Efficient local operations and classical communication extraction of quantum information encoded in stabilizer codes

We construct a protocol for extracting distributed single-qubit quantum information encoded in a stabilizer code of multiple qubits, only with single-qubit local operations and classical communication (LOCC) without global operations or entanglement resources. This protocol achieves efficient extraction within a polynomial time in terms of the number of physical qubits. We apply this protocol to a setting of quantum information splitting where a subset of spatially separated parties cooperate by classical communication to extract quantum information shared among all the parties. For this task, our LOCC extraction protocol allows designing hierarchical information access structures among the parties, where the minimum number of parties required to cooperate depends on the location of extracting the shared quantum information. Extraction of quantum information encoded in the stabilizer code appears widely in distributed quantum information processing, such as quantum secret sharing. Our results allow such distributed quantum information processing without entanglement costs.