# W-state

## Definition

In quantum information theory, W-states of n qubits are defined as

$|W\rangle = \frac{1}{\sqrt{n}}\left(|100\dots 0\rangle + |010\dots 0\rangle + \dots + |00\dots 1\rangle\right)$

## Properties

One of the main properties of the W-states is concerned with type of entanglement encoding this states.

### Three qubits

Regarding to three qubits scenario: $|W\rangle = \frac{1}{\sqrt{3}}\left(|100\rangle + |010\rangle + |001\rangle\right)$ the way of entanglement encoding in W-states can be easy illustrated on a picture.

thumb|right|type of entanglement in W-states

There is no generic three qubit entanglement in W-states and all entanglement monotones as well as their normal form will be equal to zero. However tracing out one of the parties leaves the remaing two entangled. In this sense three qubit W-states contain maximal amount of sum of two qubit entanglement.

Referring to the equivalence classes W-states define a complemetary class to the GHZ-states.

### General situation

In general case one can speak about W-type of encoding of entanglement, which means that one deals with some pure state of n qubits which contains no generic n qubit entanglement but rather m qubit, where m < n.

## Physical realizations

W-states are known to realize first excitation over the ground state of free spinless fermions confined to one dimensional chain, which are described by the following Hamiltioan

HF =  − ∑iJi, i + 1ci † ci + 1.

Due to the progress made in physics of cold atoms this kind of systems can be pretty easily produced experimentally, using for instance trapping techniques for atoms and putting those into one dimensional optical lattice.

Category:Handbook of Quantum Information