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AIP conference proceedings

Nonclassicality and Entanglement in multimode radiation fields under the action of classicality preserving devices

S. Chaturvedi, Dipankar Home, Guruprasad Kar, Archan S. Majumda

Published Jan 1, 2011• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.3635859

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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.
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.
We introduce a new quantity for describing nonclassicality of an arbitrary optical two-mode Gaussian state which remains invariant under any global photon-number preserving unitary transformation of the covariance matrix of the state. The invariant naturally splits into an entanglement monotone and local-nonclassicality quantifiers applied to the reduced states. This shows how entanglement can be converted into local squeezing and vice versa. Twin beams and their transformations at a beam splitter are analyzed as an example providing squeezed light. An extension of this approach to pure three-mode Gaussian states is given.
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.
Several definitions of classicality are considered, such as P-representability, generalized coherent states and separable states. These notions are treated under a simple and general definition based on convex sets, which enables the use of the Hahn-Banach theorem to separate classical states from nonclassical ones. Nonclassicality linear witnesses are constructed, based on the observables available in a given physical situation. Some examples of nonclassical states are considered, with detection schemes available nowadays. Reviewing the concept of entanglement potential from a different perspective, it is shown that in some contexts an arbitrary single-system nonclassical state can be converted into a bipartite entangled state, provided a generalized controlled-displacement (e.g., beam-splitter, CNOT gate) is available. Also, this entanglement can be detected in a simple way using the nonclassicality witness of the original single-system state. Finally, we extend the discussion to multipartite states, proposing alternative ways to generate and classify multipartite entanglement.
While for single-mode bosonic fields, one talks about nonclassicality of quantum states, which plays an important role in various quantum information processing tasks, for composite systems, one talks about quantum correlations such as quantum entanglement and quantum discord, which are crucial and indispensable for implementation of quantum communication and computation. Both nonclassicality and quantum correlations are quantum features of physical systems and fundamental resources which can be exploited to bring supremacy over classical systems. A natural question arises on the relation and interconversion between these two resources. In this work, we investigate the conversion of single-mode nonclassicality into quantum correlations via beamsplitters. Our work is based on the idea of Asbó et al . ( Phys. Rev. Lett. , 94 (2005) 173602) of converting nonclassicality into quantum entanglement via a balanced beamsplitter. Through a paradigmatic setup with Gaussian states fed into the beamsplitters, we demonstrate that quantum correlations between the output states, as exhibited by either entanglement or quantum discord, increase monotonically with nonclassicality of the input Gaussian states, as quantified by an information-theoretic measure in terms of the Wigner-Yanase skew information.