# A set of $4d-3$ observables to determine any pure qudit state

## TL;DR

This paper introduces a minimal tomographic method requiring only 4d-3 measurements to reconstruct any pure quantum state of dimension d, validated through experiments and simulations, and capable of certifying purity within the same measurements.

## Contribution

The paper proposes a new measurement scheme that efficiently reconstructs pure qudit states with fewer observables and includes purity certification, improving on existing methods.

## Key findings

- Achieved high fidelity (0.94) in reconstructing 7-dimensional pure states.
- Validated the method through experimental and numerical simulations.
- Demonstrated the ability to certify purity within the same measurement set.

## Abstract

We present a tomographic method which requires only $4d-3$ measurement outcomes to reconstruct \emph{any} pure quantum state of arbitrary dimension $d$. Using the proposed scheme we have experimentally reconstructed a large number of pure states of dimension $d=7$, obtaining a mean fidelity of $0.94$. Moreover, we performed numerical simulations of the reconstruction process, verifying the feasibility of the method for higher dimensions. In addition, the \emph{a priori} assumption of purity can be certified within the same set of measurements, what represents an improvement with respect to other similar methods and contributes to answer the question of how many observables are needed to uniquely determine any pure state.