Abstract. Several nonclassical effects displayed by wave packets subject to generic nonlinear Hamiltonians can be identified and assessed directly from tomograms without attempting to reconstruct the Wigner function or the density matrix explicitly. We have demonstrated this for both single-mode and bipartite systems. We have shown that a wide spectrum of effects such as the revival phenomena, quadrature squeezing and Hong-Mandel and Hillery type higherorder squeezing in both the single-mode system and the double-well Bose-Einstein condensate can be obtained from appropriate tomograms in a straightforward manner. We have investigated entropic squeezing of the subsystem state of a bipartite system as it evolves in time, solely from tomograms. Further we have identified a quantifier of the extent of entanglement between subsystems which can be readily obtained from the tomogram and which mirrors the qualitative behavior of other measures of entanglement such as the subsystem von Neumann entropy and the subsystem linear entropy. The procedures that we have demonstrated can be readily adapted to multimode systems.

Extensive theoretical and experimental investigations on multipartite systems close to an avoided energy-level crossing reveal interesting features such as the extremisation of entanglement. Conventionally, the estimation of entanglement directly from experimental observation involves either one of two approaches: uncertainty-relation-based estimation that captures the linear correlation between relevant observables, or rigorous but error-prone quantum state reconstruction on tomograms obtained from homodyne measurements. We investigate the behaviour, close to avoided crossings, of entanglement indicators that can be calculated directly from a numerically-generated tomogram. The systems we study are two generic bipartite continuous-variable systems: a Bose–Einstein condensate trapped in a double-well potential, and a multi-level atom interacting with a radiation field. We also consider a multipartite hybrid quantum system of superconducting qubits interacting with microwave photons. We carry out a quantitative comparison of the indicators with a standard measure of entanglement, the subsystem von Neumann entropy (SVNE). It is shown that the indicators that capture the nonlinear correlation between relevant subsystem observables are in excellent agreement with the SVNE.

We assess the performance of an entanglement indicator which can be obtained directly from tomograms, avoiding state reconstruction procedures. In earlier work, we have examined this tomographic entanglement indicator, and a variant obtained from it, in the context of continuous variable systems. It has been shown that, in multipartite systems of radiation fields, these indicators fare as well as standard measures of entanglement. In this paper we assess these indicators in the case of two generic hybrid quantum systems, the double Jaynes-Cummings model and the double Tavis-Cummings model using, for purposes of comparison, the quantum mutual information as a standard reference for both quantum correlations and entanglement. The dynamics of entanglement is investigated in both models over a sufficiently long time interval. We establish that the tomographic indicator provides a good estimate of the extent of entanglement both in the atomic subsystems and in the field subsystems. An indicator obtained from the tomographic indicator as an approximation, however, does not capture the entanglement properties of atomic subsystems, although it is useful for field subsystems. Our results are inferred from numerical calculations based on the two models, simulations of relevant equivalent circuits in both cases, and experiments performed on the IBM computing platform.

We investigate the advantages of extracting the degree of entanglement in bipartite systems directly from tomograms, as it is the latter that are readily obtained from experiments. This would provide a superior alternative to the standard procedure of assessing the extent of entanglement between subsystems after employing the machinery of state reconstruction from the tomogram. The latter is both cumbersome and involves statistical methods, while a direct inference about entanglement from the tomogram circumvents these limitations. In an earlier paper, we had identified a procedure to obtain a bipartite entanglement indicator directly from tomograms. To assess the efficacy of this indicator, we now carry out a detailed investigation using two nonlinear bipartite models by comparing this tomographic indicator with standard markers of entanglement such as the subsystem linear entropy and the subsystem von Neumann entropy and also with a commonly-used indicator obtained from inverse participation ratios. The two model systems selected for this purpose are a multilevel atom interacting with a radiation field, and a double-well Bose-Einstein condensate. The role played by the specific initial states of these two systems in the performance of the tomographic indicator is also examined. Further, the efficiency of the tomographic entanglement indicator during the dynamical evolution of the system is assessed from a timeseries analysis of the difference between this indicator and the subsystem von Neumann entropy.

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