Scalar compressions and asymptotically optimal tough error models
A complete candidate for the asymptotic tough-error-model problem: universal bounds differ by at most one code dimension, with explicit nearly worst-case error spaces.
13 November 2026 · Review version 1.0
If an error space has dimension e and acts on an n-dimensional quantum system, how much quantum information must remain exactly correctable, regardless of the particular errors? The tough-error-model problem asks for this worst-case guarantee and for explicit error spaces that nearly attain it.
The manuscript gives a complete proof candidate for the original question’s asymptotically matching-bounds option. Its two bounds differ by at most one code dimension. The proof has received sequential internal analytical review; independent review and comprehensive novelty assessment remain outstanding.
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The result and its scope
Let an error model be an arbitrary complex linear space \(\mathcal E\subseteq M_n(\mathbb C)\) of dimension e. A subspace C corrects it when its projection satisfies the Knill–Laflamme conditions
\[P_C A^*B P_C=\lambda(A,B)P_C \qquad\text{for every }A,B\in\mathcal E.\]Let c(e,n) be the minimum, over these error spaces, of the largest correcting-code dimension. For every integer n ≥ 1 and 1 ≤ e ≤ n², the proposed bounds are
\[\max\left\{1,\left\lfloor\frac{n+1}{e^2+1}\right\rfloor\right\} \le c(e,n)\le \left\lceil\frac{n}{e^2+1}\right\rceil.\]The upper bound is attained by explicit error spaces. Their largest codes are therefore within one dimension of the worst possible value. Uniformly,
\[\left|c(e,n)-\frac{n}{e^2+1}\right|<1,\]and the ratio of c(e,n) to n/(e²+1) tends to one whenever n/(e²+1) tends to infinity, including when e varies with n.
This is the arbitrary linear-error formulation of the original problem. The errors need not include the identity, and no trace-preserving Kraus normalization is assumed. The result concerns the dimension of a code, not the logarithm of that dimension or an efficient algorithm for finding it. Some exact finite values remain open; the displayed bounds leave 1 ≤ c(2,6) ≤ 2.
Why a large scalar compression must exist
The lower bound follows from a theorem about any D-dimensional operator system: a complex matrix space that contains the identity and is closed under adjoints. Such a system has a rank-k scalar compression whenever 2 ≤ k ≤ n and n ≥ Dk − 1. For an error space, the relevant system is
\[\mathcal V_{\mathcal E} =\operatorname{span}_{\mathbb C}\{I,A_i^*A_j:1\le i,j\le e\}, \qquad \dim\mathcal V_{\mathcal E}\le e^2+1.\]Choose a Hermitian basis containing the identity and D − 1 observables, and an n-by-k isometry X. Write
\[G_j=X^*H_jX,\qquad T_j=G_j-\frac{\operatorname{Tr}G_j}{k}I_k,\qquad F(X)=\sum_j\operatorname{Tr}(T_j^2).\]The nonnegative function F vanishes precisely at a scalar compression. It attains a minimum because the space of isometries is compact. Suppose that minimum were positive. Stationarity gives
\[\sum_jH_jXT_j=XS,\qquad S=\sum_jG_jT_j.\]Consequently, the matrices
\[M=[X,H_1X,\ldots,H_{D-1}X],\qquad B=[-S;T_1;\ldots;T_{D-1}]\]satisfy MB = 0, where the semicolons indicate vertical blocks. At least one traceless Hermitian block is nonzero, so B has rank at least two. Thus rank M ≤ Dk − 2 < n. There is a unit vector y orthogonal to every column of M.
Rotate column a of X toward y while leaving the other columns fixed. Each rotation remains an isometry. Common orthogonality makes the first variation of every compressed observable vanish. The second variations of F along the k rotations satisfy
\[\sum_{a=1}^k F_a''(0)=-4F(X)<0.\]At a minimum all those second derivatives must be nonnegative. This contradiction forces F(X) = 0. The PDF derives the differential, the rank defect, and the second-variation identity explicitly.
Explicit tough models
For e < n, split the ambient space into min(n,e²+1) balanced coordinate blocks. Label them by zero and a truncated list of vectors
\[u_i,\qquad u_i+u_j,\qquad u_i+\mathrm{i}u_j\quad(i<j),\]with all coordinate vectors u_i first. Define error A_a to be the a-th coordinate of the label on each block. The e errors are independent. Their pairwise products, together with the identity, span all the block projections: cross-products isolate pair labels, diagonal products isolate coordinate labels, and the identity supplies the zero block.
If a code scalar-compresses every block projection, at least one compressed projection is a positive multiple of the code projection. Its rank bounds the code dimension by the largest block. That block is itself a code, so the largest code has exactly ceil(n/(e²+1)) dimensions. The argument also covers subspaces whose vectors extend across several blocks. For e ≥ n, including all rank-one coordinate projections among the errors and extending to dimension e gives largest-code dimension one.
For example, on five dimensions,
\[A_1=\operatorname{diag}(0,1,0,1,1),\qquad A_2=\operatorname{diag}(0,0,1,1,\mathrm{i})\]have no two-dimensional correcting code. Replacing the five coordinates by balanced blocks gives the upper construction for two errors at larger n.
Relation to earlier work
The original collection of Krüger and Werner records the Knill–Laflamme–Viola guarantee of order n/[e²(e²+1)]. Weaver’s operator-system theorem uses diagonal compression followed by a convex partition. Allen and Kornell, Theorem 2.7 improve the small-dimensional bounds; for two errors their theorem gives ceil(n/9), compared with the asymptotic coefficient 1/5 here. Their dimension-three scalar-compression threshold is sharper than the universal threshold used in this manuscript.
The rank-one spanning vectors are inherited from Li and Tam, and the block-rank obstruction is Weaver’s Proposition 2.1. The proposed new ingredient is the variational lower bound for arbitrary operator systems. Recent work by Gupta and collaborators and Sun and collaborators studies related compression geometry for Pauli detection; the inspected statements do not imply this unconditional dimension guarantee. The manuscript also compares Reimpell’s earlier tough-model analysis.
These are comparisons with specific primary-source statements. They do not establish exhaustive priority. The full integer-threshold conjecture for general operator systems remains outside the result proved here.
Review and reproducibility
The proof and its original-problem scope were read analytically in full. Exact arithmetic checks reconstruct nine local second-variation identities and reject nine controls that violate the common-orthogonality hypothesis. A separate checker exercises the block construction, its dimension and normalization assumptions, and deliberately incorrect variants. These finite checks support calculations; the universal theorem rests on the analytic proof.
The source package includes the full manuscript, both checkers, a replay command, hashes and review scope. The PDF is the same seven-page review draft approved for this website release. The derivation, literature work, computation and drafting used AI assistance under human direction; the current proof review was sequential within the same agent. There has been no independent expert acceptance, formal proof-assistant certification or verified first-publication claim.