# Outer Product

- operates on two vectors and produces a matrix
- notation is $`\ket{x}\bra{y}`$
  - $`\ket{x}`$ is a [[ket]]
  - $`\bra{y}`$ is a [[bra]]
- [[matrix-multiplication]], associates with [[inner-product]]
- a sum $`\ket{x}\bra{0} + \ket{y}\bra{1}`$ where $`\ket{x}`$ and $`\ket{y}`$ are [[orthonormal]] produces a [[unitary-operator]] sending $`\ket{0}`$ to $`\ket{x}`$ and $`\ket{1}`$ to $`\ket{y}`$