The most entangled states of four and seven qubits
Abstract
Pure states of four and seven qubits cannot be maximally entangled across every balanced bipartition; we determine how entangled they can be on average. For four qubits we prove the conjecture of Higuchi and Sudbery: the mean von Neumann entropy of the three 2|2 cuts is at most 1+1/2log2 3 bits, with equality exactly on the local-unitary orbits of their state M4 and its complex conjugate. For seven qubits the mean purity of the 35 balanced 3|4 cuts is at least 121/917 and their mean entropy is at most 2.965356029… bits, each with a unique optimiser up to local unitaries and qubit permutations. The two optimisers are distinct members of one icosahedral family, so minimum purity and maximum entropy select different states. The proofs bound moments of the marginal spectra by positive identities on several copies of the state, some of them spin flipped, and bound the entropy by a polynomial in those moments. The four-qubit maximum has an analytic proof; the seven-qubit proofs, and some of the four-qubit refinements below, are computer-assisted, with exact arithmetic and rigorous error bounds. We also prove the Gour--Wallach conjecture for Rényi entropies of all orders, with the maximiser changing from M4 to cluster states at q*≈1.588, and obtain sharp Tsallis bounds, mixed-state bounds and quantitative stability.
Related
- Entanglement without the Entangled State: A Local Theory of Qubits with an Extended Born Rule
- Robust self-test of the maximally entangled state of two-qubits without assuming unitary observables
- Entangled logical qubits on a superconducting quantum processor
- Finite Nonsignalling Resources Suffice to Simulate Entangled Qubits
- Local Vacuum Entanglement through Most Entangled Modes
BibTeX
@misc{lewis2026most,
title = {{The most entangled states of four and seven qubits}},
author = {Antony Lewis},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05757},
eprint = {2611.05757},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05757},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The most entangled states of four and seven qubits
AU - Antony Lewis
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05757
UR - https://arxiv.org/abs/2611.05757
AB - Pure states of four and seven qubits cannot be maximally entangled across every balanced bipartition; we determine how entangled they can be on average. For four qubits we prove the conjecture of Higuchi and Sudbery: the mean von Neumann entropy of the three 2|2 cuts is at most 1+1/2log_2 3 bits, with equality exactly on the local-unitary orbits of their state M_4 and its complex conjugate. For seven qubits the mean purity of the 35 balanced 3|4 cuts is at least 121/917 and their mean entropy is at most 2.965356029… bits, each with a unique optimiser up to local unitaries and qubit permutations. The two optimisers are distinct members of one icosahedral family, so minimum purity and maximum entropy select different states. The proofs bound moments of the marginal spectra by positive identities on several copies of the state, some of them spin flipped, and bound the entropy by a polynomial in those moments. The four-qubit maximum has an analytic proof; the seven-qubit proofs, and some of the four-qubit refinements below, are computer-assisted, with exact arithmetic and rigorous error bounds. We also prove the Gour--Wallach conjecture for Rényi entropies of all orders, with the maximiser changing from M_4 to cluster states at q_*≈1.588, and obtain sharp Tsallis bounds, mixed-state bounds and quantitative stability.
ER -