Necessary and sufficient criterion for k-separability of N-qubit noisy GHZ states

TL;DR

This paper derives a necessary and sufficient criterion for k-separability of N-qubit GHZ states in white noise, providing a comprehensive understanding of their entanglement structure and implications for quantum communication.

Contribution

It introduces explicit entanglement witnesses and a criterion for k-separability, advancing the analysis of multipartite entanglement in noisy GHZ states.

Findings

Criterion covers more than half of all partial entanglement types

States become neither genuinely entangled nor fully separable at zero channel capacity

Method for sufficient conditions of separability in multipartite entanglement

Abstract

A Multipartite entangled state has many different kinds of entanglement specified by the number of partitions. The most essential example of multipartite entanglement is the entanglement of multi-qubit Greenberger-Horne-Zeilinger (GHZ) state in white noise. We explicitly construct the entanglement witnesses for these states with stabilizer generators of the GHZ states. For a $N$ qubit GHZ state in white noise, we demonstrate the necessary and sufficient criterion of separability when it is divided into $k$ parties with $N\leq 2k-1$ for arbitrary $N$ and $k$. The criterion covers more than a half of all kinds of partial entanglement for $% N $-qubit GHZ states in white noise. For the rest of multipartite entanglement problems, we present a method to obtain the sufficient conditions of separability. As an application, we consider $N$ qubit GHZ state as a codeword of the degenerate quantum code passing through depolarizing channel. We find that the output state is neither genuinely entangled nor fully separable when the quantum channel capacity reduces from positive to zero.