J. Solomon Ivan

A measure of non-Gaussianity for quantum states

We propose a measure of non-Gaussianity for quantum states of a system of n oscillator modes. Our measure is based on the quasi-probability

Entanglement and nonclassicality for multimode radiation-field states

{Q(\alpha),\alpha\in\mathcal{C}^n}

Nonclassicality breaking is the same as entanglement breaking for bosonic Gaussian channels

. Since any measure of non-Gaussianity is necessarily an attempt at making a quantitative statement on the departure of the shape of the Q function from Gaussian, any good measure of non-Gaussianity should be invariant under transformations which do not alter the shape of the Q functions, namely displacements, passage through passive linear systems, and uniform scaling of all the phase space variables: Q(α) → λ2nQ(λα). Our measure which meets this ‘shape criterion’ is computed for a few families of states, and the results are contrasted with existing measures of non-Gaussianity. The shape criterion implies, in particular, that the non-Gaussianity of the photon-added thermal states should be independent of temperature.

Operator-sum representation for bosonic Gaussian channels

The non-negativity of the density operator of a state is faithfully coded in its Wigner distribution, and this coding places on the moments of the Wigner distribution constraints arising from the non-negativity of the density operator. Working in a monomial basis for the algebra of operators on the Hilbert space of a bosonic mode, we formulate these constraints in a canonically covariant form which is both concise and explicit. Since the conventional uncertainty relation is such a constraint on the first and second moments, our result constitutes a generalization of the same to all orders. The structure constants of , in the monomial basis, are shown to be essentially the SU(2) Clebsch?Gordan coefficients. Our results have applications in quantum state reconstruction using optical homodyne tomography and, when generalized to the n-mode case, which will be done in the second part of this work, will have applications also for continuous variable quantum information systems involving non-Gaussian states.

Quantum discord plays no distinguished role in characterization of complete positivity: Robustness of the traditional scheme

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.

Generation and distillation of non-Gaussian entanglement from nonclassical photon statistics

We demonstrate a method of obtaining, from coherent light, tunable classical light where the temporal characteristics and photon number distribution can be controlled electronically. The tunability of the temporal coherence is shown through second-order correlation (G2(τ)) measurements, both in the continuous intensity measurement as well as in the photon counting regime. The generation of desired classical photon number distributions is illustrated by creating two light sources —one emitting a thermal state and the other a specific classical non-Gaussian state. Such tailored light sources with emission characteristics quite different from those of existing natural light sources are likely to be useful in quantum information processing, for example, in conjunction with photon addition, to generate tailored non-classical states of light with desired photon number distributions. As a particular application in this direction we also outline how a tailored classical non-Gaussian state generated by our technique may be mixed with a non-classical Gaussian state at a beamsplitter, to generate novel forms of non-Gaussian entanglement.

Invariant theoretic approach to uncertainty relations for quantum systems

Nonclassicality and entanglement are notions fundamental to quantum information processes involving continuous variable systems. That these two notions are intimately related has been intuitively appreciated for quite some time. An aspect of considerable interest is the behaviour of these attributes of a state under the action of a noisy channel. Inspired by the notion of entanglementbreaking channels, we define the concept of nonclassicality-breaking channels in a natural manner. We show that the notion of nonclassicality-breaking is essentially equivalent—in a clearly defined sense of the phrase ‘essentially’—to the notion of entanglement-breaking, as far as bosonic Gaussian channels are concerned. This is notwithstanding the fact that the very notion of entanglementbreaking requires reference to a bipartite system, whereas the definition of nonclassicality-breaking makes no such reference. Our analysis rests on our classification of channels into nonclassicalitybased, as against entanglement-based, types of canonical forms. Our result takes ones intuitive understanding of the close relationship between nonclassicality and entanglement a step closer.