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 $\ket{y}$
• $\ket{x}\bra{x}$ is a projection matrix onto $\ket{x}$