Recovery-Free CHSH Nonlocality with Particle Loss
Can CHSH nonlocality survive particle loss without applying an explicit recovery operation? In principle, any deterministic recovery can be absorbed into the measurement. Operationally, we show that the answer depends on the allowed measurements on the lossy system. We distinguish flagged erasure, in which each lost particle leaves a detectable record, from unflagged deletion, in which no such record remains. For flagged erasure and survival probability $\eta>1/2$, using known quantum-capacity results, we show measurements that asymptotically approach the quantum maximum $2\sqrt2$. In contrast, for $\eta\le1/2$, CHSH violation is impossible for both loss models. Then, we construct explicit recovery-free protocols using permutation-invariant encodings built from $n$-qubit Dicke states $| D_N^n\rangle$ and measurements on the surviving particles. A one-excitation $(N=1)$ encoding violates CHSH for $\eta>1/\sqrt{2}$. Increasing the excitation number $N$ yields a family of protocols that violates CHSH for $\eta>\eta_G=(\sqrt{5}-1)/2$, with the asymptotic golden ratio approached as $N \to \infty$. Finally, we present a sparse-deletion binomial PI protocol that guarantees CHSH violation for up to $O(\sqrt n)$ deletion errors. Our results distinguish fundamental limits imposed by loss from those set by explicit measurements without recovery.