Quantum Experiments · I · Observations and models · Chapter 01
How does a detector click become evidence for a physical claim?
Suppose a collimated source sends individual quantum systems toward a two-channel analyzer whose detectors register outcome \(A\) or outcome \(B\). After \(N=100\) trigger windows, the digital counter displays \(k_A=53\) clicks in channel \(A\) and \(k_B=47\) clicks in channel \(B\). Does that six-count excess establish a genuine physical asymmetry favoring \(A\)? Before computing any probability, an experimenter must specify what constitutes a valid trial: whether empty trigger windows were discarded, whether the emission rate drifted over time, and whether the two detectors had matched quantum efficiencies \(\eta_A\) and \(\eta_B\). Raw integers become physical evidence only when paired with a complete operational protocol.
Every quantum experiment divides into three distinct stages: preparation, transformation, and measurement. Preparation defines the state initialization procedure and heralding criteria; transformation specifies the controlled physical interaction or analyzer setting applied along the flight path; and measurement maps each trial to a classical record in a mutually exclusive outcome set. Repeating the identical three-stage cycle over \(N\) independent runs estimates the underlying outcome probabilities. A quantum state vector or density matrix is an operational rule for predicting those outcome frequencies across all available measurement settings, not a miniature photograph of an isolated trajectory between clicks.
The interactive model in this chapter pairs a configurable two-outcome probability parameter \(p\) with two digital event counters. In the browser, a deterministic pseudo-random generator seeded at a fixed integer produces the sequence of \(A\) and \(B\) arrivals; no physical quantum hardware is queried. Resetting the simulation replays the exact same sequence of outcomes trial by trial, making the distinction between a mathematical probability model and genuine laboratory detector data immediately visible.
Visual experiment · Section 1.1
Two-Outcome Trial Cycle: Preparation, Transformation, and Measurement
When \(N\) trials are statistically independent and each trial yields outcome \(A\) with constant probability \(p\), the total count \(k\) obeys the binomial distribution
\[P(k\mid N,p)=\binom{N}{k}p^k(1-p)^{N-k}, \qquad \langle k\rangle = Np, \qquad \sigma_k = \sqrt{Np(1-p)}.\]
Dividing the count \(k\) by the trial number \(N\) gives the sample frequency estimator \(\hat{p}=k/N\) together with its standard error \(\mathrm{SE}(\hat{p})=\sqrt{p(1-p)/N}\approx\sqrt{\hat{p}(1-\hat{p})/N}\). Because \(\mathrm{SE}(\hat{p})\) scales as \(1/\sqrt{N}\), reducing the statistical uncertainty by a factor of two requires four times as many independent trials.
For \(N=100\) balanced trials with \(p=1/2\), the expected count is \(\langle k\rangle=50\) and the binomial standard deviation is \(\sigma_k=\sqrt{100(1/2)(1/2)}=5\). Observing \(k_A=53\) and \(k_B=47\) represents a deviation of only \(0.6\sigma_k\), well within ordinary shot-to-shot statistical scatter. Even a one-standard-error interval \(\hat{p}\pm\mathrm{SE}\) is not a logical guarantee that the true parameter \(p\) lies inside that bracket; when \(N\) is small or \(p\) lies close to \(0\) or \(1\), exact binomial confidence intervals or likelihood ratios should replace symmetric Gaussian approximations.
Statistical uncertainty and systematic instrumental bias behave completely differently as \(N\) grows. Collecting millions of trials shrinks \(\mathrm{SE}(\hat{p})\) toward zero, but an uncalibrated mismatch between detector efficiencies \(\eta_A\) and \(\eta_B\) persists unchanged at any sample size. If outcome \(A\) is registered with efficiency \(\eta_A\) and outcome \(B\) with efficiency \(\eta_B\), then conditioning only on trials that produce a click shifts the observed \(A\) fraction to
\[f_A = \frac{p\,\eta_A}{p\,\eta_A + (1-p)\,\eta_B}.\]
Postselecting on detected events without correcting for \(\eta_A/\eta_B\) alters the statistical population under study and can mimic a biased source even when \(p=1/2\).
Visual experiment · Section 1.2
Binomial Fluctuations, \(1/\sqrt{N}\) Convergence, and Efficiency Bias
A persuasive experimental claim combines a documented protocol, propagated statistical and systematic uncertainties, independent calibration of detector efficiencies and dark counts, and a quantitative comparison showing why a competing model cannot account for the observations. Swapping detectors between output ports or calibrating \(\eta_A\) and \(\eta_B\) with a known reference beam separates source properties from readout asymmetries before any physical conclusion is drawn.
Throughout this course we maintain an explicit assumption ledger for every experiment: trial independence, stationary preparation, fair sampling of undetected events, and the mathematical model of the analyzer. In later chapters on Bell inequalities, contextuality, and Wigner’s friend, keeping each assumption on the ledger prevents a violation of a specific composite hypothesis from being misstated as a sweeping claim about all possible physical theories.
The interactive display below illustrates the law of large numbers for binomial sampling, not the physical origin of quantum indeterminism. Because the browser generates events directly from the user-selected value of \(p\), its histograms cannot test the Born rule, certify physical randomness, or rule out local hidden variables.
Visual experiment · Section 1.3
Port-Swap Calibration Protocol and Explicit Assumption Ledger
Set \(p=0.5\), step through individual trials to see large early fluctuations in \(\hat{p}=k_A/N\), and then run a longer sequence. Predict the expected standard error \(\mathrm{SE}=\sqrt{p(1-p)/N}\) before increasing \(N\). Pressing Reset replays the identical seeded sequence, whereas changing \(p\) initializes a fresh run.
Interactive model · Synthetic observations
On a small screen, swipe the diagram to inspect its components. Moving marks illustrate the model; they are not measured trajectories.
A nominally balanced source (\(p=1/2\)) produces \(k_A=530\) clicks in channel \(A\) out of \(N=1{,}000\) complete trials. How many standard deviations above the mean is this result, and does it establish source bias?
Under the fair binomial model \(p=1/2\), the expected count is \(\langle k_A\rangle=500\) and the standard deviation is \(\sigma_k=\sqrt{1{,}000\times(1/2)\times(1/2)}=\sqrt{250}\approx 15.81\). The observed excess of \(530-500=30\) counts corresponds to \(30/15.81\approx 1.90\) standard deviations (a two-sided \(p\)-value near \(0.058\)). By itself, a \(1.9\sigma\) fluctuation is not decisive evidence of physical bias, particularly if multiple runs were inspected; an experimenter should verify detector efficiency balance \(\eta_A/\eta_B\) and repeat with a prespecified sample size.
How many independent trials \(N\) are required to reduce the binomial standard error \(\mathrm{SE}(\hat{p})\) from \(0.05\) to \(0.01\) when \(p=1/2\)?
Use \(\mathrm{SE}(\hat{p})=1/(2\sqrt{N})\) at \(p=1/2\).
Solving \(1/(2\sqrt{N})=0.01\) gives \(\sqrt{N}=50\), so \(N=2{,}500\) trials. Because \(\mathrm{SE}=0.05\) corresponds to \(N=100\), improving the precision by a factor of \(5\) requires \(5^2=25\) times as many trials.
An unbiased source (\(p=1/2\)) is monitored by two detectors with efficiencies \(\eta_A=1\) and \(\eta_B=1/2\). What fraction \(f_A\) of the recorded clicks appear in channel \(A\)?
Substitute \(p=1/2\), \(\eta_A=1\), and \(\eta_B=1/2\) into \(f_A=p\eta_A/[p\eta_A+(1-p)\eta_B]\).
The conditioned \(A\) fraction is \(f_A=(0.5\times 1)/(0.5\times 1 + 0.5\times 0.5)=0.5/0.75=2/3\). Even though the source emits \(A\) and \(B\) with equal probability, discarding the unrecorded \(B\) trials skews the postselected dataset to \(2/3\).
Can generating \(N=10^6\) synthetic clicks in a browser simulation validate the efficiency or dark-count rate of a laboratory single-photon detector?
Ask whether the simulation couples to the physical photodiode.
No. Simulated trials only verify the software estimator and plotting pipeline under the assumed mathematical model. Validating a physical detector requires illuminating the hardware with independently calibrated reference fluxes and measuring its actual dark counts, dead time, and efficiency.
An experimental inference links a preparation protocol, a calibrated measurement model, raw counts, and an explicit assumption ledger. Increasing the sample size \(N\) shrinks random binomial scatter as \(1/\sqrt{N}\) but never corrects an uncalibrated detector asymmetry.
These readings provide historical context, experimental implementations, or deeper derivations. The interactive display above is an educational model, not a reproduction of their measured data.