Peres–Horodecki (PPT) criterion — partial transpose positivity is necessary for separability, and sufficient only in 2×2 and 2×3

Statement

Let $\rho$ be a density matrix on a bipartite Hilbert space $\mathcal H_A\otimes\mathcal H_B$. Its partial transpose $\rho^{T_B}$ is obtained by transposing only the $B$-indices: in a product basis, $\langle i_A j_B|\rho^{T_B}|k_A l_B\rangle = \langle i_A l_B|\rho|k_A j_B\rangle$. The state is PPT (has *positive partial transpose*) if $\rho^{T_B}\ge 0$ (equivalently $\rho^{T_A}\ge0$; the two are related by global transpose and share eigenvalue positivity).

Peres 1996 (necessity). If $\rho$ is separable — i.e. $\rho=\sum_k p_k\,\sigma_k^A\otimes\tau_k^B$ with $p_k\ge0$, $\sum_k p_k=1$ — then $\rho^{T_B}\ge0$. Contrapositive: a negative partial transpose (NPT) is a *sufficient witness of entanglement*. (Source: arXiv:quant-ph/9604005, Asher Peres, "Separability Criterion for Density Matrices," Phys. Rev. Lett. 77 (1996) 1413 — direct fetch: "a necessary condition for separability is that the partial transpose ... have only non-negative eigenvalues.")

Horodecki³ 1996 (sufficiency, only in low dimension). PPT is *also sufficient* for separability exactly when $\dim\mathcal H_A\cdot\dim\mathcal H_B \le 6$, i.e. in the $2\times2$ and $2\times3$ (and, by symmetry, $3\times2$) cases. In every higher dimension ($2\times4$, $3\times3$, and up) there exist PPT states that are entangled (see Bound entanglement — entangled states that are PPT, hence undistillable; the rare structured regime where a PPT² counterexample must live), so PPT is strictly weaker than separability. (Source: arXiv:quant-ph/9605038, M./P./R. Horodecki, "Separability of Mixed States: Necessary and Sufficient Conditions," Phys. Lett. A 223 (1996) 1 — direct fetch: "positivity of the partial transposition of a state is necessary and sufficient for its separability" in $2\times2$ and $2\times3$ systems.)

Facts

- The mechanism is positive-but-not-completely-positive maps. Transpose $T$ is a positive map that is not completely positive. Peres's criterion is the $\Lambda=T$ instance of the general Horodecki characterization: $\rho$ is separable iff $(\mathrm{id}\otimes\Lambda)\rho\ge0$ for *every* positive map $\Lambda$. So a single positive map (transpose) gives only a *necessary* condition; the full family of positive maps is needed for an iff. (Source: arXiv:quant-ph/9605038.) - Why 2×2 and 2×3 are special. In those dimensions every positive map is decomposable — a sum of a completely-positive map and the composition of a completely-positive map with transpose (Størmer 1963, Woronowicz 1976). Decomposability means the transpose test *already exhausts* the positive-map family, so PPT becomes equivalent to separability. From $2\times4$ / $3\times3$ onward, non-decomposable positive maps exist, opening the gap where PPT-entangled states live. (Source: arXiv:quant-ph/9605038; standard operator-algebra background, Størmer/Woronowicz.) - PPT is a semidefinite, efficiently checkable property. Given $\rho$ explicitly, one forms $\rho^{T_B}$ and checks eigenvalue signs — a polynomial-time computation. This is what makes PPT the workhorse *sufficient* test for entanglement (NPT ⇒ entangled) and the practical certificate in numerics, even though deciding separability itself is NP-hard in general (Gurvich 2003). - Naming. "Peres–Horodecki criterion," "PPT criterion," and "positive partial transpose test" all denote the same condition $\rho^{T_B}\ge0$. Peres conjectured (1996) that PPT might be equivalent to distillability; this was later disproved by the discovery of PPT bound entanglement — see Bound entanglement — entangled states that are PPT, hence undistillable; the rare structured regime where a PPT² counterexample must live. - This repo's live use case. The PPT criterion is the *definitional core* of the PPT² conjecture (PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4): a channel is "PPT" when both it and its composition-with-transpose are completely positive (CP and co-CP), and the conjecture asserts the composition of two such channels is entanglement-breaking. The n=2 (qubit) case of PPT² is a folklore corollary precisely *because* Peres–Horodecki gives PPT ⇔ separable in $2\times2$ / $2\times3$, collapsing the composed map's Choi state to separable = Entanglement-breaking channel — measure-and-prepare maps: separable Choi matrix ⇔ destroys all entanglement. (Source: this repo's ppt2-scan.md §1.)

Related

- PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — the PPT-squared conjecture is stated entirely in the PPT/co-PPT language this page defines; the low-dimension sufficiency here is exactly why the n=2 case is trivial and n≥4 is hard (the sufficiency gap *is* where a counterexample would hide). - Bound entanglement — entangled states that are PPT, hence undistillable; the rare structured regime where a PPT² counterexample must live — the phenomenon that exists *only* because PPT sufficiency fails above dimension 6; PPT-entangled states are the direct refutation of "PPT ⇒ separable" in general. - Covariance / symmetry reduction of quantum channels — THE WALL: the one recipe every partial PPT² result reuses, and nobody has escaped to the general case — every partial PPT² result restores tractability by imposing symmetry so the PPT cone block-diagonalizes; this page supplies the underlying PPT test being restricted. - Choi–Jamiołkowski isomorphism — state–channel duality: a map is CP iff its Choi matrix is PSD — turns the channel-level PPT condition into the state-level partial-transpose test of this page, letting Peres–Horodecki apply to maps via their Choi matrices. - Entanglement-breaking channel — measure-and-prepare maps: separable Choi matrix ⇔ destroys all entanglement — the target property in PPT²; a channel is EB iff its Choi state is separable, which by this page is *decided* by PPT only in the low-dimensional cases. - PPT map — a channel that is CP and completely co-positive: Choi matrix PSD with positive partial transpose — the map-level lift of the PPT property (CP and co-CP), built directly on this criterion. - Entanglement witness — a Hermitian W with Tr(Wσ)≥0 on all separable states but Tr(Wρ)<0 on some entangled ρ — the Hahn–Banach dual object; the transpose map is the canonical (decomposable) witness, and non-decomposable witnesses are exactly what detect PPT-entangled states this criterion misses.