The Finite Advantage

Entanglement distillation is the gold standard for extracting quantum correlations. Given many copies of a noisy entangled state, two parties communicate classically to distill a smaller number of high-quality Bell pairs. In the asymptotic limit — infinitely many copies — the protocol is optimal. It produces maximally entangled states that achieve the theoretical maximum of quantum nonlocality.

Høyer, Rashid, and ud Din show that for finite copies, a different strategy wins. Nonlocality distillation — which operates without classical communication between the parties — can achieve a higher CHSH violation than optimal entanglement distillation when the number of available copies is small.

This inverts the resource hierarchy. Entanglement distillation uses more resources (classical communication) and produces a more refined product (Bell states). It should dominate. And asymptotically, it does. But for a finite number of copies, the communication overhead costs more than it buys. The parties spend copies coordinating their distillation protocol rather than extracting nonlocality directly.

Nonlocality distillation avoids this overhead by operating locally. Each party manipulates their half of the shared states independently, without communication. The protocol is cruder — it cannot produce Bell states — but it extracts more CHSH violation per copy when copies are scarce. The advantage is not in the quality of the output but in the efficiency of the input usage.

The structural lesson is about the cost of coordination. Communication is a resource, not a free operation. When the primary resource (copies of the shared state) is abundant, the cost of communication is negligible and the coordinated strategy dominates. When the primary resource is scarce, the communication cost becomes significant relative to the total budget, and the uncoordinated strategy — despite its theoretical inferiority — extracts more value per unit of input.

Asymptotic optimality and finite optimality answer different questions. The strategy that wins eventually is not the strategy that wins now.