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

Abstract

We derive a class of inequalities, from the uncertainty relations of the su͑1,1͒ and the su͑2͒ algebra in conjunction with partial transposition, that must be satisfied by any separable two-mode states. These inequalities are presented in terms of the su͑2͒ operators J, 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 ͑2006͔͒, and the one by Agarwal and Biswas ͓New J. Phys. 7, 211 ͑2005͔͒. 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 su͑2͒ minimum-uncertainty states. Experimental schemes to test the optimal criterion are also discussed, especially the one using linear optical devices and photodetectors.

Multipartite entanglement criterion from uncertainty relations

“…Other criteria have been proposed so they could be tested experimentally in a direct manner, as the Bell inequalities [3,4] or the entanglement witnesses [5]. More recently, criteria based on variance measurements have been studied for continuous and discrete variable systems [6,7,8,9,10,11,12,13,14].…”

“…Other criteria have been proposed so they could be tested experimentally in a direct manner, as the Bell inequalities [3,4] or the entanglement witnesses [5]. More recently, criteria based on variance measurements have been studied for continuous and discrete variable systems [6,7,8,9,10,11,12,13,14].In [11] the Heisenberg relation has been used along with the partial transpose operation to obtain a criterion detecting entanglement condition in bipartite nongaussian states. That idea was generalized in [13,14] with use of the Schrödinger-Robertson relation instead of the Heisenberg inequality.…”

Complete conditions for legitimate Wigner distributions

“…In our formalism, for a given two-mode entangled state with the Wigner distribution W t (α 1 , α 2 ), one can take the partially transposed distribution, W t (α 1 , α * 2 ) [21,25]. If the state becomes negative under PT, then the distribution W t (α 1 , α * 2 ) must violate at least one of the nonnegative conditions for the matrix {α kk ′ }, which demonstrates the presence of entanglement.…”

“…Note that we have used the collective indices k ≡ {k 1 , k 2 } and k ′ ≡ {k ′ 1 , k ′ 2 } in Eq. (25). In constructing the matrix {α kk ′ }, these indices may be arranged in the increasings order of the…”

“…Detection of bipartite entanglement-As an illustration, let us consider the problem of detecting bipartite entanglement for continuous variables (CVs). The entanglement criteria so far known for CVs are all based on the partial transposition (PT) [21,22,23,24,25]. When a two-mode state ρ is separable, it is represented by a form…”