PPT map — a channel that is CP and completely co-positive: Choi matrix PSD with positive partial transpose

Statement

Definition. Let $T$ denote the transpose map. A quantum channel $\Phi:\mathcal L(\mathcal H_A)\to\mathcal L(\mathcal H_B)$ is a PPT map (positive-partial-transpose-preserving; also PPT channel or bi-CP map) if it is completely positive AND completely co-positive, i.e. $$ \Phi\ \text{is CP} \quad\text{and}\quad T\circ\Phi\ \text{is CP} $$ (equivalently $\Phi\circ T$ is CP). The map $T\circ\Phi$ being CP is the "co-CP" (completely co-positive) condition. (Source: WebSearch — Rahaman–Jaques–Paulsen / Prudhoe: "Φ is PPT if it is completely positive and co-completely positive, that is, Φ∘t is completely positive.")

Choi-matrix characterization. Via the Choi–Jamiołkowski isomorphism — state–channel duality: a map is CP iff its Choi matrix is PSD, $\Phi$ is a PPT map iff its Choi matrix satisfies both positivity conditions: $$ J(\Phi)\ge0 \qquad\text{and}\qquad J(\Phi)^{\Gamma}\ge0, $$ where $\Gamma$ is the partial transpose (transpose on one of the two tensor factors). The first condition is complete positivity; the second is complete co-positivity. So a PPT map is exactly a channel whose Choi matrix is a PPT operator (PSD with positive partial transpose). (Source: WebSearch — "a linear map Φ is PPT if and only if its Choi matrix J(Φ) is positive and has positive partial transpose.")

"Block-positivity" phrasing / the precise CP-vs-positive point. A map is merely positive (sends states to states) iff its Choi matrix is block-positive ($\langle a\otimes b|J(\Phi)|a\otimes b\rangle\ge0$ for all product vectors), a strictly weaker condition than PSD. "PPT" is the sharper, symmetric requirement that $J(\Phi)$ be genuinely PSD *and* stay PSD after a partial transpose — i.e. $\Phi$ is CP and $T\circ\Phi$ is CP, not just positive. Do not conflate "PPT map" (CP + co-CP) with "positive map" (block-positive Choi): the transpose $T$ itself is positive but neither CP nor a PPT map.

Hierarchy (with strict inclusions). $$ \mathrm{EB}\ \subsetneq\ \mathrm{PPT}\ \subsetneq\ \mathrm{CP}. $$ - $\mathrm{EB}\subseteq\mathrm{PPT}$: an entanglement-breaking channel has *separable* Choi matrix (Entanglement-breaking channel — measure-and-prepare maps: separable Choi matrix ⇔ destroys all entanglement), and every separable operator is PPT. Strict, because there are PPT channels whose Choi matrix is PPT-but-entangled (bound-entangled Choi state) — not EB. - $\mathrm{PPT}\subseteq\mathrm{CP}$: PPT adds the co-CP condition on top of CP. Strict, because a generic CP channel (e.g. any unitary channel in dimension $\ge2$) has a Choi matrix — the SWAP-like maximally entangled projector — that fails PPT. (Source: WebSearch — "the set of entanglement breaking channels are a subset of the PPT channels ... there are PPT channels that are not entanglement breaking.")

Facts

- **Its role in PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4: PPT is the hypothesis class. The PPT-squared conjecture takes two** PPT maps $\Phi_1,\Phi_2$ and asserts $\Phi_2\circ\Phi_1$ is entanglement-breaking. So "PPT map" is the exact object on the *input* side of the conjecture; the whole question is whether the co-CP condition, applied twice through a composition, forces the output Choi matrix all the way down from "PPT" to "separable" (EB). - PPT is closed under composition; EB is the conjectured "two steps down." Composition of two PPT maps is again PPT (CP∘CP is CP, and the co-CP conditions compose too), so $\mathrm{PPT}$ is a semigroup. The conjecture is that you do not merely stay in $\mathrm{PPT}$ but drop into the strictly smaller ideal $\mathrm{EB}$ after two factors. The difficulty is that composition acts on Choi matrices by a non-multiplicative "link product" (Choi–Jamiołkowski isomorphism — state–channel duality: a map is CP iff its Choi matrix is PSD), so PPT-ness of each factor gives no direct handle on separability of the composite's Choi matrix. - The gap PPT ∖ EB is exactly bound entanglement. A channel sits in $\mathrm{PPT}\setminus\mathrm{EB}$ iff its Choi matrix is PPT but entangled, i.e. bound-entangled (Bound entanglement — entangled states that are PPT, hence undistillable; the rare structured regime where a PPT² counterexample must live). These states exist only in dimension $\ge3\times3$ (Horodecki), which is why PPT² is trivially true for qubits (no room for bound entanglement, Peres–Horodecki gives PPT ⇔ separable in $2\times2,2\times3$) and first becomes nontrivial at qutrits — the $n=3$ case that was proven only in 2018. - Verification is cheap, refutation is a single SDP. Deciding whether a candidate $\Phi$ is a PPT map is a two-line check: form $J(\Phi)$, test $J(\Phi)\ge0$ and $J(\Phi)^\Gamma\ge0$ (two eigenvalue/SDP feasibility checks). This makes constructing and screening candidate PPT maps for a PPT² counterexample computationally trivial; the hard part is that *deciding separability* of the resulting composite (to confirm it violates EB) is the genuinely hard step, handled cheaply only in the refuting direction by an Entanglement witness — a Hermitian W with Tr(Wσ)≥0 on all separable states but Tr(Wρ)<0 on some entangled ρ. - Terminology cross-reference. In the operator-algebra literature "PPT map" = "bi-CP map" = "channel that is CP and co-CP"; Christandl–Müller-Hermes–Wolf (2019) give four equivalent formulations of the conjecture in these terms (see this wiki's ppt2-scan.md). Beware the distinct notion "bi-PPT channel" (Hirche–Leditzky: $\Phi$ and its *complementary* channel are both PPT), proved always-EB by Müller-Hermes–Singh 2022 — related to but not the same as PPT².

Related

- PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — PPT maps are the hypothesis: the conjecture is that a composition of two of them is entanglement-breaking. - Entanglement-breaking channel — measure-and-prepare maps: separable Choi matrix ⇔ destroys all entanglement — the strictly smaller class $\mathrm{EB}\subsetneq\mathrm{PPT}$ that the composition is conjectured to land in. - Choi–Jamiołkowski isomorphism — state–channel duality: a map is CP iff its Choi matrix is PSD — supplies the matricial characterization: PPT map ⇔ Choi matrix PSD with positive partial transpose. - Bound entanglement — entangled states that are PPT, hence undistillable; the rare structured regime where a PPT² counterexample must live — the PPT-but-entangled Choi states are exactly the gap $\mathrm{PPT}\setminus\mathrm{EB}$; a PPT² counterexample would be a composition landing there. - Peres–Horodecki (PPT) criterion — partial transpose positivity is necessary for separability, and sufficient only in 2×2 and 2×3 — for a Choi matrix, "PPT" is literally the Peres–Horodecki condition; it collapses PPT to separable (hence PPT map to EB) exactly in $2\times2$ and $2\times3$. - Covariance / symmetry reduction of quantum channels — THE WALL: the one recipe every partial PPT² result reuses, and nobody has escaped to the general case — the dominant proof technique restricts to PPT maps with an imposed symmetry group so the PPT cone becomes tractable; every 2020–2026 partial result is a covariance-group instance of this PPT-cone analysis.