Dimension–Complexity Tradeoffs in Quantum-Function Testing

Dimension witnesses: amplification invariance and exact anticommutation relations

Definitions and mathematical statement

Shared notation.

Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

A quantum function is represented by the source operator-valued measurements on a Hilbert space. Test query complexity counts calls to that representation; robustness measures how test performance changes with deviations from the required property.

Record scope and qualifications.

The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The supplied registry flags this extraction as unrecoverable. It is displayed as a literal source fragment; the definitions provide context rather than a replacement formal statement.

It also includes an incomplete stated theorem whose final condition is missing. The fragment is shown literally rather than supplied with an invented conclusion.

Source-record limitation: Source flags the statement as unrecoverable: The linked report was inspected: the apparent question is immediately answered by the stated corollary of $\mathrm{MIP}^*=\mathrm{RE}$, so the extraction is not an unresolved problem. The source text below is rendered literally where necessary; damaged formulas and composite extracts are not silently repaired into a new mathematical claim. The supplied catalogue marks this record as unrecoverable. Its extract and terminology are documented, but a complete original problem has not been reconstructed.

Statement recorded in the supplied source.

What about tests for larger dimensions? Are there tradeoffs between the di- mension guarantee, the complexity of the test, and the test’s robustness? Theorem 1.3 (Corollary of MIP* = RE). There exist finite sets X , A and a test T for quantum functions F : X

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to A such that $1$ the test T makes only two queries to F , $2$ if a quantum function F is finite dimensional, then Pr[T accepts F ]

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leq 21 , $3$ there exists a quantum function F such that [S2]

Source status.

The supplied catalogue labels this source record partially solved; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

The fragment asks about dimension, query complexity, and robustness tradeoffs in quantum-function tests.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30004919 (OWR-8415359-007). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Complexity Theory $2022$, Oberwolfach Reports, DOI 10.4171/owr/2021/54, pp. 2955--3005 https://doi.org/10.4171/owr/2021/54.
• S3 MIP star equals RE https://arxiv.org/abs/2001.04383.
• Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.

Verbatim supplied catalogue question

What about tests for larger dimensions? Are there tradeoffs between the di- mension guarantee, the complexity of the test, and the test’s robustness? Theorem 1.3 (Corollary of MIP* = RE). There exist finite sets X , A and a test T for quantum functions F : X

\textbackslash{}

to A such that $1$ the test T makes only two queries to F , $2$ if a quantum function F is finite dimensional, then Pr[T accepts F ]

\textbackslash{}

leq 21 , $3$ there exists a quantum function F such that

The damaged extract and the established source claim

The retained catalogue fragment stops before the last displayed condition of Theorem1.3. Reading the primary report, printed page2974, recovers that condition as acceptance probability one. The source presents it as a corollary of $\mathrm{MIP}^*=\mathrm{RE}$: a two-query test separating its stated infinite-dimensional model from finite-dimensional strategies. That deep result is existing source context, not proved or independently validated in this manuscript. The catalogue fragment must not be turned into a newly solved dimension tradeoff. We instead attempt a finite algebraic witness and determine exactly what it can certify.

All moments leave dimension amplification invisible

In the finite positive-measurement model, replace a family on $\mathbb C^d$ by $M'_{a\mid x}=M_{a\mid x}\otimes I_k$ on $\mathbb C^{dk}$ for any integer $k\ge1$. Positivity and $\sum_a(M'_{a\mid x})^2=I$ are preserved. Every product has squared Frobenius norm multiplied by $k$, because $\operatorname{tr}((C\otimes I_k)(C^*\otimes I_k)) =k\operatorname{tr}(CC^*)$. Normalization by dimension also multiplies by $k$, so all transcript probabilities, including adaptive ones, agree exactly. No test in this access model can force an upper bound on the represented Hilbert-space dimension of every passing realization. A lower bound on the minimum possible dimension is compatible with this invariance; exact represented dimension is a different target.

An exact finite relation system with an exponential witness

Suppose Hermitian matrices $U_1,\ldots,U_{2m}$ on a nonzero finite-dimensional space satisfy \[ U_i^2=I,\qquad U_iU_j=-U_jU_i\quad(i\ne j),\qquad m\ge1. \] Then the dimension is divisible by $2^m$, in particular at least $2^m$. Define $Z_j=iU_{2j-1}U_{2j}$. Anticommutation shows $Z_j^*=Z_j$ and $Z_j^2=I$. For distinct $j,k$, moving one pair of $U$'s past the other requires four sign changes, so the $Z_j$ commute. Their commuting spectral projectors \[ P_\varepsilon=\prod_{j=1}^m(I+\varepsilon_jZ_j)/2, \qquad\varepsilon\in\{-1,1\}^m, \] are orthogonal and sum to $I$: this follows by distributing the product and using the two complementary projectors for each involution. At least one has nonzero range.

Conjugation by $U_{2j-1}$ changes the sign of $Z_j$ and preserves every $Z_k$ for $k\ne j$, using respectively one and two anticommutation sign changes. Thus this unitary is a bijection between the joint eigenspaces with sign vectors differing only at coordinate $j$. By successive flips all $2^m$ joint eigenspaces have the same positive dimension. Summing their dimensions proves the assertion.

The bound is attained. With the explicit Pauli matrices \[ X=\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad Y=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\quad Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}, \] put on $m$ qubits \[ U_{2j-1}=Z^{\otimes(j-1)}\otimes X\otimes I^{\otimes(m-j)}, \quad U_{2j}=Z^{\otimes(j-1)}\otimes Y\otimes I^{\otimes(m-j)}. \] Each is Hermitian and squares to $I$. The two at the same position anticommute because $XY=-YX$. For two at different positions, the earlier $X$ or $Y$ encounters one $Z$ at its position, contributing one minus sign; every other tensor factor commutes. Thus all distinct pairs anticommute. The ambient dimension is $2^m$, as required. The case $m=0$ is the empty relation system with arbitrary positive dimension.

The failed passage from relations to a query tradeoff

The relation system has $2m$ observables and quadratically many pair relations, but this does not make it a tester with that many queries. A sequential measurement oracle supplies probabilities of products of positive measurement operators. It does not grant coherent access to the signed observables or to arbitrary polynomial matrix identities. Although an involution can be written as the difference of its two spectral projectors, expanding a relation into signed words is not by itself an accept/reject experiment with perfect completeness and controlled soundness. No such reduction is supplied here. Nor does the exact common-eigenspace proof establish stability under approximate anticommutation: its exact projectors and exact unitary bijections would require quantitative replacements. Finally, the finite relation calculation says nothing about the infinite-dimensional tracial formulation needed for the source's stated deep corollary.

Audit and status

The audit checked normalization under tensoring by an identity, the signs in the $Z_j$ construction, positivity of the common rank, and the difference between an algebraic certificate and an accessible sequential test. The recovered final line of the source theorem is contextual evidence, not a newly supplied proof of $\mathrm{MIP}^*=\mathrm{RE}$. No computation or theorem prover was used. This damaged extract remains unresolved as an original research target; the finite auxiliary witness is complete but not claimed as original-target progress.

Process and contributions

Mathematics and model-assisted adversarial self-audit: OpenAI gpt-6-astra, ultra reasoning, as dispatched for this research batch; exact serving revision not exposed. Codex (GPT-6; exact serving revision not exposed) materially assisted with orchestration and format review. No human mathematical authorship, mathematical edits, or independent human proof checking is asserted. The coordinating Codex (GPT-6; exact serving revision not exposed) also provided a separate model-assisted written-proof review of this manuscript. This is not independent human verification. Contribution prompt A was applied to the full definition and supplied source row, then prompt B was applied adversarially to the derived results. The item-specific audit above records the scope checks and rejected extrapolations. Local assembly and structural checks do not establish mathematical correctness. The source catalogue status is preserved in TOP_PROBLEM and is separate from this attempt status.

Integration review.

Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% The proved material is a rederivation of known or auxiliary facts rather than a substantiated new advance on the original target. The source fragment is incomplete and its immediate primary context states an established deep corollary. The exact finite relation witness has no proved sequential testing implementation or robustness guarantee, so no new dimension/query tradeoff is established.

References

The complete original Sources subsection above is retained, including catalogue attribution and licence notices. The additional list records primary sources actually consulted for this attempt on 11 October 2026; uncited source links reproduced above were not all rechecked.

• Henry Yuen, Noncommutative Property Testing, OWR54/2021 pp.2971–2974; full primary source passage read in cached text, and page2972 operator definition visually checked from the primary PDF. The original agenda, stated known results and convention mismatch are distinguished. https://ems.press/content/serial-article-files/46932.

Final status

Unresolved in this attempt. The source fragment is incomplete and its immediate primary context states an established deep corollary. The exact finite relation witness has no proved sequential testing implementation or robustness guarantee, so no new dimension/query tradeoff is established.