Superiority of heterodyning over homodyning: An assessment with quadrature moments

Abstract

We examine the moment-reconstruction performance of both the homodyne and heterodyne (doublehomodyne) measurement schemes for arbitrary quantum states and introduce moment estimators that optimize the respective schemes for any given data. In the large-data limit, these estimators are as efficient as the maximum-likelihood estimators. We then illustrate the superiority of the heterodyne measurement for the reconstruction of the first and second moments by analyzing Gaussian states and many other significant nonclassical states. Finally, we present an extension of our theories to two-mode sources, which can be straightforwardly generalized to all other multimode sources.

Homodyning the g(2)(0) of Gaussian states

“…Given the CM (24) it is straightforward to calculate the second-order correlation function g (2) TM (0) of the corresponding state ρ ab in Eq. (23). The analytical formula is clumsy and it is not reported explicitly in its general form.…”

“…In particular, if we set ψ = π we can still find the threshold values of the coherent amplitudes α and β and of the thermal contributions N th,1 and N th,2 , in order to have g TM of two-mode Gaussian states as a function of the displacement amplitudes α and β in the case of the TMSTS given in Eq. (23) with N th,1 = N th,2 = 0 (left panel) and N th,1 = N th,2 = 0.15 (right panel). In both the panels we set r = 0.5 and ψ = π, where ξ = r e iψ is the two-mode squeezing parameter, see the text for details.…”

Progress toward optimal quantum tomography with unbalanced homodyning

“…That HET surpasses BHOMOPT for the m = 1 case for any state is a consequence of the Heisenberg-Robertson-Schrödinger uncertainty relation [32]. The limiting case where the two methods give identical sCRBs is when the state is of minimum uncertainty.…”

“…Therefore, sCRB BHOMOPT for any state using this improved reconstruction strategy can be obtained through the Fisher information of the homodyne parameter X m ϑ . The theory for this was developed in [32]. It shall be shown that in practice, UHOM is tomographically more powerful than all other methods for moment tomography of interesting states, which forms the second main result.…”

“…Recently, the relationship between the Haar-averaged Cramér-Rao bound for state estimation and the permutation group was studied in [28,29]. In [30][31][32], we systematically analyzed the tomographic power of both BHOM and HET using the notions of the Fisher information and Cramér-Rao bound for moment estimation and found that the latter gives higher reconstruction accuracies for typically interesting states, with Gaussian states being one important class in CV quantum information processing [2,[33][34][35][36][37]. This provided irrefutable evidence of tomographic differences in parameter reconstruction for the Wigner and Husimi representations, despite their equivalence in state representation.…”