Quantum Key Distribution with Imperfections: Recent Advances in Security Proofs

TL;DR

This paper surveys analytical and numerical advances for security proofs in Quantum Key Distribution under practical imperfections. It distinguishes PM and EB paradigms, assesses imperfect devices and attack models (including DI and 1SDI scenarios), and details both asymptotic and finite-key analyses. Key contributions include SDP-based and conic-optimization methods for asymptotic key rates, Gauss–Radau expansions, and the integration of EUR, EAT, and postselection techniques to bridge theory with real-world implementations. The work underscores how combining analytical finite-size tools with numerical optimization yields robust, device-ready security guarantees for realistic QKD systems with imperfections.

Abstract

In contrast to classical cryptography, where the security of encoded messages typically relies on the inability of standard algorithms to overcome computational complexity assumptions, Quantum Key Distribution (QKD) can enable two spatially separated parties to establish an information-theoretically secure encryption, provided that the QKD protocol is underpinned by a security proof. In the last decades, security proofs robust against a wide range of eavesdropping strategies have established the theoretical soundness of several QKD protocols. However, most proofs are based on idealized models of the physical systems involved in such protocols and often include assumptions that are not satisfied in practical implementations. This mismatch creates a gap between theoretical security guarantees and actual experimental realizations, making QKD protocols vulnerable to attacks. To ensure the security of real-world QKD systems, it is therefore essential to account for imperfections in security analyses. In this article, we present an overview of recent analytical and numerical developments in QKD security proofs, which provide a versatile approach for incorporating imperfections and re-establishing the security of quantum communication protocols under realistic conditions.

Figures (12)

• Figure 1: Distribution of states in a prepare-and-measure QKD protocol. In a PM scenario, Eve intercepts the signal $\ket{\phi_{a,x}}$ emitted by the trusted source in Alice's laboratory. The malicious party can couple an auxiliary state $\ket{e}$ to it, and through an unitary operation $U_{E}$ it acquires information carried in $\ket{\Psi}_{BE}$.
• Figure 2: Distribution of states in an EB-QKD protocol. Eve controls both the source and the quantum channel and is therefore modeled as holding the purification $\ket{\Psi}_{ABE}$ of the quantum state $\rho$, the resulting distributed states to Alice and Bob.
• Figure 3: Characterization of devices in cryptographic scenarios, according to the knowledge of agents about the mechanism of their devices. In $(i)$ the devices of Alice and Bob are assumed to be fully characterized; $(ii)$ relies on the assumption that one of the laboratories cannot be characterized and the devices may be untrusted, being therefore treated as a black-box, and encoded in a steering scenario. The situation $(iii)$ depicts the protocol in which both laboratories are untrusted, matching a Bell scenario.
• Figure 4: Schematic representation of the lower bound given in Eq. (\ref{['eq: winick lower bound 1']}). In the first step, the quadratic Frank-Wolfe algorithm is used to go from an initial guess $\rho_0$ to a near-optimal variable $\rho$. The provable lower bound for the objective function evaluated in $\rho^{*}$ is achieved by considering contributions from a linearization term in the near-optimal $\rho$ together with a term appearing from duality of SDP's. The reliability of the method is guaranteed by the possibility to certify that the linearization and duality terms are always lower bounds for the primal objective function.
• Figure 5: Illustration of two methods that guarantee interior point solutions for the optimization of Eq. (\ref{['optimization alpha']}): (a) shows the perturbation method used by Winick et al. winick2018reliable, while (b) exhibits the use of a barrier method used by Lorente et al. lorente2025quantum, which we detail in Sec. \ref{['sec: Lorente']}. The purple pentagon represent the feasible set of the optimization, with the gradient increasing toward the opaque region, which represents the direction of the objective function at each step. The red dot represents the optimal variable ($\rho^{*}$), while the black dot represents the initial point of the optimization ($\rho_{0}$). The barrier function (wite dashed line) is used to avoid the convergence of the objective function to the boundary of the (reduced) feasible set. In the illustrations above, gap between the barrier and the facets of the feasible set is out of scale to evidence the action of the facial reduction methods. In practice, the reduced sets aren't restrictive enough to prevent a solution from being too far from a boundary solution.