Dicke state

In quantum optics and quantum information, a Dicke state is a quantum state defined by Robert H. Dicke in connection to spontaneous radiation processes taking place in an ensemble of two-state atoms. A Dicke state is the simultaneous eigenstate of the angular momentum operators ${\vec {J}}^{2}$ and $J_{z}.$ ^[1] Dicke states have recenly been realized with photons with up to six particles and cold atoms of more than thousands of particles. They are highly entangled, and in quantum metrology they lead to the maximal Heisenberg scaling of the precision of parameter estimation.

Defining equations

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

{\vec {J}}^{2}|j,j_{z},\alpha \rangle =j(j+1)|j,j_{z},\alpha \rangle

and

J_{z}|j,j_{z},\alpha \rangle =j_{z}|j,j_{z},\alpha \rangle .

Here, $\alpha$ 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 $\alpha$ , since for a given $j_{z},$ there is only a single simultaneous eigenstate of ${\vec {J}}^{2}$ and $J_{z}$ .

It is also common to use for the characterization of these states the quantity $n=N/2-j_{z}$ .^[2] They can be written as

|D_{n}^{(N)}\rangle ={\binom {N}{n}}^{-1/2}\sum _{k}{\mathcal {P}}_{k}(|0\rangle ^{\otimes (N-n)}\otimes |1\rangle ^{\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\rangle ={\frac {1}{\sqrt {N}}}(|1000...000\rangle +|0100...000\rangle +|0010...000\rangle +...+|0000...001\rangle )

and it equals the Dicke state $|D_{1}^{(N)}\rangle$ .

The entanglement properties of symmetric Dicke states have been studied extensively.^[3]

Symmetric Dicke states of $N$ spin- $s$ particles can easily be mapped to symmetric Dicke states of $2sN$ spin-1/2 particles.^[4]

The case of $j<N/2,$ i.e., the case of non-symmetric Dicke states in multi-qubit systems is more complicated. In this case, the simultaneous eigenstates are denoted by $|j,j_{z},\alpha \rangle$ , and we need now the $\alpha$ label to dinstinguish several eigenstates with the same eigenvalues orthogonal to each other. These states can also be obtained expclicitly.^[5]

Fidelity

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 polinomially 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,^[6]^[7] while for the 6-qubit case 21 local measuementy settings is sufficient.^[8]^[9]^[7]

Entanglement properties of Dicke states

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 $\varrho$ the fidelity with respect to W-states

{\rm {Tr}}(\varrho |W\rangle \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 $|D_{N/2}^{(N)}\rangle$ , if for the fidelity of a quantum state

{\rm {Tr}}(\varrho |D_{N/2}^{(N)}\rangle \langle D_{N/2}^{(N)}|)>{\frac {1}{2}}{\frac {N}{N-1}}

holds then the quantum state is genuine multipartite entangled.^[10] 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.^[11] It is also possible to detect multipartite entanglement or entanglement depth of such states based on collective measurements.^[12]^[13] Finally, there are efficient methods to detect multipartite entanglement of noisey Dicke states based on their density matrix.^[14]

Quantum metrological properties

For an $N$ -qubit quantum state,

F_{Q}[\varrho ,J_{l}]\leq N^{2}

holds for $l=x,y,z,$ where $J_{l}$ are the components of the collective angular momentum

J_{l}={\frac {1}{2}}\sum _{n=1}^{N}\sigma _{l}^{(n)},

and $\sigma _{l}$ are the Pauli spin matrices.

Here, $F_{Q}[\varrho ,H]$ denotes the quantum Fisher information characterizing how well the state $\varrho$ can be used to estimate the parameter $\theta$ in the unitary dynamics

U=\exp(-iH\theta ).

For separable states the bound discovered by Pezze and Smerzi ^[15]

F_{Q}[\varrho ,J_{l}]\leq N

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

F_{Q}[\varrho ,J_{x}]=F_{Q}[\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^[16]^[17]

F_{Q}[\varrho ,J_{x}]+F_{Q}[\varrho ,J_{y}]+F_{Q}[\varrho ,J_{z}]\leq N(N+2),

which is valid for any quantum state. Greenberger-Horne-Zeilinger (GHZ) states also saturate this relation.

Experiments with Dicke states

W-states of three qubits have been created in photons.^[18]

Symmetric Dicke states $|D_{N/2}^{(N)}\rangle$ have been created in a four and a six-qubit photonic experiment in which genuine four- and six-paricle entanglement, respetively, has been demonstrated.^[6]^[8]^[9]

They have also been prepared in a Bose-Einstein condensate with thousands of atoms.^[19]^[20]

Dicke states have also been used for quantum metrology in cold gasses^[19] and photonic systems.^[21] 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.^[12]^[22]^[23]

Bipartite entanglement and Einstein-Podolsky-Rosen (EPR) steering has been detected in Dicke states of an ensemble of thousands of cold atoms.^[24]^[25]

References

^ Dicke, R. H. (1 January 1954). "Coherence in Spontaneous Radiation Processes". Physical Review. 93 (1): 99–110. Bibcode:1954PhRv...93...99D. doi:10.1103/PhysRev.93.99.
^ Gühne, Otfried; Tóth, Géza (April 2009). "Entanglement detection". Physics Reports. 474 (1–6): 1–75. arXiv:0811.2803. Bibcode:2009PhR...474....1G. doi:10.1016/j.physrep.2009.02.004.
^ Stockton, John K.; Geremia, J. M.; Doherty, Andrew C.; Mabuchi, Hideo (28 February 2003). "Characterizing the entanglement of symmetric many-particle spin- 1 2 systems". Physical Review A. 67 (2): 022112. arXiv:quant-ph/0210117. Bibcode:2003PhRvA..67b2112S. doi:10.1103/PhysRevA.67.022112.
^ Vitagliano, Giuseppe; Apellaniz, Iagoba; Egusquiza, Iñigo L.; Tóth, Géza (7 March 2014). "Spin squeezing and entanglement for an arbitrary spin". Physical Review A. 89 (3): 032307. arXiv:1310.2269. Bibcode:2014PhRvA..89c2307V. doi:10.1103/PhysRevA.89.032307.
^ Cirac, J. I.; Ekert, A. K.; Macchiavello, C. (24 May 1999). "Optimal Purification of Single Qubits". Physical Review Letters. 82 (21): 4344–4347. arXiv:quant-ph/9812075. Bibcode:1999PhRvL..82.4344C. doi:10.1103/PhysRevLett.82.4344.
^ ^a ^b Kiesel, N.; Schmid, C.; Tóth, G.; Solano, E.; Weinfurter, H. (7 February 2007). "Experimental Observation of Four-Photon Entangled Dicke State with High Fidelity". Physical Review Letters. 98 (6): 063604. arXiv:quant-ph/0606234. Bibcode:2007PhRvL..98f3604K. doi:10.1103/PhysRevLett.98.063604. PMID 17358941.
^ ^a ^b Tóth, Géza; Wieczorek, Witlef; Krischek, Roland; Kiesel, Nikolai; Michelberger, Patrick; Weinfurter, Harald (4 August 2009). "Practical methods for witnessing genuine multi-qubit entanglement in the vicinity of symmetric states". New Journal of Physics. 11 (8): 083002. arXiv:0903.3910. Bibcode:2009NJPh...11h3002T. doi:10.1088/1367-2630/11/8/083002.
^ ^a ^b Wieczorek, Witlef; Krischek, Roland; Kiesel, Nikolai; Michelberger, Patrick; Tóth, Géza; Weinfurter, Harald (10 July 2009). "Experimental Entanglement of a Six-Photon Symmetric Dicke State". Physical Review Letters. 103 (2): 020504. Bibcode:2009PhRvL.103b0504W. doi:10.1103/PhysRevLett.103.020504. PMID 19659191.
^ ^a ^b Prevedel, R.; Cronenberg, G.; Tame, M. S.; Paternostro, M.; Walther, P.; Kim, M. S.; Zeilinger, A. (10 July 2009). "Experimental Realization of Dicke States of up to Six Qubits for Multiparty Quantum Networking". Physical Review Letters. 103 (2): 020503. arXiv:0903.2212. Bibcode:2009PhRvL.103b0503P. doi:10.1103/PhysRevLett.103.020503. PMID 19659190.
^ Tóth, Géza (1 February 2007). "Detection of multipartite entanglement in the vicinity of symmetric Dicke states". Journal of the Optical Society of America B. 24 (2): 275. arXiv:quant-ph/0511237. Bibcode:2007JOSAB..24..275T. doi:10.1364/JOSAB.24.000275.
^ Tóth, Géza; Knapp, Christian; Gühne, Otfried; Briegel, Hans J. (19 December 2007). "Optimal Spin Squeezing Inequalities Detect Bound Entanglement in Spin Models". Physical Review Letters. 99 (25): 250405. arXiv:quant-ph/0702219. Bibcode:2007PhRvL..99y0405T. doi:10.1103/PhysRevLett.99.250405. PMID 18233503.
^ ^a ^b Lücke, Bernd; Peise, Jan; Vitagliano, Giuseppe; Arlt, Jan; Santos, Luis; Tóth, Géza; Klempt, Carsten (17 April 2014). "Detecting Multiparticle Entanglement of Dicke States". Physical Review Letters. 112 (15): 155304. arXiv:1403.4542. Bibcode:2014PhRvL.112o5304L. doi:10.1103/PhysRevLett.112.155304. PMID 24785048.
^ Vitagliano, Giuseppe; Apellaniz, Iagoba; Kleinmann, Matthias; Lücke, Bernd; Klempt, Carsten; Tóth, Géza (20 January 2017). "Entanglement and extreme spin squeezing of unpolarized states". New Journal of Physics. 19 (1): 013027. arXiv:1605.07202. Bibcode:2017NJPh...19a3027V. doi:10.1088/1367-2630/19/1/013027.
^ Gühne, Otfried; Seevinck, Michael (5 May 2010). "Separability criteria for genuine multiparticle entanglement". New Journal of Physics. 12 (5): 053002. arXiv:0905.1349. Bibcode:2010NJPh...12e3002G. doi:10.1088/1367-2630/12/5/053002.
^ Pezzé, Luca; Smerzi, Augusto (10 March 2009). "Entanglement, Nonlinear Dynamics, and the Heisenberg Limit". Physical Review Letters. 102 (10): 100401. arXiv:0711.4840. Bibcode:2009PhRvL.102j0401P. doi:10.1103/PhysRevLett.102.100401. PMID 19392092.
^ Hyllus, Philipp; Laskowski, Wiesław; Krischek, Roland; Schwemmer, Christian; Wieczorek, Witlef; Weinfurter, Harald; Pezzé, Luca; Smerzi, Augusto (16 February 2012). "Fisher information and multiparticle entanglement". Physical Review A. 85 (2): 022321. arXiv:1006.4366. Bibcode:2012PhRvA..85b2321H. doi:10.1103/PhysRevA.85.022321.
^ Tóth, Géza (16 February 2012). "Multipartite entanglement and high-precision metrology". Physical Review A. 85 (2): 022322. arXiv:1006.4368. Bibcode:2012PhRvA..85b2322T. doi:10.1103/PhysRevA.85.022322.
^ Eibl, Manfred; Kiesel, Nikolai; Bourennane, Mohamed; Kurtsiefer, Christian; Weinfurter, Harald (18 February 2004). "Experimental Realization of a Three-Qubit Entangled W State". Physical Review Letters. 92 (7): 077901. Bibcode:2004PhRvL..92g7901E. doi:10.1103/PhysRevLett.92.077901. PMID 14995887.
^ ^a ^b Lücke, B.; Scherer, M.; Kruse, J.; Pezzé, L.; Deuretzbacher, F.; Hyllus, P.; Topic, O.; Peise, J.; Ertmer, W.; Arlt, J.; Santos, L.; Smerzi, A.; Klempt, C. (11 November 2011). "Twin Matter Waves for Interferometry Beyond the Classical Limit". Science. 334 (6057): 773–776. arXiv:1204.4102. Bibcode:2011Sci...334..773L. doi:10.1126/science.1208798. PMID 21998255.
^ Hamley, C. D.; Gerving, C. S.; Hoang, T. M.; Bookjans, E. M.; Chapman, M. S. (April 2012). "Spin-nematic squeezed vacuum in a quantum gas". Nature Physics. 8 (4): 305–308. arXiv:1111.1694. Bibcode:2012NatPh...8..305H. doi:10.1038/nphys2245.
^ Krischek, Roland; Schwemmer, Christian; Wieczorek, Witlef; Weinfurter, Harald; Hyllus, Philipp; Pezzé, Luca; Smerzi, Augusto (19 August 2011). "Useful Multiparticle Entanglement and Sub-Shot-Noise Sensitivity in Experimental Phase Estimation". Physical Review Letters. 107 (8): 080504. arXiv:1108.6002. Bibcode:2011PhRvL.107h0504K. doi:10.1103/PhysRevLett.107.080504. PMID 21929154.
^ Luo, Xin-Yu; Zou, Yi-Quan; Wu, Ling-Na; Liu, Qi; Han, Ming-Fei; Tey, Meng Khoon; You, Li (10 February 2017). "Deterministic entanglement generation from driving through quantum phase transitions". Science. 355 (6325): 620–623. arXiv:1702.03120. Bibcode:2017Sci...355..620L. doi:10.1126/science.aag1106. PMID 28183976.
^ Xin, Lin; Barrios, Maryrose; Cohen, Julia T.; Chapman, Michael S. (29 September 2023). "Long-Lived Squeezed Ground States in a Quantum Spin Ensemble". Physical Review Letters. 131 (13): 133402. arXiv:2202.12338. Bibcode:2023PhRvL.131m3402X. doi:10.1103/PhysRevLett.131.133402. PMID 37832022.
^ Lange, Karsten; Peise, Jan; Lücke, Bernd; Kruse, Ilka; Vitagliano, Giuseppe; Apellaniz, Iagoba; Kleinmann, Matthias; Tóth, Géza; Klempt, Carsten (27 April 2018). "Entanglement between two spatially separated atomic modes". Science. 360 (6387): 416–418. arXiv:1708.02480. Bibcode:2018Sci...360..416L. doi:10.1126/science.aao2035. PMID 29700263.
^ Vitagliano, Giuseppe; Fadel, Matteo; Apellaniz, Iagoba; Kleinmann, Matthias; Lücke, Bernd; Klempt, Carsten; Tóth, Géza (9 February 2023). "Number-phase uncertainty relations and bipartite entanglement detection in spin ensembles". Quantum. 7: 914. arXiv:2104.05663. Bibcode:2023Quant...7..914V. doi:10.22331/q-2023-02-09-914.

[1] Dicke, R. H. (1 January 1954). "Coherence in Spontaneous Radiation Processes". Physical Review. 93 (1): 99–110. Bibcode:1954PhRv...93...99D. doi:10.1103/PhysRev.93.99.

[2] Gühne, Otfried; Tóth, Géza (April 2009). "Entanglement detection". Physics Reports. 474 (1–6): 1–75. arXiv:0811.2803. Bibcode:2009PhR...474....1G. doi:10.1016/j.physrep.2009.02.004.

[3] Stockton, John K.; Geremia, J. M.; Doherty, Andrew C.; Mabuchi, Hideo (28 February 2003). "Characterizing the entanglement of symmetric many-particle spin- 1 2 systems". Physical Review A. 67 (2): 022112. arXiv:quant-ph/0210117. Bibcode:2003PhRvA..67b2112S. doi:10.1103/PhysRevA.67.022112.

[4] Vitagliano, Giuseppe; Apellaniz, Iagoba; Egusquiza, Iñigo L.; Tóth, Géza (7 March 2014). "Spin squeezing and entanglement for an arbitrary spin". Physical Review A. 89 (3): 032307. arXiv:1310.2269. Bibcode:2014PhRvA..89c2307V. doi:10.1103/PhysRevA.89.032307.

[5] Cirac, J. I.; Ekert, A. K.; Macchiavello, C. (24 May 1999). "Optimal Purification of Single Qubits". Physical Review Letters. 82 (21): 4344–4347. arXiv:quant-ph/9812075. Bibcode:1999PhRvL..82.4344C. doi:10.1103/PhysRevLett.82.4344.

[PRL07_Dicke-6] Kiesel, N.; Schmid, C.; Tóth, G.; Solano, E.; Weinfurter, H. (7 February 2007). "Experimental Observation of Four-Photon Entangled Dicke State with High Fidelity". Physical Review Letters. 98 (6): 063604. arXiv:quant-ph/0606234. Bibcode:2007PhRvL..98f3604K. doi:10.1103/PhysRevLett.98.063604. PMID 17358941.

[NJP09-7] Tóth, Géza; Wieczorek, Witlef; Krischek, Roland; Kiesel, Nikolai; Michelberger, Patrick; Weinfurter, Harald (4 August 2009). "Practical methods for witnessing genuine multi-qubit entanglement in the vicinity of symmetric states". New Journal of Physics. 11 (8): 083002. arXiv:0903.3910. Bibcode:2009NJPh...11h3002T. doi:10.1088/1367-2630/11/8/083002.

[PRL09_Dicke-8] Wieczorek, Witlef; Krischek, Roland; Kiesel, Nikolai; Michelberger, Patrick; Tóth, Géza; Weinfurter, Harald (10 July 2009). "Experimental Entanglement of a Six-Photon Symmetric Dicke State". Physical Review Letters. 103 (2): 020504. Bibcode:2009PhRvL.103b0504W. doi:10.1103/PhysRevLett.103.020504. PMID 19659191.

[PRL09_Dicke_B-9] Prevedel, R.; Cronenberg, G.; Tame, M. S.; Paternostro, M.; Walther, P.; Kim, M. S.; Zeilinger, A. (10 July 2009). "Experimental Realization of Dicke States of up to Six Qubits for Multiparty Quantum Networking". Physical Review Letters. 103 (2): 020503. arXiv:0903.2212. Bibcode:2009PhRvL.103b0503P. doi:10.1103/PhysRevLett.103.020503. PMID 19659190.

[10] Tóth, Géza (1 February 2007). "Detection of multipartite entanglement in the vicinity of symmetric Dicke states". Journal of the Optical Society of America B. 24 (2): 275. arXiv:quant-ph/0511237. Bibcode:2007JOSAB..24..275T. doi:10.1364/JOSAB.24.000275.

[11] Tóth, Géza; Knapp, Christian; Gühne, Otfried; Briegel, Hans J. (19 December 2007). "Optimal Spin Squeezing Inequalities Detect Bound Entanglement in Spin Models". Physical Review Letters. 99 (25): 250405. arXiv:quant-ph/0702219. Bibcode:2007PhRvL..99y0405T. doi:10.1103/PhysRevLett.99.250405. PMID 18233503.

[PRL14-12] Lücke, Bernd; Peise, Jan; Vitagliano, Giuseppe; Arlt, Jan; Santos, Luis; Tóth, Géza; Klempt, Carsten (17 April 2014). "Detecting Multiparticle Entanglement of Dicke States". Physical Review Letters. 112 (15): 155304. arXiv:1403.4542. Bibcode:2014PhRvL.112o5304L. doi:10.1103/PhysRevLett.112.155304. PMID 24785048.

[13] Vitagliano, Giuseppe; Apellaniz, Iagoba; Kleinmann, Matthias; Lücke, Bernd; Klempt, Carsten; Tóth, Géza (20 January 2017). "Entanglement and extreme spin squeezing of unpolarized states". New Journal of Physics. 19 (1): 013027. arXiv:1605.07202. Bibcode:2017NJPh...19a3027V. doi:10.1088/1367-2630/19/1/013027.

[14] Gühne, Otfried; Seevinck, Michael (5 May 2010). "Separability criteria for genuine multiparticle entanglement". New Journal of Physics. 12 (5): 053002. arXiv:0905.1349. Bibcode:2010NJPh...12e3002G. doi:10.1088/1367-2630/12/5/053002.

[15] Pezzé, Luca; Smerzi, Augusto (10 March 2009). "Entanglement, Nonlinear Dynamics, and the Heisenberg Limit". Physical Review Letters. 102 (10): 100401. arXiv:0711.4840. Bibcode:2009PhRvL.102j0401P. doi:10.1103/PhysRevLett.102.100401. PMID 19392092.

[16] Hyllus, Philipp; Laskowski, Wiesław; Krischek, Roland; Schwemmer, Christian; Wieczorek, Witlef; Weinfurter, Harald; Pezzé, Luca; Smerzi, Augusto (16 February 2012). "Fisher information and multiparticle entanglement". Physical Review A. 85 (2): 022321. arXiv:1006.4366. Bibcode:2012PhRvA..85b2321H. doi:10.1103/PhysRevA.85.022321.

[17] Tóth, Géza (16 February 2012). "Multipartite entanglement and high-precision metrology". Physical Review A. 85 (2): 022322. arXiv:1006.4368. Bibcode:2012PhRvA..85b2322T. doi:10.1103/PhysRevA.85.022322.

[18] Eibl, Manfred; Kiesel, Nikolai; Bourennane, Mohamed; Kurtsiefer, Christian; Weinfurter, Harald (18 February 2004). "Experimental Realization of a Three-Qubit Entangled W State". Physical Review Letters. 92 (7): 077901. Bibcode:2004PhRvL..92g7901E. doi:10.1103/PhysRevLett.92.077901. PMID 14995887.

[Sci2011-19] Lücke, B.; Scherer, M.; Kruse, J.; Pezzé, L.; Deuretzbacher, F.; Hyllus, P.; Topic, O.; Peise, J.; Ertmer, W.; Arlt, J.; Santos, L.; Smerzi, A.; Klempt, C. (11 November 2011). "Twin Matter Waves for Interferometry Beyond the Classical Limit". Science. 334 (6057): 773–776. arXiv:1204.4102. Bibcode:2011Sci...334..773L. doi:10.1126/science.1208798. PMID 21998255.

[20] Hamley, C. D.; Gerving, C. S.; Hoang, T. M.; Bookjans, E. M.; Chapman, M. S. (April 2012). "Spin-nematic squeezed vacuum in a quantum gas". Nature Physics. 8 (4): 305–308. arXiv:1111.1694. Bibcode:2012NatPh...8..305H. doi:10.1038/nphys2245.

[21] Krischek, Roland; Schwemmer, Christian; Wieczorek, Witlef; Weinfurter, Harald; Hyllus, Philipp; Pezzé, Luca; Smerzi, Augusto (19 August 2011). "Useful Multiparticle Entanglement and Sub-Shot-Noise Sensitivity in Experimental Phase Estimation". Physical Review Letters. 107 (8): 080504. arXiv:1108.6002. Bibcode:2011PhRvL.107h0504K. doi:10.1103/PhysRevLett.107.080504. PMID 21929154.

[22] Luo, Xin-Yu; Zou, Yi-Quan; Wu, Ling-Na; Liu, Qi; Han, Ming-Fei; Tey, Meng Khoon; You, Li (10 February 2017). "Deterministic entanglement generation from driving through quantum phase transitions". Science. 355 (6325): 620–623. arXiv:1702.03120. Bibcode:2017Sci...355..620L. doi:10.1126/science.aag1106. PMID 28183976.

[23] Xin, Lin; Barrios, Maryrose; Cohen, Julia T.; Chapman, Michael S. (29 September 2023). "Long-Lived Squeezed Ground States in a Quantum Spin Ensemble". Physical Review Letters. 131 (13): 133402. arXiv:2202.12338. Bibcode:2023PhRvL.131m3402X. doi:10.1103/PhysRevLett.131.133402. PMID 37832022.

[24] Lange, Karsten; Peise, Jan; Lücke, Bernd; Kruse, Ilka; Vitagliano, Giuseppe; Apellaniz, Iagoba; Kleinmann, Matthias; Tóth, Géza; Klempt, Carsten (27 April 2018). "Entanglement between two spatially separated atomic modes". Science. 360 (6387): 416–418. arXiv:1708.02480. Bibcode:2018Sci...360..416L. doi:10.1126/science.aao2035. PMID 29700263.

[25] Vitagliano, Giuseppe; Fadel, Matteo; Apellaniz, Iagoba; Kleinmann, Matthias; Lücke, Bernd; Klempt, Carsten; Tóth, Géza (9 February 2023). "Number-phase uncertainty relations and bipartite entanglement detection in spin ensembles". Quantum. 7: 914. arXiv:2104.05663. Bibcode:2023Quant...7..914V. doi:10.22331/q-2023-02-09-914.

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