Eq. (12) — $D(\rho\|\sigma)=D(\rho\|\mathcal P_X\rho)+D(\mathcal P_X\rho\|\sigma)$

Proved

ProjectiveMeasurementEquilibration.relEntropy_chain_rule

by Lucas · Oct 2, 2026 · Mathlib 0df444a (Lean v4.33.1)

pinchingquantum-mechanicsrelative-entropy

Throughout, $n$ is a finite index type, the Hilbert space is $\mathbb C^n$ , $X$ is a Hermitian $n\times n$ complex matrix with spectral decomposition $X=\sum_{x} x\,P_x$ ( $x$ ranging over the distinct eigenvalues, $P_x$ the orthogonal projection onto the $x$ -eigenspace, and $P_x=0$ for $x$ not an eigenvalue), and a density matrix is a positive semidefinite matrix of trace $1$ . For density matrices $\rho,\sigma$ with $[\sigma,X]=0$ ,

D(\rho\|\sigma)=D(\rho\|\mathcal P_X(\rho))+D(\mathcal P_X(\rho)\|\sigma)

as an identity in $(-\infty,+\infty]$ , where $D(\rho\|\sigma)=\operatorname{tr}(\rho\log\rho)-\operatorname{tr}(\rho\log\sigma)$ if $\operatorname{supp}\rho\subseteq\operatorname{supp}\sigma$ and $+\infty$ otherwise (for non-faithful $\sigma$ the identity is understood on its support).

Preamble

import Mathlib
import Definitions.Def_pme_quantum_basics
import Definitions.Def_pme_measurement_conditions

open Matrix Filter Topology

Formal statement

namespace ProjectiveMeasurementEquilibration

theorem relEntropy_chain_rule {n : Type} [Fintype n] [DecidableEq n] {X : Matrix n n ℂ} (hX : X.IsHermitian)
    (ρ σ : Matrix n n ℂ) (hρ : IsDensityMatrix ρ) (hσ : IsDensityMatrix σ)
    (hcomm : X * σ = σ * X) :
    relEntropy ρ σ = relEntropy ρ (pinching hX ρ) + relEntropy (pinching hX ρ) σ := by sorry

end ProjectiveMeasurementEquilibration

Source

G. Carcassi (Assumptions of Physics collaboration), "Projective measurements are equilibration processes", AoP Technical Brief 017, https://assumptionsofphysics.org/resources/briefs, proof of Theorem 1, eq. (12), p. 4

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic) - same agent as the drafter; non-blind

Non-blind read-back. This read-back was written by the same agent (Aristotle, by Harmonic) that drafted the Lean statements, at the explicit request of the account owner. It is not blind and not independent testimony: the author knew the intended meaning while writing it. Reviewers should not treat it as an independent audit.

For every finite index type $n$ , Hermitian $X\in M_n(\mathbb C)$ , and density matrices $\rho,\sigma$ with $X\sigma=\sigma X$ , the following equality holds in the extended reals $[-\infty,+\infty]$ :

D(\rho\|\sigma)=D\big(\rho\|\mathcal P(\rho)\big)+D\big(\mathcal P(\rho)\|\sigma\big),

with $\mathcal P(\rho)=\sum_{x\in\Lambda}P_x\rho P_x$ and $D(A\|B)=\operatorname{Re}\operatorname{tr}\big(A(\log A-\log B)\big)$ if $\ker B\subseteq\ker A$ , $D(A\|B)=+\infty$ otherwise ( $\log$ via functional calculus with $\log0=0$ ). Extended-real addition is used on the right (in which a finite number plus $+\infty$ is $+\infty$ ).

Human review

Endorsed by Shuze Chen · Oct 8, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 8, 2026
Confirmed by the mission captain (proposal self-audit).

View graph

Eq. (12) — D(ρ∥σ)=D(ρ∥PXρ)+D(PXρ∥σ)D(\rho\|\sigma)=D(\rho\|\mathcal P_X\rho)+D(\mathcal P_X\rho\|\sigma)D(ρ∥σ)=D(ρ∥PX​ρ)+D(PX​ρ∥σ)

Eq. (12) — $D(\rho\|\sigma)=D(\rho\|\mathcal P_X\rho)+D(\mathcal P_X\rho\|\sigma)$