Tripartite Entanglement in Qudit Stabilizer States and Application in Quantum Error Correction

TL;DR

This paper generalizes the structure of tripartite entanglement in stabilizer states from qubits to qudits of prime or squarefree dimension, and applies this to characterize related quantum channels in error correction.

Contribution

It extends a known qubit entanglement classification to qudits and uses this to analyze properties of stabilizer-based quantum channels.

Findings

States can be decomposed into unentangled, EPR, and GHZ components up to local unitaries.

Provides a complete characterization of channels associated with stabilizer error-correcting codes.

Analyzes properties of channels and their complements in the context of qudit stabilizer states.

Abstract

Consider a stabilizer state on $n$ qudits, each of dimension $D$ with $D$ being a prime or a squarefree integer, divided into three mutually disjoint sets or parts. Generalizing a result of Bravyi et al. [J. Math. Phys. \textbf{47}, 062106 (2006)] for qubits (D=2), we show that up to local unitaries on the three parts the state can be written as a tensor product of unentangled single-qudit states, maximally entangled EPR pairs, and tripartite GHZ states. We employ this result to obtain a complete characterization of the properties of a class of channels associated with stabilizer error-correcting codes, along with their complementary channels.