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Entropic uncertainty relations (EURs) have been examined in various contexts, primarily in qubit systems, including their links with entanglement, as they subsume the Heisenberg uncertainty principle. With their genesis in the Shannon entropy, EURs find applications in quantum information and quantum optics. EURs are state-dependent, and the state has to be reconstructed from tomograms (which are histograms readily available from experiments). This is a challenge when the Hilbert space is large, as in continuous variable (CV) systems and certain hybrid quantum (HQ) systems. A viable alternative approach therefore is to extract as much information as possible about the unknown quantum state directly from appropriate tomograms. Many variants of EURs can be extracted from tomograms, even for CV systems. In earlier work we have defined many tomographic entanglement indicators (TEIs) that can be readily calculated from tomograms without knowledge of the density matrix, and have reported on their efficacy as entanglement indicators in various contexts in both CV and HQ systems. The specific objectives of the present work are as follows: (i) To use the tomographic approach to investigate the links between EURs and TEIs in CV and HQ systems as they evolve in time. (ii) To identify the TEI that most closely tracks the temporal evolution of EURs. We consider two generic systems. The first is a multilevel atom modeled as a nonlinear oscillator interacting with a quantized radiation field. The second is the Λ-atom interacting with two radiation fields. The former model accomodates investigations on the role of the initial state of the field and the ratio of the strengths of interaction and nonlinearity in the connection between TEIs and EURs. The second model opens up the possibility of examining the connection between mixed state bipartite entanglement and EURs, when the number of atomic levels is finite. Since the tomogram respects the requirements of classical probability theory, this effort also sheds light on the extent to which TEIs reflect the temporal behaviour of those EURs which are rooted in the Shannon entropy.

The inherently probabilistic nature of quantum laws and the manner in which they manifest themselves in the measurement problem are typically formulated in terms of uncertainty relations. The original form of these relations, involving products of variances of incompatible observables (

Following this, Hirschman formalized the entropic uncertainty relation (EUR) (

The link between EURs and quantum entanglement is of immense interest as this is a possible route to understanding correlations and to quantify entanglement (

It is to be emphasized, however, that from both theoretical studies and experimental investigations some partial understanding about the interplay between entanglement and EURs, is only available at present for qubit systems. In the context of continuous variable (CV) systems, EURs have been examined theoretically and tight bounds obtained for quadrature observables in optics [see, e.g., (

In this work, we look at CV and hybrid quantum (HQ) systems to examine possible links between EURs and entanglement, relying only on tomograms for this purpose. More specifically, in this paper we examine the efficacy of TEIs in capturing the behaviour of EURs during dynamical evolution of two generic systems. This exercise facilitates identification of the appropriate TEIs that track EURs in the case of both pure and mixed states. We point out that this is the first and essential step in an extended program, that is, expected to shed light on the manner in which several factors affect the interplay between EURs and entanglement

The contents of this paper are arranged as follows. In

Consider a single-mode radiation field with photon creation and annihilation operators _{
θ
}, _{
θ
}, _{
θ
} as the abscissa and _{
θ
}, _{
B
}. A similar definition holds good for subsystem

Both qualitative identification and quantitative estimates of nonclassical effects such as squeezing and entanglement properties of radiation field states can be obtained solely from tomograms. In what follows, we focus on entanglement indicators. While these are not measures, it has been established that they suffice to capture the gross features of bipartite entanglement. We briefly describe, below, two of these “tomographic entanglement indicators” (TEIs) that we will use in the sequel. An interesting and useful feature of these indicators is that they can be defined for specific tomographic slices (also referred to as “sections”), by choosing appropriate values of _{
A
} and _{
B
}. Averaging over a judiciously chosen set of such indicators provides a section-independent assessment of entanglement. We will exploit this aspect in understanding the connection, if any, between EURs (which are in any case slice-dependent as they relate specific quadrature uncertainties), on the one hand, and both the slice-dependent and averaged entanglement indicator, on the other.

An interesting slice-dependent indicator denoted by _{IPR}(_{
A
}, _{
B
}) is inspired by the well known inverse participation ratio (IPR) which quantifies the delocalization of a state in a given basis. It was initially proposed to assess the extent of spatial delocalization of atomic vibrations in a specified eigenbasis of a disordered system (_{IPR}. As in the case of the example of the disordered spin chain mentioned above, entanglement indicators can now be defined based on IPR, the new feature being that it is now adapted to CV systems. It is to be noted, though, that since IPR only estimates delocalization, it is in general nonzero even for separable states. Consequently, an entanglement indicator based on IPR does not vanish for unentangled states. Despite this feature, this indicator, as also other TEIs, have been found to track entanglement dynamics effectively in a variety of systems (

It is worth pointing out that this is only an instance of how the tomographic approach is useful in calculating both slice-dependent values and averaged values of all TEIs, and not merely those corresponding to IPR. This advantage is not available in the computation of entanglement measures such as SVNE which are obtained as a single quantity from the reconstructed density matrix. Slice-dependent indicators are hence the natural choice in comparing entanglement trends with EUR trends, as the latter are defined only for specific slices—either canonically conjugate slices, or arbitrarily chosen slices defined by noncommuting operators. Hence, the SVNE is not always expected to capture trends in EURs as well as the TEIs. In the later sections we have examined the role of SVNE

We now proceed to define the TEIs. The slice-dependent entanglement indicator based on IPR is given by,_{
A
}, _{
B
}) from Eqs _{
A
} < _{
B
} < _{
A
} and _{
B
} suffices to capture the gross features.

The variance based uncertainties, entropic uncertainties and the corresponding bounds can be computed in a straightforward manner from tomograms. A useful and readily applicable procedure to compute the variance and all moments of quadrature observables (corresponding to a given tomographic slice) is given in

For any bipartite system the uncertainty relation of direct relevance to us pertains to canonically conjugate quadratures with variables (_{1}, _{1}) and (_{2}, _{2}) for the two subsystems. This EUR is given by_{1}, _{2}) is simply _{
A
} and _{
B
} equal to

In the next section we examine the reliability of bipartite TEIs in capturing the gross features of EURs in two generic systems, that are bipartite and tripartite respectively.

The Hamiltonian for a radiation field with photon creation and annihilation operators _{0}, _{
f
} ⊗|_{
a
} denoted by |

We have numerically examined how efficiently the TEIs capture these aspects of the dynamics. We illustrate these features for the atom in the ground state, _{0} = 1 and the field initially either in a Fock state or a CS |

We first consider the field initially in the Fock state |_{
A
} = _{
B
} = _{TEI} and _{TEI} do not mimic each other, _{IPR} and _{IPR} are relatively more similar in their trends (see _{
A
} = _{
B
}, while TEI has significant contributions from regions in which _{
A
} ≠ _{
B
}. This feature is more pronounced for large

_{TEI} (blue), 2 × [_{IPR}—0.507] (brown), and 0.4 × [_{IPR} = 0.507 and EU = 6.168. _{TEI} (blue), and 0.8 ×_{TEI}(0, 0) (red), 2 × [_{IPR}—0.507] (brown), and 2 × [_{IPR}(0, 0)—0.507] (orange) _{IPR}(0, 0) = 0.507.

For strong nonlinearity, the atom effectively behaves like a two-level system, with periodic exchange of energy with the field (_{IPR} with scaled time are small in comparison with the corresponding changes in _{TEI}. In this sense, _{TEI} reflects the EUR dynamics better than _{IPR}. As in the case of weak nonlinearity, _{IPR} and _{IPR} are similar in their trends, while _{TEI} and _{TEI} are dissimilar.

The EUR based on the Rényi entropy is alike in dynamics to the EUR corresponding to the Shannon entropy, independent of the extent of nonlinearity. Therefore, in what follows, we shall only examine the EUR based on the Shannon entropy.

Next, we consider the field to be initially in a CS |^{2} = 5). In this case the TEIs depend on the specific slice considered, in contrast to the preceding case. For the _{IPR} performs better than _{TEI}, on the average. Further, the dynamics of _{IPR} is similar to _{IPR}, and, overall, it captures the trends of EUR better than _{TEI} (

^{2} = 5) and weak nonlinearity (_{TEI} (blue), 1.2 × [_{IPR}—0.361] (brown), and 0.16 × [_{IPR} = 0.361 and EU = 4.289. ^{2} = 5) and weak nonlinearity (_{TEI}(0, 0) + _{TEI}(_{IPR}(0, 0) + _{IPR}(_{IPR}(0, 0) + _{IPR}(

Bipartite field-atom model. Initial state |^{2} = 5) and strong nonlinearity (_{TEI} (blue), 5 × [_{IPR}—0.361], and 0.8 × [_{IPR} = 0.361 and EU = 4.289.

From the above results we conclude that for weak nonlinearity and a pure bipartite state, the efficacy of both the TEIs in mimicking the dynamics of EUR is not very sensitive to the precise nature of the initial field state. Further, both _{IPR} and _{TEI} are comparable in their performance. In what follows, therefore, we will only consider strong nonlinearity and bipartite entanglement in a mixed state. In particular, we will compare how well the dynamics of SVNE, _{IPR} and _{TEI} follow that of EUR.

Investigation of the effect of a

We consider a tripartite model of a Λ-atom with energy levels {|_{1}⟩, |_{2}⟩, |_{3}⟩}, interacting with two radiation fields _{
i
}, with photon creation and annihilation operators _{
i
} respectively (_{
i
} mediates the |_{
i
}⟩ ↔|_{3}⟩ transition, where |_{3}⟩ is the highest energy state. The |_{1}⟩ ↔|_{2}⟩ transitions are dipole forbidden. The tripartite Hamiltonian wth zero detuning is_{
k
}} are positive constants, _{1}⟩. The subsequent temporal dynamics corresponding to this case has been investigated extensively. A spectacular bifurcation cascade has been reported as ^{2} = 15 and

_{1}⟩ (|^{2} = 15) and _{1}⟩ (|^{2} = 15) and _{TEI} (blue), 2.5 × [_{IPR}—0.361] (brown), and 0.25 × [_{IPR} = 0.361 and EU = 4.289. _{1}⟩ (|^{2} = 15) and _{TEI} (blue), and 11 × [_{IPR}—0.709] (brown) _{IPR} = 0.709, the minimum value during the collapse interval. _{1}⟩ (|^{2} = 15) and _{IPR}—0.709] (brown), _{IPR}(0, 0) + _{IPR}(

We now summarize the main results. Consider, first, the initial time interval. The broad features displayed by either field subsystem are similar (though not identical). This is because the atomic subsystem in this case is only weakly entangled with the field subsystems. Hence we have carried out the computation of SVNE with the density matrix corresponding to one of the fields. From _{TEI}, _{IPR} and EUR. However, EUR and _{IPR} resemble each other more closely in the interval considered, whereas SVNE and _{TEI} show similar roughly oscillatory behavior. We have verified that the trends in the

We now consider the dynamics during the time interval of collapse. In stark contrast to the foregoing observations, the dynamics of SVNE is not mirrored in any of the TEIs. Whereas SVNE collapses to a nearly constant value over the entire interval, the TEIs extremize in the neighbourhood of _{IPR} and that of EUR are remarkably similar. This is also reflected in the dynamics of [_{IPR}(0, 0) + _{IPR}(

We have examined the dynamics of bipartite entanglement of both pure and mixed CV states in two generic models of atom-field interaction. A primary purpose of this investigation has been to compare the manner in which TEIs on the one hand, and SVNE on the other, mimic the dynamics of EURs. Further, we have identified the TEI which closely tracks the temporal trends of EURs under different situations, such as, weak _{TEI} resemble each other in their gross features, but do not follow the trends in EURs in general. An interesting outcome of our investigation is that the efficacy of _{IPR} in mimicking EUR is very reasonable. This is an illustration of the deficiency of an entanglement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

SP carried out the numerical computations, generated the figures and wrote the first draft of the manuscript. SL, VB, and SR contributed to the conception and design of the study, and also revised and produced the final manuscript.

This work was supported in part by a grant from Mphasis to the Center for Quantum Information, Communication and Computing (CQuICC).

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.