# Motivation for the definition of k-distillability

Quantum Computing Asked on August 20, 2021

Definition of k-distillability

For a bipartite state $$rho$$, $$H=H_Aotimes H_B$$ and for an integer $$kgeq 1$$, $$rho$$ is $$k$$-distillable if there exists a (non-normalized) state $$|psiranglein H^{otimes k}$$ of Schimdt-rank at most $$2$$ such that,

$$langle psi|sigma^{otimes k}|psirangle < 0, sigma = Bbb Iotimes T(rho).$$

$$rho$$ is distillable if it is $$k$$ for some integer $$kgeq 1.$$

Source

I get the mathematical condition but don’t really understand the motivation for $$k$$-distillability in general, or more specifically the condition $$langle psi|sigma^{otimes k}|psirangle < 0$$. Could someone explain where this comes from?

Remember that the partial transpose condition is generally good for detecting entanglement, i.e. a bipartite state $$rho$$ is certainly entangled if the partial transpose is not non-negative. In other words, if there exists a state $$|psirangle$$ such that $$langlepsi|Iotimestext{T}(rho)|psirangle<0,$$ then the state is certainly entangled.

If you want to be able to distil some entanglement from $$k$$ copies then, crudely, you'd like to look at $$k$$ copies of the partially transposed state, and if that has a negative eigenvalue, you would be able to extract some entanglement.

With that level of explanation, you'd ask why looking at more than one copy is any use -- the eigenvalues of many copies of $$sigma$$ are easily related to the eigenvalues of a single copy. However, this is because of the extra condition that $$|psirangle$$ must be Schmidt rank 1 or less. I presume that this is because you can give an explicit distillation protocol based on the properties of $$|psirangle$$. Essentially, this is due to the fact that you're trying to project onto a Bell pair which, of course, is Schmidt rank 2.

For a better understanding that the very hand-wavy suggestions I've just given, you'd want to work through page 2 of https://arxiv.org/abs/quant-ph/9801069

Correct answer by DaftWullie on August 20, 2021