Remark 14: rank-one dual lower bound as a necessary condition

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QAlgorithms.Papers.Independent.Intakethird100.P95.remark14

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. The rank-one dual objective set is nonempty and bounded above. Moreover,

T_1(K;\lambda)\le T(K;\lambda),\qquad\Phi_K\to^{\rm PPT}\rho_\lambda\;\Longrightarrow\;T_1(K;\lambda)\le1.

The restricted dual therefore gives a necessary preparation condition without assuming a rank-one maximizer.

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 p.7, Remark 14: restricted SUPREMUM lower bound and necessary condition.
No general rank-one optimizer is asserted; domain facts accompany the real supremum. -/
theorem remark14 {d : ℕ} (K : ℕ) (hK : 2 ≤ K) (lam : Fin d → ℝ)
    (hprob : IsProbability lam) (hpos : ∀ i, 0 < lam i) :
    (rankOneValues K lam).Nonempty ∧ BddAbove (rankOneValues K lam) ∧
    rankOneValue K lam ≤ primalValue K lam ∧
    (Reach (maximallyEntangledDensity K) (schmidtDensity lam) → rankOneValue K lam ≤ 1) := 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, Remark 14: rank-one dual lower bound as a necessary condition

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

For every natural number $d$ , integer $K\ge2$ , and vector $\lambda_i>0$ with $\sum_{i=0}^{d-1}\lambda_i=1$ , put $R_\lambda=(\sum_i\sqrt{\lambda_i})^2$ and $B=(K R_\lambda-1)/(K^2-1)$ . Define $S_{ii}=|ii\rangle\langle ii|$ , $S_{ij}=\frac12(|ij\rangle+|ji\rangle)(\langle ij|+\langle ji|)$ for $j<i$ , and $A_{ij}=\frac12(|ij\rangle-|ji\rangle)(\langle ij|-\langle ji|)$ for $j<i$ , where the coordinate vectors are in $\mathbb C^d\otimes\mathbb C^d$ . Let $(\Gamma X)_{(i,j),(k,l)}=X_{(i,l),(k,j)}$ and $Q(s,a)=\sum_{j\le i}s_{ij}\Gamma S_{ij}+\sum_{j<i}a_{ij}\Gamma A_{ij}$ . Let $V$ be the real infimum of $\sum_{j\le i}s_{ij}+\sum_{j<i}a_{ij}$ subject to $Q(s,a)\succeq0$ , $s_{ij}\ge\sqrt{\lambda_i\lambda_j}/(K+1)$ , and $a_{ij}\ge\sqrt{\lambda_i\lambda_j}/(K-1)$ . Let $\mathcal R$ consist of all numbers

B-\sum_{j\le i}\frac{\mu_{ij}\sqrt{\lambda_i\lambda_j}}{K+1} -\sum_{j<i}\frac{t_{ij}\sqrt{\lambda_i\lambda_j}}{K-1},

where the real arrays $\mu,t$ satisfy $\mu_{ij}\le1$ , $t_{ij}\le1$ , $Q(\mu,t)\succeq0$ , and $\operatorname{rank}Q(\mu,t)=1$ ; there are no separately imposed lower bounds on these coefficients. Then $\mathcal R$ is nonempty and bounded above, and $\sup\mathcal R\le V$ . Moreover, if there exists a complex-linear, trace-preserving, completely positive map $\mathcal L$ from operators on $\mathbb C^K\otimes\mathbb C^K$ to operators on $\mathbb C^d\otimes\mathbb C^d$ , such that $\Gamma_{\rm out}\mathcal L\Gamma_{\rm in}$ is also completely positive and $\mathcal L(\Omega_K)=\rho_\lambda$ , then $\sup\mathcal R\le1$ , where $\Omega_K=K^{-1}\sum_{i,j=0}^{K-1}|ii\rangle\langle jj|$ and $(\rho_\lambda)_{(i,j),(k,l)}$ equals $\sqrt{\lambda_i\lambda_k}$ when $i=j$ and $k=l$ , and $0$ otherwise. Complete positivity means that every finite identity amplification preserves positive semidefinite matrices, including amplification size zero. The hypotheses exclude $d=0$ .

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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