# Adaptive compressive tomography: a numerical study

## TL;DR

This paper numerically investigates an adaptive compressive tomography method that efficiently reconstructs low-rank quantum states with fewer measurements, outperforming some existing measurement strategies and proposing a hybrid approach for improved speed.

## Contribution

It introduces a faster hybrid compressive tomography scheme combining random and adaptive measurements, with numerical evidence of favorable scaling and performance.

## Key findings

- Adaptive scheme performs comparably to compressed sensing methods.
- Outperforms random measurement bases for low-rank states.
- Proposes a hybrid measurement approach for efficiency.

## Abstract

We perform several numerical studies for our recently published adaptive compressive tomography scheme [D. Ahn et al. Phys. Rev. Lett. 122, 100404 (2019)], which significantly reduces the number of measurement settings to unambiguously reconstruct any rank-deficient state without any a priori knowledge besides its dimension. We show that both entangled and product bases chosen by our adaptive scheme perform comparably well with recently-known compressed-sensing element-probing measurements, and also beat random measurement bases for low-rank quantum states. We also numerically conjecture asymptotic scaling behaviors for this number as a function of the state rank for our adaptive schemes. These scaling formulas appear to be independent of the Hilbert space dimension. As a natural development, we establish a faster hybrid compressive scheme that first chooses random bases, and later adaptive bases as the scheme progresses. As an epilogue, we reiterate important elements of informational completeness for our adaptive scheme.