# Geometric interpretation of 1-distillability

Quantum Computing Asked by Sanchayan Dutta on August 20, 2021

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

Geometrical interpretation from the definition of 1-distillability

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

$$|psiranglelanglepsi|=sum_ilambda_i^2|iiranglelangle ii|, textit{ where } sum_ilambda_i^2=1tag{17}$$

• The constraint $$sum_ilambda_i^2=1$$ 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 $$H_Aotimes H_B$$. In what sense is $$|psirangle$$ an "eigenstate" of a partially transposed $$1$$-distillable state? Is for $$1$$-distillable $$rho$$‘s the (non-normalized) state $$|psirangle in H$$, such that $$langle psi|sigma|psirangle < 0$$ necessarily an eigenstate of $$rho$$? Also, can we prove that $$langle psi|sigma|psirangle < 0$$ for any eigenstate of $$rho$$?

2. I do not see how the fact that "the eigenstate $$|psirangle$$ of the partially transposed $$1$$-distillable states will have a Schmidt rank at most 2" is encapsulated within the statement "$$|psiranglelanglepsi|=sum_ilambda_i^2|iiranglelangle ii|$$ where $$sum_ilambda_i^2=1$$".

As far as I understand, the Schmidt decomposition of a pure state $$|Psirangle$$ of a composite system AB, considering an orthonormal basis $$|i_Arangle$$ for system A and $$|i_Brangle$$ for system B, is $$|Psirangle = sum_i lambda_i|i_Arangle|i_Brangle,$$ where $$lambda_i$$ are non-negative real numbers satisfying $$sum_ilambda_i^2=1$$ 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?