# Schmidt decomposition and entropy of the W state

• I
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.

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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.