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arXiv:0704.1939 · Physical Review A

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

Hyunchul Nha

Published Jul 16, 2007• 35 citations
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Abstract

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.

DOI
10.1103/physreva.76.014305

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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.
Compared with the Heisenberg uncertainty relation, Schrodinger-Robertson uncertain relation generally provides stronger bound on product of uncertainties for two noncommutation observables, so the stricter separability condition can be provided by using Schrodinger-Robertson uncertainty relation. In this paper, using the Schrodinger Robertson uncertainty relation, the separability condition previously proposed by L. M. Duan is improved into the stricter separability condition. Then we extend the above ideas to the general two-mode situation and multi-mode terms, some new separation conditions are obtained.
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.
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 ⟨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.
We address the question, does a system A being entangled with another system B, put any constraints on the Heisenberg uncertainty relation (or the Schrödinger–Robertson inequality)? We find that the equality of the uncertainty relation cannot be reached for any two noncommuting observables, for finite dimensional Hilbert spaces if the Schmidt rank of the entangled state is maximal. One consequence is that the lower bound of the uncertainty relation can never be attained for any two observables for qubits, if the state is entangled. For infinite-dimensional Hilbert space too, we show that there is a class of physically interesting entangled states for which no two noncommuting observables can attain the minimum uncertainty equality.
We address the question, does a system A being entangled with another system B, put any constraints on the Heisenberg uncertainty relation (or the Schrodinger-Robertson inequality)? We find that the equality of the uncertainty relation cannot be reached for any two noncommuting observables, for finite dimensional Hilbert spaces if the Schmidt rank of the entangled state is maximal. One consequence is that the lower bound of the uncertainty relation can never be attained for any two observables for qubits, if the state is entangled. For infinite-dimensional Hilbert space too, we show that there is a class of physically interesting entangled states for which no two noncommuting observables can attain the minimum uncertainty equality.