I came across this question as part of my quantum mathematics exam preparation:
Consider the entangled state shared between Alice and Bob:
$|\psi^{AB}\rangle = \sqrt{p_1} |1\rangle^A |1\rangle^B + \cdots + \sqrt{p_n} |n\rangle^A |n\rangle^B$
where $p_x > 0$ for all $x \in [n]$.
Prove the following theorem:
Quantum teleportation of a $d$-dimensional qudit is possible if and only if none of the Schmidt coefficients are greater than $1/d$.
I proved the direction: Schmidt coefficients are smaller equal to $1/d$ than quantum teleportation is possible, by showing that if all $p_x$ are smaller than $1/d$, than the vector $p=(p_1,...,p_n)$ is majorized by the vector $(1/d,...,1/d,0,...,0)$, and so by the Nielsen theorem $|\psi^{AB}\rangle$ can be converted to the maximally entangled state by an LOCC. Than I showed the teleportation protocol where the initial state is the maximally entangled state. In the other direction, I understand that I need to show that if teleportation is possible, than $|\psi^{AB}\rangle$ can be converted to the maximally entangled state by an LOCC. I tried to show that Alice can just preform a projection and then a local filtering on her system obtaining the maximally entangled state from $|\psi^{AB}\rangle$, but I did not use the fact that quantum teleportation is possible. I thought maybe since quantum teleportation is possible, I can use the fact that Alice can prepare any state she wants and maybe find a specific qudit that upon teleportation leaves the system in a state that includes the maximally entangled state, and than say the teleportation is all LOCC operations and so we're done, but it didn't work for me.
I would appreciate any lead you might have on that direction!
