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arXiv:quant-ph/0606154 · Physical Review A

Entanglement conditions for two-mode states: Applications

Mark S. Hillery, Muhammad Suhail Zubairy

Published Sep 27, 2006• 123 citations
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Abstract

We examine the implications of several recently derived conditions [Hillery and Zubairy, Phys. Rev. Lett. 96, 050503 (2006)] for determining when a two-mode state is entangled. We first find examples of non-Gaussian states that satisfy these conditions. We then apply the entanglement conditions to the study of several linear devices, the beam splitter, the parametric amplifier, and the linear phase-insensitive amplifier. For the first two, we find conditions on the input states that guarantee that the output states are entangled. For the linear amplifier, we determine in the limit of high and no gain, when an entangled input leads to an entangled output. Finally, we show how application of two two-mode entanglement conditions to a three-mode state can serve as a test of genuine three-mode entanglement.

DOI
10.1103/physreva.74.032333

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We examine the implications of several recently derived conditions [Hillery and Zubairy, Phys. Rev. Lett. 96, 050503 (2006)] for determining when a two-mode state is entangled. We first find examples of non-Gaussian states that satisfy these conditions. We then apply the entanglement conditions to the study of several linear devices, the beam splitter, the parametric amplifier, and the linear phase-insensitive amplifier. For the first two, we find conditions on the input states that guarantee that the output states are entangled. For the linear amplifier, we determine in the limit of high and no gain, when an entangled input leads to an entangled output. Finally, we show how application of two two-mode entanglement conditions to a three-mode state can serve as a test of genuine three-mode entanglement.
In this paper, we consider a parametric converter as a source of entangled radiation. We examine recently derived conditions (Hillery and Zubairy 2006 Phys. Rev. Lett. 96 050503, Duan et al 2000 Phys. Rev. Lett. 84 2722) for determining when the two output modes in a parametric converter are entangled. We show that for different initial field states, the two criteria give different conditions that ensure that the output states are entangled. We also present an input–output calculation for the entanglement of the output field.
In this paper, we use a criterion provided by E. Shchukin and W. Vogel [Phys. Rev. Lett. 95, 230502 (2005)] for determining when a bipartite quantum state is entangled. We first show that the entanglement criterion is necessary and sufficient condition for the partial transposition of the bipartite quantum states. Furthermore, previously known entanglement criteria are proved to be special cases of this criterion. We then apply the entanglement criterion to determine entangled property of several non-classical two-mode states. Finally, we provide the conclusion about this criterion.
Entanglement detection and quantification usually require the knowledge of correlations and thus is a global issue. For both theoretical and practical reasons, it is desirable to characterize entanglement via local quantities. Of course, this can only be achieved in special setups with some underlying structure encoding the correlations. For the often encountered bilinear interactions such as beamsplitter experiments, we propose an operational and friendly entanglement criterion for the two-mode Gaussian output states. Specifically, we show that in order to detect and evaluate the entanglement of the output states of a bilinear interaction involving uncorrelated Gaussian input states, we do not need to know the joint covariance matrix of the whole two-mode output states. Instead, we only need the respective covariances of the reduced states of the two input modes and that of the two output modes, which are all local quantities pertaining to a single mode and can be easily measured locally without explicitly considering the correlations between the modes. This is achieved by exploiting the input-output characteristics of bilinear interactions. The result recasts the celebrated entanglement criteria of Simon ( Phys. Rev. Lett. , 84 (2000) 2726) and Duan et al . ( Phys. Rev. Lett. , 84 (2000) 2722) into a more convenient and simple form.
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.
A beam splitter is a simple, readily available device which can act to entangle output optical fields. We show that a necessary condition for the fields at the output of the beam splitter to be entangled is that the pure input states exhibit nonclassical behavior. We generalize this proof for arbitrary (pure or impure) Gaussian input states. Specifically, nonclassicality of the input Gaussian fields is a necessary condition for entanglement of the field modes with the help of a beam splitter. We conjecture that this is a general property of beam splitters: Nonclassicality of the inputs is a necessary condition for entangling fields in a beam splitter.
Entangling two systems at distant locations using a separable mediating ancilla is a counterintuitive phenomenon proposed for qubits by T. Cubitt et al. [Phys. Rev. Lett. 91, 037902 (2003)]. We show that such entanglement distribution is possible with Gaussian states, using a certain three-mode fully separable mixed Gaussian state and linear optics elements readily available in experiments. Two modes of the state become entangled by sequentially mixing them on two beam splitters, while the third one remains separable in all stages of the protocol.