Reduction criterion — separable (and PPT) states obey ρ_A⊗I ⪰ ρ; the eigenvalue cap that killed our CCNR fitness

Statement

For any separable bipartite state ρ on A⊗B:

$$\rho_A \otimes I - \rho \;\succeq\; 0 \qquad\text{and}\qquad I \otimes \rho_B - \rho \;\succeq\; 0.$$

Violation ⇒ entangled (and moreover distillable). The criterion is implied by PPT: the reduction map Λ(X) = Tr(X)·I − X is decomposable (it equals a completely positive map composed with transposition), so every PPT state passes it. Strictly weaker than PPT for n ≥ 3; equivalent to PPT for 2⊗2 and 2⊗3.

Facts

- The eigenvalue cap for channels (the use that mattered here): a channel's Choi state has A-marginal exactly I/n (trace preservation). For a PPT channel, reduction then gives I/n ⊗ I ⪰ ρ, i.e. λ_max(ρ_Choi) ≤ 1/n. Hence ‖ρ‖₂ ≤ 1/√n, and since the reshuffle preserves the Frobenius norm, the transfer matrix obeys ‖T‖₂ ≤ √n. One Cauchy–Schwarz later: ‖T_Ψ T_Φ‖₁ ≤ n — the composition of two PPT channels always passes the realignment test (PPT² iterations 3–4 — composed CCNR pins at 1; resolved: realignment is PROVABLY blind to PPT² (shadow theorem, 3-line proof)). - Reduction-violating states are distillable (Horodecki–Horodecki 1999): the criterion detects only "strong" entanglement; all Bound entanglement — entangled states that are PPT, hence undistillable; the rare structured regime where a PPT² counterexample must live passes it — which is exactly why it caps PPT objects so effectively. - Meta-lesson for this wiki (Bisociation (Koestler 1964) — creative insight as the collision of two habitually unrelated frames of thought, negative direction): our retrieval could not surface this proof because the concept was NOT IN THE CORPUS. A verified-memory system's ceiling is its coverage; the oracle→prior-art→ingest loop is what raises it. This page closes that gap.

Related

- Peres–Horodecki (PPT) criterion — partial transpose positivity is necessary for separability, and sufficient only in 2×2 and 2×3 — the stronger sibling (PPT). - Bound entanglement — entangled states that are PPT, hence undistillable; the rare structured regime where a PPT² counterexample must live — invisible to reduction, by design. - PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — capped consequence: realignment can never witness a counterexample. - PPT² iterations 3–4 — composed CCNR pins at 1; resolved: realignment is PROVABLY blind to PPT² (shadow theorem, 3-line proof) — the campaign page this resolves. - Entanglement witness — a Hermitian W with Tr(Wσ)≥0 on all separable states but Tr(Wρ)<0 on some entangled ρ — where the hunt's certification must move (with DPS). - Dual-oracle verification — certify every claimed object by two independent oracles, count-certify exhaustive runs, and scan all prior-art branches before claiming novelty — the scan agent's numeric re-check followed this discipline.