The log log jam in Gaussian state tomography

2026-07-14

AI Summary

This work addresses the sample complexity of Gaussian state tomography in continuous-variable quantum systems, where existing methods typically exhibit an unavoidable dependence on the system energy \(E\), even growing unboundedly as \(\log\log E\). The authors establish that any Gaussian measurement scheme inherently suffers from this energy dependence. They further demonstrate that adaptive strategies enable a smooth trade-off between the number of measurement rounds and the strength of this energy dependence. Crucially, the paper introduces the first energy-independent protocol for pure Gaussian state learning by leveraging non-Gaussian entangled measurements together with a single-copy Holevo–Helstrom phase POVM. This approach achieves a sample complexity of \(O(n^2/\varepsilon^2)\) for \(n\) modes and \(O(1/\varepsilon^2)\) in the single-mode case, entirely eliminating dependence on the system energy \(E\).

Abstract

Unlike in finite dimensions, quantum information in continuous-variable systems has the peculiar feature that without imposing physical constraints, the sample complexity of state tomography can be unbounded. Remarkably, this is even the case for state-of-the-art protocols for learning Gaussian states, which have finite-dimensional descriptions: the best known rates scale with $\log \log E$, where $E$ is the energy of the system. We prove this is not an artifact of existing analyses, but a fundamental limitation of the measurements used. We show: (1) Any protocol that uses Gaussian measurements, even entangled or adaptively chosen ones, must incur a $\log \log E$ dependence. This answers an open question posed by a number of previous works. (2) There is a smooth tradeoff between the number of rounds of adaptivity and the energy dependence, and we give a matching protocol achieving this interpolated rate. (3) With highly entangled, non-Gaussian measurements, one can learn $n$-mode pure Gaussian states with $O(n^2 / ε^2)$ samples, independent of $E$. This answers an open question posed by Chen et al. (4) A simple protocol based on the single-copy canonical phase POVM of Holevo and Helstrom learns single-mode pure Gaussian states with $O(1/ε^2)$ samples, again independent of $E$. Our results clarify the role of energy in bosonic state tomography and shed new light on the intriguing interplay between adaptivity, entanglement, and magic in quantum learning.

Problem

• Gaussian state tomography
• sample complexity
• energy dependence
• continuous-variable systems
• quantum learning

Innovation

• Gaussian state tomography
• sample complexity
• non-Gaussian measurements
• adaptivity
• energy dependence