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arXiv:2512.17031

Comparing Homodyne and Heterodyne Tomography of Quantum States of Light

Rhea P. Fernandes, Andrew J. Pizzimenti, Christos N. Gagatsos, Joseph M. Lukens

quant-phphysics.opticsSubmitted Dec 18, 2025• Announced Dec 22, 2025• 0 citations
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Abstract

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.

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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.
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Balanced homodyning, heterodyning, and unbalanced homodyning are three well-known sampling techniques used in quantum optics to characterize photonic sources in the continuous-variable regime. We show that for all quantum states and all observable-parameter tomography schemes, which includes reconstructions of arbitrary operator moments and phase-space quasidistributions, localized sampling with unbalanced homodyning is always tomographically more powerful (gives more accurate estimators) than delocalized sampling with heterodyning. The latter is recently known to often give more accurate parameter reconstructions than conventional marginalized sampling with balanced homodyning. This result also holds for realistic photodetectors with subunit efficiency. With examples from first- through fourth-moment tomography, we demonstrate that unbalanced homodyning can outperform balanced homodyning when heterodyning fails to do so. This new benchmark takes us one step towards optimal continuous-variable tomography with conventional photodetectors and minimal experimental components.
Non-Gaussian quantum states and operations are essential tools for multiple quantum information protocols exploiting light as information career. In this context, a key role is played by schemes operating with continuous wave light, in which non-Gaussian states are obtained by photon subtraction/addition and eventually reconstructed by quantum state tomography. In these configurations, the temporal resolution of the homodyne detection and the digital data processing critically affect our ability to faithfully reconstruct the produced non-Gaussian states. In this work, we apply digital data processing to experimental data to study how the temporal performances of the detection chain affect the acquisition and treatment of tomographic data. This allows understanding how these features impact the quality of quantum states observed by non-ideal detection chains. By doing so, we discuss the actual constraints on the acquisition and reconstruction of non-Gaussian states by taking into account the limitations of realistic experimental resources.
Continuous-variable (CV) quantum states are naturally characterized by their moments, defined as expectation values of products of single- or multimode ladder operators. Many CV Hamiltonians and quantum algorithms are formulated directly in terms of these moments, and therefore an efficient procedure to estimate moments with limited state measurements is necessary. In this paper, we present a protocol for shadow tomography of moment-generating functions (MGFs) of CV states based on randomized homodyne and heterodyne measurements. The resulting shadows enable an efficient and concurrent estimation of many multimode moments. Our complexity analysis shows that the number of measurements required to estimate a certain moment to a given precision grows exponentially in the order of the moments. Finally, we assess the precision of these moment estimators by applying them to two tasks: detecting entanglement in both Gaussian and non-Gaussian states via the Shchukin-Vogel protocol, and characterizing optical loss in a photonic chip. We demonstrate that both tasks can be accomplished with only a few thousand randomized measurements. The small number of required measurements, combined with efficient sample processing, makes our protocol applicable to a wide range of CV simulation tasks.