Euler's thirty-six officers need not all be entangled: real AME(4,6), families and classification

AI summary

Overview

Research area: Quantum information theory and quantum combinatorics — specifically absolutely maximally entangled (AME) states, perfect tensors, and 2-unitary (real orthogonal) matrices.

Technical level: Advanced. The paper assumes familiarity with multipartite entanglement, reduced density matrices, unitary 2-designs/2-unitary matrices, orthogonal arrays, and finite-group phase congruences.

Scope (one sentence): The paper constructs the first explicit real AME(4,6) state with 60 non-zero coefficients, derives two closed-form families of AME(4,6) states that meet at it, and classifies the 4096 real states on the two 60-term supports into exactly 20 classes.

What This Paper Is About

Euler's problem of the thirty-six officers asks for a pair of orthogonal Latin squares of order 6, and it has no solution. Its quantum analogue asks for an AME(4,6) state — a state of four six-level systems in which every reduction to two parties is maximally mixed — and such states do exist. Every previously reported solution had complex coefficients, which led to the conjecture that no real orthogonal 2-unitary matrix of order 36 exists (listed as open problem T2 in a recent review). This paper answers that question in the affirmative: it exhibits real AME(4,6) states, shows where they come from (an eigenspace of a local symmetry), writes down two continuous families in closed form, and classifies the real states on the two 60-term supports.

Key Contributions

1. An explicit real AME(4,6) state with 60 non-zero coefficients in {±1, √2}/√72, described as the sparsest AME(4,6) state the author knows of, whose 2-unitary representation is a direct sum of 1×1 and 2×2 blocks. Twelve of its thirty-six officers are classical (single basis states with coefficient √2/√72 = 1/6) and 24 are entangled (two-term superpositions with coefficients ±1/√72).

2. A symmetry explanation and a linearization lemma. The state lies in the −1 eigenspace of the order-4 local symmetry u^⊗4 with u = diag(i, −1, −1, −i, −i, −i); on the relevant block supports the AME conditions reduce to linear equations in squared moduli plus linear congruences in the phases (Lemma 1), including a criterion over 𝔽₂ for the existence of a real state.

3. Two closed-form families that meet at the real state: a 10-parameter family supported on 72 basis states and an 11-parameter family on 60 basis states; the first family loses 12 coefficients at each of its two ends, and the AME states on the remaining 60 basis states form an 11-parameter family (the second family at one end, an analogous family at the other). The families also contain uniform-amplitude states with all 72 coefficients of modulus 1/√72, which the paper identifies as new.

4. A classification of the real states on the two 60-term supports: 4096 real AME states up to diagonal local unitaries, 1024 of which are limits of the 72-term family, falling into exactly 20 classes under local unitaries and permutations of the parties, with separation certified by interval arithmetic.

Main Findings

• The real state and its structure: The state is written as an array of Euler's officers, |ψ⟩ = (1/√72) Σ_{a,b} |ab⟩ ⊗ |φ_ab⟩, with rows labeled by rank and columns by regiment. Its support has 12 + 2·24 = 60 basis states. Proposition 1 proves it is AME by showing that U_{(ij),(kl)} = 6ψ_{ijkl} is a real orthogonal 2-unitary matrix of order 36; after permuting rows and columns, U, its realignment U^R, and its partial transpose U^Γ are each direct sums of 12 blocks (1) and 12 blocks of the form (1/√2)[[s₁, s₂], [s₃, s₄]] with sᵢ = ±1.

• Exact verification: All one- and two-party marginals were verified in exact arithmetic over ℚ(√2). For every product state |v⟩, |⟨v|ψ⟩|² ≤ 1/36, so the overlap is at most 1/6, with equality at the 12 states of P — the closest an AME(4,6) state can come to a product state.

• Classical officers are forced below 72 coefficients: Of the 36 officers, 12 are classical, and the paper states some must be classical in any AME(4,6) state with fewer than 72 non-zero coefficients. For x ∈ P all four indices lie in {0,3,4,5} with exactly one equal to 0; for x ∈ E exactly two indices lie in {1,2}.

• Origin of the state: It was found by searching eigenspaces of local symmetries u^⊗4 with u = VDV†. With V = 𝕀 and u = diag(i^{k₀}, …, i^{k₅}), k = (1,2,2,3,3,3), the −1 eigenspace is spanned by 360 basis states satisfying k_{x₁} + ⋯ + k_{x₄} ≡ 2 (mod 4), and a random-start optimization finds AME states within a few restarts. Most have a 10-dimensional space of first-order deformations modulo local unitaries; the others a 3-dimensional one.

• Sparsification and combinatorial structure: Minimizing a smoothed ℓ₁ norm over local unitaries commuting with u (a U(1)×U(2)×U(3) action on each party) brings a solution to exactly 72 non-zero coefficients; a subsequent Gauss–Newton optimization makes it AME to machine precision. Those 72 basis states form an orthogonal array of strength 2 and index 2, consisting of the 60 states E ∪ P plus 12 further states P′. Modulo an overall phase per party, the diagonal local unitaries acting as a scalar on this support form ℤ₂×ℤ₂×ℤ₄, in which u^⊗4 has order 4. Each reshaping splits into 18 full 2×2 blocks — 12 E-blocks and six mixed blocks — giving 36 E-blocks and 18 mixed blocks over the three reshapings.

• Two components of the 72-term family: Up to diagonal local unitaries, AME states with support exactly E ∪ P ∪ P′ are the states |Ψ_δ(α,β)⟩ with |δ| < 1 satisfying α_x + α_y ≡ β_{x′} + β_{y′} + π (mod 2π) on each mixed block. They form two components ℱ₇₂^±, exchanged by complex conjugation, with 10 parameters each. The quantity ω = Σ_{x∈P′, x₁≠0} φ_x − Σ_{x∈P, x₁≠0} φ_x satisfies 2ω ≡ π, so every such state has ω ≡ ±π/2, which distinguishes the components.

• The 60-term faces: AME states with support exactly E ∪ P are the states |Ψ₁(α)⟩ with 11 parameters, and those with support exactly E ∪ P′ are the states |Ψ₋₁(β)⟩. A face state is a limit of ℱ₇₂^± if and only if the sums of α (respectively β) over the quartets P₁, P₂, P₃ (respectively P₁′, P₂′, P₃′) agree modulo 2π. No state of ℱ₇₂^± is real, even up to diagonal local unitaries.

• Branches and new real states: The eight straight lines δ ↦ |Ψ_δ(0,β)⟩ are the branches through |ψ⟩ = |Ψ₁(0)⟩, four in each component. Up to diagonal local unitaries they end at four real states: |ψ₋⟩ = |Ψ₋₁(0)⟩ and the three states obtained from it by negating the coefficients on P₁′, P₂′, or P₃′ (classes 9, 9, 10, 9 of Theorem 1). No local unitary and permutation of the parties maps |ψ⟩ to |ψ₋⟩.

• Uniform states: At δ = 0 all 72 coefficients have modulus 1/√72; for constant β = ±π/2 they are ε_x, 1, and ±i times 1/√72 on E, P, and P′, and W = √2·U satisfies WW† = 2𝕀 with entries in {±1, ±i} — a 2-unitary complex weighing matrix. The paper notes that previously reported AME(4,6) states have at least 112 non-zero coefficients, while the real states of Refs. [20,22] have 60 and 96. An almost-real uniform state in ℱ₇₂⁺ has exactly four imaginary coefficients (on P₁′), and four is shown to be the minimum.

• Singularity of the real state: The dimension of first-order deformations modulo the local-unitary orbit is 10 for a generic member of ℱ₇₂^±, 11 for a generic face state, and 14 at |ψ⟩ (11 directions along the face and 3 spanned by the eight branches; numerical ranks with gaps of 10⁻¹⁵ against 10⁻²). The real state is thus a singular point where the 11-dimensional face and both 10-dimensional components meet.

• Classification result: None of the real states lies in ℱ₇₂^±. The two 60-term families contain 4096 real AME states up to diagonal local unitaries, 1024 of which are limits of the first family, falling into exactly 20 classes under local unitaries and permutations of the parties; the states are isolated (numerically), and their separation is certified with interval arithmetic. The full theorem statement (Theorem 1) lies beyond the truncated text provided.

• Existence corollary: Together with the permutation solutions for d ≠ 2, 6, a real AME(4,d) state exists if and only if d ≥ 3, exactly when a complex one does; AME(4,2) states do not exist at all.

Methodology in Plain English

Rather than searching the full space of four-qudit states, the author searches inside eigenspaces of simple local symmetries — transformations of the form u^⊗4 applied to all four parties at once. Two cheap necessary conditions narrow the choice of symmetry: reshaping a state along a bipartition must connect eigenspaces of equal (or suitably ordered) dimensions, and the squared magnitudes of the coefficients must form a probability distribution with uniform two-party marginals, which is a linear feasibility problem.

Within a promising eigenspace (the −1 eigenspace of an order-4 diagonal symmetry), a random-start numerical optimization locates AME states quickly. To sparsify a solution, the author minimizes a smoothed ℓ₁ norm over the local basis changes that preserve the eigenspace, landing on exactly 72 non-zero coefficients, then polishes the state with Gauss–Newton to machine precision.

The key structural insight is that on such block-structured supports the AME conditions become linear: each reshaping splits into 1×1 and 2×2 blocks, unitarity of a 2×2 block reduces to linear equations in the squared moduli and one linear congruence in the phases, and the real case reduces to linear equations over 𝔽₂. This lemma produces the closed-form families and turns the classification into counting solutions of finite congruence systems (via ranks and Smith normal form data). All statements about the states are checked by self-contained scripts, with exact arithmetic over ℚ(√2) and interval arithmetic used to certify that distinct classes really are distinct.

Why This Matters

The paper settles a question that had been stated as an open problem (T2) in a recent review and had been conjectured in the negative: there is no obstruction to real AME(4,6) states, and hence no gap between the real and complex existence thresholds for four-party AME states. It also supplies the sparsest known AME(4,6) states, uniform-amplitude states with fewer non-zero coefficients than earlier constructions (72 versus at least 112 reported previously), and a complete classification of the real states on two natural supports — a template for classifying AME states on block-structured orthogonal arrays.

Applications cited in the paper (these are motivations from the literature, not demonstrations performed here):

• Quantum secret sharing, where AME states underlie (4,2) threshold schemes and perfect tensors.
• Holographic quantum error-correcting codes, where perfect tensors are the building blocks of tensor networks.
• Dual-unitary circuits and gates of maximal entangling power, since 2-unitary matrices are exactly dual-un

Authors’ abstract

Absolutely maximally entangled states of four six-level systems, AME$(4,6)$, are the quantum solutions of Euler's problem of the thirty-six officers. Until recently all reported solutions were complex, and it was conjectured that no real orthogonal 2-unitary matrix of order $36$ exists. We give real AME$(4,6)$ states with $60$ non-zero coefficients in $\{\pm1,\sqrt2\}/\sqrt{72}$. Of their thirty-six officers, $12$ are classical, as some must be whenever fewer than $72$ coefficients are non-zero. They lie in the $-1$ eigenspace of an order-$4$ local symmetry, and on their support the AME conditions become linear. We derive two closed-form families of AME$(4,6)$ states that meet at one of these states: a $10$-parameter family that is supported on $72$ basis states, and an $11$-parameter family on $60$ basis states. At each of its two ends the first family loses $12$ coefficients, and the AME states on the remaining $60$ basis states form an $11$-parameter family: the second family at one end, and an analogous family at the other. These two $60$-term families contain $4096$ real AME states up to diagonal local unitaries, $1024$ of which are limits of the first family. They fall into exactly $20$ classes under local unitaries and permutations of the parties, and we certify their separation with interval arithmetic.