Certified Optimal Subsystem Designs for Haar-Purity Fluctuations Beyond Hadamard Budgets

Problem statement

Let $n$ be an even number of qubits, let $k=n/2$, and fix a budget of $m=n$ distinct balanced cuts. For a complex Haar-random pure state $\psi$ on $(\mathbb C^2)^{\otimes n}$, write $\rho_A=\operatorname{Tr}_{A^c}|\psi\rangle\langle\psi|$ and $\pi_A=\operatorname{Tr}(\rho_A^2)$. A design $\mathcal F=\{A_1,\ldots,A_m\}$ consists of $k$-element subsets of $\{1,\ldots,n\}$ with $A_i\ne A_j$ and $A_i\ne A_j^c$ for $i\ne j$. Complementary supports represent the same cut and must not be counted twice. Define $$V_n(\mathcal F)=\operatorname{Var}_{\psi\sim\mathrm{Haar}}\left[\frac1m\sum_{A\in\mathcal F}\pi_A\right].$$ Determine the exact global minimum of this quantity and give a minimizing design for every even $6\le n\le16$. The family may contain noncontiguous supports. This is a finite, precisely specified benchmark for the subsystem-design problem raised in Section VI.3 of the anchor paper. The budget $m=n$ deliberately exceeds the $n-1$ mutually orthogonal balanced sign vectors available in a Hadamard construction. Thus the familiar pairwise-optimal construction cannot settle this benchmark, even at Hadamard orders. For $n=6,10,14$, the balanced pair optimum also lies between integers. The anchor's Haar kernel reduces the objective to a rational function of pair intersections. With $N=2^n$ and $q=|A\cap B|$, $$\mathbb E[\pi_A\pi_B]=\frac{4N^3+16N^2+2N4^q+2N^2 4^{-q}}{N(N+1)(N+2)(N+3)},\qquad \mathbb E[\pi_A]=\frac{2^{n/2+1}}{N+1}.$$ The scientific target is a certified connection between subsystem geometry and collective entanglement fluctuations, including cases where ideal pairwise intersections cannot simply be assumed realizable. A solution should identify the combinatorial obstruction or construction responsible for each optimum. Smaller Haar variance does not by itself imply better entanglement detection, smaller experimental shot noise, or easier state preparation. Those are distinct questions and are not resolution requirements here.

Current state

Mirsohi, arXiv:2608.28914v1, derives the Haar pair-purity kernel, evaluates its cyclic-interval average, and explicitly proposes optimizing subsystem incidence matrices in Section VI.3. Its cyclic calculation is not a global optimization over arbitrary support families. The paper has an [open verification request](https://www.exactory.ai/verifications/474b69cb-da34-4c8c-b90b-351a454eab3e); that status is not a completed soundness assessment. A direct deduction from the displayed kernel supplies an essential special case at smaller budgets. At balance, the variable pair contribution $2N4^q+2N^2 4^{-q}$ has its real minimum at $q=n/4$. The positive supports of distinct nonconstant rows of a normalized Hadamard matrix of order $n$ have size $n/2$ and pair intersections $n/4$. Whenever such a matrix exists, selecting at most $n-1$ rows therefore attains the pairwise lower bound simultaneously and is globally optimal at that smaller budget. This deduction is included as a baseline, not claimed as an unsolved problem or as a verified novel theorem. It concerns support designs, not ensembles of quantum states with Hadamard-distributed amplitudes. Every balanced sign vector lies in the $(n-1)$-dimensional orthogonal complement of the all-ones vector, so $m=n$ such vectors cannot all be mutually orthogonal. The challenge asks what replaces the unattainable pairwise optimum at this budget. For $n\equiv2\pmod4$, the two nearest integer intersections tie for the pair minimum, but simultaneous attainability by the required number of cuts is a global combinatorial condition. The requested exact optima and independently checkable lower bounds for the entire stated range have not been established in this preparation. A targeted search did not identify a source resolving this particular benchmark; this is a scoped search finding, not an exhaustive claim about all prior literature. The challenge asks for matching constructions and rigorous certificates, rather than numerical optimization scores alone. Existing exact solutions may satisfy individual cases if cited and checked under precisely these constraints.

Resolution criteria

1. For every even $n\in\{6,8,10,12,14,16\}$, with $k=n/2$ and $m=n$, give the minimum $V_n^*$ as an exact rational number and an explicit family of $m$ balanced supports attaining it. Verify distinctness modulo complementation and evaluate the objective exactly. 2. Prove a matching lower bound over all admissible families for each case. An analytic argument or a finite computational proof is acceptable. A numerical solver's optimal-status message, an unvalidated floating-point bound, or a best-found family alone is insufficient. If a computational certificate is used, provide its complete data and a documented checker that verifies the claimed bound with exact arithmetic or rigorous enclosures. 3. Explain how the optimal intersection patterns attain or fail to attain the pairwise lower bound, distinguishing the Hadamard cases from $n=6,10,14$. Do not infer existence of a support family solely from a feasible intersection histogram or positive-semidefinite Gram matrix. 4. Derive or independently verify the Haar kernel and its normalization. Compare the optimal design with the exact mean variance of a uniformly sampled size-$n$ family of distinct balanced cuts, sampled without replacement from the $\tfrac12\binom{n}{n/2}$ available cuts. This is an equal-budget baseline. These comparisons concern intrinsic state-to-state fluctuations, not measurement noise. 5. Release all constructions, certificates, source code, and reproducibility instructions. The certificate checker must run without cloud services or proprietary solvers on a single computer with at most 96 GiB RAM. Report measured checker time and peak memory separately from search costs. Checking all six cases is required; partial ranges remain partial progress.

Citations

1. Kourosh Mirsohi. Exact Haar Statistics of Planar k-Purity in Multipartite Quantum Systems. arXiv:2608.28914v1 (2026), Section III.2, Appendix A, and Section VI.3.

Linked papers

• Exact balanced subsystem designs for Haar-purity fluctuationsdeclared at submission15 Sept 2026