Schmidt decomposition and entropy of the W state

  • #1
52
0
Hello,

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

Let [itex]H _ { 1 } [/itex] and [itex]H _ { 2 } [/itex] be Hilbert spaces of dimensions n and m respectively. Assume [itex]{\displaystyle n\geq m} [/itex].For any vector [itex]w[/itex] in the tensor product [itex]H _ { 1 } \otimes H _ { 2 }[/itex] , there exist orthonormal sets [itex]\left\{ u _ { 1 } , \ldots , u _ { m } \right\} \subset H _ { 1 } [/itex] and [itex]\left\{ v _ { 1 } , \ldots , v _ { m } \right\} \subset H _ { 2 } [/itex] such that [itex]w = \sum _ { i = 1 } ^ { m } \alpha _ { i } u _ { i } \otimes v _ { i } [/itex] where the scalars [itex] {\displaystyle \alpha _{i}}[/itex]are real, non-negative, and, as a (multi-)set, uniquely determined by [itex] w [/itex].
What would the Schmidt decomposition be for [itex]| W \rangle[/itex] ?
I am also intersted in writing the reduced density matrix but I need the basis from the Schmidt decomposition.

Thank you.
 

Answers and Replies

  • #2
kith
Science Advisor
1,358
453
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.
 

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