# Proving that a positive partial transpose (PPT) state is non-distillable

I am attempting to prove the following statement: A positive partial transpose (PPT) state cannot be distilled.

I would like to do so using the following facts:

• Let $$\rho$$ represent a bipartite quantum state. We may write $$\rho$$ via matrix elements in a product basis, $$\rho_{m\mu, n\nu} = \langle m\mu \;|\rho \;| n\nu \rangle$$. We define the partial transposition map, $$T_B$$, as follows: $$T_B(\rho)_{m\mu, n\nu} = \rho^{T_B}_{m\mu,n\nu} = \rho_{m\nu,n\mu} = \langle m\nu\;|\rho\;|n\mu\rangle$$. A state is said to have a positive partial transpose (PPT) if and only if all eigenvalues of its partial transpose are non-negative or, equivalently, if and only if $$\rho^{T_B}\geq 0$$
• Suppose a bipartite quantum state, $$\rho$$, is separable. Then, $$\rho$$ must be PPT, i.e. $$\rho^{T_B}\geq 0$$
• A state $$\rho$$ is distillable if and only if for some two-dimensional projectors P, Q and for some number n, the state $$(P\bigotimes Q̺)\rho^{\bigotimes n} (P\bigotimes Q)$$ is entangled.

My rough attempt:

Let $$\rho$$ denote a PPT bipartite state. Therefore, $$\rho^{T_B}\geq 0$$. By the second fact listed above, if a state is NPT it cannot be separable therefore it is entangled. Therefore $$\rho$$ is distillable only if $$\sigma = (P\bigotimes Q̺)\rho^{\bigotimes n} (P\bigotimes Q)$$ is NPT (negative partial transpose).

Let us consider the partial transpose of $$\sigma$$: $$(P\bigotimes Q̺)[\rho^{T_B}]^{\bigotimes n} (P\bigotimes Q)$$. Side note: how would I prove more formally that this is the partial transpose of $$\sigma$$?

It's somewhat clear to me that the partial transpose of $$\sigma$$ must be positive, since $$\rho$$ is PPT. This is logical, and I believe it would complete the proof. However, I have two queries:

1. How to formalize the fact that the partial transpose of $$\sigma$$ is $$(P\bigotimes Q̺)[\rho^{T_B}]^{\bigotimes n} (P\bigotimes Q)$$, or is this immediately clear?

2. How to rigorously state that $$\rho^{T_B}\geq 0\rightarrow (P\bigotimes Q̺)[\rho^{T_B}]^{\bigotimes n} (P\bigotimes Q) \geq 0$$.

Perhaps the key to the second issue lies in the fact that $$P,Q$$ are co - positive maps, but is this necessarily true?

Any suggestions or corrections would be greatly appreciated!

(2) That should be obvious, since (a) $$X\ge0\ \Rightarrow X^{\otimes n}\ge 0$$, and (b) $$Y\ge0\ \Rightarrow\ AYA^\dagger\ge0$$.