= Estimating a quantum expectation value
{title2=$\widehat\mu_N=N^{-1}\sum_{i=1}^NX_i$}
Prepare a fresh copy of the same <quantum state> for each trial and measure the same <observable>. If preparations and readouts are independent, the outcomes $X_i$ are <independent and identically distributed random variables> with the <Born rule> <probability distribution>. Their <sample mean> is an <unbiased estimator> of the <expectation value>:
$$
\widehat\mu_N=\frac1N\sum_{i=1}^NX_i,\qquad
\mathbb E[\widehat\mu_N]=\langle A\rangle,\qquad
\operatorname{Var}(\widehat\mu_N)=\frac{(\Delta A)^2}{N},\qquad
\operatorname{SE}(\widehat\mu_N)=\frac{\Delta A}{\sqrt N}.
$$
The <law of large numbers> explains convergence of the <sample mean>, while the <standard error> describes its finite-sample spread. It takes four times as many independent trials to halve that <standard error>. This reduces estimation error; it does not reduce the <quantum uncertainty> of a single prepared copy.
For the <spin expectation value>, let $n_+$ and $n_-$ be the two outcome counts, with $N=n_++n_-$. Then
$$
\widehat\mu_N=\frac{\hbar}{2}\frac{n_+-n_-}{N},\qquad
n_+\sim\operatorname{Binomial}(N,p_+),\qquad
\Pr\left(\widehat\mu_N=\hbar\left(\frac{k}{N}-\frac12\right)\right)
=\binom Nk p_+^k(1-p_+)^{N-k}.
$$
The <binomial distribution> gives the exact sampling model. For $\theta=\pi/3$, the <standard error> is $\hbar\sqrt3/(4\sqrt N)$: approximately $0.1083\hbar$ for $N=16$ and $0.03827\hbar$ for $N=128$. A run can fluctuate away from the predicted mean even as its typical spread decreases.
\Image[/quantum-expectation-repeated-sampling.png]
{title=Simulated running <spin> means and exact <sample mean> <probabilities>}
{description=Original independent-trial model with $p_+=3/4$. Left: four seeded simulated sequences of fresh preparations, with one-theoretical-<standard error> curves about $\hbar/4$. These curves are not simultaneous bounds on trajectories. Right: exact <binomial distribution> masses for the <sample mean> at $N=16$ and $N=128$; each set sums to one. The possible sample means become more closely spaced as $N$ increases. All points are simulations or calculations, not measured observations.}
{source=quantum-expectation-repeated-sampling.py}
Repeatedly measuring one surviving system is a different protocol. Under the ideal <Lüders rule>, outcome $a_r$ leaves $P_r|\psi\rangle/\sqrt{p_r}$ when $p_r>0$. With no intervening change of <observable> or dynamics moving the state out of that <eigenspace>, the next <projective measurement> returns $a_r$ with certainty. Such a record is correlated and does not sample the original <Born rule> distribution independently. The <quantum nondemolition measurement> discussion describes this repeatability; fresh <quantum state preparations> supply the ensemble needed for the estimator above.
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