Singular value decomposition in bra-ket notation

In linear algebra, and therefore quantum information, the singular value decomposition (SVD) is elementary, ubiquitous, and beautiful. However, I only recently realized that its expression in bra-ket notation is very elegant. The SVD is equivalent to the statement that any operator \hat{M} can be expressed as

(1)   \begin{align*} \hat{M} = \sum_i \vert A_i \rangle \lambda_i \langle B_i \vert \end{align*}

where \vert A_i \rangle and \vert B_i \rangle are orthonormal bases, possibly in Hilbert spaces with different dimensionality, and the \lambda_i \ge 0 are the singular values.

That’s it.

Bookmark the permalink.

One Comment

  1. A thing of beauty! This gorgeous formula made my day 🙂

Leave a Reply

Include [latexpage] in your comment to render LaTeX equations with $'s. (More.)

Your email address will not be published. Required fields are marked with a *.