Proposition 13: nonnegativity and exact reference equality conditions

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QAlgorithms.Papers.Independent.Intakethird100.P52.renyi_nonnegative_iff_reference

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

prove2quantum

All Hilbert spaces below are finite dimensional and nonzero over $\mathbb C$ . A density operator is positive semidefinite with trace one. Every spectral power is zero on the kernel, including zero and negative powers. Information quantities use base-two logarithms, with $\log_2 0=-\infty$ ; the operator logarithms in exponential formulas are natural logarithms.

Let $\rho$ be a density operator on $H$ , let $\sigma\ge0$ be an operator on $H$ without a trace constraint, and let $\mathcal N:\mathcal B(H)\to\mathcal B(K)$ be CPTP. Its adjoint $\mathcal N^\dagger$ is the Hilbert–Schmidt adjoint. Write

\begin{aligned} K_a&=\sigma^{(1-a)/2}\mathcal N^\dagger\!\left(\mathcal N(\sigma)^{(a-1)/2}\mathcal N(\rho)^{1-a}\mathcal N(\sigma)^{(a-1)/2}\right)\sigma^{(1-a)/2},\\ L_a&=\sigma^{(1-a)/(2a)}\mathcal N^\dagger\!\left(\mathcal N(\sigma)^{(a-1)/(2a)}\mathcal N(\rho)^{(1-a)/a}\mathcal N(\sigma)^{(a-1)/(2a)}\right)\sigma^{(1-a)/(2a)},\\ R_a&=K_a^{1/(1-a)},\qquad S_a=L_a^{a/(1-a)},\\ \Delta_a&=\frac{\log_2\operatorname{Tr}(\rho^aK_a)}{a-1},\qquad \widetilde\Delta_a=\frac{\log_2\operatorname{Tr}[(\rho^{1/2}L_a\rho^{1/2})^a]}{a-1}. \end{aligned}

For the indicated ranges,

\begin{aligned} 0<a<1 &: \quad\Delta_a\ge0,\qquad (\Delta_a=0\iff\rho=R_a),\\ 1/2<a<1 &: \quad\widetilde\Delta_a\ge0,\qquad (\widetilde\Delta_a=0\iff\rho=S_a). \end{aligned}

If $\rho,\sigma,\mathcal N(\rho),\mathcal N(\sigma)>0$ , the first pair of conclusions also holds for $1<a<2$ , and the second for every $a>1$ .

Nonnegativity and exact equality to the computed reference are both parts of the result.

Preamble

import Definitions.Def_QAlgorithms_Papers_Independent_Intakethird100_P52

Formal statement

noncomputable section

open scoped Matrix ComplexOrder

namespace QAlgorithms.Papers.Independent.Intakethird100.P52

attribute [local instance] Classical.propDecidable Classical.decEq

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

open QAlgorithms.Papers.Independent.Dependency50.P21 (channelAdjoint petz marginalAB)

open QAlgorithms.Papers.Independent.Dependency50.P35 (tr elog maxRelative)

open QAlgorithms.Papers.Independent.Information100.P71 (supportPower)

open QAlgorithms.Papers.Independent.Renew46.P35 (HasMarkovBlocks)

/-- Proposition 13, PDF p.14, equations (3.56)–(3.59). Each equality condition
is a conclusion. No support containment or strictness is added below one. -/
theorem renyi_nonnegative_iff_reference {H K : Type} [Fintype H] [Fintype K]
    [DecidableEq H] [DecidableEq K] [Nonempty H] [Nonempty K]
    (rho sigma : Mat H) (hrho : IsDensityMatrix rho)
    (hsigma : sigma.PosSemidef) (N : Superop H K) (hN : IsChannel N) :
    (∀ a : ℝ, 0 < a ∧ a < 1 →
      0 ≤ renyiDifference a rho sigma N ∧
      (renyiDifference a rho sigma N = 0 ↔ rho = renyiReference a rho sigma N)) ∧
    (∀ a : ℝ, 1 / 2 < a ∧ a < 1 →
      0 ≤ deltaTilde a rho sigma N ∧
      (deltaTilde a rho sigma N = 0 ↔ rho = sandwichedReference a rho sigma N)) ∧
    ((rho.PosDef ∧ sigma.PosDef ∧ (N rho).PosDef ∧ (N sigma).PosDef) →
      (∀ a : ℝ, 1 < a ∧ a < 2 →
        0 ≤ renyiDifference a rho sigma N ∧
        (renyiDifference a rho sigma N = 0 ↔ rho = renyiReference a rho sigma N)) ∧
      (∀ a : ℝ, 1 < a →
        0 ≤ deltaTilde a rho sigma N ∧
        (deltaTilde a rho sigma N = 0 ↔ rho = sandwichedReference a rho sigma N))) := by sorry

end QAlgorithms.Papers.Independent.Intakethird100.P52

end

Source

Quantum Markov chains, sufficiency of quantum channels, and Renyi information measures, arXiv:1501.05636v4, PDF p. 14, Proposition 13: nonnegativity and exact reference equality conditions

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

Let $H,K$ be finite nonempty types with decidable equality, let $\rho$ be a positive semidefinite complex matrix on $H$ with trace $1$ , let $\sigma$ be any positive semidefinite complex matrix on $H$ , and let $N(X)=\sum_jE_jXE_j^\dagger$ be a channel with a finite Kraus family satisfying $\sum_jE_j^\dagger E_j=I_H$ . Define $(N^\dagger Z)_{ij}=\sum_{k,l}\overline{N(E_{ij})_{kl}}Z_{kl}$ , and let $S_t(X)$ be continuous functional calculus for the function zero at zero and $x^t$ otherwise. Put

\begin{aligned} K_a&=S_{(1-a)/2}(\sigma) N^\dagger\!\left(S_{(a-1)/2}(N\sigma)S_{1-a}(N\rho)S_{(a-1)/2}(N\sigma)\right) S_{(1-a)/2}(\sigma),\\ \widetilde K_a&=S_{(1-a)/(2a)}(\sigma) N^\dagger\!\left(S_{(a-1)/(2a)}(N\sigma)S_{(1-a)/a}(N\rho)S_{(a-1)/(2a)}(N\sigma)\right) S_{(1-a)/(2a)}(\sigma),\\ R_a&=S_{1/(1-a)}(K_a),\qquad \widetilde R_a=S_{a/(1-a)}(\widetilde K_a),\\ D_a&=\frac{\ell_2(\operatorname{Re}\operatorname{Tr}(S_a(\rho)K_a))}{a-1},\\ \widetilde D_a&=\frac{\ell_2(\operatorname{Re}\operatorname{Tr}S_a(S_{1/2}(\rho)\widetilde K_aS_{1/2}(\rho)))}{a-1}, \end{aligned}

where $\ell_2(x)=\log_2x$ for $x>0$ and $-\infty$ otherwise, and $D_a,\widetilde D_a$ are extended real. For every $0<a<1$ , the assertion is $0\le D_a$ and $D_a=0$ if and only if $\rho=R_a$ ; for every $1/2<a<1$ , it is $0\le\widetilde D_a$ and $\widetilde D_a=0$ if and only if $\rho=\widetilde R_a$ . If all four matrices $\rho,\sigma,N\rho,N\sigma$ are positive definite, the first pair of assertions also holds for every $1<a<2$ , and the second for every $a>1$ . No trace normalization of $\sigma$ or support-inclusion hypothesis is imposed; the first two parameter ranges include singular matrices and $\sigma=0$ , using the stated zero-value convention for powers and extended logarithms.

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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