive users B1B_1B1​ and B2B_2B2​ are the legitimate lightweight parties that establish a shared secret key by using Alice as an active quantum preparation-and-measurement infrastructure. A single active station prepares and measures BB84 states, while two remote users apply only local polarization rotations. We show that the passive-user pair can be externally encapsulated as an effective Loop-Back node whose observable action is a balanced mixture of conjugate-basis dephasings, equivalently represented as an anisotropic Pauli channel with identity, XXX, and ZZZ components and no YYY component. This structure differs from isotropic depolarization and recovers the ideal conclusive-event probability Pconc=1/4P_{\mathrm{conc}}=1/4Pconc​=1/4. The model also clarifies why non-orthogonal intermediate states are necessary for passive-user security. This channel-level description characterizes Alice's observable statistics and provides a compact basis for subsequent analysis of passive-user Loop-Back QKD under realistic optical and adversarial conditions.
Server-aided Bell-state networks with sequential local encoding are an established approach to mediated key distribution. Against this background, we study a Loop-Back architecture in which two source-free, detector-free users transform the same traveling qubit. Alice privately samples r∈Z∗22r\in\mathbb{Z}*2^2r∈Z∗22, prepares ∣βr⟩|β_r\rangle∣βr​⟩ over the complete Bell basis, retains one qubit, and uses rrr as the reference for the returned pair. A deterministic Pauli-composition mode, U∗eff=U2U1U*{\mathrm{eff}}=U_2U_1U∗eff=U2​U1​, serves as a reference to prior serial-unitary protocols. The main extension instead uses the two same-axis settings R(±α)R(\pmα)R(±α) and a reference-versus-complement Bell test. Opposite rotations cancel, whereas equal rotations add, so P(C∣disagreement)=0P(C\mid\mathrm{disagreement})=0P(C∣disagreement)=0 and P(C∣agreement)=sin⁡2(2α)P(C\mid\mathrm{agreement})=\sin^2(2α)P(C∣agreement)=sin2(2α). At α=π/8α=π/8α=π/8, agreement is certified with probability 1/21/21/2, and the overall conclusive-key probability is 1/41/41/4 for unbiased choices. The same event supplies one shared raw-key bit. For the Pauli mode, a one-qubit Pauli twirl hides either user's factor from an endpoint mediator that may prepare an arbitrary qubit--ancilla state and choose the final measurement, provided the other Pauli is uniform and the traveler is inaccessible between users. The rotational guarantee is narrower: common-sign privacy holds only for the prescribed Bell source and coarse-grained instrument. A polarization implementation can normalize the private reference to the singlet and use a beam splitter with verified two-photon detection; loss produces a third, discarded outcome. The proposal is theoretical and does not claim composable security or experimental validation.
The Loop-Back Quantum Key Distribution (LB-QKD) protocol establishes a bidirectional architecture in which a single photon travels forth and back through the same optical channel. Unlike conventional one-way schemes such as BB84, Alice performs both state preparation and measurement, while Bob acts as a passive polarization modulator and reflector. This design eliminates detectors at Bob's side, minimizes synchronization requirements, and enables compact, low-power implementations suitable for quantum-mobile and IoT platforms. An extended three-basis configuration {X,Y,Z} is introduced, preserving the simplicity of the two-basis scheme while improving noise tolerance through enhanced orthogonality-based filtering. Analytical modeling shows that the effective protocol error decreases from Eprotocol(2)=e/2 to Eprotocol(3)=e/3, achieving a 33% improvement in noise resilience. Despite its slightly lower sifting efficiency (η=1/6), the total information gain reaches G=0.26 bits per pulse, maintaining post-sifting throughput comparable to BB84. The protocol doubles the tolerable QBER of conventional QKD, sustaining secure operation up to 22% for two bases and approximately 47.58% for three bases. Its passive, self-verifying architecture enhances resistance to man-in-the-middle, photon-number-splitting, and side-channel attacks, providing a scalable and energy-efficient framework for secure key distribution and authentication in next-generation mobile and distributed quantum networks.
Establishing scalable, secure quantum networks requires advancing beyond conventional point-to-point quantum key distribution (QKD) protocols toward point-to-multipoint QKD protocols. Here, we generalize a well-established discrete-modulated continuous-variable (CV) QKD protocol from the point-to-point to the point-to-multipoint setting. We present a comprehensive security analysis across different scenarios and derive secret key rates for loss-only and noisy channels, both in the asymptotic and the composable finite-size regime. Experimentally, we validate the protocol in a passive optical network with 10 km access links, achieving a composable secure key rate of 2.185×10 −3 bits per symbol (0.273 Mbit/s) against independent and identically distributed collective attacks. Our results demonstrate that discrete-modulated CV-QKD can support access networks with multiple users while relying solely on cost-efficient, off-the-shelf telecommunication components, paving the way toward practical, scalable, and secure quantum networks.
Bidirectional quantum key distribution (QKD) protocols face persistent challenges related to classical disclosure, confinement of the signal space to predictable subspaces, and limited detectability under substitution or entanglement-swapping attacks. In this work, we present a Bell-state extension of the Loop-Back QKD architecture that improves efficiency and detectability while preserving its defining feature of a simplified, measurement-free remote terminal. The protocol employs entangled Bell states together with deterministic local Pauli encoding at the remote node. A central element is that Alice privately prepares and knows the initial Bell state, which serves as a hidden reference enabling her to interpret the Bell-state transition induced by Bob, while preventing an adversary from reconstructing the encoding without access to this reference. By exploiting both intra- and inter-family Bell transitions, the scheme expands the effective signal space beyond the subspace restrictions of earlier two-way protocols. Alice performs a Bell-state measurement to deterministically infer Bob's operation without any basis sifting. Although the traveling subsystem remains locally maximally mixed, concealing the initial Bell family amplifies disturbance under separable substitution strategies, yielding an intrinsic detection probability of approximately 3/4 per round. From an efficiency perspective, the protocol lifts the intrinsic post-selection limitation of single-qubit Loop-Back schemes: the effective throughput is bounded only by the Bell-state measurement success probability, reaching up to 50% in linear-optical implementations. These features make the proposed scheme particularly suitable for mobile or edge-based QKD scenarios, where passive remote nodes must operate under high loss and limited interaction times.
Abstract Purpose : Practical quantum key distribution (QKD) deployments remain constrainedby finite-key effects, hardware noise, and limited adversarial attackmodeling. This work aims to close the gap between theoretical composable securityguarantees and realistic hardware performance by developing noise-resilient,multi-basis QKD protocols with provable finite-key security bounds. Methods : A generalized multi-basis BB84 protocol (MB-BB84) and a phaseengineeredE91 entanglement-based scheme (PE-E91) are developed with composablesecurity guarantees under depolarizing and amplitude-damping noisechannels. Finite-key security bounds are derived using the smooth min-entropyframework. Zero-noise extrapolation (ZNE) error mitigation is integrated intothe composable security analysis. Experimental validation is performed usinghardware-calibrated noise models derived from superconducting quantum processors. Results : The 3-basis MB-BB84 achieves a composable finite-key rate 7.4%higher than standard BB84 at a block size of one million signals, while increasingintercept-resend eavesdropper detection probability from 75.0% to 83.3%.PE-E91 improves the CHSH Bell parameter by 4.2% over standard E91. ZNE mitigationextends the practical security threshold by 30–40% in terms of maximumtolerable gate error rate on superconducting hardware. All analytical predictionsare confirmed within 3–5% of semidefinite-programming-computed key ratebounds. Conclusion : The proposed unified framework bridges theoretical composablesecurity and practical hardware constraints, providing explicit design guidelinesfor scalable, noise-resilient QKD architectures suitable for next-generationquantum communication networks.