Complexity of quantum tomography from genuine non-Gaussian entanglement

Quantum state tomography, a fundamental tool for quantum physics, usually requires a number of state copies that scale exponentially with the system size, owing to the intricate quantum correlations between subsystems. We show that, in bosonic systems, the nature of correlations indeed fully determines this scaling. Motivated by the Hong-Ou-Mandel effect and Boson-sampling, we define Gaussian-entanglable (GE) states, produced by generalized interference between separable bosonic modes. GE states greatly extend the Gaussian family, encompassing arbitrary separable states, multi-mode Gottesman-Kitaev-Preskill codes, entangled cat states, and Boson-sampling outputs – resources for error correction and quantum advantage. Nonetheless, we prove that an m-mode pure GE state is learnable with only poly(m) copies, by providing an explicit protocol involving only heterodyne detection and classical post-processing. For states outside GE, we introduce an operational monotone – the minimum number of ancillary modes required to render them GE – and prove that it exactly captures the exponential overhead in tomography. As a by-product, we show that deterministic generation of NOON states with N>=3 photons by two-mode interference is impossible.

💡 Research Summary

The paper investigates the fundamental relationship between the type of quantum correlations present in bosonic (continuous‑variable) systems and the sample complexity required for quantum state tomography. The authors introduce a new class of states, called Gaussian‑entanglable (GE) states, defined as those that can be generated by applying a Gaussian protocol (Gaussian unitaries, partial trace, vacuum ancillas, and probabilistic mixing) to a separable multimode input. This definition captures not only all Gaussian states and their convex mixtures but also a wide variety of non‑Gaussian yet experimentally accessible states such as multi‑mode Gottesman‑Kitaev‑Preskill (GKP) codes, entangled cat states, and the output of Boson‑sampling experiments.

The first major result (Theorem 1) shows that any pure GE state admits a simple decomposition: it is exactly a Gaussian unitary acting on a product of two single‑mode pure states, \