Bound entanglement — entangled states that are PPT, hence undistillable; the rare structured regime where a PPT² counterexample must live

Statement

A bipartite state $\rho$ on $\mathcal H_A\otimes\mathcal H_B$ is bound entangled if it is entangled yet undistillable: no LOCC (local operations + classical communication) protocol acting on any number of copies $\rho^{\otimes n}$ can extract even one pure maximally-entangled pair. This splits entanglement into two regimes:

- Free (distillable) entanglement — usable as a resource; can be concentrated into pure Bell pairs. - Bound entanglement — genuinely entangled (not separable) but *locked*: the entanglement is present but cannot be pumped out.

Key structural theorem (Horodecki³ 1998). Every distillable state has non-positive partial transpose (is NPT). Contrapositive: every PPT entangled state is undistillable, i.e. bound entangled. (Source: arXiv:quant-ph/9801069, M./P./R. Horodecki, "Mixed-state entanglement and distillation: is there a 'bound' entanglement in nature?", Phys. Rev. Lett. 80 (1998) 5239 — direct fetch: distillable mixed states must violate the partial-transposition criterion, establishing a "bound" undistillable category.)

Existence. PPT-entangled states exist in every dimension above $2\times3$ (they must, since the Peres–Horodecki (PPT) criterion — partial transpose positivity is necessary for separability, and sufficient only in 2×2 and 2×3 sufficiency fails there; arXiv:quant-ph/9605038). The first explicit constructions and the range criterion detecting them are due to P. Horodecki (1997); the 1998 paper then shows these PPT-entangled states are exactly the bound-entangled ones. So bound entanglement is not hypothetical — it is a proven, if rare, feature of $3\times3$, $2\times4$, and higher systems.

Facts

- PPT ⊊ separable is the whole story. Separable ⇒ PPT ⇒ undistillable, and all inclusions are strict above dimension 6. Bound entangled = the gap between PPT and separable. Below dimension 6 the gap is empty (Peres–Horodecki), so there is no bound entanglement in $2\times2$ or $2\times3$. (Source: arXiv:quant-ph/9605038, arXiv:quant-ph/9801069.) - NPT bound entanglement is a separate open question. Whether *NPT* undistillable (bound) states also exist is a long-standing open problem (DiVincenzo–Shor–Smolin–Terhal–Thapliyal, and independently the Horodeckis, 1999); the *PPT* bound entangled states of this page are the proven, canonical case. This page's use in PPT² concerns the PPT variety. - Why bound entanglement is "rare and structured." PPT-entangled states form a measure-zero-adjacent, thin, non-generic set: random/generic PPT states are overwhelmingly separable (hence entanglement-breaking under Choi–Jamiołkowski), which is exactly why free-probability "generic" arguments (Collins–Yin–Zhong 2018) succeed while a *targeted* construction remains the hard open question. A blind random search does not find them; they must be built from specific structure (unextendible product bases, range-criterion constructions, symmetric families). (Source: this repo's ppt2-scan.md §6.) - Cryptographic reality. Bound entanglement is not useless: Horodecki⁴ (2005, quant-ph/0309110) showed secret key can be drawn from certain bound-entangled states even though no pure entanglement is distillable — the "private states" line that motivated Christandl's interest in composed PPT maps. (Source: ppt2-scan.md §0, background refs.) - This repo's live use case — where a PPT² counterexample must hide. The PPT² conjecture (PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4) says the composition of two PPT channels is entanglement-breaking, i.e. its Choi state is separable. A counterexample would be two PPT channels whose composition's Choi state is PPT but entangled — precisely a bound-entangled state produced by a composed PPT map. Since bound entanglement only exists above dimension 6 and is rare/structured, any counterexample must live in exactly this thin regime, at $n\ge4$ — which is why the low-$n$ cases are provable and blind numerical search (Jin 2020) fails. (Source: ppt2-scan.md §1, §6.)

Related

- PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 — a counterexample to the PPT-squared conjecture *is* a bound-entangled Choi state emitted by a composition of two PPT channels; this page pins down where in state-space to look (rare, structured, dimension ≥ 7 total). - Peres–Horodecki (PPT) criterion — partial transpose positivity is necessary for separability, and sufficient only in 2×2 and 2×3 — bound entanglement exists precisely because PPT sufficiency fails above 2×3; without that gap there would be no PPT-entangled states. - 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 partial results dodge bound entanglement by restricting to symmetry classes where the invariant PPT cone is provably separable; escaping the wall means confronting genuine bound entanglement in the general cone. - Entanglement-breaking channel — measure-and-prepare maps: separable Choi matrix ⇔ destroys all entanglement — EB ⇔ separable Choi state; a channel *fails* to be EB exactly when its Choi state carries (free or bound) entanglement, and PPT² is the claim that composition kills it down to at most separable. - Choi–Jamiołkowski isomorphism — state–channel duality: a map is CP iff its Choi matrix is PSD — the bridge translating "composed PPT map" into a bipartite Choi state on which the PPT-vs-separable (bound-entanglement) question is asked. - Entanglement witness — a Hermitian W with Tr(Wσ)≥0 on all separable states but Tr(Wρ)<0 on some entangled ρ — non-decomposable witnesses are the only objects that *detect* PPT bound entanglement (ordinary/decomposable witnesses and the PPT test cannot); constructing one is often how a bound-entangled state is certified.