CHSH Constrained Variational Attacks on the E91 Quantum Key Distribution Protocol

We investigate Quantum Circuit Learning (QCL) as a variational framework for exploring eavesdropping attacks on the entanglement-based E91 quantum key distribution protocol. Eve's attack is modeled by a hardware-efficient parameterized quantum circuit optimized with a loss function combining the Clauser-Horne-Shimony-Holt (CHSH) parameter, the Alice-Bob quantum bit error rate (QBER), and Eve's conditional bit- inference probability. We consider two attack models: a two- sided attack, in which Eve has coherent access to both distributed qubits before their delivery to Alice and Bob, and a one-sided attack restricted to an Alice-Eve interaction. In the noiseless two-sided scenario, the optimized circuit reaches PEve ≈ 0.79, S ≈ 2.57, and QBER ≈ 4.6%, while the corresponding CHSH- based lower bound on the asymptotic secret-key rate remains positive. Under depolarizing noise with probability p = 0.1, the two-sided attack remains inside the exploratory region |S| > 2 and QBER < 8%, but the resulting key-rate lower bound be- comes non-positive. In contrast, the one-sided attack reaches unit conditional inference probability only by causing a substantial QBER and eliminating Bell violation. These results show that variational circuits can explore information, disturbance regimes beyond fixed attack models, while the quantitative CHSH-based key-rate bound, rather than a binary Bell-violation or QBER threshold, is required to assess whether positive asymptotic secret-key extraction is certified.