Real-life secrecy capacity of quantum secure direct communication in the Wyner wiretap framework: finite block lengths, imperfect devices, and capacity-approaching forward codes

Quantum secure direct communication (QSDC) conveys a secret message directly over a quantum channel, so its security must be quantified by a directly achievable secure rate rather than by the key-generation accounting of quantum key distribution. Wu, Lin, Yin and Long (2019) applied Wyner's wiretap channel theory to QSDC, deriving secrecy-capacity lower bounds for the two-step and DL04 protocols under ideal devices and asymptotically large blocks, and left open the passage to realistic conditions: imperfect sources and detectors, finite block lengths, explicit forward coding schemes achieving the computed rates, and improved noise tolerance. Later work by the same and other groups discharged substantial fragments — finite-length collective-attack security proofs in a private-dense-coding formulation, finite-block-length secrecy-capacity bounds for decoy-state weak-coherent-pulse protocols, joint-encryption-and-error-control codes approaching the secrecy capacity of a quantum-memory-free DL04, masking-based capacity increase, and measurement-device-independent variants closing detector loopholes — but, as the field's own 2024 survey states, the secrecy capacity considering finite-length effects and practical system parameters, and the security analysis of imperfect devices, remain to be determined. The surviving open problem is the real-life secrecy-capacity characterization of QSDC: the extension of the wiretap-framework analysis from idealized asymptotic conditions to finite block lengths and realistic device imperfections, together with provably capacity-approaching forward codes and improved noise tolerance of the wiretap-based bounds.

Background

Quantum secure direct communication (QSDC) transmits a confidential message directly through a quantum channel, without first generating and post-processing a shared cryptographic key as in quantum key distribution (QKD). In the canonical two-step protocol, Alice and Bob share blocks of EPR pairs (entangled two-particle states); part of each block checks for eavesdropping and the remainder carries the message. In the DL04 protocol, Bob sends Alice a random sequence of single photons and Alice encodes her message by applying Pauli operations (a 'quantum one-time pad', i.e. bitwise random Pauli flips and phase shifts known only to her) before returning the sequence, so that the message never appears on a classical channel and eavesdropping on the quantum states disturbs detectable error-rate checks. Because the message bits are carried deterministically by the quantum states and cannot be discarded or reshaped by privacy amplification over a sacrificial key, QSDC needs a direct measure of the maximum achievable secure transmission rate.

Wyner's classical wiretap channel theory supplies such a measure: the legitimate link is modeled as a main channel from sender to receiver, and everything the eavesdropper Eve can obtain is modeled as a wiretap channel; the secrecy capacity $C_s$, roughly $\max\{I(A:B)-I(A:E)\}$ where $I(A:B)$ and $I(A:E)$ are mutual informations of the legitimate and eavesdropping links, is the largest rate at which the receiver can decode reliably while Eve's information about the message vanishes, and it is achieved by classical wiretap codes. Wu, Lin, Yin and Long (2019) applied this theory to QSDC: treating the ideal protocol as the main channel and eavesdropping as the wiretap channel, they derived secrecy-capacity lower bounds for the entanglement-based two-step protocol and, through the equivalence between entanglement-based and single-photon protocols, for DL04 — for instance a bound of the form $C_s \ge Q_{\mathrm{Bob}}[1-h(e)]-Q_{\mathrm{Eve}}h(e_x+e_z)$, where $Q_{\mathrm{Bob}}$ and $Q_{\mathrm{Eve}}$ are Bob's and Eve's maximal photon reception rates, $e$ is the quantum bit error rate (QBER, the fraction of transmitted quantum states that decode incorrectly) of the message-carrying transmission, $e_x$ and $e_z$ are the QBERs measured in the two conjugate bases during the eavesdropping check, and $h$ is the binary entropy function. They also proposed a modified DL04 with higher secrecy capacity.

These bounds were derived under idealized conditions — perfect entanglement or single-photon sources, perfect measurement devices, asymptotically many quantum systems per block (the infinite-key regime), and a restricted attack model — and, like all secrecy-capacity results, they assert only the existence of coding schemes at rate $C_s$ without constructing them. The paper therefore left open the extension of the wiretap analysis to realistic conditions, which the field has since consolidated under the goal of a 'real-life secrecy capacity' of QSDC: (i) imperfect sources and detectors, including detector inefficiency, bias, and side channels (unintended information leaks through the physical apparatus, e.g. light back-reflected to Eve); (ii) finite block length, where estimated error rates fluctuate statistically and finite-block coding penalties apply to the forward-coded deterministic message; (iii) explicit classical forward error-correcting codes achieving rates near the computed $C_s$ under realistic loss and noise; and (iv) the comparatively low tolerable QBER (noise tolerance) of the wiretap-based bounds relative to entanglement-distillation constructions, which require far more complex quantum operations.

Problem Statement

Determine the real-life secrecy capacity of quantum secure direct communication in the Wyner wiretap framework, and construct codes that provably approach it. For QSDC protocols of the two-step/DL04 family and their decoy-state and measurement-device-independent variants, characterize the achievable secure direct-transmission rate as a function of measurable channel and device parameters by extending the wiretap-framework secrecy-capacity analysis from idealized asymptotic conditions to realistic conditions, namely (i) imperfect sources and measurement devices, including detector inefficiency, bias, and mismatch, source flaws, and side channels; (ii) finite block lengths, where estimated error rates suffer statistical fluctuations and finite-block coding penalties must be quantified for the forward-coded deterministic message; (iii) explicit classical forward encoding/decoding schemes (wiretap codes, joint encryption and error control, or equivalent) that are proved to achieve rates approaching the computed secrecy capacity under realistic loss and noise; and (iv) improved noise tolerance (tolerable QBER and loss thresholds) of the wiretap-based secrecy-capacity bounds. The four conditions (i)-(iv) are jointly required rather than alternative routes: a resolution must characterize the achievable rate under (i), (ii) and (iv) together and supply the forward codes of (iii) proved to approach it, while a result addressing only a single listed condition (for example a finite-block-length bound alone) counts as partial progress toward, not a resolution of, the stated problem.

The 2019 source stated this objective as four open directions (imperfect devices, finite-size analysis, explicit forward codes, improved noise tolerance). Later literature has discharged restricted fragments of each: finite-length collective-attack proofs in a private-dense-coding formulation (Wu, Long and Hayashi, 2022), finite-block-length decoy-state bounds for weak coherent pulses (Sun et al., 2024), JEEC codes approaching the secrecy capacity of a quantum-memory-free DL04 (Sun et al., 2020), masking-based capacity increase (Long and Zhang, 2021), and MDI variants closing detector loopholes (Niu et al., 2020; Sun et al., 2023). The surviving problem, as the field's 2024 survey itself states, is the extension of the secrecy-capacity analysis to realistic conditions: the security analysis of imperfect devices and the secrecy capacity considering finite-length effects and practical system parameters, together with provably capacity-approaching forward codes. The same literature separately documents as open the analytical capacity of the relevant non-degraded wiretap channels, the removal of n-fold i.i.d. assumptions, and security beyond collective attacks (Wu, Long and Hayashi, 2022; the 2024 survey); these are related open items rather than requirements of the present problem.

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