bibkey: gourwallach2010fourqubits authors: Gilad Gour; Nolan R. Wallach year: 2010 title: “All Maximally Entangled Four Qubits States” doi: 10.1063/1.3511477 url: https://arxiv.org/abs/1006.0036v2 claim: “Four-qubit marginal entropy: the Higuchi–Sudbery candidate, its known restrictions, and the unrestricted maximum question.” strata_touched:

• D5/S3/Quantum/Entanglement/HiguchiSudbery/HiguchiSudberyEntropyMaximum license: citation-only triage: anchor

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DOI: https://doi.org/10.1063/1.3511477

Source: https://arxiv.org/abs/1006.0036v2

All Maximally Entangled Four Qubits States

Purity and entropy boundaries

The discussion before Theorem 5:

Hence, since ${\tau }_1$ is bounded by $1$, it follows that ${\tau }_2\le 4/3<3/2$. That is, there are no 4-qubit states for which all the 3 reduced density matrices, obtained by tracing out two qubits, are proportional to the identity.

With the paper’s tangle normalization, ${\tau }_2\le 4/3$ is the bound ${\sum }_c\mathrm{Tr}{\rho }_c^2\ge 1$ over the three inequivalent two-versus-two cuts.

Theorem 10(a), p. 8, has the hypothesis $\psi \in \mathcal{A}$:

$$E_2^{(\alpha )}(\psi )\le E_2^{(\alpha )}(|M⟩)\text{for }0<\alpha <2.$$

with equality if and only if up to local unitaries $\psi =|M⟩$.

Theorem 12, p. 9:

Let $\psi \in {\mathcal{H}}_4$ and $\alpha \ge 2$. Then,

$$E_2^{(\alpha )}(\psi )\le 5/3,$$

with equality if and only if, up to local unitaries, $\psi$ is one of the cluster states given in Eqs.(3,4,5).

Theorem 12 concerns average Rényi entropy. Theorem 10’s restricted-family conclusion does not provide the unrestricted von Neumann bound, and the cluster-state result for $\alpha \ge 2$ does not settle the analogous unrestricted question for $1<\alpha <2$.