Theorem 1. Let and be Hermitian matrices and a state vector. Then
Proof. Let and be two Hermitian operators and a unit vector. Notice that so that . Thus, By the Cauchy-Schwartz Inequality, and so Now if and are Hermitian operators, take and . Note that and so Now is the expected outcome of measuring qubits in state and so is the variance. Writing (resp. ) for the standard deviation, we obtain and the result follows.
If is a Hermitian operator on (finite dimensional) Hilbert space, then where and are projections.