Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0512180 · Physical Review A

Entanglement criteria via the uncertainty relations in su(2) and su(1,1) algebras: Detection of non-Gaussian entangled states

Hyunchul Nha, Jaewan Kim

Published Jul 21, 2006• 70 citations
arXivHTMLPDF

Abstract

We derive a class of inequalities, from the uncertainty relations of the su(1,1) and the su(2) algebra in conjunction with partial transposition, that must be satisfied by any separable two-mode states. These inequalities are presented in terms of the su(2) operators ${J}_{x}=({a}^{\ifmmode\dagger\else\textdagger\fi{}}b+a{b}^{\ifmmode\dagger\else\textdagger\fi{}})∕2$, ${J}_{y}=({a}^{\ifmmode\dagger\else\textdagger\fi{}}b\ensuremath{-}a{b}^{\ifmmode\dagger\else\textdagger\fi{}})∕2i$, and the total photon number $⟨{N}_{a}+{N}_{b}⟩$. They include as special cases the inequality derived by Hillery and Zubairy [Phys. Rev. Lett. 96, 050503 (2006)], and the one by Agarwal and Biswas [New J. Phys. 7, 211 (2005)]. In particular, optimization over the whole inequalities leads to the criterion obtained by Agarwal and Biswas. We show that this optimal criterion can detect entanglement for a broad class of non-Gaussian entangled states, i.e., the su(2) minimum-uncertainty states. Experimental schemes to test the optimal criterion are also discussed, especially the one using linear optical devices and photodetectors.

DOI
10.1103/physreva.74.012317

Related papers

Entanglement Criteria for Non-Gaussian States

Prasoon Kumar Shandilya, Supervisor Prof. Prasanta K. Panigrahi

Entanglement for continuous variable states opens up an avenue for experimental utilization of the entanglement properties of quantum physics very easily. Non- Gaussian continuous variable state improves the efficiency and robustness of a quantum teleportation and communication. Thus, it is important to understand the entanglement properties of non-Gaussian states. For a class of non-Gaussian continuous variable states, we derive a general criterion for SU(2) operators which incidentally coincides with the work of Agarwal and Biswas and another entanglement criteria for SU(1,1) operators for a two-mode non-Gaussian state using uncertainty relation and Cauchy-Schwarz inequality.

•Apr 1, 2017•0 citations

Entanglement conditions for four-mode states via the uncertainty relations in conjunction with partial transposition

Lijun Song, Xiaoguang Wang, Dong Yan +1 more

Abstract From the Heisenberg uncertainty relation in conjunction with partial transposition, we derive a class of inequalities for detecting entanglements in four-mode states. The sufficient conditions for bipartite entangled states are presented. We also discuss the generalization of the entanglement conditions via the Schrödinger-Robertson indeterminacy relation, which are in general stronger than those based on the Heisenberg uncertainty relation.

Open Physics•Mar 26, 2008•1 citation

Detection of non-Gaussian entangled states with an improved continuous-variable separability criterion

Anaelle Hertz, Evgueni Karpov, Aikaterini Mandilara +1 more

Currently available separability criteria for continuous-variable states are generally based on the covariance matrix of quadrature operators. The well-known separability criterion of Duan et al. [L. M. Duan et al., Phys. Rev. Lett. 84, 2722 (2000)] and Simon [R. Simon, Phys. Rev. Lett. 84, 2726 (2000)], for example, gives a necessary and sufficient condition for a two-mode Gaussian state to be separable, but leaves many entangled non-Gaussian states undetected. Here we introduce an improvement of this criterion that enables a stronger entanglement detection. The improved condition is based on the knowledge of an additional parameter, namely, the degree of Gaussianity, and exploits a connection with Gaussianity-bounded uncertainty relations [A. Mandilara and N. J. Cerf, Phys. Rev. A 86, 030102 (2012)]. We exhibit families of non-Gaussian entangled states whose entanglement remains undetected by the Duan-Simon criterion.

arXiv:1511.06621•Physical Review A•Mar 22, 2016•22 citations

Entanglement condition via su(2) and su(1,1) algebra using Schrödinger-Robertson uncertainty relation

Hyunchul Nha

The Schr\"odinger-Robertson inequality generally provides a stronger bound on the product of uncertainties for two noncommuting observables than the Heisenberg uncertainty relation, and as such it can yield a stricter separability condition in conjunction with partial transposition. In this paper, using the Schr\"odinger-Robertson uncertainty relation, the separability condition previously derived from the su(2) and su(1,1) algebra is made stricter and refined to a form invariant with respect to local phase shifts. Furthermore, a linear optical scheme is proposed to test this invariant separability condition.

arXiv:0704.1939•Physical Review A•Jul 16, 2007•35 citations

Entanglement Condition for W Type Multimode States and a Scheme for Experimental Realization

M. Karthick Selvan, Prasanta Kumar Panigrahi

We derive a class of inequality relations, using both the sum uncertainty relations of su(2) algebra operators and the Schrodinger-Robertson uncertainty relation of partially transposed su(1, 1) algebra operators, to detect the three-mode entanglement of non-Gaussian states of electromagnetic field. These operators are quadratic in mode creation and annihilation operators. The inseparability condition obtained using su(2) algebra operators is shown to guarantee the violation of stronger separability condition provided by Schrodinger-Robertson uncertainty relation of partially transposed su(1, 1) algebra operators. The obtained inseparability condition is also shown to be a necessary condition for W type entangled states and it is used to derive the general form for a family of such inseparability conditions. An experimental scheme is proposed to test the violation of separability condition. The results derived for three-mode systems are generalized to multimode systems.

•Jul 1, 2018•0 citations

Entanglement conditions for tripartite systems via indeterminacy relations

Lijun Song, Xiaoguang Wang, Dong Yan +1 more

Based on the S-R indeterminacy relations in conjugation with the partial transposition, we derive a class of inequalities for detecting entanglement in several tripartite systems, including bosonic, SU(2), and SU(1,1) systems. These inequalities are in general stronger than those based on the usual Heisenberg relations for detecting entanglement. We also discuss the reduction from SU(2) and SU(1,1) to bosonic systems and the generalization to multipartite case.

arXiv:0705.2264•Journal of Physics B Atomic Molecular and Optical Physics•Dec 20, 2007•7 citations
We derive a class of inequalities, from the uncertainty relations of the su(1,1) and the su(2) algebra in conjunction with partial transposition, that must be satisfied by any separable two-mode states. These inequalities are presented in terms of the su(2) operators Jx=(a\ifmmode†\else\textdagger\fib+ab\ifmmode†\else\textdagger\fi)∕2{J}_{x}=({a}^{\ifmmode\dagger\else\textdagger\fi{}}b+a{b}^{\ifmmode\dagger\else\textdagger\fi{}})∕2Jx​=(a\ifmmode†\else\textdagger\fib+ab\ifmmode†\else\textdagger\fi)∕2, Jy=(a\ifmmode†\else\textdagger\fib\ensuremath−ab\ifmmode†\else\textdagger\fi)∕2i{J}_{y}=({a}^{\ifmmode\dagger\else\textdagger\fi{}}b\ensuremath{-}a{b}^{\ifmmode\dagger\else\textdagger\fi{}})∕2iJy​=(a\ifmmode†\else\textdagger\fib\ensuremath−ab\ifmmode†\else\textdagger\fi)∕2i, and the total photon number ⟨Na+Nb⟩⟨{N}_{a}+{N}_{b}⟩⟨Na​+Nb​⟩. They include as special cases the inequality derived by Hillery and Zubairy [Phys. Rev. Lett. 96, 050503 (2006)], and the one by Agarwal and Biswas [New J. Phys. 7, 211 (2005)]. In particular, optimization over the whole inequalities leads to the criterion obtained by Agarwal and Biswas. We show that this optimal criterion can detect entanglement for a broad class of non-Gaussian entangled states, i.e., the su(2) minimum-uncertainty states. Experimental schemes to test the optimal criterion are also discussed, especially the one using linear optical devices and photodetectors.
Entanglement for continuous variable states opens up an avenue for experimental utilization of the entanglement properties of quantum physics very easily. Non- Gaussian continuous variable state improves the efficiency and robustness of a quantum teleportation and communication. Thus, it is important to understand the entanglement properties of non-Gaussian states. For a class of non-Gaussian continuous variable states, we derive a general criterion for SU(2) operators which incidentally coincides with the work of Agarwal and Biswas and another entanglement criteria for SU(1,1) operators for a two-mode non-Gaussian state using uncertainty relation and Cauchy-Schwarz inequality.
Currently available separability criteria for continuous-variable states are generally based on the covariance matrix of quadrature operators. The well-known separability criterion of Duan et al. [L. M. Duan et al., Phys. Rev. Lett. 84, 2722 (2000)] and Simon [R. Simon, Phys. Rev. Lett. 84, 2726 (2000)], for example, gives a necessary and sufficient condition for a two-mode Gaussian state to be separable, but leaves many entangled non-Gaussian states undetected. Here we introduce an improvement of this criterion that enables a stronger entanglement detection. The improved condition is based on the knowledge of an additional parameter, namely, the degree of Gaussianity, and exploits a connection with Gaussianity-bounded uncertainty relations [A. Mandilara and N. J. Cerf, Phys. Rev. A 86, 030102 (2012)]. We exhibit families of non-Gaussian entangled states whose entanglement remains undetected by the Duan-Simon criterion.
The Schr\"odinger-Robertson inequality generally provides a stronger bound on the product of uncertainties for two noncommuting observables than the Heisenberg uncertainty relation, and as such it can yield a stricter separability condition in conjunction with partial transposition. In this paper, using the Schr\"odinger-Robertson uncertainty relation, the separability condition previously derived from the su(2) and su(1,1) algebra is made stricter and refined to a form invariant with respect to local phase shifts. Furthermore, a linear optical scheme is proposed to test this invariant separability condition.
We derive a class of inequality relations, using both the sum uncertainty relations of su(2) algebra operators and the Schrodinger-Robertson uncertainty relation of partially transposed su(1, 1) algebra operators, to detect the three-mode entanglement of non-Gaussian states of electromagnetic field. These operators are quadratic in mode creation and annihilation operators. The inseparability condition obtained using su(2) algebra operators is shown to guarantee the violation of stronger separability condition provided by Schrodinger-Robertson uncertainty relation of partially transposed su(1, 1) algebra operators. The obtained inseparability condition is also shown to be a necessary condition for W type entangled states and it is used to derive the general form for a family of such inseparability conditions. An experimental scheme is proposed to test the violation of separability condition. The results derived for three-mode systems are generalized to multimode systems.