Cauchy-Schwarz Inequality Given two Vector ∣u⟩,∣v⟩∈Cn\left\lvert u \right\rangle,\left\lvert v \right\rangle\in \mathbb{C}^n∣u⟩,∣v⟩∈Cn, ∣⟨u∣v⟩∣≤∥u∥∥v∥|\left\langle u|v \right\rangle|\leq \|u\|\|v\|∣⟨u∣v⟩∣≤∥u∥∥v∥ It’s squared form can be written as ∣⟨u∣v⟩∣2≤⟨u∣u⟩⟨v∣v⟩|\left\langle u|v \right\rangle|^2\leq \left\langle u|u \right\rangle\left\langle v|v \right\rangle∣⟨u∣v⟩∣2≤⟨u∣u⟩⟨v∣v⟩Conditional Probability