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 $|\phi_{\{\kappa\}}\rangle_{C}$ 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 ^[3] .^[4] They have been created also in optical lattices of cold atoms .^[5]

See also

• Bell state
• Graph state

References

1. H. J. Briegel and R. Raussendorf (2001). "Persistent Entanglement in arrays of Interacting Particles". Physical Review Letters 86 (5): 910–3. arXiv:quant-ph/0004051. 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:quant-ph/0503126. Bibcode:2005Natur.434..169W. doi:10.1038/nature03347. PMID 15758991.
4. N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin, and H. Weinfurter (2005). "Experimental Analysis of a 4-Qubit Cluster State". Phys. Rev. Lett. 95: 210502. arXiv:quant-ph/0508128. Bibcode:2005PhRvL..95u0502K. doi:10.1103/PhysRevLett.95.210502. PMID 16384122.
5. O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hänsch, and I. Bloch (2003). "Controlled collisions for multi-particle entanglement of optically trapped atoms". Science 425: 937–940. arXiv:quant-ph/0308080. Bibcode:2003Natur.425..937M. doi:10.1038/nature02008. PMID 14586463.