I Schmidt decomposition and entropy of the W state

[yamata1]

Hello,

The state | W \rangle = \frac { 1 } { \sqrt { 3 } } ( | 001 \rangle + | 010 \rangle + | 100 \rangle ) is entangled.
The Schmidt decomposition is :

Let H _ { 1 } and H _ { 2 } be Hilbert spaces of dimensions n and m respectively. Assume {\displaystyle n\geq m}.For any vector w in the tensor product H _ { 1 } \otimes H _ { 2 } , there exist orthonormal sets \left\{ u _ { 1 } , \ldots , u _ { m } \right\} \subset H _ { 1 } and \left\{ v _ { 1 } , \ldots , v _ { m } \right\} \subset H _ { 2 } such that w = \sum _ { i = 1 } ^ { m } \alpha _ { i } u _ { i } \otimes v _ { i } where the scalars {\displaystyle \alpha _{i}}are real, non-negative, and, as a (multi-)set, uniquely determined by w.

What would the Schmidt decomposition be for | W \rangle ?
I am also intersted in writing the reduced density matrix but I need the basis from the Schmidt decomposition.

Thank you.

 

[kith]

The Schmidt decomposition refers to a tensor product of two spaces while your state vector is an element of a tensor product of three spaces.

A quick search yielded this paper which talks about generalizing the Schmidt decomposition and this post on stackexchange which talks about counterexamples in the tripartite case.