Distributed entanglement

TLDR

Three qubits A, B, and C may be entangled with each other. The study demonstrates a trade‑off between a qubit’s entanglement with two others. The authors express this trade‑off using concurrence and define a permutation‑invariant three‑way entanglement measure. The squared concurrence between A and B plus that between A and C cannot exceed the squared concurrence between A and the pair BC, and this bound is tight.

Abstract

Consider three qubits A, B, and C which may be entangled with each other. We show that there is a trade-off between A's entanglement with B and its entanglement with C. This relation is expressed in terms of a measure of entanglement called the concurrence, which is related to the entanglement of formation. Specifically, we show that the squared concurrence between A and B, plus the squared concurrence between A and C, cannot be greater than the squared concurrence between A and the pair BC. This inequality is as strong as it could be, in the sense that for any values of the concurrences satisfying the corresponding equality, one can find a quantum state consistent with those values. Further exploration of this result leads to a definition of an essential three-way entanglement of the system, which is invariant under permutations of the qubits.