Estimation of entanglement in bipartite systems directly from tomograms

Abstract

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.

Signatures of avoided energy-level crossings in entanglement indicators obtained from quantum tomograms

“…ξ IPR , however, remains closer to ξ SVNE than the other indicators. This fact is consistent with inferences drawn from our earlier work [4] about the relation between the Hamming distance [34] and the efficacy of ξ IPR . We recall that the Hamming distance between two bipartite qudits |u 1 ⊗ |u 2 and |v 1 ⊗ |v 2 attains its maximum value of 2 when u 1 |v 1 = 0 and u 2 |v 2 = 0.…”

“…Indicators based on the IPR are also found to be good candidates for estimating the extent of entanglement [4,25]. The IPR corresponding to a bipartite system in the basis of the rotated quadrature operators is defined as…”

“…We denote the corresponding averaged value by ξ. In the context of bipartite CV models, we have shown in earlier work [3,4] that averaging ε TEI (θ A , θ B ) over 25 different values of (θ A , θ B ) selected at equal intervals in the range [0, π) yields a reliable entanglement indicator ξ TEI . A similar averaging of each of the quantities ε IPR , ε PCC and ε BD yields ξ IPR , ξ PCC and ξ BD , respectively.…”

“…In earlier work [3] we have proposed a tomographic entanglement indicator ξ TEI based on mutual information, that is obtained directly from the relevant tomograms. We have tested its efficacy in bipartite CV systems evolving unitarily under nonlinear Hamiltonians, by comparing it both with ξ SVNE and with an entanglement indicator ξ IPR based on inverse participation ratios (IPRs) [4]. (The participation ratio is a measure of delocalization in a given basis).…”

A tomographic approach to the sum uncertainty relation and quantum entanglement in continuous variable systems

“…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 (Sharmila et al, 2019).…”