Five-Dimensional Compatibility and Stratified Local-Unitary Structure of Pure Three-Qubit States
Abstract
We investigates the joint compatibility problem of the three pairwise concurrences, the three-tangle, and the Kempe invariant in arbitrary pure three-qubit states, and gives necessary and sufficient conditions for these five quantities to be simultaneously realizable by the same pure state. This five-dimensional compatibility region can be viewed as a fiber structure over the previously derived four-dimensional reachable region. We further find that the residual continuous local-unitary freedom depends on the position of the state within this region. In the generic interior, one continuous degree of freedom remains. It is fixed if the three-tangle vanishes or a pairwise concurrence becomes zero, and also at the boundary of the four-dimensional tangle region. The Kempe invariant gives the simplest algebraic parametrization of the fifth coordinate, but the same degree of freedom may be described by a known pure-state entanglement monotone. The compatibility conditions can also be expressed in terms of concurrence-based bipartite measures and multipartite quantities fixed by the pairwise concurrences and the three-tangle.
Related
- Unitary fault-tolerant encoding of Pauli states in surface codes
- Characterizing local unitary equivalence of multipartite states by local unitary Bargmann invariants
- Quantum states with the same entanglement and local unitary equivalence
- Bilinear correlations in Fluctuating Gaussian States with anti-unitary symmetries
- Fast Unitary Preparation of Surface-Code Logical States on Neutral-Atom Hardware
BibTeX
@misc{song2026fivedimensional,
title = {{Five-Dimensional Compatibility and Stratified Local-Unitary Structure of Pure Three-Qubit States}},
author = {Wei Song and Xiao-Lan Zong and Ming Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03864},
eprint = {2609.03864},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03864},
}
RIS
TY - UNPB
TI - Five-Dimensional Compatibility and Stratified Local-Unitary Structure of Pure Three-Qubit States
AU - Wei Song
AU - Xiao-Lan Zong
AU - Ming Yang
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.03864
UR - https://arxiv.org/abs/2609.03864
AB - We investigates the joint compatibility problem of the three pairwise concurrences, the three-tangle, and the Kempe invariant in arbitrary pure three-qubit states, and gives necessary and sufficient conditions for these five quantities to be simultaneously realizable by the same pure state. This five-dimensional compatibility region can be viewed as a fiber structure over the previously derived four-dimensional reachable region. We further find that the residual continuous local-unitary freedom depends on the position of the state within this region. In the generic interior, one continuous degree of freedom remains. It is fixed if the three-tangle vanishes or a pairwise concurrence becomes zero, and also at the boundary of the four-dimensional tangle region. The Kempe invariant gives the simplest algebraic parametrization of the fifth coordinate, but the same degree of freedom may be described by a known pure-state entanglement monotone. The compatibility conditions can also be expressed in terms of concurrence-based bipartite measures and multipartite quantities fixed by the pairwise concurrences and the three-tangle.
ER -