The Low-Individual-Degree Test Without the Diagonal-Lines Test Is Not Quantum-Sound

arXiv:2609.12346 · quant-ph, cs.CC · Submitted 2026-09-11 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "The Low-Individual-Degree Test Without the Diagonal-Lines Test Is Not Quantum-Sound".

Mira: Here's the gist of the paper. The low-individual-degree test without diagonal lines is not quantum-sound because axis-parallel consistency alone cannot enforce the global compatibility required by quantum soundness, necessitating an explicit diagonal-lines test for perfect acceptance Low-Individual-Degree Test Components The paper recalls two components of the low-individual-degree test used in this work 1.

Kai: Axis-parallel lines test: This compares a value reported at a point with a degree-d polynomial reported on a coordinate-parallel line 2.

Mira: Self-consistency test: This compares the two provers’ answers at the same point Failure of Reduced Test The core finding is that the reduced test, which chooses between axis-parallel lines or self-consistency with probability 1/2, does not possess the required quantum soundness property

JNV+20: found an example that requires the diagonal-lines test when (m, d, q) = (2, 2, 4); we give an example when (m, d) = (2, 2) and q is any odd prime.

Kai: The Obstruction to Global Quantum Soundness The paper demonstrates that for every odd prime q and parameters (m, d) = (2, 2), there exists a two-prover projective strategy for the reduced test that passes with probability 1

JNV+20: found an example that requires the diagonal-lines test when (m, d, q) = (2, 2); we give an example when (m, d) = (2, 2) and q is any odd prime.

Title and authors: Kai: So we're looking at this paper, "The Low-Individual-Degree Test Without the Diagonal-Lines Test Is Not Quantum-Sound." It tackles whether you can get quantum soundness just by using the axis-parallel lines test alone.

Mira: It’s a fundamental question about what it takes to guarantee global compatibility in these kinds of tests when you're dealing with quantum measurements. The authors are essentially asking if we can drop the diagonal-lines test without losing that necessary connection between local answers and the global structure.

Lev: From an error correction side, this is huge because it tells us exactly what kind of consistency we need to enforce across different parts of a quantum system. If you only check axis-parallel lines, you might miss correlations that are actually present globally >

Kai: Exactly. It points out that axis-parallel consistency alone isn't enough to force the global compatibility required for quantum soundness, which is what we need when we talk about proving things like MIP equals RE >

Mira: That means there’s a missing piece in how those local checks interact globally, and the paper shows that this obstruction isn't just an artifact of a specific setup; it persists even in the large-field regime where the degree of the polynomial is very small compared to the field size >

Lev: And that’s what worries me for real hardware. If we try to build a test based only on axis-parallel lines, we risk having local answers that look consistent but are globally incompatible, which means our error correction overhead wouldn't be enough to catch them >

The paper's summary: Kai: The paper summarizes the low-individual-degree test by recalling the two components they used: the axis-parallel lines test and the self-consistency test. They’re saying these are what we have, but they’re showing that even with these two, you still need that third component, the diagonal-lines test >

Mira: It goes deeper than just saying "we need a third thing." The paper shows that the reduced test—which just chooses between axis-parallel lines or self-consistency with a one/two probability—doesn't have the quantum soundness property they are trying to prove > <ref:2609.12346#pg1>

Lev: That’s a strong result because it means the standard way we might reduce these checks, by mixing them randomly, fails. It shows that you can't just pick one of those two options without losing the guarantee >

Kai: They found a specific example where this failure happens when the parameters are (m, d) equal to two and q is any odd prime number >

Mira: That's what makes it concrete. They have an example for when (m, d) is two and q is any odd prime number, which shows that the reduced test fails with probability one in that specific case >

Lev: So, for us to run a reliable protocol based on this test structure, we actually have to be explicit about using the diagonal-lines component or another explicit compatibility mechanism >

The paper's improvements: Kai: The paper outlines what they did next by showing how that diagonal-lines test actually supplies the necessary compatibility guarantee between point measurements associated with arbitrary pairs of points in a quantum setting >

Mira: They prove that this diagonal-lines component is exactly what enforces the missing compatibility; it’s how you bridge the gap between axis-parallel checks and global consistency, which is automatic for classical functions but not for quantum ones >

Lev: The paper also establishes an exact commutator calculation showing that every global function-valued projective measurement has average point-consistency with their constructed measurements at most one minus one over fifty-four >

Kai: That bound, one minus one over fifty-four, is important. It quantifies the approximation error for any general quantum output when compared against these structured low-degree measurements >

Mira: They calculate the exact averaged commutator as Eu∼F2qXa∈FqXg:g(u)=a⟨ψ A u a ⊗ G g ψ⟩ is less than or equal to one minus one over fifty-four >

Lev: It means that even if we use the best possible global measurement, there's still a measurable error bound tied to this constant, so we can’t just hope for perfect consistency >

Conclusion: Kai: So, to wrap up this paper, the main point is that axis-parallel consistency alone cannot enforce the global compatibility required by quantum soundness in this setting >

Mira: They proved that even with the best possible constructions, like those involving Hilbert spaces of dimension q cubed and specific measurement operators A u a = one/q X t∈Fq ω-t a W(t w(u)) or B l g = X λ:h l λ=g P l λ=Xλ:λ(w(u))=a P l λ=one/Sl X v∈Sl ω-λ(v)W(v), the consistency is bounded away from one by a universal constant > <ref:2609.12346#pg1>

Lev: For my perspective on hardware, this means that when we actually build something, we need to account for that one over fifty-four error bound in our noise models if we want to trust these low-degree tests >

Kai: It confirms the necessity of the diagonal-lines test for quantum soundness here. This work establishes that the diagonal-lines test is necessary for quantum soundness in this setting >

Mira: And they mentioned that this example can be generalized in a few ways, like q being any prime power, or m being larger than two >

Lev: It’s important to know it scales up and doesn't just disappear when you move to more complex systems; the obstruction is robust across those generalizations >

Kai: So, to recap the paper "The Low-Individual-Degree Test Without the Diagonal-Lines Test Is Not Quantum-Sound," we see that axis-parallel consistency misses global compatibility, and we need that diagonal test >

Mira: We also see explicit bounds on how far away general measurements are from low degree polynomials, down to one minus one over fifty-four >

Lev: And the implication for us is that if you're trying to build a real protocol, you have to account for this error in your analysis >

Stony Brook University

quant-ph, cs.CC

Submitted: 2026-09-11

Updated: 2026-10-19

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 89/100

The gist: The low-individual-degree test without diagonal lines is not quantum-sound because axis-parallel consistency alone cannot enforce the global compatibility required by quantum soundness, necessitating

Key concepts

Axis-parallel lines test
This component compares a value reported at a single point against the degree-d polynomial reported on a line that runs parallel to one of the coordinate axes. It checks consistency along straight lines in two dimensions, but it is insufficient for global quantum soundness.
Self-consistency test
This test verifies whether the answers provided by two different provers are identical at the exact same point. While useful for checking local agreement, this check alone cannot enforce the necessary global compatibility required by quantum soundness in this context.
Quantum Soundness
This is a property of a test that ensures if a prover gives an answer consistent with a low-degree polynomial, then the answer must be correct with high probability. The paper investigates whether simpler tests, like the axis-parallel line test, possess this required quantum soundness property.
Diagonal-lines test
This is an additional consistency check that compares values along lines that are not parallel to the coordinate axes (i.e., diagonal lines). The paper demonstrates that this specific type of test is necessary to enforce the global compatibility needed for quantum soundness in certain scenarios.

Summary

The low-individual-degree test without diagonal lines is not quantum-sound because axis-parallel consistency alone cannot enforce the global compatibility required by quantum soundness, necessitating an explicit diagonal-lines test for perfect acceptance

Low-Individual-Degree Test Components

The paper recalls two components of the low-individual-degree test used in this work

  1. Axis-parallel lines test: This compares a value reported at a point with a degree-d polynomial reported on a coordinate-parallel line

  2. Self-consistency test: This compares the two provers’ answers at the same point

Failure of Reduced Test

The core finding is that the reduced test, which chooses between axis-parallel lines or self-consistency with probability 1/2, does not possess the required quantum soundness property

[JNV+20] found an example that requires the diagonal-lines test when (m, d, q) = (2, 2, 4); we give an example when (m, d) = (2, 2) and q is any odd prime.

The Obstruction to Global Quantum Soundness

The paper demonstrates that for every odd prime q and parameters (m, d) = (2, 2), there exists a two-prover projective strategy for the reduced test that passes with probability 1

[JNV+20] found an example that requires the diagonal-lines test when (m, d, q) = (2, 2); we give an example when (m, d) = (2, 2) and q is any odd prime.

The Commutator Obstruction

The global measurement obstruction is established by showing that every global function-valued projective measurement has average point-consistency with the constructed measurements at most 1 − 1/54

[JNV+20] found an example that requires the diagonal-lines test when (m, d, q) = (2, 2); we give an example when (m, d) = (2, 2) and q is any odd prime.

Exact Commutator Calculation

The exact averaged commutator is calculated as Eu∼F2qXa∈FqXg:g(u)=a⟨ψ A u a ⊗ G g ψ⟩ ≤ 1 − 1/54

[JNV+20] found an example that requires the diagonal-lines test when (m, d, q) = (2, 2); we give an example when (m, d) = (2, 2) and q is any odd prime.

Conclusion

The result shows that axis-parallel consistency alone cannot enforce the global compatibility required by the usual quantum soundness conclusion

[JNV+20] found an example that requires the diagonal-lines test when (m, d, q) = (2, 2); we give an example when (m, d) = (2, 2) and q is any odd prime.

The paper proves that the reduced test does not have the corresponding quantum soundness property

The construction involves a Hilbert space H = C F 3 q where dim H = q cubed

The point measurement construction uses A u a = 1/q X t∈Fq ω-t a W(t w(u))

The line measurement construction defines B l g = X λ:h l λ=g P l λ=Xλ:h l λ=g P l λ=Xλ:λ(w(u))=a P l λ=1/Sl X v∈Sl ω-λ(v)W(v)

The final result shows that for every projective measurement G = a projective measurement whose outcomes are functions g: F 2 q → Fq, the consistency is bounded away from 1 by a universal constant

[JNV+20] found an example that requires the diagonal-lines test when (m, d, q) = (2, 2); we give an example when (m, d) = (2, 2) and q is any odd prime.

This work establishes that the diagonal-lines test is necessary for quantum soundness in this setting

[JNV+20] found an example that requires the diagonal-lines test when (m, d, q) = (2, 2); we give an example when (m, d) = (2, 2) and q is any odd prime.

The paper mentions that the above example can be generalized in three directions: q can be chosen as any prime power; m can be larger than 2; and d can be larger than 2

[JNV+20] found an example that requires the diagonal-lines test when (m, d, q) = (2, 2); we give an example when (m, d) = (2, 2) and q is any odd prime.

The paper's findings are important because they correct a previous analysis error and supply an ingredient for MIP∗ = RE<ref:2609.12346n, "Ji, Natarajan, Vidick, Wright, and Yuen identified that error and established quantum soundness of the classical low individual degree test by a different analysis [JNV+20]. This corrected theorem supplies an important ingredient for MIP∗ = RE">

[JNV+21] Communications of the ACM, 64(11):131–138.

The paper's findings are important because they correct a previous analysis error and supply an ingredient for MIP∗ = RE<ref:2609.

Improvements for AI systems

  1. Fundamental Soundness Verification for Low-Degree Polynomials (LDP): A Quantum-Sound Test for MIP=RE and Beyond. This AI system can rigorously verify whether local answers arise from a single global polynomial by employing a quantum soundness analysis, specifically testing the low-individual-degree test using the constructed projective strategy. The system can prove that axis-parallel consistency alone cannot enforce the global compatibility required by the usual quantum soundness conclusion, highlighting where classical tests fail and necessitating more robust checks like the diagonal-lines test.

  2. Enhanced Consistency Bounds for Quantum Measurements: Improving Global Function Approximation. The system can establish explicit bounds on how close a general projective measurement G is to a low-degree polynomial measurement, showing that every global function-valued projective measurement has average point-consistency with the constructed measurements at most 1 − 1/54. This allows the AI to quantify the approximation error for any quantum output against classical low-degree functions, providing a concrete measure of quantum advantage or limitations in approximating global functions.

  3. Guaranteed Perfect Acceptance for Structured Queries: Reliable Verification of Axis-Parallel Line Answers. The system can guarantee perfect acceptance (probability 1) for the axis-parallel lines test and self-consistency test when the underlying structure is defined by the constructed Hilbert space and measurements. This capability ensures that if a quantum prover adheres to these specific structural constraints, their answers are perfectly consistent with the expected low-degree polynomial on those lines.

  4. Explicit Commutator Obstruction Analysis: Detecting Non-Commuting Global Operators. The system can precisely calculate the average commutator error for independent random points u and v, showing that Eu,vΓ(u, v) = 2 − 1/q3, which is a direct obstruction to global quantum soundness. This allows the AI to detect when two different point measurements fail to commute in a way that violates expected classical behavior, providing a mathematical tool for identifying non-local correlations or errors in quantum protocols.

Abstract

To prove the quantum soundness of the classical low-individual-degree test, the authors of defined three subtests, namely the axis-parallel lines test, the self-consistency test, and the diagonal-lines test. An interesting question is whether the diagonal-lines test can be removed. In this paper, we show that the diagonal-lines test cannot simply be removed without another compatibility mechanism. Consequently, replacing the "conditional linear functions" by "coordinate deletion functions" in the proof of MIP*=RE, as mentioned in, does not by itself preserve the required soundness. The authors of found an example that requires the diagonal-lines test when (m, d, q) = (2, 2, 4); we give an example when (m, d) = (2, 2) and q is any odd prime.

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