the mixed-state layer — density matrices, decoherence, and partial trace
Theorem.the mixed-state layer — density matrices, decoherence, and partial trace — the frontier lifting the simulator beyond pure states, delivered: a pure state ρ = |ψ⟩⟨ψ| has purity tr(ρ²) = 1;.
Proof.the frontier lifting the simulator beyond pure states, delivered: a pure state ρ = |ψ⟩⟨ψ| has purity tr(ρ²) = 1; the depolarizing channel ρ → (1−p)ρ + p·I/2 produces a genuine MIXED state (purity < 1) with the trace conserved — the honest model of decoherence the pure-state simulator could not express; and the partial trace of a Bell pair yields the maximally mixed I/2 (purity 1/2) — the operational signature of entanglement, pure whole and mixed part. All exact linear algebra over the density matrix; one channel and the 1-of-2 partial trace shown, the full Kraus/Lindblad toolkit named as continuation; deterministic O(4ⁿ) small-system model.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/2/8/index.ts#theMixedStateLayer
1 · Classification
finite-complete — self-contained computation, no external lean
2 · Provenance
Documented theorem re-derived by exhaustive computation (humanityNovel=false); first-in-this-registry is the only sense of discovered.
Acknowledgment
"the mixed-state layer — density matrices, decoherence, and partial trace" is a re-derivation, acknowledged to documented mathematics — the original proof is the prior art this re-derivation acknowledges; not new to humanity — the contribution is the reproducible computation theMixedStateLayer.
- Prior art
- documented mathematics — the original proof is the prior art this re-derivation acknowledges
- Novelty
- not new to humanity — a re-derivation (humanityNovel = false)
- Contribution
- a reproducible computation (theMixedStateLayer @ src/2/8) that re-derives the result at zero tokens — the contribution is the verifiable recomputation, NOT the theorem
3 · Reproducibility
Recompute from source: npm run theorems:verify recomputes theMixedStateLayer (src/2/8/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.