Main result: a PPT bound entangled state with strictly positive distillable key

Let $\varrho_s$ and $\varrho_a$ be the extreme Werner states (9) on $\mathbb C^d\otimes\mathbb C^d$, $\tau_0 = \varrho_s^{\otimes l}$ and $\tau_1 = [(\varrho_a+\varrho_s)/2]^{\otimes l}$ the hiding states (10), and $\rho = \rho(p,d,l)$ the state (11) on key qubits $A,B$ and shields $A' = B' = (\mathbb C^d)^{\otimes l}$: in the basis $|00\rangle,|01\rangle,|10\rangle,|11\rangle$ of $AB$,

For every real $p$ with $\frac14<p\le\frac13$ there are integers $l\ge 1$ and $d\ge 2$ such that, across the cut $AA':BB'$,

1. $\rho$ is a state;
2. $\rho$ has positive partial transpose;
3. $\rho$ is not separable;
4. $E_D(\rho) = 0$;
5. $K_D(\rho) > 0$.

Clauses 1–4 say that $\rho$ is bound entangled (p.1: "states which require entanglement to create, but from which no pure entanglement can be distilled"), and clause 5 that a secure key can nevertheless be distilled from it. This overturns the expectation that key distillation from a quantum state is possible exactly when entanglement distillation is.

Formalization Note $\rho$ is the published Dependency50.P24.hidingFamily p d l ($\tau_0$ = hidingTwo, $\tau_1$ = hidingOne); PPT is transposeBob, the transpose of $B$ and $B'$ together. The rates are the published operational Gap48.P42.bipartiteDistillable ($E_D$) and Gap48.P42.distillableKey ($K_\gamma$), valued in $[0,\infty]$, on the cut finCut.

Differences from the source:

• reading: distillable key K_D (key drawn by local operations and public communication; not defined formally in the paper) → K_γ, the operational private-dit rate Gap48.P42.distillableKey under deterministic finite adaptive LOCC (ℝ≥0∞); reason: p.2 identifies the two rates ("We thus have equality of the two rates", with "Clearly K_γ ≤ K_D") and the proof on p.3 distils γ states "at the same rate as K_D"; the mission uses this one key rate throughout
• reading: "a bound entangled state which can be shown to have K_D > 0" (no parameters named) → for every p with 1/4 < p ≤ 1/3 there are l ≥ 1 and d ≥ 2 such that ρ(p,d,l) is a PPT, non-separable state with E_D = 0 and K > 0; reason: the paragraph derives exactly this range (p ≤ 1/3 for PPT, p > 1/4 for (13) → 1/2) and chooses l, then d; the family form implies the single-state claim
• reading: "bound entangled" → state ∧ PPT ∧ not separable ∧ E_D = 0; reason: p.1 defines bound entangled states as states which require entanglement to create (not separable) and from which no pure entanglement can be distilled (E_D = 0); PPT is the property the paper derives for them ("PPT … being therefore bound entangled")
• encoding: the Alice:Bob cut AA′:BB′ of a state on A B A′ B′ → finCut: partyCut followed by a fixed relabelling of Alice's register Fin 2 × A′ as Fin (2|A′|) and Bob's likewise; reason: the published rates take matrices on Fin a × Fin b; rates, separability and partial transposition are invariant under local relabellings
• encoding: real-valued rates E_D and K_D → ℝ≥0∞-valued suprema; reason: avoids the junk value of a real supremum of an unbounded or empty set; same instances