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Sandbox Physics
Time-frequency quantum optics

L2 · Chapter 3

How one intensity map can hide two different purities

Sandbox Physics ·

Four unchanged joint intensities can accompany a heralded purity that falls from one to one half. Phase controls whether the contributions to signal coherence add or cancel. The idler can retain information that an unresolved herald never reads.

A click without a mode label

A broadband herald reports an idler photon without identifying its mode. For each possible idler mode, one column of the pair matrix gives the signal amplitudes. Amplitudes within that column remain coherent. Ignoring distinct idler outcomes means mixing their probabilities. Matrix multiplication performs exactly this sum.

Complex conjugation reverses the imaginary part and is written with a star. Transposition followed by conjugation gives the adjoint †\dagger. The signal matrix entry j,ℓj,\ell multiplies signal amplitudes from the same idler column, then sums over columns. Diagonal entries are probabilities; off-diagonal entries describe coherence.

(ρs)jℓ=∑kFjkFℓk∗,ρs=FF†,Tr⁡ρs=1(\rho_s)_{j\ell}=\sum_kF_{jk}F_{\ell k}^{*},\qquad \rho_s=FF^\dagger,\qquad \operatorname{Tr}\rho_s=1
This herald has equal efficiency over the retained modes and does not resolve them. Mode-dependent detection needs its own measurement operator.

Reading: A. Bhattacharjee et al.

Rotate one contribution to coherence

For F(θ)F(\theta), the first column contributes one quarter to signal coherence. The second contributes e−iθ/4e^{-i\theta}/4. At zero phase they point together; after half a turn they cancel. The two diagonal probabilities remain one half each.

At zero phase the columns are identical, so learning the idler mode would not alter the signal wave packet. At half a turn the columns describe orthogonal signal superpositions. The idler can identify which one accompanied it. Ignoring that information leaves an equal mixture.

ρs(θ)=14(21+e−iθ1+eiθ2)\rho_s(\theta)=\frac14\begin{pmatrix}2&1+e^{-i\theta}\\1+e^{i\theta}&2\end{pmatrix}
The joint intensity map keeps only the squared magnitudes of the four pair coefficients. It has lost the relative phase needed for coherence.
P=Tr⁡(ρs2)=∑j,ℓ∣(ρs)jℓ∣2=3+cos⁡θ4P=\operatorname{Tr}(\rho_s^2)=\sum_{j,\ell}|(\rho_s)_{j\ell}|^2=\frac{3+\cos\theta}{4}
Diagonal terms contribute one half. The off-diagonal terms contribute (1+cos⁡θ)/4(1+\cos\theta)/4. Purity is one at zero phase and one half at half a turn.

All phases aligned

FA=12(1111)F_A=\frac12\begin{pmatrix}1&1\\1&1\end{pmatrix}
P=1,Pcoin(0)=0P=1,\quad P_{\mathrm{coin}}(0)=0

One arrow reversed

FB=12(111−1)F_B=\frac12\begin{pmatrix}1&1\\1&-1\end{pmatrix}
P=12,Pcoin(0)=14P=\frac12,\quad P_{\mathrm{coin}}(0)=\frac14
Equal cell brightness represents equal joint intensities. Arrow direction is complex phase. Coincidences assume independent identical heralded inputs and otherwise perfect matching. Original two-mode model figure.

Find the paired modes with Schmidt decomposition

A suitable choice of modes puts the pair state into a sum of matched signal-idler pairs. Each signal mode then accompanies just one idler mode. This is Schmidt decomposition. Its nonnegative weights are eigenvalues of the signal density matrix: the matrix scales each corresponding mode without changing its direction.

The singular-value decomposition writes F=UΣV†F=U\Sigma V^\dagger. Signal Schmidt modes are columns of UU; idler modes are columns of V∗V^*. Their weights are squared singular values. The conjugation matters for rebuilding a complex biphoton amplitude.

∣Ψ⟩=∑nλn∣un⟩s∣vn⟩i,ρs=∑nλn∣un⟩⟨un∣,P=∑nλn2|\Psi\rangle=\sum_n\sqrt{\lambda_n}|u_n\rangle_s|v_n\rangle_i,\qquad \rho_s=\sum_n\lambda_n|u_n\rangle\langle u_n|,\qquad P=\sum_n\lambda_n^2
Weights sum to one. A single nonzero weight gives a pure signal. Several weights leave a mixture when idler outcomes are ignored.
λ±=1±∣cos⁡(θ/2)∣2,K=1P\lambda_\pm=\frac{1\pm|\cos(\theta/2)|}{2},\qquad K=\frac{1}{P}
The effective Schmidt number runs from one to two. Its entanglement interpretation here assumes a pure biphoton input.

Reading: A. Eckstein

A different basis leaves purity unchanged

Changing mode basis resembles changing coordinates on a map. A unitary change preserves lengths and inner products. Matrix entries change, but eigenvalues and purity do not. A mixture cannot become pure through relabeling modes.

This gives useful independent checks. Direct coefficient multiplication, eigenvalues of the reduced matrix and singular values of the pair matrix should agree. Unit trace and nonnegative eigenvalues are required before the weights can represent probabilities.

Try the prediction

Predict purity before reading it

  1. Compare zero phase and half a turn, first predicting all four joint intensities.
  2. Stop at a quarter turn and predict purity three quarters.
  3. Follow the two Schmidt weights and explain why more equal weights mean a more mixed signal.

Exercises and explanations

  1. Find the purity and Schmidt weights at θ=π/2\theta=\pi/2.

    Read the explanation

    Purity is three quarters. The weights are (1±1/2)/2(1\pm1/\sqrt2)/2. Their squared sum is again three quarters, even though every joint intensity remains one quarter.

  2. Can a change of signal basis turn the mixed state into a pure state?

    Read the explanation

    No. A unitary basis change preserves eigenvalues. Conditional selection requires a physical measurement and discarding other outcomes. Chapter five calculates the associated success probability.

Sources and calculation scope

  1. A. Eckstein, B. Brecht and C. Silberhorn, A Quantum Pulse Gate based on Spectrally Engineered Sum Frequency Generation, arXiv:1007.3215v2; Optics Express 19, 13770–13778 (2011)

    Equations (1)–(4), Figure 2 and the heralding application define the pump-times-phase-matching kernel, its Schmidt channels and ideal mode selection. Our finite two-channel Schmidt kernel is a dimensionless teaching reduction, not the paper’s PPLN parameter calculation.

  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)

    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.

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