Separability criteria with angular and Hilbert space averages
The practically useful criteria of separable states ρ=∑k wkρk in d=2×2 are discussed. The equality G(a,b)=4[〈ψ|P(a)⊗P(b)|ψ〉−〈ψ|P(a)⊗1|ψ〉〈ψ|1⊗P(b)|ψ〉]=0 for any two projection operators P(a) and P(b) provides a necessary and sufficient separability criterion in the case of a separable pure state ρ=|ψ...
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| Format: | Journal Article |
| Language: | English |
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2017
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| Online Access: | https://hdl.handle.net/10356/80717 http://hdl.handle.net/10220/43432 |
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| author | Fujikawa, Kazuo Oh, Choo Hiap Umetsu, Koichiro Yu, Sixia |
| author2 | Institute of Advanced Studies |
| author_facet | Institute of Advanced Studies Fujikawa, Kazuo Oh, Choo Hiap Umetsu, Koichiro Yu, Sixia |
| author_sort | Fujikawa, Kazuo |
| collection | NTU |
| description | The practically useful criteria of separable states ρ=∑k wkρk in d=2×2 are discussed. The equality G(a,b)=4[〈ψ|P(a)⊗P(b)|ψ〉−〈ψ|P(a)⊗1|ψ〉〈ψ|1⊗P(b)|ψ〉]=0 for any two projection operators P(a) and P(b) provides a necessary and sufficient separability criterion in the case of a separable pure state ρ=|ψ〉〈ψ|. We propose the separability criteria of mixed states, which are given by Trρ{a⋅σ⊗b⋅σ}=(1/3)Ccosφ for two spin 1/2 systems and 4Trρ{P(a)⊗P(b)}=1+(1/2)Ccos2φ for two photon systems, respectively, after taking a geometrical angular average of a and b with fixed cosφ=a⋅b. Here −1≤C≤1, and the difference in the numerical coefficients 1/2 and 1/3 arises from the different rotational properties of the spinor and the transverse photon. If one instead takes an average over the states in the d=2 Hilbert space, the criterion for two photon systems is replaced by 4Trρ{P(a)⊗P(b)}=1+(1/3)Ccos2φ. Those separability criteria are shown to be very efficient using the existing experimental data of Aspect et al. in 1981 and Sakai et al. in 2006. When the Werner state is applied to two photon systems, it is shown that the Hilbert space average can judge its inseparability but not the geometrical angular average. |
| first_indexed | 2024-10-01T06:17:13Z |
| format | Journal Article |
| id | ntu-10356/80717 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T06:17:13Z |
| publishDate | 2017 |
| record_format | dspace |
| spelling | ntu-10356/807172020-09-26T21:55:52Z Separability criteria with angular and Hilbert space averages Fujikawa, Kazuo Oh, Choo Hiap Umetsu, Koichiro Yu, Sixia Institute of Advanced Studies Separability Entanglement The practically useful criteria of separable states ρ=∑k wkρk in d=2×2 are discussed. The equality G(a,b)=4[〈ψ|P(a)⊗P(b)|ψ〉−〈ψ|P(a)⊗1|ψ〉〈ψ|1⊗P(b)|ψ〉]=0 for any two projection operators P(a) and P(b) provides a necessary and sufficient separability criterion in the case of a separable pure state ρ=|ψ〉〈ψ|. We propose the separability criteria of mixed states, which are given by Trρ{a⋅σ⊗b⋅σ}=(1/3)Ccosφ for two spin 1/2 systems and 4Trρ{P(a)⊗P(b)}=1+(1/2)Ccos2φ for two photon systems, respectively, after taking a geometrical angular average of a and b with fixed cosφ=a⋅b. Here −1≤C≤1, and the difference in the numerical coefficients 1/2 and 1/3 arises from the different rotational properties of the spinor and the transverse photon. If one instead takes an average over the states in the d=2 Hilbert space, the criterion for two photon systems is replaced by 4Trρ{P(a)⊗P(b)}=1+(1/3)Ccos2φ. Those separability criteria are shown to be very efficient using the existing experimental data of Aspect et al. in 1981 and Sakai et al. in 2006. When the Werner state is applied to two photon systems, it is shown that the Hilbert space average can judge its inseparability but not the geometrical angular average. NRF (Natl Research Foundation, S’pore) MOE (Min. of Education, S’pore) Accepted version 2017-07-25T04:58:25Z 2019-12-06T13:57:25Z 2017-07-25T04:58:25Z 2019-12-06T13:57:25Z 2016 Journal Article Fujikawa, K., Oh, C. H., Umetsu, K., & Yu, S. (2016). Separability criteria with angular and Hilbert space averages. Annals of Physics, 368, 248-257. 0003-4916 https://hdl.handle.net/10356/80717 http://hdl.handle.net/10220/43432 10.1016/j.aop.2016.02.006 en Annals of Physics © 2016 Elsevier. This is the author created version of a work that has been peer reviewed and accepted for publication by Annals of Physics, Elsevier. It incorporates referee’s comments but changes resulting from the publishing process, such as copyediting, structural formatting, may not be reflected in this document. The published version is available at: [http://dx.doi.org/10.1016/j.aop.2016.02.006]. 14 p. application/pdf |
| spellingShingle | Separability Entanglement Fujikawa, Kazuo Oh, Choo Hiap Umetsu, Koichiro Yu, Sixia Separability criteria with angular and Hilbert space averages |
| title | Separability criteria with angular and Hilbert space averages |
| title_full | Separability criteria with angular and Hilbert space averages |
| title_fullStr | Separability criteria with angular and Hilbert space averages |
| title_full_unstemmed | Separability criteria with angular and Hilbert space averages |
| title_short | Separability criteria with angular and Hilbert space averages |
| title_sort | separability criteria with angular and hilbert space averages |
| topic | Separability Entanglement |
| url | https://hdl.handle.net/10356/80717 http://hdl.handle.net/10220/43432 |
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