The most entangled states of four and seven qubits

The paper in detail

Split four qubits into two pairs in one of three ways: \(12|34\), \(13|24\), \(14|23\). For a pure state \(\psi\), the entropy of the two-qubit marginal \(\rho_{1j}\) measures entanglement across that cut, and is at most 2 bits. The quantity studied is \[E(\psi)=\tfrac13\sum_{j=2}^4S(\rho_{1j}).\] Reaching \(E=2\) would need all three marginals maximally mixed, which is impossible. The state \(M_4\) puts phases \(1,\omega,\omega^2\) on the three complementary pairs of weight-two strings, and each of its marginals has spectrum \((\tfrac12,\tfrac16,\tfrac16,\tfrac16)\).

Purity alone cannot settle the question. The cluster state \(C_4\) and \(M_4\) both minimise the summed purity \(P=\sum_j\operatorname{tr}\rho_{1j}^2\) at \(P=1\), but \(E(C_4)=5/3\) while \(E(M_4)\approx1.792\). The proof therefore also bounds the summed third moment \(T\). A two-copy identity gives \(P\ge1\). A three-copy sum of squares gives \(F=17P-12T-12\ge0\). A cubic that lies above \(-x\ln x\) and touches it at \(\tfrac16\) and \(\tfrac12\) turns these two moment bounds into the entropy bound, and equality forces the \(M_4\) spectrum on every cut.

At seven qubits there are 35 cuts of type \(3|4\). The candidates form a family \(\psi_x\) with a distinguished qubit and six qubits labelled by the axes of an icosahedron, where the amplitude signs come from a symmetric conference matrix. Along this family the mean purity is a quadratic in \(x\), minimised at \(x=37/524\) with value \(121/917\approx0.131952\). The mean entropy peaks at a different point \(x_*\approx0.0698\). Five-copy certificates show that these are the global optima over all pure states.

The same method yields Rényi and Tsallis bounds at four qubits. For Rényi entropies the maximiser switches from \(M_4\) to cluster states at order \(q_*\approx1.588\). It also gives mixed-state and stability results. Stabilizer states do worse at seven qubits, with mean purity at least \(19/140\). Open questions include an analytic seven-qubit certificate and the scope beyond qubits.

Transferable methods

1. Moment bounds as positive multi-copy identities

   Entropies are not polynomials in the state, but the moments \(\operatorname{tr}\rho_S^k\) are, and a degree-\(m\) polynomial in \(|\psi\rangle\langle\psi|\) is an expectation value on \(m\) copies. The paper writes each gap as a sum of manifestly nonnegative terms, so the optimisers are exactly the states on which every term vanishes.

   Lemma 2 (two copies). Spin flip acts on a qubit operator as \(\sigma_yA^T\sigma_y=(\operatorname{tr}A)I-A\). Tensoring over four qubits gives \(\tilde\rho=\sum_S(-1)^{|S|}\rho_S\otimes I\), hence \(\operatorname{tr}(\rho\tilde\rho)=\sum_S(-1)^{|S|}\operatorname{tr}\rho_S^2\). For a pure state the left side is the four-tangle \(\tau=|\psi^T\sigma_y^{\otimes4}\psi|^2\), and complementary purities agree, so \[P=1+\tfrac12\Bigl(\sum_i\|\mathbf r_i\|^2+\tau\Bigr)\ge1,\] where \(\mathbf r_i\) are the one-qubit Bloch vectors.

   Theorem 3 (three copies) is a degree-six identity expressing \(8F\) as \(6NW_1\) plus squared norms of contractions of \(z\otimes z\otimes\tilde z\) with the antisymmetric tensor and a term measuring how differently the one-qubit marginals act on the state. Here \(W_1\) sums the squared one-qubit Pauli expectations. It gives \(F\ge\tfrac34W_1\ge0\), that is, \(T\le\tfrac5{12}+\tfrac{17}{12}(P-1)\). The slope \(17/12\) cannot be lowered: \(M_4\) has \((P,T)=(1,\tfrac5{12})\) and two Bell pairs have \((\tfrac32,\tfrac98)\), and both lie on the line.

   Editorial check: on 2,000 random four-qubit states the purity identity held to \(2\times10^{-15}\), and the smallest ratio \(F/W_1\) was 1.42, above the guaranteed \(\tfrac34\). We confirmed \((P,T)\) for \(M_4\) and for two Bell pairs.

   How to reuse it

   To bound a spectral function over pure states, first bound the moments it depends on by sums of squares in a few copies, with time-reversed copies where the symmetry group allows. The equality cases can then be read off from the vanishing terms. When a few copies admit moment data that no pure state realises, as at seven qubits, add copies.

2. A polynomial majorant that touches at the optimiser's spectrum

   Moment bounds become an entropy bound through a polynomial \(h\ge-x\ln x\) on \([0,1]\) that equals the entropy function at the conjectured optimiser's eigenvalues. Equality then forces every eigenvalue onto the touching points.

   Let \(h\) be the cubic Hermite interpolant of \(f(x)=-x\ln x\) at \(\tfrac16\) and \(\tfrac12\), matching value and slope at both, so \(h\ge f\) on \([0,1]\). Summing \(h\) over the twelve eigenvalues of the three cuts gives a combination of \(P\) and \(T\) only, since each spectrum sums to 1. With \(\Delta_H\ge0\) the total gap \(\sum(h-f)\), the paper's identity is \[3\ln2\,\bigl[E(M_4)-E(\psi)\bigr]=\bigl(\tfrac{15}{2}-6\ln3\bigr)(P-1)+\tfrac32(\ln3-1)F+\Delta_H.\] Both coefficients are positive (\(\tfrac{15}2-6\ln3\approx0.908\)), so each term is nonnegative and \(E\le E(M_4)\).

   Editorial check: we built \(h\) numerically and found \(\min_{[0,1]}(h-f)=0\), attained at the nodes. On 20,000 random states the displayed identity held to \(5\times10^{-15}\). The largest sampled \(E\) was 1.677, below \(1+\tfrac12\log_23=1.7925\), and \(E(C_4)=5/3\).

   Editorial check of the seven-qubit family: we built \(\psi_x\) from the stated signs (the conference matrix satisfies \(C^2=5I\)) and computed all 35 three-qubit marginals. At \(x=37/524\) the mean purity is \(0.1319520174\ldots=121/917\). The root of \[2^{32}(1-2x)^{15}=3^8(1+8x)^8(1+28x)^7\] is \(x_*=0.0698288\), where the mean entropy is \(2.9653560295\) bits, exceeding its value at \(x=37/524\) by \(7.16\times10^{-6}\) bits, as stated.

   How to reuse it

   When an entropy maximiser is conjectured, interpolate \(-x\ln x\) from above at its eigenvalues. The number of distinct eigenvalues sets the polynomial degree, hence how many moments, and so how many copies, the proof needs.

Source: Lemma 2, Theorem 3 and Section 2.3 (four qubits); Section 5 (seven qubits). Based on the linked paper version. Derivations marked as editorial, and suggested uses in other problems, are editorial guidance rather than results of the paper.

Context & limitations

Why it matters. The four-qubit question had been open for a quarter of a century, and the proof is short and analytic. The seven-qubit result shows that "most entangled" depends on the measure: the state of least mean purity does not have the greatest mean entropy, although the two values differ only in the sixth decimal place. The method bounds the moments of marginal spectra by positive multi-copy identities, some using spin-flipped copies, and converts them into entropy bounds with a polynomial majorant. The paper also proves the Gour–Wallach conjecture for Rényi entropies.

Limitations. The four-qubit maximum and its equality case are proved analytically. The seven-qubit bounds rest on five-copy identities verified in floating point with rigorous error bounds, followed by exact equality analyses, and some four-qubit refinements are computer-assisted as well. The spin-flip argument relies on a property of \(SU(2)\) and does not extend directly to qudits. The paper states that all results, documents and scripts were generated by AI agents (mainly Claude Opus and Sonnet 5.5, and OpenAI's GPT 6.1 sol) with human supervision and many cross-checks, including AI-generated Lean verification of most of the four-qubit results.

Description based on the abstract, Sections 1, 2.1–2.3 and 5, part of Section 6 and the AI-use statement of the linked arXiv version, read 2026-11-05. The appendices, the supplement and the five-copy certificates were not checked.