Bound Entanglement

Some entanglement is stuck: it took entanglement to create the state, yet no entanglement can ever be pumped back out by LOCC. Such bound entangled states are the dramatic demonstration that mixed-state entanglement is irreversible — the entanglement cost can strictly exceed the distillable entanglement.

Distillable vs. bound

Recall the two operational rates from earlier in the module:

Entanglement cost $E_C(\rho)$ — Bell pairs consumed per copy to manufacture $\rho$ by LOCC.
Distillable entanglement $E_D(\rho)$ — Bell pairs extractable per copy from $\rho$ by LOCC.

For pure states these are equal; entanglement is a fully reversible resource. For mixed states a gap opens, $E_D(\rho) \leq E_C(\rho)$ , and a state is called bound entangled when it is entangled ( $E_C > 0$ ) yet has zero distillable entanglement,

E_D(\rho) = 0 \quad\text{while}\quad \rho\ \text{is entangled}.

You can pour entanglement in, but you can never get a single clean Bell pair back. The resource is locked.

The PPT route to bound entanglement

The key structural fact links distillation to the partial transpose:

Any PPT state has zero distillable entanglement. No LOCC protocol can distil Bell pairs from a state whose partial transpose is positive.

The reason: distillation can only ever produce NPT output from NPT input — the PPT property is preserved under LOCC and under tensoring, so $\rho^{\otimes n}$ stays PPT, and a PPT state can never be mapped close to the (NPT) maximally entangled state. Combine this with what we learned in the PPT lesson:

In $2\times2$ and $2\times3$ , PPT $\Leftrightarrow$ separable, so a PPT state is unentangled and the bound is vacuous — no bound entanglement exists in these smallest systems.
In higher dimensions there exist entangled PPT states. Being PPT, they are undistillable; being entangled, they cost entanglement to make. These are bound entangled.

The Horodecki family constructed the first explicit examples in $3\times3$ and $2\times4$ , often via the range criterion (a separable state's range is spanned by product vectors whose conjugates span the range of $\rho^{T_B}$ — a condition entangled PPT states violate).

Two flavors of undistillable entanglement

It is worth separating two claims:

PPT bound entanglement — proven to exist. All entangled PPT states are bound entangled.
NPT bound entanglement — conjectured. It is a long-standing open problem whether some NPT states are also undistillable. The Werner-state analyses strongly suggest such states exist, but a rigorous proof is still missing.

Why it matters

Bound entanglement is not a mere curiosity:

Irreversibility. It shows the resource theory of mixed-state entanglement is irreversible: $E_C$ and $E_D$ genuinely differ, so "amount of entanglement" is not a single number even asymptotically.
Activation. Bound entangled states are not useless — combined with a small amount of free entanglement they can boost otherwise-impossible tasks (entanglement activation), and they enable certain forms of secure key distribution despite yielding no distillable pairs.
Detection. Because PPT no longer certifies separability, detecting entanglement in high dimensions requires stronger tools — entanglement witnesses and the range/realignment criteria — motivating the witness machinery developed elsewhere in the curriculum.