slug: li-theil-harrow-chuang-2026-purify-and-clone-optimality
bibkey: litheilharrowchuang2026coherent
doi: 10.48550/arXiv.2605.21457
url: https://arxiv.org/abs/2605.21457v2
triage: theorem
motivation_gids:
  - D5/S3/Quantum/Information/LiTheilHarrowChuangPurifyCloneRefutation.result

Purify-and-clone fails one-site worst-case optimality

Problem

Z. Li, E. Theil, A. W. Harrow and I. Chuang, An Exponential Sample-Complexity Advantage for Coherent Quantum Inference, arXiv:2605.21457v2, Section S3 A, p. 9, states:

This channel is optimal for m ≤ n (see Ref. [S2]), and we conjecture that it is also optimal for m > n.

Preregistration #19410 fixes $n=1$, $m=2$ and the one-site worst-case squared-infidelity reading. The universal optimality assertion is refuted by $d=r=2$. A strict-improvement theorem for every $d\ge r\ge 2$ is not part of this settlement. The Library note identifies the source and conventions.

Statement echo

Motivation

The settling GID is D5/S3/Quantum/Information/LiTheilHarrowChuangPurifyCloneRefutation.result. Cloning the entire system-environment register and cloning the system before adjoining a symmetric environment have different one-site risks. The latter supplies a strictly better admissible channel in dimension two.

Gap

The source and literature screen recorded in #19410 includes source versions 1 and 2, Werner’s cloning literature and fidelity monotonicity. No settlement was located in that searched scope. Literature completeness is ASSUMED-UNVERIFIED. The source’s unqualified word “optimal” admits different losses; this dossier settles precisely the one-site worst-case reading.

Route

For $d=r=2$, apply the system Werner cloner, tensor with one third of the symmetric projector on the two environment registers, and reorder the four registers into two joint sites. The consumed private theorems competitor_channel and competitor_support certify complete positivity, trace preservation and both-sided symmetric support. Both marginals equal the same system depolarization tensored with $I_2/2$.

A four-element Pauli average puts $\rho \otimes I_2/2$ in the common target convex hull. competitor_risk_upper bounds its worst-case risk by $1/6$. At the pure zero input, purifyAndClone_risk_lower bounds the purify-and-clone worst-case risk below by $1/5$. result combines these inequalities to refute the universal assertion.

Falsifier

The comparison requires the complete rank-at-most-two input domain, both registers at each site, a common target chosen before the maximum, squared fidelity, genuine CPTP competitors and both-sided total symmetric support. A theorem for another loss, a sampled input set or swap conjugation alone would not settle this statement.

Evidence

Canonical Lean source and its Blueprint mirror. Admission basis: open-problem-resolution (#19410; Refuted). The new module’s declaration classifications are bind-only; its private helpers have named consumers, and the settling theorem is admitted as the externally preregistered named-problem resolution. No escape witness is claimed.

The module has utility: none. Its public result therefore has no designated-refutation registration exemption. Its escape audit, including the generated sqFidelity.congr_simp theorem, and that of the published owner helpers BroadcastingChannelExtremaRefutation.density_le_one, WassersteinComplexityNonconvex.trace_sqrt_sq_le_two and WassersteinComplexityNonconvex.pure_diag_coupling are unfinished under #19490. No validated registration is asserted.

Experiment: pinned experiment directory. Command: python3 check.py in that directory; exit 0, RESULT PASS; script SHA-256 3d272d84388b46f3f85274842c8eea91c119b107774a6fd2f066f22d23511cbd. The deterministic experiment tests exactly $(d,r)\in \{(2,2),(3,2),(3,3),(4,2)\}$, with 60 sampled states for each pair. It checks marginal formulas, symmetric support and sampled worst-case risks against the displayed formulas with tolerance $10^{-7}$. The sampled risk pairs are $(0.2000000000,0.1666666667)$, $(0.2857142857,0.2500000000)$, $(0.3000000000,0.2500000000)$ and $(0.3333333333,0.2999999955)$. This is a finite numerical reading; it proves neither a uniform family nor an exact optimum.

Declaration classification and consumption

All source declarations are classified bind-only after expansion of same-delivery helpers. This is a proof-shape classification; the named external problem supplies the admission basis. Private identifiers below are source-qualified names; compiler-generated congruence support is identified separately. Consumers come from the elaborated declaration values. The independent consumption checks establish the live use of each source helper.

Triage

Tier 1 externally published conjecture; resolution Refuted for the universal one-site worst-case reading by the instance $n=1$, $m=2$, $d=r=2$.

What the settlement shows

ASSUMED-UNVERIFIED

Literature completeness and worldwide priority are unverified. The optimal competitor, all-site and average risks, other copy numbers and the arbitrary-dimension strict-improvement family remain open here. Numerical sampled maxima are not exact worst-case proofs. The four-slot escape audit remains unfinished under #19490.