Threshold (Q, P) Quantum Distillation

Quantum distillation is the task of concentrating quantum correlations present in ‘N’ imperfect copies using free operations by involving all ‘P’ parties sharing the quantum correlations. We present a threshold quantum distillation task where the same objective is achieved but using fewer parties ‘Q’. In particular, we give exact local filtering operations by the participating parties sharing a high-dimension multipartite GHZ or W state to distil the perfect quantum correlation. Specifically, an arbitrary GHZ state can be distilled using just one party in the network, as both the success probability of the distillation protocol and the fidelity after the distillation are independent of the number of parties. However, for a general W-state, at least ‘P-1’ parties are required for the distillation, indicating a strong relationship between the distillation and the separability of such states. Further, we connect threshold entanglement distillation and quantum steering distillation.

💡 Research Summary

The paper introduces the concept of threshold quantum distillation, a framework that allows a multipartite quantum network to concentrate imperfect quantum correlations into perfect ones while involving only a subset Q of the total P parties. Inspired by the classical (Q, P) secret‑sharing scheme, the authors define two concrete protocols: Threshold Entanglement Distillation (TED) and Threshold Steering Distillation (TSD). Both rely solely on free operations—LOCC for entanglement and one‑way LOCC for steering—and use local filtering (dichotomic POVM) performed by the participating parties (PP). After each round, the parties broadcast their binary outcomes; only runs where every PP obtains the “0” outcome are kept, yielding a post‑selected state (or assemblage) that is a convex mixture of the perfect target and the original imperfect resource. The overall success probability is (P_s = 1-(1-P_{us})^{N-1}), where (P_{us}) is the per‑copy success probability of the local filter.

GHZ states. For a high‑dimensional GHZ state (|\psi_{\text{GHZ}}\rangle = \sum_{i=0}^{d-1}\alpha_i |i\rangle^{\otimes P}), the authors prove that a single party (Q = 1) suffices to achieve optimal distillation. The per‑copy success probability is (P_{us}=d\alpha_0^2) and the fidelity after distillation, \