Dicke state

Dicke states are defined in a system of $N$ spin-$s$ particles as the simultaneous eigenstates of the angular momentum operators $\{\backslash vec\; J\}^2$ and $J\_z$ by the equations

: $\{\backslash vec\; J\}^2|j,j\_z,\backslash alpha\backslash rangle\; =\; j(j+1)|j,j\_z,\backslash alpha\backslash rangle$

and

: $J\_z\; |j,j\_z,\backslash alpha\backslash rangle\; =\; j\_z|j,j\_z,\backslash alpha\backslash rangle.$

Here, $\backslash alpha$ is a label used to distinguish several states orthogonal to each other, for which the two eigenvalues are the same.

It is worth to consider the $s=1/2$ case, namely an $N$-qubit system. For $j=N/2$, Dicke states are symmetric. In this case, we do not need the additional parameter $\backslash alpha$, since for a given $j\_z,$ there is only a single simultaneous eigenstate of $\{\backslash vec\; J\}^2$ and $J\_z$.

It is also common to use for the characterization of these states the quantity $n=N/2-j\_z$.{{cite journal |last1=Gühne |first1=Otfried |last2=Tóth |first2=Géza |title=Entanglement detection |journal=Physics Reports |date=April 2009 |volume=474 |issue=1–6 |pages=1–75 |doi=10.1016/j.physrep.2009.02.004|arxiv=0811.2803 |bibcode=2009PhR...474....1G }} They can be written as

: $|D\_n^\{(N)\}\backslash rangle\; =\; \backslash binom\{N\}\{n\}^\{-1/2\}\backslash sum\_\{k\}\; \{\backslash mathcal\{P\}\}\_\{k\}(|0\backslash rangle^\{\backslash otimes(N-n)\}\; \backslash otimes\; |1\backslash rangle^\{\backslash otimes\; n\}),$

where $n=0,1,...,N$ is the number of 1's, and the summation is over all distinct permutations.

A W-state is given as

: $|W\backslash rangle\; =\; \backslash frac1\{\backslash sqrt\{N\}\}(|1000...000\backslash rangle+|0100...000\backslash rangle+|0010...000\backslash rangle+...$

+|0000...001\rangle)

and it equals the Dicke state $|D\_1^\{(N)\}\backslash rangle$.

The entanglement properties of symmetric Dicke states have been studied extensively.{{cite journal |last1=Stockton |first1=John K. |last2=Geremia |first2=J. M. |last3=Doherty |first3=Andrew C. |last4=Mabuchi |first4=Hideo |title=Characterizing the entanglement of symmetric many-particle spin- 1 2 systems |journal=Physical Review A |date=28 February 2003 |volume=67 |issue=2 |page=022112 |doi=10.1103/PhysRevA.67.022112|arxiv=quant-ph/0210117 |bibcode=2003PhRvA..67b2112S }}

Symmetric Dicke states of $N$ spin-$s$ particles can easily be mapped to symmetric Dicke states of $2sN$ spin-1/2 particles.{{cite journal |last1=Vitagliano |first1=Giuseppe |last2=Apellaniz |first2=Iagoba |last3=Egusquiza |first3=Iñigo L. |last4=Tóth |first4=Géza |title=Spin squeezing and entanglement for an arbitrary spin |journal=Physical Review A |date=7 March 2014 |volume=89 |issue=3 |page=032307 |doi=10.1103/PhysRevA.89.032307|arxiv=1310.2269 |bibcode=2014PhRvA..89c2307V }}

The case of

In an experiment, determining the fidelity with respect to pure quantum states is not an easy task in general. However, for states in the symmetric (bosonic subspace) the necessary measuement effort increases only polynomially with the number of particles. For instance, for $N$ qubits it is upper bounded by $N^2/2+3N/2+1$ local measurement settings, which is known from the theory of Permutationally invariant quantum state tomography. It is also a valid bound for measuring the fidelity with respect to symmetric Dicke states.

For the 4-qubit case, 7 local measurement settings is sufficient,{{cite journal |last1=Kiesel |first1=N. |last2=Schmid |first2=C. |last3=Tóth |first3=G. |last4=Solano |first4=E. |last5=Weinfurter |first5=H. |title=Experimental Observation of Four-Photon Entangled Dicke State with High Fidelity |journal=Physical Review Letters |date=7 February 2007 |volume=98 |issue=6 |page=063604 |doi=10.1103/PhysRevLett.98.063604|pmid=17358941 |arxiv=quant-ph/0606234 |bibcode=2007PhRvL..98f3604K }}{{cite journal |last1=Tóth |first1=Géza |last2=Wieczorek |first2=Witlef |last3=Krischek |first3=Roland |last4=Kiesel |first4=Nikolai |last5=Michelberger |first5=Patrick |last6=Weinfurter |first6=Harald |title=Practical methods for witnessing genuine multi-qubit entanglement in the vicinity of symmetric states |journal=New Journal of Physics |date=4 August 2009 |volume=11 |issue=8 |pages=083002 |doi=10.1088/1367-2630/11/8/083002|doi-access=free |arxiv=0903.3910 |bibcode=2009NJPh...11h3002T }} while for the 6-qubit case 21 local measuementy settings is sufficient.{{cite journal |last1=Wieczorek |first1=Witlef |last2=Krischek |first2=Roland |last3=Kiesel |first3=Nikolai |last4=Michelberger |first4=Patrick |last5=Tóth |first5=Géza |last6=Weinfurter |first6=Harald |title=Experimental Entanglement of a Six-Photon Symmetric Dicke State |journal=Physical Review Letters |date=10 July 2009 |volume=103 |issue=2 |page=020504 |doi=10.1103/PhysRevLett.103.020504|pmid=19659191 |doi-access=free |bibcode=2009PhRvL.103b0504W }}{{cite journal |last1=Prevedel |first1=R. |last2=Cronenberg |first2=G. |last3=Tame |first3=M. S. |last4=Paternostro |first4=M. |last5=Walther |first5=P. |last6=Kim |first6=M. S. |last7=Zeilinger |first7=A. |title=Experimental Realization of Dicke States of up to Six Qubits for Multiparty Quantum Networking |journal=Physical Review Letters |date=10 July 2009 |volume=103 |issue=2 |page=020503 |doi=10.1103/PhysRevLett.103.020503|pmid=19659190 |url=https://pure.qub.ac.uk/en/publications/ef31d425-24f2-4ebf-a40d-59e249374db6 |arxiv=0903.2212 |bibcode=2009PhRvL.103b0503P }}

When a Dicke states has been prepared in an experiment, it is important to verify that the state has been prepared with a good quality. Apart from obtaining the fidelity, a usual goal is to show that the quantum state was highly entangled.

If for a quantum state $\backslash varrho$ the fidelity with respect to W-states

: $\{\backslash rm\; Tr\}(\backslash varrho|W\backslash rangle\backslash langle\; W|)>\; 1-1/N$

holds then the quantum state is genuine multipartite entangled. This means that all the particles are entangled with each other, and the quantum state

cannot be put together with entangled quantum states of smaller units by trivial operations such as making a tensor product and mixing.

Note that the bound is approaching 1 for a large $N$, which can make experiments with large systems difficult.

For the symmetric Dicke state $|D\_\{N/2\}^\{(N)\}\backslash rangle$, if for the fidelity of a quantum state

: $\{\backslash rm\; Tr\}(\backslash varrho|D\_\{N/2\}^\{(N)\}\backslash rangle\backslash langle\; D\_\{N/2\}^\{(N)\}|)>\; \backslash frac\{1\}\{2\}\; \backslash frac\{N\}\{N-1\}$

holds then the quantum state is genuine multipartite entangled.{{cite journal |last1=Tóth |first1=Géza |title=Detection of multipartite entanglement in the vicinity of symmetric Dicke states |journal=Journal of the Optical Society of America B |date=1 February 2007 |volume=24 |issue=2 |pages=275 |doi=10.1364/JOSAB.24.000275|arxiv=quant-ph/0511237 |bibcode=2007JOSAB..24..275T }} Now the bound approaches 1/2 for large $N$, which makes experiments for detecting genuine multipartite entanglement feasible even for a large $N$.

Unlike in the case of GHZ states, the entanglement of Dicke states can be detected by measuring collective observables.{{cite journal |last1=Tóth |first1=Géza |last2=Knapp |first2=Christian |last3=Gühne |first3=Otfried |last4=Briegel |first4=Hans J. |title=Optimal Spin Squeezing Inequalities Detect Bound Entanglement in Spin Models |journal=Physical Review Letters |date=19 December 2007 |volume=99 |issue=25 |page=250405 |doi=10.1103/PhysRevLett.99.250405|pmid=18233503 |arxiv=quant-ph/0702219 |bibcode=2007PhRvL..99y0405T }} It is also possible to detect multipartite entanglement or entanglement depth of such states based on collective measurements.{{cite journal |last1=Lücke |first1=Bernd |last2=Peise |first2=Jan |last3=Vitagliano |first3=Giuseppe |last4=Arlt |first4=Jan |last5=Santos |first5=Luis |last6=Tóth |first6=Géza |last7=Klempt |first7=Carsten |title=Detecting Multiparticle Entanglement of Dicke States |journal=Physical Review Letters |date=17 April 2014 |volume=112 |issue=15 |page=155304 |doi=10.1103/PhysRevLett.112.155304|pmid=24785048 |arxiv=1403.4542 |bibcode=2014PhRvL.112o5304L }}{{cite journal |last1=Vitagliano |first1=Giuseppe |last2=Apellaniz |first2=Iagoba |last3=Kleinmann |first3=Matthias |last4=Lücke |first4=Bernd |last5=Klempt |first5=Carsten |last6=Tóth |first6=Géza |title=Entanglement and extreme spin squeezing of unpolarized states |journal=New Journal of Physics |date=20 January 2017 |volume=19 |issue=1 |pages=013027 |doi=10.1088/1367-2630/19/1/013027|arxiv=1605.07202 |bibcode=2017NJPh...19a3027V }} Finally, there are efficient methods to detect multipartite entanglement of noisey Dicke states based on their density matrix.{{cite journal |last1=Gühne |first1=Otfried |last2=Seevinck |first2=Michael |title=Separability criteria for genuine multiparticle entanglement |journal=New Journal of Physics |date=5 May 2010 |volume=12 |issue=5 |pages=053002 |doi=10.1088/1367-2630/12/5/053002|arxiv=0905.1349 |bibcode=2010NJPh...12e3002G }}

For an $N$-qubit quantum state,

: $F\_Q[\backslash varrho,J\_l]\backslash le\; N^2$

holds for $l=x,y,z,$ where $J\_l$ are the components of the collective angular momentum

: $J\_l=\backslash frac\; 1\; 2\; \backslash sum\_\{n=1\}^N\; \backslash sigma\_l^\{(n)\},$

and $\backslash sigma\_l$ are the Pauli spin matrices.

Here, $F\_Q[\backslash varrho,H]$ denotes the quantum Fisher information characterizing how well the state $\backslash varrho$

can be used to estimate the parameter $\backslash theta$ in the unitary dynamics

: $U=\backslash exp(-iH\backslash theta).$

For separable states the bound discovered by Pezze and Smerzi {{cite journal |last1=Pezzé |first1=Luca |last2=Smerzi |first2=Augusto |title=Entanglement, Nonlinear Dynamics, and the Heisenberg Limit |journal=Physical Review Letters |date=10 March 2009 |volume=102 |issue=10 |page=100401 |doi=10.1103/PhysRevLett.102.100401|pmid=19392092 |arxiv=0711.4840 |bibcode=2009PhRvL.102j0401P }}

: $F\_Q[\backslash varrho,J\_l]\backslash le\; N$

holds, which is relevant for linear interferometers, a very large class of interferometers used in experiments. For the Dicke state $|D\_\{N/2\}^\{(N)\}\backslash rangle$

: $F\_Q[\backslash varrho,J\_x]=F\_Q[\backslash varrho,J\_y]=N(N+2)/2$

holds, which corresponds to a quadratic scaling in the particle number, that is, a Heisenberg scaling.

Such Dicke states also saturate the relation{{cite journal |last1=Hyllus |first1=Philipp |last2=Laskowski |first2=Wiesław |last3=Krischek |first3=Roland |last4=Schwemmer |first4=Christian |last5=Wieczorek |first5=Witlef |last6=Weinfurter |first6=Harald |last7=Pezzé |first7=Luca |last8=Smerzi |first8=Augusto |title=Fisher information and multiparticle entanglement |journal=Physical Review A |date=16 February 2012 |volume=85 |issue=2 |page=022321 |doi=10.1103/PhysRevA.85.022321|arxiv=1006.4366 |bibcode=2012PhRvA..85b2321H }}{{cite journal |last1=Tóth |first1=Géza |title=Multipartite entanglement and high-precision metrology |journal=Physical Review A |date=16 February 2012 |volume=85 |issue=2 |page=022322 |doi=10.1103/PhysRevA.85.022322|arxiv=1006.4368 |bibcode=2012PhRvA..85b2322T }}

: $F\_Q[\backslash varrho,J\_x]+F\_Q[\backslash varrho,J\_y]+F\_Q[\backslash varrho,J\_z]\backslash le\; N(N+2),$

which is valid for any quantum state.

Greenberger-Horne-Zeilinger (GHZ) states also saturate this relation.

W-states of three qubits have been created in photons.{{cite journal |last1=Eibl |first1=Manfred |last2=Kiesel |first2=Nikolai |last3=Bourennane |first3=Mohamed |last4=Kurtsiefer |first4=Christian |last5=Weinfurter |first5=Harald |title=Experimental Realization of a Three-Qubit Entangled W State |journal=Physical Review Letters |date=18 February 2004 |volume=92 |issue=7 |page=077901 |doi=10.1103/PhysRevLett.92.077901|pmid=14995887 |bibcode=2004PhRvL..92g7901E }}

Symmetric Dicke states $|D\_\{N/2\}^\{(N)\}\backslash rangle$ have been created in a four and a six-qubit photonic experiment in which genuine four- and six-paricle entanglement, respectively, has been demonstrated.

They have also been prepared in a Bose-Einstein condensate with thousands of atoms.{{cite journal |last1=Lücke |first1=B. |last2=Scherer |first2=M. |last3=Kruse |first3=J. |last4=Pezzé |first4=L. |last5=Deuretzbacher |first5=F. |last6=Hyllus |first6=P. |last7=Topic |first7=O. |last8=Peise |first8=J. |last9=Ertmer |first9=W. |last10=Arlt |first10=J. |last11=Santos |first11=L. |last12=Smerzi |first12=A. |last13=Klempt |first13=C. |title=Twin Matter Waves for Interferometry Beyond the Classical Limit |journal=Science |date=11 November 2011 |volume=334 |issue=6057 |pages=773–776 |doi=10.1126/science.1208798|pmid=21998255 |arxiv=1204.4102 |bibcode=2011Sci...334..773L }}{{cite journal |last1=Hamley |first1=C. D. |last2=Gerving |first2=C. S. |last3=Hoang |first3=T. M. |last4=Bookjans |first4=E. M. |last5=Chapman |first5=M. S. |title=Spin-nematic squeezed vacuum in a quantum gas |journal=Nature Physics |date=April 2012 |volume=8 |issue=4 |pages=305–308 |doi=10.1038/nphys2245|arxiv=1111.1694 |bibcode=2012NatPh...8..305H }}

Dicke states have also been used for quantum metrology in cold gases and photonic systems.{{cite journal |last1=Krischek |first1=Roland |last2=Schwemmer |first2=Christian |last3=Wieczorek |first3=Witlef |last4=Weinfurter |first4=Harald |last5=Hyllus |first5=Philipp |last6=Pezzé |first6=Luca |last7=Smerzi |first7=Augusto |title=Useful Multiparticle Entanglement and Sub-Shot-Noise Sensitivity in Experimental Phase Estimation |journal=Physical Review Letters |date=19 August 2011 |volume=107 |issue=8 |page=080504 |doi=10.1103/PhysRevLett.107.080504|pmid=21929154 |arxiv=1108.6002 |bibcode=2011PhRvL.107h0504K }} In these experiments it has been demonstrated that the experimentally created Dicke states outperform separable states in metrology.

Multipartite entanglement and the depth of entanglement has been detected in Dicke states in an ensemble of cold atoms.{{cite journal |last1=Luo |first1=Xin-Yu |last2=Zou |first2=Yi-Quan |last3=Wu |first3=Ling-Na |last4=Liu |first4=Qi |last5=Han |first5=Ming-Fei |last6=Tey |first6=Meng Khoon |last7=You |first7=Li |title=Deterministic entanglement generation from driving through quantum phase transitions |journal=Science |date=10 February 2017 |volume=355 |issue=6325 |pages=620–623 |doi=10.1126/science.aag1106|pmid=28183976 |arxiv=1702.03120 |bibcode=2017Sci...355..620L }}{{cite journal |last1=Xin |first1=Lin |last2=Barrios |first2=Maryrose |last3=Cohen |first3=Julia T. |last4=Chapman |first4=Michael S. |title=Long-Lived Squeezed Ground States in a Quantum Spin Ensemble |journal=Physical Review Letters |date=29 September 2023 |volume=131 |issue=13 |page=133402 |doi=10.1103/PhysRevLett.131.133402|pmid=37832022 |arxiv=2202.12338 |bibcode=2023PhRvL.131m3402X }}

Bipartite entanglement and Einstein-Podolsky-Rosen (EPR) steering has been detected in Dicke states of an ensemble of thousands of cold atoms.{{cite journal |last1=Lange |first1=Karsten |last2=Peise |first2=Jan |last3=Lücke |first3=Bernd |last4=Kruse |first4=Ilka |last5=Vitagliano |first5=Giuseppe |last6=Apellaniz |first6=Iagoba |last7=Kleinmann |first7=Matthias |last8=Tóth |first8=Géza |last9=Klempt |first9=Carsten |title=Entanglement between two spatially separated atomic modes |journal=Science |date=27 April 2018 |volume=360 |issue=6387 |pages=416–418 |doi=10.1126/science.aao2035|pmid=29700263 |arxiv=1708.02480 |bibcode=2018Sci...360..416L }}{{cite journal |last1=Vitagliano |first1=Giuseppe |last2=Fadel |first2=Matteo |last3=Apellaniz |first3=Iagoba |last4=Kleinmann |first4=Matthias |last5=Lücke |first5=Bernd |last6=Klempt |first6=Carsten |last7=Tóth |first7=Géza |title=Number-phase uncertainty relations and bipartite entanglement detection in spin ensembles |journal=Quantum |date=9 February 2023 |volume=7 |pages=914 |doi=10.22331/q-2023-02-09-914|arxiv=2104.05663 |bibcode=2023Quant...7..914V }}

• Bell state
• Graph state
• Cluster state
• Optical cluster state
• Greenberger-Horne-Zeilinger (GHZ) state
• Dicke model
• Jaynes–Cummings model

{{reflist}}

{{Quantum computing}}

Category:Quantum information science

Category:Quantum states