# 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.

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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.

"Schmidt decomposition and entropy of the W state"

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