Non-Gaussian quantum states are critical resources in photonic quantum information processing, rendering their generation and characterization of increasing importance in quantum optics. In this work, we theoretically and numerically analyze the relative efficiency of homodyne versus heterodyne measurements for reconstructing non-Gaussian states, a major outstanding question in continuous-variable tomography. Combining a Fisher information-based formalism with simulated experiments, we find homodyne tomography to outperform heterodyne measurements for all non-Gaussian states tested, although the separation between the two modalities proves significantly narrower than suggested by the asymptotic Cramer-Rao lower bound. Our results should find use for optimizing measurement strategies in practical continuous-variable quantum systems.
We examine the moment-reconstruction performance of both the homodyne and heterodyne (double-homodyne) 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.
In this work, we consider the estimation of single-mode Gaussian states using four different measurement schemes, namely, (1) homodyne measurement, (2) heterodyne measurement, (3) sequential measurement scheme, and (4) the Arthurs-Kelly measurement scheme, with a view to compare their relative performance. To this end, we work in the phase space formalism, specifically at the covariance matrix level, which provides an elegant and intuitive way to explicitly carry out the involved calculations. We show that the optimal performance of the Arthurs-Kelly and sequential measurement schemes is equal to the heterodyne measurement. While the heterodyne measurement outperforms the homodyne measurement in the mean estimation of squeezed state ensembles, the homodyne measurement outperforms the heterodyne measurement for variance estimation of squeezed state ensembles up to a certain range of the squeezing parameter. We then modify the Hamiltonian in the Arthurs-Kelly measurement scheme, such that the two meters can have correlations and show that the optimal performance is achieved when the meters are uncorrelated. We expect that these results will be useful in various quantum information and quantum communication protocols.
Neural networks can identify an optical quantum state from an image of its phase-space function, but a real measurement gives a finite record of
N detection events. Heterodyne detection samples the Husimi
Q function directly, whereas homodyne detection gives the Wigner function only after a reconstruction from several phases. We ask which scheme better identifies one of seven families of states, including Gottesman-Kitaev-Preskill (GKP) states, under loss, dephasing and Gaussian noise. We simulate finite records on identical states for two sets of families, one with equal mean photon number and one built from the code of Ahmed et al. For convolutional networks, heterodyne detection outperforms homodyne detection with filtered back-projection by 14.1 percentage points at
N=100 events and 20% dephasing in the first set (adjusted
p=0.003 over
24 cells across training runs), and it depends on the number of phases and on the reconstruction. At
N=103 and
104 no difference is significant, but the intervals are wide. At
N=100, a homodyne network without reconstruction does not differ significantly from heterodyne detection in the first set and is nominally worse in the second. Near thermal states we show to leading order that heterodyne detection damps phase-space structure of order
n exponentially, whereas homodyne detection shares unit weight among
n+1 angular components, a local result that does not explain the finite-record differences. Experimental records of a GKP state, photon-subtracted squeezed states and one mode of an Einstein-Podolsky-Rosen source are classified like simulated ones, but these low-power tests probe our counting-noise model, not the seven-class accuracy. A likelihood-ratio observer shows no such gap at
N=100, so our evidence concerns a reconstruction-based pipeline, not homodyne detection in general.
Pulsed homodyne quantum tomography usually requires a high detection efficiency, limiting its applicability in quantum optics. Here, it is shown that the presence of low detection efficiency (<50%) does not prevent the tomographic reconstruction of quantum states of light, specifically, of Gaussian states. This result is obtained by applying the so-called minimax adaptive reconstruction of the Wigner function to pulsed homodyne detection. In particular, we prove, by both numerical and real experiments, that an effective discrimination of different Gaussian quantum states can be achieved. Our finding paves the way to a more extensive use of quantum tomographic methods, even in physical situations in which high detection efficiency is unattainable.