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arXiv:2609.16490

A complete classification of the existence of finite-dimensional quantum solutions to inconsistent linear constraint systems

Markus Frembs

quant-phmath-phSubmitted Sep 15, 2026• Announced Sep 16, 2026• 0 citations
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Abstract

Linear constraint systems (LCS) provide a compact algebraic language for nonlocal games and state-independent contextuality. In the binary case the Mermin--Peres square shows that inconsistent LCS can be solved by promoting variables to local Pauli operators, whereas a series of no-go results excludes analogous Pauli-, Clifford- and monomial unitary-based constructions at odd prime modulus. We first isolate the reason for this difference: for tensor-product quantum solutions, the obstruction to a classical solution of a LCS over $\mathbb{Z}_n$ decomposes into obstruction classes carried by the individual factors. As a consequence, a genuinely global obstruction cannot arise solely by tensoring locally unobstructed systems. The argument is specific to odd modulus; at even modulus the reordering phase need not vanish, as in Pauli-based examples. Given this distinction, we instead associate LCS to finite arrangements of rank-one projectors and resolutions of the identity in finite-dimensional vector spaces. Quantum solvability for such LCS is automatic and the search for a quantum-classical gap rests entirely on proving classical unsatisfiability of the underlying modular incidence problem of the arrangement. By relating such LCS with group-valued frame functions, we identify a family of examples in the work of [Harding, Jager, and Smith, Int. J. Theor. Phys. 44, 539 (2005)]. To establish the (non)existence of classically unsatisfiable LCS over $\mathbb{Z}_n$ with quantum solutions in all dimensions $d$ and for all $n\in\mathbb{N}$, we further construct explicit examples for the cases $d=n$ prime. Our main result thus positively resolves the existence problem of such LCS beyond the binary case: such systems exist if and only if $d\geq 3$ and $\gcd(n,d)>1$. Moreover, we establish the existence of LCS with a quantum-classical gap over finite fields if and only if $d\geq 3$ and $p\mid d$.

Comments
20+10 pages, comments welcome

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Linear constraint systems (LCS) provide a compact algebraic language for nonlocal games and state-independent contextuality. In the binary case the Mermin--Peres square shows that inconsistent LCS can be solved by promoting variables to local Pauli operators, whereas a series of no-go results excludes analogous Pauli-, Clifford- and monomial unitary-based constructions at odd prime modulus. We first isolate the reason for this difference: for tensor-product quantum solutions, the obstruction to a classical solution of a LCS over Zn\mathbb{Z}_nZn​ decomposes into obstruction classes carried by the individual factors. As a consequence, a genuinely global obstruction cannot arise solely by tensoring locally unobstructed systems. The argument is specific to odd modulus; at even modulus the reordering phase need not vanish, as in Pauli-based examples.
Given this distinction, we instead associate LCS to finite arrangements of rank-one projectors and resolutions of the identity in finite-dimensional vector spaces. Quantum solvability for such LCS is automatic and the search for a quantum-classical gap rests entirely on proving classical unsatisfiability of the underlying modular incidence problem of the arrangement. By relating such LCS with group-valued frame functions, we identify a family of examples in the work of [Harding, Jager, and Smith, Int. J. Theor. Phys. 44, 539 (2005)]. To establish the (non)existence of classically unsatisfiable LCS over Zn\mathbb{Z}_nZn​ with quantum solutions in all dimensions ddd and for all n∈Nn\in\mathbb{N}n∈N, we further construct explicit examples for the cases d=nd=nd=n prime. Our main result thus positively resolves the existence problem of such LCS beyond the binary case: such systems exist if and only if d≥3d\geq 3d≥3 and gcd⁡(n,d)>1\gcd(n,d)>1gcd(n,d)>1. Moreover, we establish the existence of LCS with a quantum-classical gap over finite fields if and only if d≥3d\geq 3d≥3 and p∣dp\mid dp∣d.
Linear constraint systems (LCS) have proven to be a surprisingly prolific tool in the study of non-classical correlations and various related issues in quantum foundations. Many results are known for the Boolean case, yet the generalisation to systems of odd dimension is largely open. In particular, it is not known whether there exist LCS in odd dimension, which admit finite-dimensional quantum, but no classical solutions. In recent work, \cite{QassimWallman2020} have shown that unlike in the Boolean case, where the n -qubit Pauli group gives rise to quantum solutions of LCS such as the Mermin-Peres square, the n-qudit Pauli group never gives rise to quantum solutions of a LCS in odd dimension. Here, we generalise this result towards the Clifford hierarchy. More precisely, we consider tensor products of groups generated by (single-qudit) Pauli and diagonal Clifford operators.
We combine the study of resources in measurement-based quantum computation (MBQC) with that of quantum solutions to linear constraint systems (LCS). Contextuality of the input state in MBQC has been identified as a key resource for quantum advantage, and in a stronger form, underlies algebraic relations between (measurement) operators which obey classically unsatisfiable (linear) constraints. Here, we compare these two perspectives on contextuality, and study to what extent they are related. More precisely, we associate a LCS to certain MBQC which exhibit strong forms of state-dependent contextuality, and ask if the measurement operators in such MBQC give rise to state-independent contextuality in the form of quantum solutions of its associated LCS. Our main result rules out such quantum solutions for a large class of MBQC. This both sharpens the distinction between state-dependent and state-independent forms of contextuality, and further generalises results on the non-existence of quantum solutions to LCS in finite odd (prime) dimension.
The Mermin-Peres magic square provides a simple example of a system of linear equations over Z/2Z\mathbb{Z}/2\mathbb{Z}Z/2Z which has no classical solutions but does have a finite-dimensional operator solution. For a long time, it was not known how to construct similar examples over Z/dZ\mathbb{Z}/d\mathbb{Z}Z/dZ with ddd odd. In this paper, we construct, for every integer d≥2d \geq 2d≥2, a linear system over Z/dZ\mathbb{Z}/d\mathbb{Z}Z/dZ that has a finite-dimensional operator solution but no classical solution. For an odd prime ppp, our operators act on two ppp-dimensional qudits and generate a finite ppp-group obtained by adjoining diagonal polynomial phase operators to the generalized Pauli group, thereby forming a natural analogue of the Mermin--Peres magic square. Classical inconsistency follows from an elementary linearity argument comparing assignments on abelian subgroups. Furthermore, we prove a sharp threshold for the required degree of the added polynomial phase operators: our construction uses polynomials up to degree p−1p - 1p−1, and we show that the group formed by polynomials up to degree p−2p - 2p−2 is noncontextual.
We prove that the quantum monad in dimension 2n2n2n admits no natural transformation to the polymorphism clone of linear equations modulo nnn. Consequently, for every n≥2n\geq 2n≥2, there exists an unsatisfiable system of linear equations over Zn\mathbb{Z}_nZn​ whose constraint system game admits a perfect finite-dimensional quantum strategy. As a corollary, we completely characterise pseudo-telepathic constraint languages in finite dimension. The proof combines a result of Harding, Jager, and Smith on group-valued measures on subspaces of Hilbert spaces with the polymorphism-minion characterisation of quantum pseudo-telepathy.
Abstract A system of linear constraints can be unsatisfiable and yet admit a solution in the form of quantum observables whose correlated outcomes satisfy the constraints. Recently, it has been claimed that such a satisfiability gap can be demonstrated using tensor products of generalized Pauli observables in odd dimensions. We provide an explicit proof that no quantum-classical satisfiability gap in any linear constraint system can be achieved using these observables. We prove a few other results for linear constraint systems modulo d > 2. We show that a characterization of the existence of quantum solutions when d is prime, due to Cleve et al , holds with a small modification for arbitrary d . We identify a key property of some linear constraint systems, called phase-commutation, and give a no-go theorem for the existence of quantum solutions to constraint systems for odd d whenever phase-commutation is present. As a consequence, all natural generalizations of the Peres–Mermin magic square and pentagram to odd prime d do not exhibit a satisfiability gap.
We study linear constraint system (LCS) games over the ring of arithmetic modulo ddd. We give a new proof that certain LCS games (the Mermin--Peres Magic Square and Magic Pentagram over binary alphabets, together with parallel repetitions of these) have unique winning strategies, where the uniqueness is robust to small perturbations. In order to prove our result, we extend the representation-theoretic framework of Cleve, Liu, and Slofstra (Journal of Mathematical Physics 58.1 (2017): 012202.) to apply to linear constraint games over Zd\mathbb{Z}_dZd​ for d≥2d\geq 2d≥2. We package our main argument into machinery which applies to any nonabelian finite group with a ''solution group'' presentation. We equip the nnn-qubit Pauli group for n≥2n\geq 2n≥2 with such a presentation; our machinery produces the Magic Square and Pentagram games from the presentation and provides robust self-testing bounds. The question of whether there exist LCS games self-testing maximally entangled states of local dimension not a power of 222 is left open. A previous version of this paper falsely claimed to show self-testing results for a certain generalization of the Magic Square and Pentagram mod d≠2d\neq 2d=2. We show instead that such a result is impossible.