PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4

Statement

A PPT channel is a completely positive, trace-preserving (CPTP) map $\Phi: M_n \to M_n$ whose composition with the transpose $T$ is also completely positive — i.e. both $\Phi$ and $T\circ\Phi$ are CP ($\Phi$ is CP and co-CP, "completely copositive"). Equivalently, via the Choi–Jamiołkowski isomorphism, the Choi matrix $C_\Phi = (\mathrm{id}\otimes\Phi)(|\Omega\rangle\langle\Omega|)$ has positive partial transpose ($C_\Phi^{T_A}\ge 0$). A channel is entanglement-breaking (EB) iff its Choi matrix is separable (equivalently $\Phi(\rho)=\sum_k p_k\,\sigma_k$ collapses every input to a fixed measure-and-prepare form).

Conjecture (Christandl, Banff 2012). For any two PPT channels $\Phi,\Psi$ on $M_n$, the composition $\Phi\circ\Psi$ is entanglement-breaking.

Named the PPT² ("PPT-squared") conjecture; catalogued as Problem 38 on the IQOQI Open Quantum Problems tracker (posed via M.B. Ruskai's 2012 Banff open-problem list), still listed there as "prove or disprove."

Dimensional status.

| $n$ | Status | Basis | |---|---|---| | $n=2$ (qubit) | Proven | Folklore: every positive map on $M_2$ is decomposable (Størmer) + Peres–Horodecki (PPT ⇔ separable for $2{\times}2$). Appears only as a preliminary remark in the $n=3$ papers, not as a stand-alone result. | | $n=3$ (qutrit) | Proven (2018, two independent proofs) | Christandl–Müller-Hermes–Wolf, arXiv:1807.01266 (Schmidt-number technique). Chen–Yang–Tang, arXiv:1807.03636 (two-qutrit PPT states have Schmidt number ≤ 2). | | $n\ge 4$ | OPEN as of 2026-07 | No general proof and no counterexample in any dimension ≥ 4. Chen–Yang–Tang abstract: "The PPT square conjecture in the case $n\ge4$ is still open." All 2020–2026 progress is restricted-symmetry-class only. |

Facts

- Verification is CHEAP. A counterexample is a finite, closed-form certificate: two explicit Choi matrices $C_\Phi, C_\Psi$ (each PSD with PSD partial transpose) plus one entanglement witness $W$ with $\mathrm{Tr}(W\,C_{\Phi\circ\Psi}) < 0$ detecting entanglement in the composition's Choi state. Checking it is a bounded PSD/eigenvalue computation — no large-scale compute. Composing PPT maps is itself just a Choi-matrix operation. - Finding is HARD, and requires INTELLECT not brute force. Random PPT states are entanglement-breaking with probability → 1 in the relevant asymptotics (Collins–Yin–Zhong, arXiv:1803.00143, free-probability / Weingarten calculus — the conjecture holds *generically* for independent random channels). So a genuine counterexample cannot be stumbled on by sampling: it must be a rare, structured, adversarially-built bound-entangled state living in a measure-zero region. Blind numerical search accordingly failed — Jin, arXiv:2010.15554, calls the direct Choi-matrix + witness search "extremely difficult since it is the genuine part of the separability problem" and returns no counterexample. - THE WALL: the covariance-reduction recipe. Every 2020–2026 partial result uses the *same* move — restrict the channel to a symmetry/covariance class, characterize its invariant Choi/PPT cone, and show that cone collapses to entanglement-breaking. Diagonal-unitary → Singh–Nechita ("Choi-type maps", arXiv:2011.03809), which introduced the technique; Cartan-covariant → Prudhoe (arXiv:2501.03959); symplectic / conjugate-symplectic → Park (arXiv:2602.09860); random diagonal-orthogonal-covariant → Nechita–Park (arXiv:2403.03667); assorted small symmetric families (U(n)-, DU(2)-, DU(3)-equivariant) → García-Velo–Ibort ("Schwarz maps with symmetry", arXiv:2601.02282). The recipe has an intrinsic ceiling: it only ever proves the conjecture for channels *with imposed symmetry*, and each paper merely swaps the symmetry group. Nobody has broken the pattern for the unrestricted $n\ge4$ case in 14 years.

Related

Choi–Jamiołkowski isomorphism — state–channel duality: a map is CP iff its Choi matrix is PSD, Entanglement-breaking channel — measure-and-prepare maps: separable Choi matrix ⇔ destroys all entanglement, PPT map — a channel that is CP and completely co-positive: Choi matrix PSD with positive partial transpose, Entanglement witness — a Hermitian W with Tr(Wσ)≥0 on all separable states but Tr(Wρ)<0 on some entangled ρ, Covariance / symmetry reduction of quantum channels — THE WALL: the one recipe every partial PPT² result reuses, and nobody has escaped to the general case, Bound entanglement — entangled states that are PPT, hence undistillable; the rare structured regime where a PPT² counterexample must live, Peres–Horodecki (PPT) criterion — partial transpose positivity is necessary for separability, and sufficient only in 2×2 and 2×3 — the operator-theoretic scaffolding: the Choi correspondence turns the channel question into a state question, EB ⇔ separable Choi, PPT is the Peres–Horodecki necessary condition, and covariance-reduction is the wall every recent partial proof leans on.

Cross-domain edges (the wikiscale bet — importing technique from a distant field to crack the covariance wall): - Simulated-annealing extremal search — Metropolis local search for rare, structured witness/counterexample objects — the Erdős-campaign technique of annealing toward rare, structured *extremal* witness-objects (as opposed to blind uniform sampling, which Collins–Yin–Zhong shows is doomed here); a counterexample is exactly a rare structured extremal object, so a targeted extremal search seeded from known bound-entangled families is the natural transplant. - Bisociation (Koestler 1964) — creative insight as the collision of two habitually unrelated frames of thought — the mechanism (Koestler's cross-frame collision) by which a tool from an unrelated field could bypass covariance-reduction entirely: the whole field is stuck because it keeps re-instantiating one frame (symmetry-group + representation theory); breaking out requires colliding it with a genuinely foreign frame rather than adding a new symmetry group to the same recipe.