Threshold ( Q , P ) distillation of multipartite quantum correlations

We introduce a threshold quantum distillation protocol that enables the extraction of perfect quantum correlations from multiple imperfect copies of a multipartite quantum state using only a subset of the parties ( <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>Q</a:mi> <a:mo><</a:mo> <a:mi>P</a:mi> </a:mrow> </a:math> ) in the network. In particular, we show that for high-dimensional Greenberger–Horne–Zeilinger states, entanglement can be distilled by local filtering operations applied by just a single party, independent of the total number of parties in the network. In contrast, for the <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>W</b:mi> </b:mrow> </b:math> states, at least <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mi>P</c:mi> <c:mo>−</c:mo> <c:mn>1</c:mn> </c:mrow> </c:math> parties are required for successful distillation, highlighting a strong connection between distillability and separability. We provide explicit protocols for threshold entanglement distillation and threshold steering distillation (TSD), and establish a direct link between the two: Any assemblage that can be distilled via TSD necessarily corresponds to a distillable entangled state. Our framework introduces a scalable and experimentally viable approach to distillation in quantum networks, with potential applications in distributed quantum computing, communication, and cryptography.