# John Lindsay Orr

$\newcommand{\CC}{\mathbb{C}} \newcommand{\NN}{\mathbb{N}} \newcommand{\RR}{\mathbb{R}} \newcommand{\H}{\mathcal{H}} \newcommand{\e}{\epsilon} \newcommand{\x}{\mathbf{x}} \newcommand{\y}{\mathbf{y}}$

# Heisenberg's Inequality

Theorem 1. Let $S$ and $T$ be Hermitian matrices and $x$ a state vector. Then

Proof. Let $A$ and $B$ be two Hermitian operators and $x$ a unit vector. Notice that so that $\langle [A, B]x, x\rangle = 2 \mathop{Im}\langle ABx, x\rangle$. Thus, By the Cauchy-Schwartz Inequality, $|\langle Bx, Ax\rangle| \le \|Ax\| \|Bx\|$ and so Now if $S$ and $T$ are Hermitian operators, take $A = S - \langle Sx, x \rangle I$ and $B = T - \langle Tx, x \rangle I$. Note that $[S, T] = [A, B]$ and so Now $\langle Sx, x \rangle$ is the expected outcome of measuring qubits in state $x$ and so $\langle A^2x, x \rangle$ is the variance. Writing $\Delta(S)$ (resp. $\Delta(T)$) for the standard deviation, we obtain and the result follows.

## Hermitian Operators and Mean Values

If $H$ is a Hermitian operator on (finite dimensional) Hilbert space, then $H = \sum_{i=1}^n m_i P_i$ where $m_i\in\RR$ and $P_i$ are projections.