The log log jam in Gaussian state tomography
2026-07-14 • Data Structures and Algorithms
Data Structures and Algorithms
AI summaryⓘ
The authors show that when trying to learn quantum states in systems with continuous variables, like light modes, the difficulty depends strongly on the energy in the system. They prove that if the measurements are limited to Gaussian types, no matter how clever or adaptive the strategy is, you still need a number of samples that grows with the double logarithm of the energy. However, if you use more complex, non-Gaussian measurements with strong quantum properties, you can learn these states efficiently without the energy affecting the sample size. Their work clarifies how measurement choices and quantum effects influence the ability to learn continuous-variable quantum states.
continuous-variable quantum systemsstate tomographyGaussian statessample complexityGaussian measurementsadaptivityentanglementnon-Gaussian measurementsenergy dependencephase POVM
Authors
Sitan Chen, Weiyuan Gong, Qi Ye, Zhihan Zhang
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.