A single leak bound now secures quantum key distribution

Abhishek Mishra of Université libre de Bruxelles (ULB) and colleagues demonstrate a path toward securing quantum key distribution (QKD) with a significantly simplified set of assumptions. The work shows that a single bound on information leakage, quantified by the probability of guessing the input from everything the source emits, is sufficient to prove security against collective attacks, replacing the need to model the quantum source in detail.

This analysis finds that widely used protocols like B92 and BB84 do not guarantee key security under this minimal leakage bound. But related protocols can achieve one secret bit per key round in ideal conditions and tolerate visibilities down to 0.84, corresponding to an 8% quantum bit error rate.

Prepare-and-Measure QKD: The Challenge of Information Leakage

A leakage bound, quantified by the probability of correctly guessing the input from everything emitted by the sender’s device, now proves sufficient to guarantee the security of quantum key distribution (QKD) protocols against collective attacks, a simplification that bypasses the need for detailed modeling of the source itself. This finding challenges the conventional reliance on detailed source models, which have traditionally been necessary to prove the security of QKD.

Real-world implementations inevitably introduce imperfections, which can either alter the states without affecting the leakage bound or potentially increase it. The work accounts for these imperfections, computing key rates using semidefinite programming relaxations to assess the robustness of the protocols.

Leakage Bound as a Minimal Security Assumption for QKD

This approach defines security through a “leakage bound,” measured as the optimal probability of correctly guessing the sender’s input based on all emitted signals, and represents a shift toward simpler security proofs for prepare-and-measure QKD systems. The analysis places no restrictions on the emitted states’ dimension, energy, or purity, or on the receiver’s measurement device beyond standard data protection requirements.

The work demonstrates that these widely implemented methods do not guarantee security when relying solely on this leakage bound, a surprising result given their foundational status in the field. This suggests a direct question for prepare-and-measure QKD: can a quantitative bound on information leaving the source replace a detailed model of the preparations and measurements?

At a leakage bound of 0.08, one calculation yielded $v_c$ = 0.883 and ${\eta }_c$ = 0.830, a result comparable to 0.879 and 0.827 obtained with full statistics. The work concludes that a bound on the information that the source emits about Alice’s input is enough to prove security of suitably designed prepare-and-measure protocols.

Quantifying Leakage: The Probability of Guessing Alice’s Input

A quantifiable limit on information escaping a quantum source, measured as the probability of correctly guessing Alice’s input from all emitted signals, now provides a sufficient basis for secure communication, bypassing the need for detailed source modeling. This simplification is critical for practical implementation of prepare-and-measure quantum key distribution (QKD) systems, as it removes constraints on the precise form, dimension, energy, or purity of the emitted states.

This approach establishes that a “leakage bound” alone is enough to prove security against collective attacks, a finding that challenges established protocols. The widely used B92 and BB84 protocols, for example, fail to certify key security when relying solely on this minimal leakage bound assumption. Researchers demonstrated this by quantifying the leakage operationally, calculating the optimal probability of guessing Alice’s input, denoted as $P_g$, from everything the source emits.

For instance, with two states ${\rho }_0$ = and ${\rho }_1$ =, a $P_g$ value of 3/4 was observed, while any purification attempts resulted in a $P_g$ greater than or equal to 0.85. Key rate calculations using a (4,3) code reveal minimal performance loss compared to full statistics, with a visibility of 0.883 and an efficiency of 0.830 at a leakage bound of 0.08. The work concludes that this single bound on information leakage yields security without requiring a detailed model of the source itself, a significant step toward practical, secure quantum communication networks.

Standard Protocols, B92 and BB84, Fail Under Leakage Bounds

This finding contrasts with device-independent QKD, where minimal assumptions are made about the devices themselves, but offers a practical alternative for prepare-and-measure systems. Researchers found that if observed statistics can be replicated through classical communication carrying equivalent information about the input, Eve could potentially learn everything Bob does, mirroring a similar principle in device-independent key distribution. The best protocols tested tolerated visibilities down to 0.84 and detection efficiencies to 0.76, surpassing the 0.78 visibility threshold of standard BB84 protocols with a characterized receiver.

“Relative to device independence, little is lost as our bounds certify, within the precision of our relaxations, as much as the DI analysis along the whole curves, provided leakage is bounded by its honest value,” the paper states. Future work will focus on extending the analysis to non-collective attacks and refining the asymptotic bounds established in this study.

Rotating Measurement Bases for Security in P&M QKD

Rotating Bob’s measurement bases relative to Alice’s preparation bases can restore security in prepare-and-measure quantum key distribution (QKD), even when a strict model of the source is unavailable. This approach generates non-classical statistics while adhering to a defined limit on information leakage, a quantifiable bound on how much information about the input leaves the sender’s laboratory. The work investigates protocols employing three, four, or six preparations, including a prepare-and-measure analogue of a device-independent (DI) protocol based on the CHSH inequality.

In these schemes, Alice prepares eigenstates of observables, with a bias towards key bases to maximize the fraction of key rounds. Every decomposition of the mixed state into two states with equal weights is induced by a projective measurement on this auxiliary system. This measurement, defined by observables, leaves the communicating parties in specific states dependent on the measurement outcome.

The resulting states, denoted ${\rho }_{\theta ,a}$ for key bases and ${\rho }_x$ for preparations outside the key bases, are central to establishing security under the leakage bound. This approach offers a pathway to secure QKD with fewer assumptions about the source device than traditional methods.

Achieving One Secret Bit Per Round with Optimized Protocols

This advancement stems from a refined approach to information leakage, the unavoidable loss of data from the sender’s laboratory, and a focus on minimizing its impact on security. The analysis establishes a lower bound on the conditional von Neumann entropy, minimizing the information available to potential eavesdroppers and ensuring secure communication. This bound was obtained by adapting existing variational methods and incorporating a relaxation of correlations under a defined leakage limit. The researchers state, highlighting the potential for highly efficient and secure QKD systems.

Key Rate Computation Using Semidefinite Programming Relaxations

Semidefinite programming relaxations now allow for the computation of key rates in quantum key distribution (QKD) protocols, offering improved performance over standard methods. The analysis establishes that these optimized protocols can tolerate signal degradation, functioning with visibilities down to 0.84. Researchers obtained these results by combining a variational bound on relative entropy with a semidefinite relaxation of correlations, all while adhering to the leakage bound.

This approach minimizes assumptions about the quantum source, requiring only that the receiver’s measurement device does not leak Bob’s data. The team’s work computes lower bounds on the conditional von Neumann entropy, a measure of information uncertainty, using these relaxations. These bounds are minimized over all states and measurements consistent with observed data and the established leakage limit.

Curves representing these bounds demonstrate the performance of the protocols, ending at the threshold where a positive key rate is still certified. The analysis also includes comparisons to device-independent rates, showing how closely the leakage bound approach can match the security offered by fully device-independent QKD.

Tolerance to Noise: Visibility Threshold of 0.84

A visibility threshold of 0.84 defines the tolerance to noise for optimized quantum key distribution (QKD) protocols, a figure achieved through a new analysis focusing on information leakage. This surpasses the 11% threshold of standard device-dependent BB84. The curves representing lower bounds under the leakage bound, computed using semidefinite programming relaxations, provide a visual representation of the achievable key rates.

Leakage Bound Compared to Device-Independent and Semi-Device-Independent Approaches

This alignment suggests that the leakage bound offers a practical alternative to fully device-independent protocols without sacrificing significant security assurances. The analysis extends beyond simply matching device-independent rates. It also establishes a relationship with semi-device-independent protocols. The work demonstrates that a device-independent attack on the entanglement-based twin of a prepare-and-measure protocol induces a corresponding attack on the original protocol that respects the leakage bound.

The observed guessing probability of the key bit attains the device-independent value of 1/2 + 1/2 $\sqrt{2–S^2/4}$, to within 5· ${10}^{–4}$, further validating the approach. For the (4,3) protocol, the device-independent twin is the CHSH-based protocol detailed in a prior reference, and the analysis confirms that the induced prepare-and-measure attack reproduces the same key state at the honest level.

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