# Heisenberg's Inequality

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.

## Hermitian Operators and Mean Values

If is a Hermitian operator on (finite dimensional) Hilbert space, then
where and are projections.