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arXiv:2504.15401

Thirty-six officers, artisanally entangled

Gross, David, Paulina L. A. Goedicke

Published Apr 21, 2025• 0 citations
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Abstract

A perfect tensor of order $d$ is a state of four $d$-level systems that is maximally entangled under any bipartition. These objects have attracted considerable attention in quantum information and many-body theory. Perfect tensors generalize the combinatorial notion of orthogonal Latin squares (OLS). Deciding whether OLS of a given order exist has historically been a difficult problem. The case $d=6$ proved particularly thorny, and was popularized by Leonhard Euler in terms of a putative constellation of "36 officers". It took more than a century to show that Euler's puzzle has no solution. After yet another century, its quantum generalization was resolved in the affirmative: 36 entangled officers can be suitably arranged. However, the construction and verification of known instances relies on elaborate computer codes. In this paper, we present the first human-made order-$6$ perfect tensors. We decompose the Hilbert space $(\mathbb{C}^6)^{\otimes 2}$ of two quhexes into the direct sum $(\mathbb{C}^3)^{\otimes 2}\oplus(\mathbb{C}^3)^{\otimes 3}$ comprising superpositions of two-qutrit and three-qutrit states. Perfect tensors arise when certain Clifford unitaries are applied separately to the two sectors. Technically, our construction realizes solutions to the perfect functions ansatz recently proposed by Rather. Generalizing an observation of Bruzda and Życzkowski, we show that any solution of this kind gives rise to a two-unitary complex Hadamard matrix, of which we construct infinite families. Finally, we sketch a formulation of the theory of perfect tensors in terms of quasi-orthogonal decompositions of matrix algebras.

DOI
10.48550/arxiv.2504.15401

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A perfect tensor of order ddd is a state of four ddd-level systems that is maximally entangled under any bipartition. These objects have attracted considerable attention in quantum information and many-body theory. Perfect tensors generalize the combinatorial notion of orthogonal Latin squares (OLS). Deciding whether OLS of a given order exist has historically been a difficult problem. The case d=6d=6d=6 proved particularly thorny, and was popularized by Leonhard Euler in terms of a putative constellation of "36 officers". It took more than a century to show that Euler's puzzle has no solution. After yet another century, its quantum generalization was resolved in the affirmative: 36 entangled officers can be suitably arranged. However, the construction and verification of known instances relies on elaborate computer codes. In this paper, we present the first human-made order-666 perfect tensors. We decompose the Hilbert space (C6)⊗2(\mathbb{C}^6)^{\otimes 2}(C6)⊗2 of two quhexes into the direct sum (C3)⊗2⊕(C3)⊗3(\mathbb{C}^3)^{\otimes 2}\oplus(\mathbb{C}^3)^{\otimes 3}(C3)⊗2⊕(C3)⊗3 comprising superpositions of two-qutrit and three-qutrit states. Perfect tensors arise when certain Clifford unitaries are applied separately to the two sectors. Technically, our construction realizes solutions to the perfect functions ansatz recently proposed by Rather. Generalizing an observation of Bruzda and Życzkowski, we show that any solution of this kind gives rise to a two-unitary complex Hadamard matrix, of which we construct infinite families. Finally, we sketch a formulation of the theory of perfect tensors in terms of quasi-orthogonal decompositions of matrix algebras.
The negative solution to the famous problem of 36 officers of Euler implies that there are no two orthogonal Latin squares of order six. We show that the problem has a solution, provided the officers are entangled, and construct orthogonal quantum Latin squares of this size. As a consequence, we find an example of the long-elusive Absolutely Maximally Entangled state AME(4,6) of four subsystems with six levels each, equivalently a 2-unitary matrix of size 36, which maximizes the entangling power among all bipartite unitary gates of this dimension, or a perfect tensor with four indices, each running from one to six. This special state deserves the appellation golden AME state, as the golden ratio appears prominently in its elements. This result allows us to construct a pure nonadditive quhex quantum error detection code ((3,6,2))_{6}, which saturates the Singleton bound and allows one to encode a six-level state into a triplet of such states.
A Latin square can be extended by replacing its symbols with probabilities or quantum states. We relate these two extensions through explicit examples, from classical and fractional Latin squares to quantum Latin squares and a real solution of Euler's quantum 36 officers problem. A simultaneous cyclic-shift ansatz on a four-torus of tensor coefficients gives a real absolutely maximally entangled (AME) state of four six-level systems. A local orthogonal change of basis yields a representative with 96 nonzero amplitudes. Two single-party arrays obtained from the corresponding quantum orthogonal Latin square yield fractional Latin squares. Signs matter: comparison with the independent real construction of Bruzda and its sign variant distinguishes three local-unitary classes, two of which have identical probabilities. We extend this probability representation to existing seven- and nine-party AME states with six levels each. Their amplitude matrices reveal 48 complex Hadamard blocks of order 333^333 and six of order 646^464, respectively. These constructions separate what can be read from a probability picture from the coherent conditions responsible for maximal entanglement.
There exist pairs of orthogonal Latin squares of any order n except if n=2 or n=6 [Bose et al., Further results on the construction of mutually orthogonal latin squares and the falsity of Euler's conjecture, Can. J. Math. 12, 189 (1960)CJMAAB0008-414X10.4153/CJM-1960-016-5]. In particular, the problem of Euler's thirty-six officers does not have a solution. However, it has a "quantum solution": there exist so-called entangled quantum Latin squares of order 6 [Rather et al., Thirty-six entangled officers of Euler: Quantum solution to a classically impossible problem, Phys. Rev. Lett. 128, 080507 (2022)PRLTAO0031-900710.1103/PhysRevLett.128.080507]. We prove that mutually orthogonal quantum Latin squares of order 6 do not exist if entanglement is not allowed.
Absolutely maximally entangled states of four six-level systems, AME(4,6)(4,6)(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 363636 exists. We give real AME(4,6)(4,6)(4,6) states with 606060 non-zero coefficients in {±1,2}/72\{\pm1,\sqrt2\}/\sqrt{72}{±1,2​}/72​. Of their thirty-six officers, 121212 are classical, as some must be whenever fewer than 727272 coefficients are non-zero. They lie in the −1-1−1 eigenspace of an order-444 local symmetry, and on their support the AME conditions become linear. We derive two closed-form families of AME(4,6)(4,6)(4,6) states that meet at one of these states: a 101010-parameter family that is supported on 727272 basis states, and an 111111-parameter family on 606060 basis states. At each of its two ends the first family loses 121212 coefficients, and the AME states on the remaining 606060 basis states form an 111111-parameter family: the second family at one end, and an analogous family at the other. These two 606060-term families contain 409640964096 real AME states up to diagonal local unitaries, 102410241024 of which are limits of the first family. They fall into exactly 202020 classes under local unitaries and permutations of the parties, and we certify their separation with interval arithmetic.
This dissertation investigates well-known (quantum) combinatorial structures, such as (quantum) designs and (quantum) orthogonal Latin squares through the lenses of matrix algebras, group theory, and category theory. A central focus is placed on perfect tensors that are examined using representation theory and quasi-orthogonal subalgebras of matrix algebras. In particular, it is shown that the existence of a 2-unitary in dimension d-square is equivalent to the existence of four mutually quasi-orthogonal subalgebras of a d-square dimensional matrix algebra that are each isomorphic to a d-dimensional matrix algebra. This correspondence is further interpreted in terms of groups and their representations, relating the existence of perfect tensors to the existence of a group with a d-square dimensional irreducible representation and four subgroups on which the respective character “factorises“. A search algorithm implemented in the algebra software GAP is presented that was used to find examples of such groups. Furthermore, additional construction schemes for other dimensions, leading to 2-unitary complex Hadamard matrices, are developed using doubly perfect bi-unimodular sequences. Using the same ansatz, an analytical resolution of Euler’s 36 officers problem is provided, which differs from the earlier constructions of perfect tensors in dimension 36 via computer-assisted methods. Beyond this, the thesis introduces an abstract notion of categorical designs formulated via arrow categories. When applied to the categories Mat(N) and CP[FHilb], this framework produces a category-theoretic description of balanced incomplete block designs and of quantum designs, respectively. The construction generalises earlier definitions of quantum designs and extends it to a broader concept that can also be interpreted as combinatorial superoperators. These new concepts are used to define a category of mutually unbiased bases and a category of so-called combinatorial quantum channels.