outer products

outer-product.md

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+# 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 $`\ket{y}`$