Complexity of quantum tomography from genuine non-Gaussian entanglement

TL;DR

This paper investigates the complexity of quantum tomography in bosonic systems, showing that the nature of correlations determines whether the number of copies needed scales polynomially or exponentially, with implications for quantum resources and state generation.

Contribution

It introduces Gaussian-entanglable (GE) states, provides a protocol for efficiently learning pure GE states, and defines an operational monotone that captures the exponential overhead for non-GE states.

Findings

Pure GE states are learnable with polynomial copies.

States outside GE require exponential copies for tomography.

Deterministic NOON state generation with N≥3 photons is impossible.

Abstract

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.