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 can be expressed as

(1)

where and are orthonormal sets of vectors, possibly in Hilbert spaces with different dimensionality, and the are the singular values.

That’s it.

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