---
title: "Reconstruct coherence from clicks and predict a new measurement"
description: "Reconstruct a two-mode state from three complementary counting bases, then test an unused coherent projection with sampling uncertainty."
canonical_url: "https://sandboxphysics.com/stories/time-frequency-quantum-optics/tomography-and-holdouts/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/time-frequency-quantum-optics/tomography-and-holdouts/"
last_modified: "2026-10-28"
---

[Time-frequency quantum optics](https://sandboxphysics.com/stories/time-frequency-quantum-optics/)

L3 · Chapter 6

# Reconstruct coherence from clicks and predict a new measurement

Sandbox Physics · 2026-10-28

Frequency counts alone cannot recover relative phase. Coherent superposition measurements supply the missing information. Three directions determine a two-mode state; a fourth direction tests a prediction that reconstruction did not use.

## Before you begin

Density matrices from chapter three and mode projections from chapter five. The Bloch vector is introduced here.

[Review the preceding chapter](https://sandboxphysics.com/stories/time-frequency-quantum-optics/coherent-heralding/)

- Identify what frequency counts cannot determine.
- Derive a density matrix from saved counts.
- Distinguish holdout agreement, sampling error and calibration error.

## Ask in three different mode bases

The $Z$ basis asks whether the photon occupies mode zero or one. The $X$ basis uses their equal in-phase and opposite-phase sums. The $Y$ basis first gives the second mode a quarter-turn phase. The latter measurements convert off-diagonal coherence into countable probability differences.

Each basis has two exclusive outcomes. Subtract negative counts from positive counts and divide by their total. The resulting three averages form a Bloch vector, a set of coordinates for a two-mode density matrix.

$$
|\pm x\rangle=\frac{|0\rangle\pm|1\rangle}{\sqrt2},\qquad |\pm y\rangle=\frac{|0\rangle\pm i|1\rangle}{\sqrt2}
$$

Quarter-turn phase projections reveal imaginary coherence. Projections with only real coefficients miss it.

## Build the matrix from count ratios

Let $n_{a,+}$ and $n_{a,-}$ denote the saved counts along direction $a$. Reconstruction uses these integers and trial totals. Reading the simulator’s hidden state is not reconstruction.

Finite sampling can put the estimated vector outside the unit ball, producing a negative eigenvalue. The teaching experiment radially shortens such a vector to the boundary. This restores positivity but introduces boundary bias, especially near pure states. Experimental analysis often uses constrained likelihood with calibrated detection effects.

$$
\widehat r_a=\frac{n_{a,+}-n_{a,-}}{n_{a,+}+n_{a,-}},\qquad \widehat\rho=\frac12\begin{pmatrix}1+\widehat r_z&\widehat r_x-i\widehat r_y\\\widehat r_x+i\widehat r_y&1-\widehat r_z\end{pmatrix}
$$

A physical Bloch vector has length at most one. Retain the uncorrected estimate as well so that any positivity correction is traceable.

$$
\widehat P=\frac{1+|\widehat{\boldsymbol r}|^2}{2}
$$

This purity comes from the reconstructed matrix. Both sampling and the physicality correction affect its statistical behavior.

Reading: A. Bhattacharjee et al. · L. Serino et al.

## Work one example and reserve another question

Measure each basis a thousand times. Suppose positive counts in the three directions are eight hundred, five hundred and five hundred. The Bloch vector has first component 0.6 and the other two zero. The diagonal matrix entries are one half, the off-diagonal entries 0.3, and purity is 0.68.

Choose the unused direction $\boldsymbol n=(1,1,1)/\sqrt3$. The reconstructed state predicts positive-outcome probability about 0.673. Independently collect its outcomes. These holdout counts must stay outside the reconstruction; feeding them back would remove their role as a new prediction test.

$$
\widehat p_{\boldsymbol n,+}=\frac{1+\boldsymbol n\cdot\widehat{\boldsymbol r}}2,\qquad \operatorname{SE}(\widehat p)\approx\sqrt{\frac{\widehat p(1-\widehat p)}{N}}
$$

The binomial standard error describes fluctuations at fixed probability. A comparison also needs uncertainty in the reconstructed prediction.

## Let records leave the live controls

Save source settings, measurement bases, trial totals, seed, counts and model version with an acquisition. A seed makes synthetic sampling reproducible. Real experiments also need times, calibration and background. Changing controls must leave an old record tied to its original conditions.

A holdout checks compatibility within the chosen model and measurements. A common phase-calibration error can bias training and holdout together. Independent instrumentation, a wider scan or a renewed detector calibration can address that possibility.

## Try the prediction

### [Save a reconstruction before examining its holdout](https://sandboxphysics.com/labs/quantum-pulse-gate/)

1. Fix source and herald settings, then acquire synthetic three-basis and independent holdout counts.
2. Export the record and verify that reconstruction uses only the three basis measurements.
3. Repeat with another seed, compare purity and holdout variation, then move controls and confirm the saved record stays fixed.

## Exercises and explanations

1. Why do mode-zero and mode-one counts alone fail to determine purity?

   Read the explanation

   They determine only diagonal probabilities. Off-diagonal coherence can vanish or take nonzero values consistent with positivity. Those states have different purities. Superposition measurements reveal the missing coherence.

2. Does a holdout discrepancy of two standard errors prove the model wrong?

   Read the explanation

   An individual discrepancy can arise from sampling. Prediction uncertainty also matters. A predefined comparison, repeated independent records and calibration checks support a stronger conclusion.

[Previous: Preparing a partner photon by selecting one wave packet](https://sandboxphysics.com/stories/time-frequency-quantum-optics/coherent-heralding/) [Next: How a pulse gate approaches a single-mode measurement](https://sandboxphysics.com/stories/time-frequency-quantum-optics/conversion-kernels/)

## Sources and calculation scope

1. [L. Serino et al., Programmable time-frequency mode-sorting of single photons with a multi-output quantum pulse gate, arXiv:2410.03606v1; DOI 10.1364/OE.544206](https://arxiv.org/html/2410.03606v1)

   Section 2 and Figure 2 specify the transfer kernel and the ideal/projective versus nonideal POVM description. Sections 3–4 explain detector tomography and output resolution. This work has an Optics Express journal DOI; the version read here is the 2024 preprint.

2. [A. Bhattacharjee et al., Coherent temporal filtering of multimode parametric down-conversion using a quantum pulse gate, arXiv:2608.12544v1 (12 August 2026 preprint)](https://arxiv.org/html/2608.12544v1)

   Main equations (1)–(5), Figures 1–4 and supplementary sections 1–7 were read for conditional filtering and reconstruction. The bounded reproduction is supplementary equations (12)–(13); (15)–(16) are evaluated analytically. We do not reproduce the measured Figure 3 points, tomography algorithm or full device.