Cluster state

Here are some examples of one-dimensional cluster states (d=1), for ${\displaystyle n=2,3,4}$, where ${\displaystyle n}$ is the number of qubits. We take ${\displaystyle \kappa _{a}=0}$ for all ${\displaystyle a}$, which means the cluster state is the unique simultaneous eigenstate that has corresponding eigenvalue 1 under all correlation operators. In each example the set of correlation operators ${\displaystyle \{K^{(a)}\}_{a}}$and the corresponding cluster state is listed.

• ${\displaystyle n=2}$
  ${\displaystyle \{\sigma _{x}\sigma _{z},\ \sigma _{z}\sigma _{x}\}}$

${\displaystyle |\phi \rangle ={\frac {1}{\sqrt {2}}}(|0+\rangle +|1-\rangle )}$
This is an EPR-pair (up to local transformations).

• ${\displaystyle n=3}$

${\displaystyle \{\sigma _{x}\sigma _{z}I,\ \sigma _{z}\sigma _{x}\sigma _{z},\ I\sigma _{z}\sigma _{x}\}}$

${\displaystyle |\phi \rangle ={\frac {1}{\sqrt {2}}}(|+0+\rangle +|-1-\rangle )}$
This is the GHZ-state (up to local transformations).

• ${\displaystyle n=4}$

${\displaystyle \{\sigma _{x}\sigma _{z}II,\ \sigma _{z}\sigma _{x}\sigma _{z}I,\ I\sigma _{z}\sigma _{x}\sigma _{z},\ II\sigma _{z}\sigma _{x}\}}$

${\displaystyle |\phi \rangle ={\frac {1}{2}}(|+0+0\rangle +|+0-1\rangle +|-1-0\rangle +|-1+1\rangle )}$.

This is not a GHZ-state and can not be converted to a GHZ-state with local operations.

In all examples ${\displaystyle I}$ is the identity operator, and tensor products are omitted. The states above can be obtained from the all zero state ${\displaystyle |0\ldots 0\rangle }$ by first applying a Hadamard gate to every qubit, and then a controlled-Z gate between all qubits that are adjacent to each other.

Cluster states can be realized experimentally. One way to create a cluster state is by encoding logical qubits into the polarization of photons, one common encoding is the following:

${\displaystyle {\begin{cases}|0\rangle _{\rm {L}}\longleftrightarrow |{\rm {H\rangle }}\\|1\rangle _{\rm {L}}\longleftrightarrow |{\rm {V\rangle }}\end{cases}}}$

This is not the only possible encoding, however it is one of the simplest: with this encoding entangled pairs can be created experimentally through spontaneous parametric down-conversion.[3][4] The entangled pairs that can be generated this way have the form

${\displaystyle |\psi \rangle ={\frac {1}{\sqrt {2}}}{\big (}|{\rm {H\rangle |{\rm {H\rangle +e^{i\phi }|{\rm {V\rangle |{\rm {V\rangle {\big )}}}}}}}}}}$

equivalent to the logical state

${\displaystyle |\psi \rangle ={\frac {1}{\sqrt {2}}}{\big (}|0\rangle |0\rangle +e^{i\phi }|1\rangle |1\rangle {\big )}}$

for the two choices of the phase ${\displaystyle \phi =0,\pi }$ the two Bell states ${\displaystyle |\Phi ^{+}\rangle ,|\Phi ^{-}\rangle }$ are obtained: these are themselves two examples of two-qubits cluster states. Through the use of linear optic devices as beam-splitters or wave-plates these Bell states can interact and form more complex cluster states.[5] Cluster states have been created also in optical lattices of cold atoms.[6]

After a cluster state was created in an experiment, it is important to verify that indeed, an entangled quantum state has been created and obtain the fidelity with respect to an ideal cluster state. There are efficient conditions to detect entanglement close to cluster states, that need only the minimal two local measurement settings.[7] Similar conditions can also be used to estimate the fidelity with respect to an ideal cluster state.[8] Bell inequalities have also been developed for cluster states.[9] [10] [11] All these entanglement conditions and Bell inequalities are based on the stabilizer formalism.[12]

• Bell state
• Graph state
• Optical cluster state