W state

Properties

The W state is the representative of one of the two non-biseparable classes of three-qubit states (the other being the GHZ state) which cannot be transformed (not even probabilistically) into each other by local quantum operations. Thus | W ⟩ and | G H Z ⟩ represent two very different kinds of tripartite entanglement.

This difference is, for example, illustrated by the following interesting property of the W state: if one of the three qubits is lost, the state of the remaining 2-qubit system is still entangled. This robustness of W-type entanglement contrasts strongly with the Greenberger-Horne-Zeilinger state which is fully separable after loss of one qubit.

The states in the W class can be distinguished from all other three-qubit states by means of multipartite entanglement measures. In particular, W states have non-zero entanglement across any bipartition while the 3-tangle vanishes, which is also non-zero for GHZ-type states.

Generalization

The notion of W state has been generalized for n qubits and then refers to the quantum superpostion with equal expansion coefficients of all possible pure states in which exactly one of the qubits in an "excited state" | 1 ⟩ , while all other ones are in the "ground state" | 0 ⟩

Both the robustness against particle loss and the LOCC-inequivalence with the (generalized) GHZ state also hold for the n -qubit W state.

Applications

In systems in which a single qubit is stored in an ensemble of many two level systems the logical "1" is often represented by the W state while the logical "0" is represented by the state | 00...0 ⟩ . Here the W state's robustness against particle loss is a very beneficial property ensuring good storage properties of these ensemble based quantum memories.