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arXiv:1009.6104 · Physical Review A

Entanglement and nonclassicality for multimode radiation-field states

J. Solomon Ivan, S. Chaturvedi, Elisa Ercolessi, Giuseppe Marmo, G. Morandi, Narasimhaiengar Mukunda, R. Simon

Published Mar 25, 2011• 35 citations
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Abstract

Nonclassicality in the sense of quantum optics is a prerequisite for entanglement in multimode radiation states. In this work we bring out the possibilities of passing from the former to the latter, via action of classicality preserving systems like beam splitters, in a transparent manner. For single-mode states, a complete description of nonclassicality is available via the classical theory of moments, as a set of necessary and sufficient conditions on the photon number distribution. We show that when the mode is coupled to an ancilla in any coherent state, and the system is then acted upon by a beam splitter, these conditions turn exactly into signatures of negativity under partial transpose (NPT) entanglement of the output state. Since the classical moment problem does not generalize to two or more modes, we turn in these cases to other familiar sufficient but not necessary conditions for nonclassicality, namely the Mandel parameter criterion and its extensions. We generalize the Mandel matrix from one-mode states to the two-mode situation, leading to a natural classification of states with varying levels of nonclassicality. For two-mode states we present a single test that can, if successful, simultaneously show nonclassicality as well as NPT entanglement. We also develop a test for NPT entanglement after beam-splitter action on a nonclassical state, tracing carefully the way in which it goes beyond the Mandel nonclassicality test. The result of three-mode beam-splitter action after coupling to an ancilla in the ground state is treated in the same spirit. The concept of genuine tripartite entanglement, and scalar measures of nonclassicality at the Mandel level for two-mode systems, are discussed. Numerous examples illustrating all these concepts are presented.

DOI
10.1103/physreva.83.032118

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Nonclassicality in the sense of quantum optics is a prerequisite for entanglement in multimode radiation states. In this work we bring out the possibilities of passing from the former to the latter, via action of classicality preserving systems like beam splitters, in a transparent manner. For single-mode states, a complete description of nonclassicality is available via the classical theory of moments, as a set of necessary and sufficient conditions on the photon number distribution. We show that when the mode is coupled to an ancilla in any coherent state, and the system is then acted upon by a beam splitter, these conditions turn exactly into signatures of negativity under partial transpose (NPT) entanglement of the output state. Since the classical moment problem does not generalize to two or more modes, we turn in these cases to other familiar sufficient but not necessary conditions for nonclassicality, namely the Mandel parameter criterion and its extensions. We generalize the Mandel matrix from one-mode states to the two-mode situation, leading to a natural classification of states with varying levels of nonclassicality. For two-mode states we present a single test that can, if successful, simultaneously show nonclassicality as well as NPT entanglement. We also develop a test for NPT entanglement after beam-splitter action on a nonclassical state, tracing carefully the way in which it goes beyond the Mandel nonclassicality test. The result of three-mode beam-splitter action after coupling to an ancilla in the ground state is treated in the same spirit. The concept of genuine tripartite entanglement, and scalar measures of nonclassicality at the Mandel level for two-mode systems, are discussed. Numerous examples illustrating all these concepts are presented.
In this work we examine the possibilities of converting quantum optical nonclassicality into entanglement in multimode under the action of classicality preserving devices such as beamsplitters. While the single mode case is amenable to a complete analysis, non availability of certain crucial results in the classical theory of moments in the multimode situations forces us to treat these cases with lesser degree of generality by taking recourse to the familiar Mandel matrix and its extensions. We generalize the Mandel matrix from one‐mode states to the two‐mode situation, leading to a natural classification of states with varying levels of nonclassicality. For two‐mode states we present a single test which, if successful, simultaneouly witnesses nonclassicality as well as NPT entanglement. We develop a test for NPT entanglement after beamsplitter action on a nonclassical state, designed in such a way that it remains ‘close’ to that for nonclassicality. In the same spirit we analyse the result of three‐mode ‘beamsplitter’ action after coupling to an ancilla in the Fock ground state. The concept of genuine tripartite entanglement and scalar measures of nonclassicality at the Mandel level for two mode systems are discussed and illustrated with the help of several examples.
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.
With a product state of the form $ρ_{in} = ρ_a\otimes|0>_b_b< 0|$ as input, the output two-mode state ρoutρ_{\rm out}ρout​, of the beam splitter is shown to be NPT whenever the photon number distribution (PND) statistics {p(na)}\{p(n_a) \}{p(na​)} associated with the possibly mixed state ρaρ_aρa​ of the a_mode is antibunched or otherwise nonclassical, i.e., if {p(na)}\{p(n_a)\}{p(na​)} fails to respect any one of an infinite of classicality conditions.
Abstract Nonclassicality of a single-mode state can be quantified by the entanglement (potential) the state creates at the output of a beam splitter. Thus, beam splitter transformation can be utilized for obtaining single-mode nonclassicality conditions (quantifications) from two-mode entanglement criteria (measures). Here, (i) we employ this relation to quantify the single-mode nonclassicality (SMNc) of a nonlinear crystal field generating second harmonic signals. (ii) We also manage to witness the SMNc of superradiant emission, a non-Gaussian state, in a Dicke phase transition. Entanglement potential, at the beam splitter output, is obtained in terms of measurable single-mode quantities. Thus, presence of a beam splitter in an experiment is not necessary. Employing the beam splitter method, one can also utilize hybrid entanglement criteria as SMNc conditions working well in different classes of single-mode states. (iii) We also work on essentials of the entanglement-SMNc relation: one obtains the generalized squeezing condition ⟨ â † â ⟩ < |⟨ â 2 ⟩| as the SMNc condition when an optimized form of Duan–Giedke–Cirac–Zoller criterion [PRL 84, 2722] is used at the output of the beam splitter.
We consider a way to generate operational inequalities to test nonclassicality (or quantumness) of multimode bosonic fields (or multiparty bosonic systems) that unifies the derivation of many known inequalities and allows to propose new ones. The nonclassicality criteria are based on Vogel's criterion corresponding to analyzing the positivity of multimode PPP functions or, equivalently, the positivity of matrices of expectation values of, e.g., creation and annihilation operators. We analyze not only monomials but also polynomial functions of such moments, which can sometimes enable simpler derivations of physically relevant inequalities. As an example, we derive various classical inequalities which can be violated only by nonclassical fields. In particular, we show how the criteria introduced here easily reduce to the well-known inequalities describing (a) multimode quadrature squeezing and its generalizations, including sum, difference, and principal squeezing; (b) two-mode one-time photon-number correlations, including sub-Poisson photon-number correlations and effects corresponding to violations of the Cauchy-Schwarz and Muirhead inequalities; (c) two-time single-mode photon-number correlations, including photon antibunching and hyperbunching; and (d) two- and three-mode quantum entanglement. Other simple inequalities for testing nonclassicality are also proposed. We have found some general relations between the nonclassicality and entanglement criteria, in particular those resulting from the Cauchy-Schwarz inequality. It is shown that some known entanglement inequalities can be derived as nonclassicality inequalities within our formalism, while some other known entanglement inequalities can be seen as sums of more than one inequality derived from the nonclassicality criterion. This approach enables a deeper analysis of the entanglement for a given nonclassicality.