Preparing a partner photon by selecting one wave packet

Bring two idler alternatives into one outcome

Choose the normalized idler mode $|q\rangle=\cos a|0\rangle+e^{i\phi}\sin a|1\rangle$ . This measurement retains relative phase between frequency alternatives. Both contribute to the same selected-mode outcome. Taking the idler inner product conjugates the mode coefficients, so the signal amplitude is $Fq^*$ .

First compute the unnormalized signal vector. Its squared length is the success probability. Normalize only when success is nonzero. A zero-probability outcome has no conditional output to normalize.

v=Fq^*,\qquad p_q=v^\dagger v,\qquad \rho_{s|q}=\frac{vv^\dagger}{p_q}\quad(p_q>0)

The outer product of a single nonzero vector has rank one, hence unit purity. This requires a pure pair input and an ideal rank-one effect.

Reading: A. Eckstein · A. Bhattacharjee et al.

Transmit frequencies and ignore their labels

An intensity filter transmits frequencies with their own probabilities. A broadband detector then combines the transmitted frequency outcomes. The signal state is a weighted sum of column outer products. Coherence within each column remains; cross terms between distinct idler outcomes are absent.

Write intensity transmissions as $t_0,t_1$ , each between zero and one. A linear amplitude filter can itself carry phase. In this unresolved herald, its measurement effect retains only intensity transmission. The mixture arises when the idler output frequency is ignored.

\widetilde\rho_s=F\begin{pmatrix}t_0&0\\0&t_1\end{pmatrix}F^\dagger,\qquad p_t=\operatorname{Tr}\widetilde\rho_s,\qquad \rho_{s|t}=\frac{\widetilde\rho_s}{p_t}

This is diagonal unresolved idler heralding. Selecting a single nonzero column can also produce a pure signal.

Reading: A. Bhattacharjee et al.

Calculate the whole trade-off

Take the half-turn matrix $F_B$ . Its columns are orthogonal and equally long. Every normalized coherent idler projection succeeds with probability one half and prepares a pure signal. Transmitting both frequencies without resolving them leaves the equal mixture.

Let the first frequency transmit fully and the second with intensity transmission $t$ . Normalized column weights become $1/(1+t)$ and $t/(1+t)$ . Blocking the second column raises purity to one and reduces herald probability from one to one half. Coherent selection additionally permits programmable superpositions across the full two-frequency space.

p_t=\frac{1+t}{2},\qquad P_t=\frac{1+t^2}{(1+t)^2}

For

t=1/2

, success is three quarters and purity is five ninths. For

t=0

, success is one half and purity is one.

Remote preparation uses conditional selection

Changing the selected idler mode changes the conditional signal packet. Identifying that subset requires communicating the herald outcome. Averaging all idler outcomes leaves the unconditional signal state unchanged, so remote preparation does not transmit a message faster than light.

Ideal projection is a device-design limit. A real pulse gate may convert several input modes into the accepted output window, and detection can miss photons. Chapter seven includes these processes in one conversion-kernel measurement model.

Exercises and explanations

For the half-turn source, find success and purity for transmissions one and one quarter.

Read the explanation

Success is $5/8$ . Normalized weights are four fifths and one fifth, giving purity $17/25$ . Squaring unnormalized weights would omit conditioning on successful detection.

Does a narrowband detector make coherent pulse gating unnecessary?

Read the explanation

Narrowband selection can raise purity in a particular model but restricts the selected spectral packet. A coherent selector can distinguish broadband modes with overlapping intensities through their relative phases. This programmability has a different purpose from a universal claim of better purity and rate.

Sources and calculation scope

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.

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.