Entanglement quantification with randomized measurements is maximally difficult

TL;DR

This paper investigates the fundamental difficulty of certifying entanglement using randomized measurements, establishing the complexity of accessing invariants and extending the analysis to multi-qubit systems.

Contribution

It determines the minimal measurement settings for two-qubit invariants and shows entanglement certification is maximally difficult, revealing a hierarchy among invariants.

Findings

Entanglement certification requires the most demanding invariants.

Minimal measurement settings for all two-qubit invariants are identified.

Extended analysis to three-qubit systems improves measurement protocols.

Abstract

The certification of quantum systems is essential for emerging quantum technologies, particularly in quantum communication, networks, and distributed computing, where maintaining a common reference frame across distant nodes poses significant challenges. Reference frame independent approaches, such as randomized measurement schemes, offer a promising route by reducing experimental demands while granting access to basis-independent quantities, including entanglement. However, the efficiency of such schemes in measuring such local invariants has remained unclear. In this work, we determine the minimal number of measurement settings required to access all two-qubit invariants. We further demonstrate that entanglement certification necessarily involves the most demanding invariants, establishing it as a maximally difficult task. Our results reveal a fundamental hierarchy among invariants, with direct implications for experimental feasibility and theoretical understanding of quantum certification. Finally, we extend our analysis beyond bipartite systems by applying it to the Kempe invariant in three-qubit systems, improving known measurement protocols and providing a first step toward uncovering similar hierarchies in higher dimensions.