slug: kiem-kye-2026-local-distinguishability
bibkey: kiemkye2026localgeometry
doi: 10.48550/arXiv.2608.14199
url: https://arxiv.org/abs/2608.14199v1
triage: theorem
motivation_gids:
  - D5/S3/Quantum/Entanglement/LocalDistinguishability/KiemKyeLocalDistinguishability.result

Local distinguishability at every boundary projection state

Problem

Young-Hoon Kiem and Seung-Hyeok Kye, Local geometry for Schmidt number witnesses, arXiv:2608.14199v1, Section 5, states:

Nevertheless, we conjecture that the inclusion BP_ℓ ⊋ BP_{ℓ+1} is locally distinguishable around ϱ_{E⊥} for ℓ ≥ k, whenever ϱ_{E⊥} is on the boundary of the convex set BP_k.

For a nonzero proper complex subspace $E\subset {\mathbb{C}}^m\otimes {\mathbb{C}}^n$, write $\rho =P_{E^{\perp }}/\dim E^{\perp }$. The result says that if $\rho \in {\partial }_{H}B{P}_k$, then every neighbourhood of $\rho$ meets $B{P}_{\ell }\setminus B{P}_{\ell +1}$ whenever $1\le k\le \ell <\min (m,n)$. Here $H$ is the trace-one Hermitian affine space, and $B{P}_j$ tests nonnegativity against all vectors of Schmidt rank at most $j$.

Statement echo

All public definition bodies are the literal conventions of #17787, with the matrix abbreviation expanded. The imported Schmidt rank is the rank of the coefficient matrix.

Motivation

The settling GID is D5/S3/Quantum/Entanglement/LocalDistinguishability/KiemKyeLocalDistinguishability.result, with admission basis open-problem-resolution (#17787; Proved). This is an external named problem with no digestion atom. One OpenProblemResolutionClaim records Proved on the settling declaration. The source is documented in the Library note.

Gap

Sections 3–4 of the source establish generic cases using principal supporting hyperplanes. Section 5 asks for local distinguishability when that method does not give witnesses. The result covers non-generic kernels by moving the kernel vector and then perturbing the state.

Route

The following statements are proved in the settling module; the named helpers are private.

1. boundary_kernel and boundary_subspace_vector obtain a unit vector $v\in E$ with Schmidt rank at most $k$. Compactness of unit low-rank vectors makes the projection of the nonpositive quadratic-form tests closed. A boundary limit therefore has a zero quadratic-form test, which is a kernel vector for a positive semidefinite matrix.
2. rank_one_raise constructs a rank-one perturbation of a rectangular coefficient map: pair a dual functional nonzero on its kernel with a vector outside its range. For nonzero perturbation parameter, the new kernel is the intersection with that functional’s kernel. Rank–nullity raises the rank by one. mem_closure_exact_rank and unit_schmidtRank_density iterate this and normalize, approximating $v$ by unit vectors $u$ of rank $\ell +1$.
3. closure_of_low_rank_kernel compresses $A$ by $I-u{u}^{†}$. The compression is positive semidefinite, annihilates $u$, and equals $A$ at the original unit kernel vector. Its trace is continuous and tends to one, so normalization is defined nearby and tends to $A$.
4. compact_cone_overlap, overlap_schmidtRank and overlap_witness obtain $0\le \alpha <1$ bounding squared overlap with unit vectors of rank at most $\ell$. Thus $Q=\alpha I-u{u}^{†}$ is nonnegative on those vectors and has value $\alpha -1<0$ at $u$.
5. kernel_state_in_closure adds a sufficiently small positive multiple of $Q$ to the compressed state and normalizes its trace. The resulting matrices lie in $B{P}_{\ell }\setminus B{P}_{\ell +1}$. The two closure steps and closure_locallyDistinguishable give result.

Falsifier

A counterexample would consist of dimensions, a nonzero proper subspace, indices in the stated chain, its boundary projection state, and a neighbourhood missing $B{P}_{\ell }\setminus B{P}_{\ell +1}$. result rules out this complete configuration under the displayed hypotheses. A finite sampling failure to locate a witness is not such a neighbourhood certificate.

Evidence

D5.S3.Quantum.Entanglement.LocalDistinguishability.KiemKyeLocalDistinguishability.result is the settling theorem. Its public definitions and its proof are compiled with the scoped build and freeze door. The axiom closure of every public declaration is contained in $\{\mathrm{propext},Classical.choice,Quot.sound\}$.

The escape audit is unfinished under #18546. The targets are the settling result and the published owner D5.S3.Quantum.Information.SeparableStateLocalUnitaryStabilizerObstruction.quadratic_projector. No Reg mirror is delivered. The dependent telescope for the first target and the universe-polymorphic family for the second still lack complete source reconstruction, occurrence bridges, whole-family variation, sensitivity and observational-dependence evidence. No declared_validated registration is asserted.

Numerical experiment

Experiment directory at d771f467de03d12ca5206f1d061e345356b3a5a1. The check.py SHA-256 is 785a92069698f8a98003b872c8813e82a385b03fd5a1533327990e276fbccb03; the source at that commit and the executed file have the same hash.

From the experiments repository, the commands are OPENBLAS_NUM_THREADS=1 python3 docs/reports/kiem-kye-2026-local-schmidt-witnesses/check.py 20261016 20 300 (exit 1) and OPENBLAS_NUM_THREADS=1 python3 docs/reports/kiem-kye-2026-local-schmidt-witnesses/check.py 20261016 60 3000 (exit 0). The first misses preregistered criterion (a): the numerical overlap maximum differs from the top-$\ell$ Schmidt formula by more than $10^{-4}$ in some instances. The second meets that tolerance and all witness tests in all twelve instances. This reproduces the miss and pass recorded in #17787.

The exact tested scope is four configurations at $t\in \{0.2,0.05,0.01\}$: a product-vector kernel in $3\times 3$ with $k=1$ and $\ell =1,2$; a product-plus-entangled-vector kernel in $3\times 3$ with $k=1,\ell =2$; and a Schmidt-rank-two kernel in $3\times 4$ with $k=\ell =2$. The checks measure overlap optimization, the negative test at the constructed vector, a sampled low-rank quadratic-form minimum, Hermiticity, and distance from the projection state. These finite numerical checks do not establish universal blockpositivity.

Triage

What the settlement shows

The source’s generic-case theorems keep their stated hypotheses and conclusions. The settling result supplies the unrestricted local-distinguishability conclusion asked for in Section 5; it adds no hypothesis to those earlier theorems. The proof gives a sufficient criterion for arbitrary density matrices, while its converse and effective rates remain separate questions.

ASSUMED-UNVERIFIED

Literature completeness is ASSUMED-UNVERIFIED. The source v1, Sections 2, 3 and 5, and the literature screen are recorded in #17787. That screen reports no later author-listed paper and no OpenAlex record, a Semantic Scholar rate-limit response, and seat-reported comparisons with Han–Kye arXiv:2505.10288 and arXiv:2506.03733, Song–Chen arXiv:2508.11308, and Kye 1996/1997. Tomiyama, Linear Algebra and its Applications 69 (1985), 169–177, remains unread in that screen. No claim of exhaustive literature coverage is made.

The sufficient density-matrix closure criterion is proved; the converse and quantitative rates are open here. The escape audit remains unfinished under #18546. The module’s generality: I follows its existing I imports, including the current frozen Schmidt-rank owner, despite the dimension-uniform conclusion.