Pure State Tomography with Pauli Measurements

TL;DR

This paper introduces an efficient protocol for pure state tomography using a minimal set of Pauli measurements, significantly reducing measurement requirements for two- and three-qubit systems while maintaining high fidelity.

Contribution

The authors present a novel measurement scheme that reduces the number of Pauli measurements needed for pure state tomography, outperforming random sampling methods and applicable to quantum optical systems.

Findings

Only 11 Pauli measurements needed for two-qubit pure states.

Only 31 Pauli measurements needed for three-qubit pure states.

Protocol is robust under depolarizing errors and validated experimentally.

Abstract

We examine the problem of finding the minimum number of Pauli measurements needed to uniquely determine an arbitrary $n$-qubit pure state among all quantum states. We show that only $11$ Pauli measurements are needed to determine an arbitrary two-qubit pure state compared to the full quantum state tomography with $16$ measurements, and only $31$ Pauli measurements are needed to determine an arbitrary three-qubit pure state compared to the full quantum state tomography with $64$ measurements. We demonstrate that our protocol is robust under depolarizing error with simulated random pure states. We experimentally test the protocol on two- and three-qubit systems with nuclear magnetic resonance techniques. We show that the pure state tomography protocol saves us a number of measurements without considerable loss of fidelity. We compare our protocol with same-size sets of randomly selected Pauli operators and find that our selected set of Pauli measurements significantly outperforms those random sampling sets. As a direct application, our scheme can also be used to reduce the number of settings needed for pure-state tomography in quantum optical systems.