W state

The W state is an entangled quantum state of three qubits which has the following shape

|W\rangle = \frac{1}{\sqrt{3}}(|001\rangle + |010\rangle + |100\rangle)

and which is remarkable for representing a specific type of multipartite entanglement and for occurring in several applications in quantum information theory.

Properties

The W state is the representative of one of the two non-biseparable^[1] 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.^[2] Thus $|W\rangle$ and $|GHZ\rangle$ 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^[3] while the 3-tangle vanishes, which is also non-zero for GHZ-type states.^[2]

Generalization

The notion of W state has been generalized for $n$ qubits^[2] 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" $|1\rangle$ , while all other ones are in the "ground state" $|0\rangle$

|W\rangle = \frac{1}{\sqrt{n}}(|100...0\rangle + |010...0\rangle + ... + |00...01\rangle)

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 $|00...0\rangle$ . Here the W state's robustness against particle loss is a very beneficial property ensuring good storage properties of these ensemble based quantum memories.^[4]

References

↑ A pure state $|\psi\rangle$ of $N$ parties is called biseparable, if one can find a partition of the parties in two disjoint subsets $A$ and $B$ with $A\cup B=\{1,...,N\}$ such that $|\psi\rangle = |\phi\rangle_A\otimes|\gamma\rangle_B$ , i.e. $|\psi\rangle$ is a product state with respect to he partition $A|B$ .
↑ 2.0 2.1 2.2 W. Dür, G. Vidal, and J.I. Cirac (2000). "Three qubits can be entangled in two inequivalent ways". Phys. Rev. A 62: 062314. arXiv:quant-ph/0005115. Bibcode:2000PhRvA..62f2314D. doi:10.1103/PhysRevA.62.062314.
↑ A bipartition of the three qubits $1,2,3$ is any grouping $(12) 3, 1 (23)$ and $(13) 2$ in which two qubits are considered to belong to the same party. The three qubit state can then be considered as a state on $\mathbb{C}^4\otimes \mathbb{C}^2$ and studied with bipartite entanglement measures.
↑ M. Fleischhauer and M. D. Lukin (2002). "Quantum memory for photons: Dark-state polaritons" 65. p. 022314. arXiv:quant-ph/0106066. Bibcode:2002PhRvA..65b2314F. doi:10.1103/PhysRevA.65.022314.