outer products

unitary-operator.md

@@ -0,0 +1,12 @@
+# Unitary Operator
+
+- [[quantum-computing]]
+- a matrix $`\mathcal{U}`$ operating on a [[quantum-state]] to produce another [[quantum-state]]
+  - maps unit vectors to unit vectors
+  - [[norm]]
+- other equivalent definitions:
+  - columns of $`\mathcal{U}`$ are [[orthonormal]]
+  - rows of $`\mathcal{U}`$ are [[orthonormal]]
+  - $`\mathcal{U}^{-1} = \mathcal{U}^\dagger = \overline{\mathcal{U}^\intercal} = \overline{\mathcal{U}}^\intercal`$
+  - preserves [[norm]]
+  - preserves [[inner-product]]