Cluster state

In quantum information and quantum computing, a cluster state is a type of highly entangled state of multiple qubits. Cluster states are generated in lattices of qubits with Ising type interactions. A cluster C is a connected subset of a d-dimensional lattice, and a cluster state is a pure state of the qubits located on C. They are different from other types of entangled states such as GHZ states or W states in that it is more difficult to eliminate quantum entanglement (via projective measurements) in the case of cluster states. Another way of thinking of cluster states is as a particular instance of graph states, where the underlying graph is a connected subset of a d-dimensional lattice. Cluster states are especially useful in the context of the one-way quantum computer. For a comprehensible introduction to the topic see.

Formally, cluster states $|\phi_{\{\kappa\}}\rangle_{C}$ are states which obey the set eigenvalue equations:

$K^{(a)} {\left|\phi_{\{\kappa\}}\right\rangle_{C}} =(-1)^{\kappa_{a}} {\left|\phi_{\{\kappa\}}\right\rangle_{C}}$

where $K^{(a)}$ are the correlation operators

$K^{(a)} = \sigma_x^{(a)} \bigotimes_{b\in \mathrm{N}(a)} \sigma_z^{(b)}$

with $\sigma_x$ and $\sigma_z$ being Pauli matrices, $N(a)$ denoting the neighbourhood of $a$ and $\{\kappa_a\in\{0,1\}|a\in C\}$ being a set of binary parameters specifying the particular instance of a cluster state.

Cluster states have been realized expertimentally. They have been obtained in photonic experiments using parametric downconversion . They have been created also in optical lattices of cold atoms .

See also

• Bell state
• Graph state