Eq. (2) — block decomposition $\rho=\sum_{x,y}P_x\rho P_y$

Proved

ProjectiveMeasurementEquilibration.block_decomposition

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

pinchingquantum-mechanics

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 every $n\times n$ complex matrix $\rho$ ,

\rho=\sum_{x,y}P_x\,\rho\,P_y,

the double sum running over pairs of distinct eigenvalues of $X$ (block-matrix decomposition of $\rho$ with respect to the eigenspaces of $X$ ).

Preamble

import Mathlib
import Definitions.Def_pme_quantum_basics
import Definitions.Def_pme_measurement_conditions

open Matrix Filter Topology

Formal statement

namespace ProjectiveMeasurementEquilibration

theorem block_decomposition {n : Type} [Fintype n] [DecidableEq n] {X : Matrix n n ℂ} (hX : X.IsHermitian)
    (ρ : Matrix n n ℂ) :
    ∑ x ∈ eigenvalueSet hX, ∑ y ∈ eigenvalueSet hX, eigenproj hX x * ρ * eigenproj hX y = ρ := 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. (2), p. 2

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 $n$ , every Hermitian matrix $X\in M_n(\mathbb C)$ and every matrix $\rho\in M_n(\mathbb C)$ (not required to be a density matrix):

\sum_{x\in\Lambda}\sum_{y\in\Lambda}P_x\,\rho\,P_y=\rho,

where $\Lambda$ is the finite set of distinct eigenvalues of $X$ and $P_x=U\,\mathrm{diag}(\mathbf 1[\lambda_i=x])\,U^*$ for the library's eigen-decomposition $X=U\mathrm{diag}(\lambda)U^*$ . For $n$ empty both sides are the empty matrix.

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. (2) — block decomposition ρ=∑x,yPxρPy\rho=\sum_{x,y}P_x\rho P_yρ=∑x,y​Px​ρPy​

Eq. (2) — block decomposition $\rho=\sum_{x,y}P_x\rho P_y$