Identifying the $3$-qubit $W$ state with quantum uncertainty relation

TL;DR

This work addresses the problem of identifying the 3-qubit $W$ state without full quantum state tomography. It builds a framework in the $XXZ$ Heisenberg model for three spin-1/2 particles and derives additive and multiplicative tripartite uncertainty relations among the observables $H_{12}$, $H_{23}$, and $H_{31}$. The key finding is that saturation of both uncertainty relations uniquely identifies $W$-state configurations, with explicit forms $|\psi\rangle_c1'$ and $|\psi\rangle_c2'$ arising under specific anisotropy values $\gamma=1$ or $\gamma=-2$. This provides a tomography-free method for entanglement discrimination and suggests extensions to $N$-qubit $W$ states, with potential practical impact on quantum networks and fault-tolerant computation.

Abstract

The $W$ state, a canonical representative of multipartite quantum entanglement, plays a crucial role in quantum information science due to its robust entanglement properties. Quantum uncertainty relations, on the other hand, are a fundamental cornerstone of quantum mechanics. This paper introduces a novel approach to Identifying tripartite $W$ states by leveraging tripartite quantum uncertainty relations. By employing a specific set of non-commuting observables, we formulate an uncertainty-based criterion for identifying $W$ states and rigorously demonstrate its generality in distinguishing them from other tripartite entangled states, such as the Greenberger-Horne-Zeilinger state. Our approach bypasses the need for complete quantum state tomography, as it requires only the verification of a set of uncertainty inequalities for efficient $W$-state identification. This work provides a new theoretical tool for identifying multipartite entangled states and underscores the significant role of quantum uncertainty relations in entanglement characterization.