---
title: "Weak Measurement & Conditional Pointer Interactive Lab | Sandbox Physics"
description: "To make the pointer shift bigger in a weak measurement, how many events must you give up? Most of them."
canonical_url: "https://sandboxphysics.com/labs/weak-measurement-pointer/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/weak-measurement-pointer/"
last_modified: "2026-10-27"
---

Q007 · A bigger shift from fewer kept events

# Weak Measurement & Conditional Pointer

**To make the pointer shift bigger in a weak measurement, how many events must you give up?** Most of them. Turning the final analyzer nearly orthogonal to the input makes the pointer shift of the selected events large, but only a small fraction of events get through. A fair cost count includes every trial: those that left by the other port and those that never clicked, not only the shifted ones.

Prepare a Gaussian beam, pull its two polarization components apart in a birefringent crystal, and choose an analyzer axis. Record both position sensors, compare the center of the selected spot with that of the whole beam, and look at the events that were left out.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The model is one-dimensional paraxial optics with exactly shifted Gaussian amplitudes. The crystal's separation is defined in the imaged sensor plane. Both analyzer ports and the trials with no click count toward the total. The finite detector window, position noise and pixels act during acquisition. Classical coherent light shows the same effect, and a large shift alone does not mean better precision.

What the same clicks show

Simulated detector events

## How many clicks the shift costs

Acquire events to begin.

Top: each observed group normalized to compare shape; gray is the ideal selected shape. Bottom: counts per emitted trial per millimeter to keep the sample cost visible; orange plus blue equals the white total, gray is the ideal total. Both gray references precede loss, window, noise and pixels.

Selected per emitted trial, 95% Wilson range

$\text{—}$

Selected centroid and empirical standard error

$\text{—}$

All-click centroid and empirical standard error

$\text{—}$

Nothing to estimate until events are acquired.

Centroids use pixel records only. Standard errors use the sample variance and require at least 20 clicks in the respective group to be shown; they are not exact confidence intervals or a precision comparison. Finite-window selection can bias a centroid.

Clicks, centroids and standard errors recomputed from pixel records

| Group | Clicks | Centroid (mm) | Standard error (mm) |
| --- | --- | --- | --- |

Latest 12 raw trials; CSV contains every trial

| Trial | Analyzer port | Position (mm) |
| --- | --- | --- |

Physics tutorial

## A large centroid is not a free measurement

**Background** A birefringent crystal shifts the two polarization components sideways in opposite directions. The analyzer then combines their amplitudes into two output ports. When the analyzer is nearly crossed, destructive interference can leave a faint selected spot that has moved.

**Why it matters** The selected position distribution, its spread and the fraction of photons that survive all matter. An exactly crossed analyzer also shows where the usual weak-value formula stops being defined.

Start with the essentials

- **Focus question**: What happens when the selection is exactly orthogonal to the preparation?
- **One-sentence intuition**: At every coupling strength, use the full shifted waves. Count the other port and the trials with no click, and compute the selected centroid only from positions that were actually observed.

### Core mathematical model

#### Normalized input pointer

$$
\psi(x)=(2\pi\sigma^2)^{-1/4}e^{-x^2/(4\sigma^2)}
$$

The width is the standard deviation of the input intensity. All transverse positions, the separation and the detector widths are in millimeters, in a sensor plane imaged at unit magnification.

#### Exact selected amplitude

$$
f_+(x)=a\psi(x-d/2)+b e^{i\varphi}\psi(x+d/2),\quad a=\cos\alpha\cos\beta,\quad b=\sin\alpha\sin\beta
$$

The preparation sets the polarization and relative phase. The crystal shifts the two components in opposite directions, and the orthogonal analyzer row gives the other port. The sampled distribution uses no small-coupling expansion.

#### Probability and exact ideal centroid

$$
P_+=a^2+b^2+2ab\cos\varphi\,e^{-d^2/(8\sigma^2)},\quad \langle x\rangle_+=\frac{d(a^2-b^2)}{2P_+}
$$

These values assume an ideal, infinitely large sensor and serve only as references. If the selected probability is zero, the centroid is undefined. A finite sensor or losses change the observed fraction, and clipping at the sensor edge can shift the observed centroid.

#### The two ports add up to the same total

$$
|f_+(x)|^2+|f_-(x)|^2=\cos^2\alpha\,|\psi(x-d/2)|^2+\sin^2\alpha\,|\psi(x+d/2)|^2
$$

The total profile does not depend on the analyzer basis. Its two Gaussian components are used only to draw samples; no hidden polarization path is recorded.

#### Where the weak-value formula applies

$$
A_w=\frac{a-b e^{i\varphi}}{a+b e^{i\varphi}},\quad \langle x\rangle_+\approx\frac d2\operatorname{Re}A_w
$$

The formula needs a nonzero denominator and a distortion that is small compared with the beam width. The Lab always uses the exact model. When preparation and selection are exactly orthogonal and the coupling is not zero, the selected profile can be symmetric with two lobes, a finite chance of selection and a centroid of zero.

#### Measured centroid and its cost in events

$$
\widehat x_+=\frac1{N_+}\sum_{i\in +}x_i,\quad \mathrm{SE}(\widehat x_+)=\frac{s_+}{\sqrt{N_+}},\quad \widehat p_+=\frac{N_+}{N_{\mathrm{emit}}}
$$

Positions are detector coordinates rounded to pixels. With fewer than 20 selected detections, no standard error is shown. The standard errors are not exact confidence intervals, and the Lab establishes no advantage in fixed-resource estimation or in Fisher information.

### Common difficulties

#### The spot never moves infinitely far

**Typical misconception** An undefined weak value means the detector sees an infinite displacement.

**Better mental model** The actual profile comes from the wave at finite coupling. With orthogonal preselection and postselection, the selected distribution can be symmetric instead.

#### A bigger shift does not give free precision

**Typical misconception** A larger centroid shift alone proves better sensitivity per emitted photon.

**Better mental model** To claim better precision you need a specific estimator, the same resources on both sides and a competing measurement. This Lab shows the shift and its cost but does not make that comparison.

#### Classical optics shows the same effect

**Typical misconception** This centroid amplification alone proves a nonclassical light source.

**Better mental model** Classical coherent polarized light produces the same interference. Reading the data as photon counts is the sampling interpretation this Lab states.

### Run the experiment

1. 01

   #### Compare against the full beam

   Acquire with Nearly crossed, then compare with Aligned analyzer at the same number of emitted trials.

   **What to observe:** The selected centroid can move farther than either component was shifted, but far fewer events are selected.

2. 02

   #### Keep track of what the shift costs

   Look at the counts in both ports, the number of no-click trials and the standard errors. Export all trials.

   **What to observe:** A large shift does not by itself compare measurement precision. The surviving fraction, the width, nuisance parameters and the information thrown away all matter.

3. 03

   #### Test exact orthogonality

   Choose Exactly crossed and acquire several batches. Then choose No coupling, crossed.

   **What to observe:** With finite coupling, a faint, symmetric profile with two lobes remains. With zero coupling nothing is selected, so there is no centroid to estimate.

4. 04

   #### Go beyond the weak approximation

   Try Strong separation, then imperfect sensors or a narrow window.

   **What to observe:** The exact distribution is still well defined. The ideal centroid on an infinite sensor need not match the centroid of clipped, noisy pixel data.

Physical conclusion

Look at the exact wave, the selected sample and the total number of trials together; a large shift in the selected events is only one part of the measurement.

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)