Geometric interpretation of 1-distillability

This is a sequel to Motivation for the definition of k-distillability

Geometrical interpretation from the definition of 1-distillability

• The eigenstate $$|\psi\rangle$$ of the partially transposed $$1$$-distillable states will have Schmidt rank at most 2, i.e.,

$$|\psi\rangle\langle\psi|=\sum_i\lambda_i^2|ii\rangle\langle ii|, \textit{ where } \sum_i\lambda_i^2=1\tag{17}$$

• The constraint $$\sum_i\lambda_i^2=1$$ gives rise to a geometric structure in arbitrary $$N$$ dimensions.

Source

Questions:

1. In the definition of $$k$$-distillability (cf. here) we were talking about bipartite density matrices $$\rho$$ in $$H_A\otimes H_B$$. In what sense is $$|\psi\rangle$$ an "eigenstate" of a partially transposed $$1$$-distillable state? Is for $$1$$-distillable $$\rho$$'s the (non-normalized) state $$|\psi\rangle \in H$$, such that $$\langle \psi|\sigma|\psi\rangle < 0$$ necessarily an eigenstate of $$\rho$$? Also, can we prove that $$\langle \psi|\sigma|\psi\rangle < 0$$ for any eigenstate of $$\rho$$?

2. I do not see how the fact that "the eigenstate $$|\psi\rangle$$ of the partially transposed $$1$$-distillable states will have a Schmidt rank at most 2" is encapsulated within the statement "$$|\psi\rangle\langle\psi|=\sum_i\lambda_i^2|ii\rangle\langle ii|$$ where $$\sum_i\lambda_i^2=1$$".

As far as I understand, the Schmidt decomposition of a pure state $$|\Psi\rangle$$ of a composite system AB, considering an orthonormal basis $$|i_A\rangle$$ for system A and $$|i_B\rangle$$ for system B, is $$|\Psi\rangle = \sum_i \lambda_i|i_A\rangle|i_B\rangle,$$ where $$\lambda_i$$ are non-negative real numbers satisfying $$\sum_i\lambda_i^2=1$$ known as Schmidt co-efficients. Now the number of non-zero $$\lambda_i$$'s in the Schmidt decomposition is called Schmidt rank or Schmidt number. So I don't quite understand the geometric constraint they're talking about; if the Schmidt rank is at most 2, then we'd be restricted to only two cases i.e. $$\lambda_1^2=1$$ and $$\lambda_1^2+\lambda_2^2 = 1$$...which aren't so interesting. Am I missing something?

• I agree with your statement in question 2. I don't see it either. (This is often the problem with following slides rather than papers. Is this a typo? Is this an error in a non-refereed document?) Jul 20 '20 at 7:52
• actually, it doesn't look like a typo because that structure is being used throughout the rest of the slides. It does not look right to me. Jul 20 '20 at 8:26