Corollary 1.2

Corollary 1.2: Let the state vector be as follows:

• For the marked state S_m such that C(S_m) = 1, the amplitude is k
• For all other (N - 1) states the amplitude is l

Then after the application of A:

k² + (N - 1)l² = k^'2 + (N - 1)l^'2

Proof: This follows directly from the fact that A is unitary, and that unitary transformations preserve normalization of the state vector. That means precisely that the sum of the absolute squares of the components is the same before and after the operation. Since we never deal with any complex amplitudes in the processing of Grover's algorithm, corollary 1.2 follows directly. [Grover96]