Proposition 11: dual SDP equality and maximum attainment

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QAlgorithms.Papers.Independent.Intakethird100.P95.proposition11

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 $(|ij\rangle\pm|ji\rangle)(\langle ij|\pm\langle ji|)/2$ . For a finite order $t$ , $S_t$ is the Rényi entropy, with $S_0$ the log support size, $S_1=-\sum_i\lambda_i\log_2\lambda_i$ (using $0\log_2 0=0$ ), and $S_t=(1-t)^{-1}\log_2\sum_i\lambda_i^t$ otherwise. The separate infinity order is $S_\infty=-\log_2\max_i\lambda_i$ . Let $\mathcal R$ consist of real coordinates $\mu_{ij}\le1$ and $t_{ij}\le1$ whose matrix $\sum\mu_{ij}\sigma_{ij}^\Gamma+\sum t_{ij}\alpha_{ij}^\Gamma$ is positive semidefinite. There are no nonnegative lower bounds on these coordinates. Then the maximum is attained and

T(K;\lambda)=\max_{(\mu,t)\in\mathcal R}\left[\frac{K2^{S_{1/2}(\lambda)}-1}{K^2-1}-\left(\sum_{i\ge j}\frac{\mu_{ij}\sqrt{\lambda_i\lambda_j}}{K+1}+\sum_{i>j}\frac{t_{ij}\sqrt{\lambda_i\lambda_j}}{K-1}\right)\right].

This provides an attained dual characterization of the primal value.

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.6, Proposition 11: equality with the dual MAXIMUM, including attainment. -/
theorem proposition11 {d : ℕ} (K : ℕ) (hK : 2 ≤ K) (lam : Fin d → ℝ)
    (hprob : IsProbability lam) (hpos : ∀ i, 0 < lam i) :
    IsGreatest (schmidtDualValues K lam) (primalValue K 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. 6, Proposition 11: dual SDP equality and maximum attainment

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

Let $d,K\in\mathbb N$ , with $K\ge2$ , and let $\lambda=(\lambda_i)_{i=0}^{d-1}$ have strictly positive entries summing to one, so $d=0$ is impossible. Define $S_{ii}=|i,i\rangle\langle i,i|$ , $S_{ij}=\tfrac12(|i,j\rangle+|j,i\rangle)(\langle i,j|+\langle j,i|)$ for $j<i$ , and $A_{ij}=\tfrac12(|i,j\rangle-|j,i\rangle)(\langle i,j|-\langle j,i|)$ for $j<i$ . Let $\Gamma$ transpose the second subsystem and put $C(u,t)=\sum_{j\le i}u_{ij}\Gamma S_{ij}+\sum_{j<i}t_{ij}\Gamma A_{ij}$ . Define $p_K(\lambda)$ as the real infimum of $\sum_{j\le i}s_{ij}+\sum_{j<i}a_{ij}$ over real families satisfying $s_{ij}\ge\sqrt{\lambda_i\lambda_j}/(K+1)$ , $a_{ij}\ge\sqrt{\lambda_i\lambda_j}/(K-1)$ , and $C(s,a)\succeq0$ . The declaration says that $p_K(\lambda)$ is the greatest member of the set of real numbers

\frac{K\,2^{\,2\log_2(\sum_i\sqrt{\lambda_i})}-1}{K^2-1} -\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}

obtained from real families $\mu,t$ satisfying $\mu_{ij}\le1$ , $t_{ij}\le1$ , and $C(\mu,t)\succeq0$ . Thus some such families attain $p_K(\lambda)$ , and every such value is at most $p_K(\lambda)$ . No individual lower bounds on $\mu$ or $t$ are required. Here $\log_2x=\log x/\log2$ , the power of $2$ is a real power, and the antisymmetric families are empty when $d=1$ .

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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