Cluster State
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Cluster State

In quantum information and quantum computing, a cluster state[1] 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.[2]

Formally, cluster states ${\displaystyle |\phi _{\{\kappa \}}\rangle _{C}}$ are states which obey the set eigenvalue equations:

${\displaystyle K^{(a)}{\left|\phi _{\{\kappa \}}\right\rangle _{C}}=(-1)^{\kappa _{a}}{\left|\phi _{\{\kappa \}}\right\rangle _{C}}}$

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

${\displaystyle K^{(a)}=\sigma _{x}^{(a)}\bigotimes _{b\in \mathrm {N} (a)}\sigma _{z}^{(b)}}$

with ${\displaystyle \sigma _{x}}$ and ${\displaystyle \sigma _{z}}$ being Pauli matrices, ${\displaystyle N(a)}$ denoting the neighbourhood of ${\displaystyle a}$ and ${\displaystyle \{\kappa _{a}\in \{0,1\}|a\in C\}}$ being a set of binary parameters specifying the particular instance of a cluster state.

Cluster states have been realized experimentally. They have been obtained in photonic experiments using parametric downconversion [3] .[4] They have been created also in optical lattices of cold atoms .[5]

## References

1. ^ H. J. Briegel; R. Raussendorf (2001). "Persistent Entanglement in arrays of Interacting Particles". Physical Review Letters. 86 (5): 910-3. arXiv:. Bibcode:2001PhRvL..86..910B. doi:10.1103/PhysRevLett.86.910. PMID 11177971.
2. ^ Briegel, Hans J. "Cluster States". In Greenberger, Daniel; Hentschel, Klaus & Weinert, Friedel. Compendium of Quantum Physics - Concepts, Experiments, History and Philosophy. Springer. pp. 96-105. ISBN 978-3-540-70622-9.
3. ^ P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer and A. Zeilinger (2005). "Experimental one-way quantum computing". Nature. 434 (7030): 169-76. arXiv:. Bibcode:2005Natur.434..169W. doi:10.1038/nature03347. PMID 15758991.
4. ^ N. Kiesel; C. Schmid; U. Weber; G. Tóth; O. Gühne; R. Ursin; H. Weinfurter (2005). "Experimental Analysis of a 4-Qubit Cluster State". Phys. Rev. Lett. 95: 210502. arXiv:. Bibcode:2005PhRvL..95u0502K. doi:10.1103/PhysRevLett.95.210502. PMID 16384122.
5. ^ O. Mandel; M. Greiner; A. Widera; T. Rom; T. W. Hänsch; I. Bloch (2003). "Controlled collisions for multi-particle entanglement of optically trapped atoms". Nature. 425: 937-940. arXiv:. Bibcode:2003Natur.425..937M. doi:10.1038/nature02008. PMID 14586463.