cluster state

Here are some examples of one-dimensional cluster states (d=1), for $n=2,3,4$, where $n$ is the number of qubits. We take $\backslash kappa\_a=0$ for all $a$, 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 $\backslash \{K^\{(a)\}\backslash \}\_a$and the corresponding cluster state is listed.

• $n=2$
  $\backslash \{\backslash sigma\_x\backslash sigma\_z,\backslash \; \backslash sigma\_z\backslash sigma\_x\backslash \}$

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

• $n=3$

:$\backslash \{\; \backslash sigma\_x\backslash sigma\_z\; I,\backslash \; \backslash sigma\_z\backslash sigma\_x\; \backslash sigma\_z,\backslash \; I\backslash sigma\_z\backslash sigma\_x\backslash \}$

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

• $n=4$

:$\backslash \{\; \backslash sigma\_x\backslash sigma\_z\; I\; I,\backslash \; \backslash sigma\_z\backslash sigma\_x\; \backslash sigma\_z\; I,\backslash \; I\backslash sigma\_z\backslash sigma\_x\backslash sigma\_z,\backslash \; II\; \backslash sigma\_z\backslash sigma\_x\; \backslash \}$

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

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

In all examples $I$ is the identity operator, and tensor products are omitted. The states above can be obtained from the all zero state $|0\backslash ldots\; 0\; \backslash rangle$ 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:

$\backslash begin\{cases\}$

|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.{{cite journal | author=P. Walther, K. J. Resch, T. Rudolph, E. Schenck, H. Weinfurter, V. Vedral, M. Aspelmeyer and A. Zeilinger| title=Experimental one-way quantum computing| journal=Nature| year=2005| volume= 434| pages=169–76 |doi=10.1038/nature03347 | pmid=15758991 | issue=7030|arxiv = quant-ph/0503126 |bibcode = 2005Natur.434..169W | s2cid= 119329998}}{{cite journal |author1=N. Kiesel |author2=C. Schmid |author3=U. Weber |author4=G. Tóth |author5=O. Gühne |author6=R. Ursin |author7=H. Weinfurter | title=Experimental Analysis of a 4-Qubit Cluster State| journal=Phys. Rev. Lett.| year=2005| volume= 95|issue=21 | pages=210502 |doi=10.1103/PhysRevLett.95.210502 |pmid=16384122|

arxiv = quant-ph/0508128 |bibcode = 2005PhRvL..95u0502K |s2cid=5322108 }} The entangled pairs that can be generated this way have the form

$|\backslash psi\backslash rangle\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}\backslash big(|\backslash rm\; H\backslash rangle|\backslash rm$

H\rangle+e^{i\phi}|\rm V\rangle|\rm V\rangle\big)

equivalent to the logical state

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

for the two choices of the phase $\backslash phi\; =\; 0,\; \backslash pi$ the two Bell states $|\backslash Phi^+\backslash rangle,\; |\backslash Phi^-\backslash rangle$ 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.{{Cite journal|last1=Zhang|first1=An-Ning|last2=Lu|first2=Chao-Yang|last3=Zhou|first3=Xiao-Qi|last4=Chen|first4=Yu-Ao|last5=Zhao|first5=Zhi|last6=Yang|first6=Tao|last7=Pan|first7=Jian-Wei|date=2006-02-17|title=Experimental construction of optical multiqubit cluster states from Bell states|url=https://link.aps.org/doi/10.1103/PhysRevA.73.022330|journal=Physical Review A|language=en|volume=73|issue=2|pages=022330| arxiv=quant-ph/0501036|doi=10.1103/PhysRevA.73.022330|bibcode=2006PhRvA..73b2330Z |s2cid=118882320 |issn=1050-2947}} Cluster states have been created also in optical lattices of

cold atoms.{{cite journal |author1=O. Mandel |author2=M. Greiner |author3=A. Widera |author4=T. Rom |author5=T. W. Hänsch |author6=I. Bloch | title=Controlled collisions for multi-particle entanglement of optically trapped atoms| journal=Nature| year=2003| volume= 425|issue=6961 | pages=937–940 |doi=10.1038/nature02008 |pmid=14586463|arxiv = quant-ph/0308080 |bibcode = 2003Natur.425..937M |s2cid=4408587 }}

After a cluster state was created in an experiment, it is important to verify that indeed, an entangled quantum state has been created. The fidelity with respect to the $N$-qubit cluster state $|C\_N\backslash rangle$ is given by

$F\_\{CN\}=\{\backslash rm\; Tr\}(\backslash rho\; |C\_N\backslash rangle\backslash langle\; C\_N|),$

It has been shown that if $F\_\{CN\}>1/2$, then the state $\backslash rho$ has genuine multiparticle entanglement.{{cite journal |last1=Tóth |first1=Géza |last2=Gühne |first2=Otfried |title=Detecting Genuine Multipartite Entanglement with Two Local Measurements |journal=Physical Review Letters |date=17 February 2005 |volume=94 |issue=6 |page=060501 |doi=10.1103/PhysRevLett.94.060501|pmid=15783712 |arxiv=quant-ph/0405165 |bibcode=2005PhRvL..94f0501T |s2cid=13371901 }}

Thus, one can obtain an entanglement witness detecting entanglement close the cluster states as

$W\_\{CN\}=\backslash frac1\; 2\; \{\backslash rm\; Identity\}-\; |C\_N\backslash rangle\backslash langle\; C\_N|.$

where signals genuine multiparticle entanglement.

Such a witness cannot be measured directly. It has to be decomposed to a sum of correlations terms, which can then be measured. However, for large systems this approach can be difficult.

There are also entanglement witnesses that work in very large systems, and they also detect genuine multipartite entanglement close to cluster states. They need only the minimal two local measurement settings. Similar conditions can also be used to put a lower bound on the fidelity with respect to an ideal cluster state.{{cite journal |last1=Tóth |first1=Géza |last2=Gühne |first2=Otfried |title=Entanglement detection in the stabilizer formalism |journal=Physical Review A |date=29 August 2005 |volume=72 |issue=2 |page=022340 |doi=10.1103/PhysRevA.72.022340|arxiv=quant-ph/0501020 |bibcode=2005PhRvA..72b2340T |s2cid=56269409 }} These criteria have been used first in an experiment realizing four-qubit cluster states with photons. These approaches have also been used to propose methods for detecting entanglement in a smaller part of a large cluster state or graph state realized in optical lattices.{{cite journal |last1=Alba |first1=Emilio |last2=Tóth |first2=Géza |last3=García-Ripoll |first3=Juan José |title=Mapping the spatial distribution of entanglement in optical lattices |journal=Physical Review A |date=21 December 2010 |volume=82 |issue=6 |page=062321 |doi=10.1103/PhysRevA.82.062321|arxiv=1007.0985 }}

Bell inequalities have also been developed for cluster states.{{cite journal |last1=Scarani |first1=Valerio |last2=Acín |first2=Antonio |last3=Schenck |first3=Emmanuel |last4=Aspelmeyer |first4=Markus |title=Nonlocality of cluster states of qubits |journal=Physical Review A |date=18 April 2005 |volume=71 |issue=4 |page=042325 |arxiv=quant-ph/0405119| doi=10.1103/PhysRevA.71.042325|bibcode=2005PhRvA..71d2325S |s2cid=4805039 |url=https://archive-ouverte.unige.ch/unige:47355 }}{{cite journal |last1=Gühne |first1=Otfried |last2=Tóth |first2=Géza |last3=Hyllus |first3=Philipp |last4=Briegel |first4=Hans J. |title=Bell Inequalities for Graph States |journal=Physical Review Letters |date=14 September 2005 |volume=95 |issue=12 |page=120405 |doi=10.1103/PhysRevLett.95.120405|pmid=16197057 |arxiv=quant-ph/0410059 |bibcode=2005PhRvL..95l0405G |s2cid=5973814 }}{{cite journal |last1=Tóth |first1=Géza |last2=Gühne |first2=Otfried |last3=Briegel |first3=Hans J. |title=Two-setting Bell inequalities for graph states |journal=Physical Review A |date=2 February 2006 |volume=73 |issue=2 |pages=022303 |doi=10.1103/PhysRevA.73.022303|arxiv=quant-ph/0510007 |bibcode=2006PhRvA..73b2303T |s2cid=108291031 }} All these entanglement conditions and Bell inequalities are based on the stabilizer formalism.{{cite journal |last1=Gottesman |first1=Daniel |title=Class of quantum error-correcting codes saturating the quantum Hamming bound |journal=Physical Review A |date=1 September 1996 |volume=54 |issue=3 |pages=1862–1868 |doi=10.1103/PhysRevA.54.1862|pmid=9913672 |arxiv=quant-ph/9604038 |bibcode=1996PhRvA..54.1862G |s2cid=16407184 }}

• Bell state
• Graph state
• Optical cluster state
• Dicke state

{{cite book |last=Briegel |first=Hans J. |editor-last1=Greenberger |editor-first1=Daniel |editor-last2=Hentschel |editor-first2=Klaus |editor-last3=Weinert |editor-first3=Friedel |title=Compendium of Quantum Physics - Concepts, Experiments, History and Philosophy |publisher=Springer |pages=96–105 |chapter=Cluster States |date=12 August 2009 |isbn=978-3-540-70622-9|name-list-style=amp}}

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Category:Quantum states