LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generated

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Density mixing preserves admissibility but need not preserve idempotence

Statement

If $D \in D_{r}$ and the gapped map $Φ (D)$ of The occupied-projector response map on its strict-gap domain is defined, then for every $α \in [0, 1]$ , $D^{+} = (1 - α) D + α Φ (D)$ belongs to $D_{r}$ . The conclusion does not assert $D^{+} \in P_{r}$ .

Facts & Assumptions

Given: $D \in D_{r}$ , a defined occupied projector $Φ (D)$ of rank $r$ , and $0 \leq α \leq 1$ .

[F1]

$D_{r}$ is the set of Hermitian positive contractions of trace $r$ (The fixed-particle admissible Hermitian density domain).

[F2]

The gapped response construction gives a unique orthogonal projector $Φ (D) \in P_{r}$ (The occupied-projector response map on its strict-gap domain, A strict occupied gap defines a unique rank-r spectral projector).

[F4]

Every rank- $r$ orthogonal projector is an admissible density (Every rank-r orthogonal projector is an admissible density).

Proof

1.1F1F2F4givenalgebra

Since $D$ and $Φ (D)$ are Hermitian, $D^{+}$ is Hermitian. For every $x$ , $x^{*} D^{+} x = (1 - α) x^{*} D x + α x^{*} Φ (D) x \geq 0$ . Likewise $x^{*} (I - D^{+}) x = (1 - α) x^{*} (I - D) x + α x^{*} (I - Φ (D)) x \geq 0$ , since $D \in D_{r}$ and $Φ (D) \in D_{r}$ by [F1], [F2], [F4].

2.1step 1.1F1F2algebra

The trace is linear, so $tr D^{+} = (1 - α) r + α r = r$ . The two positive-semidefinite inequalities and this trace identity give $D^{+} \in D_{r}$ .

3.1step 2.1F2givenalgebra∎

To verify that idempotence need not be preserved, take $n = 2$ , $r = 1$ , the constant Hermitian response $F (X) = diag (0, 1)$ on $U = H_{2}$ , $D = diag (0, 1)$ , and $α = 1 / 2$ . Directly, $D$ is Hermitian, lies between $0$ and $I_{2}$ , and has trace $1$ , so $D \in D_{1}$ . The response matrix has gap $1$ , so [F2] gives $Φ (D) = diag (1, 0)$ . Then $D^{+} = \frac{1}{2} I_{2}$ and $(D^{+})^{2} = \frac{1}{4} I_{2} \neq D^{+}$ . Thus the update belongs to $D_{1}$ by step 2.1 but is not an orthogonal projector.

Depends on

Used by

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Sources

Eric Cancès, Gaspard Kemlin, and Antoine Levitt, Convergence analysis of direct minimization and self-consistent iterations, §4.4 and Appendix Lemma 4.1 (standard reference, not scraped)