# Entanglement depth

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In [quantum physics](/source/Quantum_physics), **entanglement depth** characterizes the strength of multiparticle [entanglement](/source/Quantum_entanglement). An entanglement depth k means that the quantum state of a particle ensemble cannot be described under the assumption that particles interacted with each other only in groups having fewer than k particles. It has been used to characterize the quantum states created in experiments with cold gases.

## Definition

Entanglement depth appeared in the context of [spin squeezing](/source/Spin_squeezing). It turned out that to achieve larger and larger spin squeezing, and thus larger and larger precision in parameter estimation, a larger and larger entanglement depth is needed.[1]

Later it was formalized in terms of [convex sets](/source/Convex_set) of [quantum states](/source/Quantum_state), independent of spin squeezing as follows.[2] Let us consider a [pure state](/source/Quantum_state#Pure_states) that is the [tensor product](/source/Tensor_product) of multi-particle quantum states

|\Psi\rangle=|\phi_1\rangle\otimes|\phi_2\rangle\otimes ... \otimes|\phi_n\rangle.

The pure state |\Psi\rangle is said to be k-producible if all \phi_i are states of at most k particles. A mixed state is called k-producible, if it is a mixture of pure states that are all at most k-producible. The k-producible mixed states form a convex set.

A quantum state contains at least multiparticle entanglement of k+1 particles, if it is not k-producible. A N-particle state with N-entanglement is called genuine multipartite entangled.

Finally, a quantum state has an entanglement depth k, if it is k-producible, but not (k-1)-producible.

It was possible to detect the entanglement depth close to states different from spin-squeezed states. Since there is not a general method to detect [multipartite entanglement](/source/Multipartite_entanglement), these methods had to be tailored to experiments with various relevant quantum states.

Thus, entanglement criteria has been developed to detect entanglement close to symmetric [Dicke states](/source/Dicke_state) with \langle J_z\rangle=0.[3][4][5] They are very different from spin-squeezed states, since they do not have a large [spin polarization](/source/Spin_polarization). They can provide Heisenberg limited [metrology](/source/Metrology), while they are more robust to particle loss than Greenberger-Horne-Zeilinger (GHZ) states.

There are also criteria for detecting the entanglement depth in *planar-squeezed state*s.[6] Planar squeezed states are quantum states that can be used to estimate a rotation angle that is not expected to be small.[7]

Finally, multipartite entanglement can be detected based on the metrological usefulness of the quantum state.[8][9] The criteria applied are based on bounds on the [quantum Fisher information](/source/Quantum_Fisher_information).

## Experiments

The entanglement criterion in Ref.[1] has been used in many experiments with cold gases in spin-squeezed states.[10][11][12][13][14]

There have also been experiments in cold gases for detecting multipartite entanglement in symmetric Dicke states.[4][15]

There have been also experiments with Dicke states that detected entanglement based on metrological usefulness in cold gases[16] and in photons.[17]

## References

1. Sørensen, Anders S. & Mølmer, Klaus (14 May 2001). "Entanglement and Extreme Spin Squeezing". *Physical Review Letters*. **86** (20): 4431–4434. [arXiv:quant-ph/0011035](https://arxiv.org/abs/quant-ph/0011035). [Bibcode:2001PhRvL..86.4431S](https://ui.adsabs.harvard.edu/abs/2001PhRvL..86.4431S). [doi:10.1103/PhysRevLett.86.4431](https://doi.org/10.1103/PhysRevLett.86.4431). [PMID 11384252](https://pubmed.ncbi.nlm.nih.gov/11384252). [S2CID 206327094](https://api.semanticscholar.org/CorpusID:206327094)

1. Gühne, Otfried; Tóth, Géza; Briegel, Hans J (4 November 2005). "Multipartite entanglement in spin chains". *New Journal of Physics*. **7**: 229. [arXiv:quant-ph/0502160](https://arxiv.org/abs/quant-ph/0502160). [doi:10.1088/1367-2630/7/1/229](https://doi.org/10.1088/1367-2630/7/1/229)

1. Duan, L.-M. (27 October 2011). "Entanglement Detection in the Vicinity of Arbitrary Dicke States". *Physical Review Letters*. **107** (18). [arXiv:1107.5162](https://arxiv.org/abs/1107.5162). [Bibcode:2011PhRvL.107r0502D](https://ui.adsabs.harvard.edu/abs/2011PhRvL.107r0502D). [doi:10.1103/PhysRevLett.107.180502](https://doi.org/10.1103/PhysRevLett.107.180502). [PMID 22107616](https://pubmed.ncbi.nlm.nih.gov/22107616)

1. 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). [arXiv:1403.4542](https://arxiv.org/abs/1403.4542). [Bibcode:2014PhRvL.112o5304L](https://ui.adsabs.harvard.edu/abs/2014PhRvL.112o5304L). [doi:10.1103/PhysRevLett.112.155304](https://doi.org/10.1103/PhysRevLett.112.155304). [PMID 24785048](https://pubmed.ncbi.nlm.nih.gov/24785048)

1. 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). [arXiv:1605.07202](https://arxiv.org/abs/1605.07202). [Bibcode:2017NJPh...19a3027V](https://ui.adsabs.harvard.edu/abs/2017NJPh...19a3027V). [doi:10.1088/1367-2630/19/1/013027](https://doi.org/10.1088/1367-2630/19/1/013027)

1. Vitagliano, G.; Colangelo, G.; Martin Ciurana, F.; Mitchell, M. W.; Sewell, R. J.; Tóth, G. (15 February 2018). "Entanglement and extreme planar spin squeezing". *Physical Review A*. **97** (2). [arXiv:1705.09090](https://arxiv.org/abs/1705.09090). [Bibcode:2018PhRvA..97b0301V](https://ui.adsabs.harvard.edu/abs/2018PhRvA..97b0301V). [doi:10.1103/PhysRevA.97.020301](https://doi.org/10.1103/PhysRevA.97.020301)

1. He, Q. Y.; Peng, Shi-Guo; Drummond, P. D.; Reid, M. D. (11 August 2011). "Planar quantum squeezing and atom interferometry". *Physical Review A*. **84** (2). [arXiv:1101.0448](https://arxiv.org/abs/1101.0448). [Bibcode:2011PhRvA..84b2107H](https://ui.adsabs.harvard.edu/abs/2011PhRvA..84b2107H). [doi:10.1103/PhysRevA.84.022107](https://doi.org/10.1103/PhysRevA.84.022107)

1. Hyllus, Philipp (2012). "Fisher information and multiparticle entanglement". *Physical Review A*. **85** (2). [arXiv:1006.4366](https://arxiv.org/abs/1006.4366). [Bibcode:2012PhRvA..85b2321H](https://ui.adsabs.harvard.edu/abs/2012PhRvA..85b2321H). [doi:10.1103/physreva.85.022321](https://doi.org/10.1103/physreva.85.022321). [S2CID 118652590](https://api.semanticscholar.org/CorpusID:118652590)

1. Tóth, Géza (2012). "Multipartite entanglement and high-precision metrology". *Physical Review A*. **85** (2). [arXiv:1006.4368](https://arxiv.org/abs/1006.4368). [Bibcode:2012PhRvA..85b2322T](https://ui.adsabs.harvard.edu/abs/2012PhRvA..85b2322T). [doi:10.1103/physreva.85.022322](https://doi.org/10.1103/physreva.85.022322). [S2CID 119110009](https://api.semanticscholar.org/CorpusID:119110009)

1. Gross, C.; Zibold, T.; Nicklas, E.; Estève, J.; Oberthaler, M. K. (April 2010). "Nonlinear atom interferometer surpasses classical precision limit". *Nature*. **464** (7292): 1165–1169. [arXiv:1009.2374](https://arxiv.org/abs/1009.2374). [Bibcode:2010Natur.464.1165G](https://ui.adsabs.harvard.edu/abs/2010Natur.464.1165G). [doi:10.1038/nature08919](https://doi.org/10.1038/nature08919). [PMID 20357767](https://pubmed.ncbi.nlm.nih.gov/20357767). [S2CID 4419504](https://api.semanticscholar.org/CorpusID:4419504)

1. Riedel, Max F.; Böhi, Pascal; Li, Yun; Hänsch, Theodor W.; Sinatra, Alice; Treutlein, Philipp (April 2010). "Atom-chip-based generation of entanglement for quantum metrology". *Nature*. **464** (7292): 1170–1173. [arXiv:1003.1651](https://arxiv.org/abs/1003.1651). [Bibcode:2010Natur.464.1170R](https://ui.adsabs.harvard.edu/abs/2010Natur.464.1170R). [doi:10.1038/nature08988](https://doi.org/10.1038/nature08988). [PMID 20357765](https://pubmed.ncbi.nlm.nih.gov/20357765). [S2CID 4302730](https://api.semanticscholar.org/CorpusID:4302730)

1. Bohnet, J. G.; Cox, K. C.; Norcia, M. A.; Weiner, J. M.; Chen, Z.; Thompson, J. K. (September 2014). "Reduced spin measurement back-action for a phase sensitivity ten times beyond the standard quantum limit". *Nature Photonics*. **8** (9): 731–736. [arXiv:1310.3177](https://arxiv.org/abs/1310.3177). [Bibcode:2014NaPho...8..731B](https://ui.adsabs.harvard.edu/abs/2014NaPho...8..731B). [doi:10.1038/nphoton.2014.151](https://doi.org/10.1038/nphoton.2014.151). [S2CID 67780562](https://api.semanticscholar.org/CorpusID:67780562)

1. Cox, Kevin C.; Greve, Graham P.; Weiner, Joshua M.; Thompson, James K. (4 March 2016). "Deterministic Squeezed States with Collective Measurements and Feedback". *Physical Review Letters*. **116** (9). [arXiv:1512.02150](https://arxiv.org/abs/1512.02150). [Bibcode:2016PhRvL.116i3602C](https://ui.adsabs.harvard.edu/abs/2016PhRvL.116i3602C). [doi:10.1103/PhysRevLett.116.093602](https://doi.org/10.1103/PhysRevLett.116.093602). [PMID 26991175](https://pubmed.ncbi.nlm.nih.gov/26991175)

1. Mitchell, Morgan W & Beduini, Federica A (17 July 2014). "Extreme spin squeezing for photons". *New Journal of Physics*. **16** (7). [arXiv:1304.2527](https://arxiv.org/abs/1304.2527). [Bibcode:2014NJPh...16g3027M](https://ui.adsabs.harvard.edu/abs/2014NJPh...16g3027M). [doi:10.1088/1367-2630/16/7/073027](https://doi.org/10.1088/1367-2630/16/7/073027)

1. Zou, Yi-Quan; Wu, Ling-Na; Liu, Qi; Luo, Xin-Yu; Guo, Shuai-Feng; Cao, Jia-Hao; Tey, Meng Khoon; You, Li (19 June 2018). "Beating the classical precision limit with spin-1 Dicke states of more than 10,000 atoms". *Proceedings of the National Academy of Sciences*. **115** (25): 6381–6385. [arXiv:1802.10288](https://arxiv.org/abs/1802.10288). [Bibcode:2018PNAS..115.6381Z](https://ui.adsabs.harvard.edu/abs/2018PNAS..115.6381Z). [doi:10.1073/pnas.1715105115](https://doi.org/10.1073/pnas.1715105115). [PMC 6016791](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6016791). [PMID 29858344](https://pubmed.ncbi.nlm.nih.gov/29858344)

1. 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](https://arxiv.org/abs/1204.4102). [Bibcode:2011Sci...334..773L](https://ui.adsabs.harvard.edu/abs/2011Sci...334..773L). [doi:10.1126/science.1208798](https://doi.org/10.1126/science.1208798). [PMID 21998255](https://pubmed.ncbi.nlm.nih.gov/21998255)

1. 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). [arXiv:1108.6002](https://arxiv.org/abs/1108.6002). [Bibcode:2011PhRvL.107h0504K](https://ui.adsabs.harvard.edu/abs/2011PhRvL.107h0504K). [doi:10.1103/PhysRevLett.107.080504](https://doi.org/10.1103/PhysRevLett.107.080504). [PMID 21929154](https://pubmed.ncbi.nlm.nih.gov/21929154)

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