Comparing Homodyne and Heterodyne Tomography of Quantum States of Light

TL;DR

The paper investigates whether homodyne or heterodyne tomography more efficiently reconstructs non-Gaussian single-mode CV quantum states. It develops a Fisher-information-based framework to compute the classical Fisher information for both modalities, uses finite-dimensional truncation and maximum-likelihood estimation on simulated data from random and tailored states, and compares performance against the CRLB. The main finding is that homodyne tomography is more efficient across tested states, though finite-sample deviations reduce the apparent advantage relative to asymptotic theory, especially in high-dimensional or pure-state regimes. The work provides a practical toolkit for optimizing CV tomography and highlights limitations of asymptotic bounds in finite experiments.

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.

Figures (4)

• Figure 1: Schematic of (a) homodyne detection and (b) heterodyne detection.
• Figure 2: Simulation results for random non-Gaussian states with Hilbert space dimension $d\in\{2,3,4,5,6\}$. (a) Ground truth Wigner functions and (b) corresponding estimation errors. Circles denote the Frobenius error for a specific trial, solid lines trace the mean Frobenius error of ten trials, and dashed lines give the CRLB.
• Figure 3: Simulation results for random non-Gaussian states with Hilbert space dimension $d\in\{7,8,9,10,11\}$. (a) Ground truth Wigner functions and (b) corresponding estimation errors. Circles denote the Frobenius error for a specific trial, solid lines trace the mean Frobenius error of ten trials, and dashed lines give the CRLB.
• Figure 4: Simulation results for common optical states truncated to Hilbert space dimension $d=11$. (a) Ground truth Wigner functions. (b) Estimation errors for ten simulated experiments on each state. See Sec. \ref{['sec:tailored']} for state definitions.