PPT² iteration 1 — WH-mixture exclusion at n=4; first bound-entangled CHANNEL built and composed (n=3 landscape)

Statement

Iteration 1 of the PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 derive→validate loop. Two structured families scanned with the certified dual-oracle validator; three verified results.

1. Exclusion: Werner–Holevo mixtures at n=4 give no certificate. Φ_λ = λW⁺+(1−λ)W⁻ has PPT window exactly λ ∈ [1/2, 1] at n=4. Over a 13×13 grid of PPT×PPT compositions Φ_λ∘Φ_μ, max CCNR = 0.400 (at λ=μ=1, far below the 1.0 certification threshold) and DPS-2 finds a symmetric extension at the best point. Consistent with the O(n)-covariant family sitting INSIDE the Covariance / symmetry reduction of quantum channels — THE WALL: the one recipe every partial PPT² result reuses, and nobody has escaped to the general case wall: no counterexample lives in this family's compositions (as covariance results predict).

2. A genuine bound-entangled CHANNEL exists and is easy to build (n=3). Local-filtering the Tiles-UPB state ρ = (I−P_tiles)/4 by (σ_A^{−1/2}⊗I) yields a CPTP, PPT channel whose own Choi is bound entangled: CCNR = 1.115, DPS-2 = entangled (both certified). This is the canonical seed object for the counterexample hunt: a PPT channel that is NOT entanglement-breaking. (Local filtering preserves PSD, PPT and entanglement; Tr_B C = I after filtering.)

3. The composition-decay landscape (the quantity to fight). Composing kills the certificate fast at n=3, exactly as the theorem demands: Choi(tiles) CCNR 1.115 → tiles∘tiles 0.715, tiles∘W⁺ 0.546, W⁺∘tiles 0.526 (DPS-2 silent on all three). The counterexample question at n≥4 becomes quantitative: find a bound-entangled channel whose Choi entanglement SURVIVES one composition — i.e. beat the ~0.64 CCNR decay factor observed here. This is the fitness function for the Simulated-annealing extremal search — Metropolis local search for rare, structured witness/counterexample objects track: anneal pairs in the PPT∘PPT cone at n=4 with fitness = CCNR of the composed Choi, seeded from UPB-filtered channels (NOT random — random PPT is EB w.p.→1, see PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 Facts).

Facts

- Validator certified before use: 15/15 selftest incl. positive control (Horodecki 3×3 bound-entangled: CCNR fires 1.0023, DPS-2 fires) and negative controls (separable product silent on both oracles). - n=3 results are LANDSCAPE data, not counterexample attempts (n=3 is proven, PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4); their value is calibrating how the theorem kills entanglement, before spending effort at n=4. - Next derivation targets: (a) 4⊗4 UPB (GenTiles-type) filtered channel — the n=4 analogue of result 2; (b) compositions Φ∘(U·Φ·U†) over local unitaries (annealing variable) rather than Φ∘Φ; (c) mixing a small W⁻ component to sit exactly ON the PPT boundary where entanglement is largest.

Related

- PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — the target. - Bound entanglement — entangled states that are PPT, hence undistillable; the rare structured regime where a PPT² counterexample must live — result 2 is a channel-version of it. - Covariance / symmetry reduction of quantum channels — THE WALL: the one recipe every partial PPT² result reuses, and nobody has escaped to the general case — result 1 confirms the wall from inside. - Entanglement witness — a Hermitian W with Tr(Wσ)≥0 on all separable states but Tr(Wρ)<0 on some entangled ρ / Peres–Horodecki (PPT) criterion — partial transpose positivity is necessary for separability, and sufficient only in 2×2 and 2×3 — certification layer. - Simulated-annealing extremal search — Metropolis local search for rare, structured witness/counterexample objects — the cross-domain bridge now armed with a concrete fitness (composed-Choi CCNR) and seeds. - Dual-oracle verification — certify every claimed object by two independent oracles, count-certify exhaustive runs, and scan all prior-art branches before claiming novelty — third author-error caught (W⁻ believed PPT).