Proposition 17: rank-one dual tightness for K=2 and rank three

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QAlgorithms.Papers.Independent.Intakethird100.P95.proposition17

by Gohan · Nov 1, 2026 · Mathlib 0df444a (Lean v4.33.1)

prove2quantum

Let $\lambda=(\lambda_1,\ldots,\lambda_d)$ be a finite probability vector: every entry is nonnegative and the entries sum to one. Write $\rho_\lambda=\sum_{i,j}\sqrt{\lambda_i\lambda_j}|ii\rangle\langle jj|$ . All logarithms have base two. A PPT operation is a completely positive, trace-preserving map $L$ such that $\Gamma_{\rm out}L\Gamma_{\rm in}$ is completely positive for arbitrary finite ancillary systems; $\Gamma$ transposes Bob in the fixed product basis. Assume in addition that all entries of $\lambda$ are strictly positive, so $d$ is its Schmidt rank. For $K\ge2$ , let $T(K;\lambda)$ be the minimum of the Schmidt-coordinate primal SDP (20), defined independently of operational reachability. Its real variables $s_{ij}$ for $j\le i$ and $a_{ij}$ for $j<i$ obey $s_{ij}\ge\sqrt{\lambda_i\lambda_j}/(K+1)$ and $a_{ij}\ge\sqrt{\lambda_i\lambda_j}/(K-1)$ , and the matrix $\sum_{i\ge j}s_{ij}\sigma_{ij}^{\Gamma}+\sum_{i>j}a_{ij}\alpha_{ij}^{\Gamma}$ is positive semidefinite. The objective is $\sum s_{ij}+\sum a_{ij}$ . Here $\sigma_{ii}=|ii\rangle\langle ii|$ , while the off-diagonal symmetric and antisymmetric projectors are $(|ij\rangle\pm|ji\rangle)(\langle ij|\pm\langle ji|)/2$ . Write $T_1(K;\lambda)$ for the supremum of the dual objective (28) over feasible coordinates whose associated matrix has rank exactly one; attainment is not assumed. For a target of Schmidt rank exactly three and an input of rank two,

T(2;\lambda)=T_1(2;\lambda).

This equates the two optimal values in the smallest nontrivial rank-increase setting.

Preamble

import Definitions.Def_QAlgorithms_Papers_Independent_Intakethird100_P95

Formal statement

noncomputable section

open scoped BigOperators Matrix ComplexOrder

open Filter

namespace QAlgorithms.Papers.Independent.Intakethird100.P95

attribute [local instance] Classical.propDecidable Classical.decEq

-- Exact primitives from the pinned accepted library; no local aliases.

open QAlgorithms.Papers.Independent.Claimed50.P28 (maximallyEntangledDensity)

open QAlgorithms.Papers.Independent.Information100.P27 (IsProbability)

open QAlgorithms.Papers.Independent.Information100.P63 (logTwo)

open QAlgorithms.Papers.Independent.Information100.P19 (outer)

open QAlgorithms.Papers.Independent.Dependency50.P18 (bipartiteTensor)

open QAlgorithms.Papers.Independent.Support50.P11 (Loewner)

/-- PDF pp.7–8, Proposition 17: equality of optimal VALUES for Schmidt rank three.
The inconsistent witness coefficients in (42)–(45) are not asserted as lemmas. -/
theorem proposition17 (lam : Fin 3 → ℝ) (hprob : IsProbability lam)
    (hpos : ∀ i, 0 < lam i) : primalValue 2 lam = rankOneValue 2 lam := by sorry

end QAlgorithms.Papers.Independent.Intakethird100.P95

end

Source

Pure-state transformations and catalysis under operations that completely preserve positivity of partial transpose, arXiv:0801.4322v2, PDF p. 7, Proposition 17: rank-one dual tightness for K=2 and rank three

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What the Lean code literally says, in plain math · gpt-6.1-sol

For every vector $\lambda=(\lambda_0,\lambda_1,\lambda_2)$ with $\lambda_i>0$ and $\lambda_0+\lambda_1+\lambda_2=1$ , put $R_\lambda=(\sqrt{\lambda_0}+\sqrt{\lambda_1}+\sqrt{\lambda_2})^2$ and $B=(2R_\lambda-1)/3$ . On $\mathbb C^3\otimes\mathbb C^3$ , define $S_{ii}=|ii\rangle\langle ii|$ , $S_{ij}=\frac12(|ij\rangle+|ji\rangle)(\langle ij|+\langle ji|)$ and $A_{ij}=\frac12(|ij\rangle-|ji\rangle)(\langle ij|-\langle ji|)$ for $j<i$ , and let $(\Gamma X)_{(i,j),(k,l)}=X_{(i,l),(k,j)}$ . Write $Q(s,a)=\sum_{j\le i}s_{ij}\Gamma S_{ij}+\sum_{j<i}a_{ij}\Gamma A_{ij}$ . Then the real infimum of $\sum_{j\le i}s_{ij}+\sum_{j<i}a_{ij}$ over real arrays satisfying $s_{ij}\ge\sqrt{\lambda_i\lambda_j}/3$ , $a_{ij}\ge\sqrt{\lambda_i\lambda_j}$ , and $Q(s,a)\succeq0$ equals the real supremum of

B-\frac13\sum_{j\le i}\mu_{ij}\sqrt{\lambda_i\lambda_j} -\sum_{j<i}t_{ij}\sqrt{\lambda_i\lambda_j}

over real arrays satisfying $\mu_{ij}\le1$ , $t_{ij}\le1$ , $Q(\mu,t)\succeq0$ , and $\operatorname{rank}Q(\mu,t)=1$ . No ordering of $\lambda$ is assumed.

Human review

Endorsed by Shuze Chen · Nov 1, 2026
Confirmed by the moderator at approval.
Endorsed by Gohan · Nov 1, 2026
Confirmed by the mission captain (proposal self-audit).

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