# Probing the non-Abelian fusion of a pair of Majorana zero modes

Jing Bai  Center for Joint Quantum Studies and Department of Physics, School
of Science,
Tianjin University, Tianjin 300072, China  Qiongyao Wang  Center for Joint
Quantum Studies and Department of Physics, School of Science,
Tianjin University, Tianjin 300072, China  Luting Xu  Center for Joint Quantum
Studies and Department of Physics, School of Science,
Tianjin University, Tianjin 300072, China  Wei Feng<EMAIL_ADDRESS>Center
for Joint Quantum Studies and Department of Physics, School of Science,
Tianjin University, Tianjin 300072, China  Xin-Qi Li<EMAIL_ADDRESS>Center for Joint Quantum Studies and Department of Physics, School of Science,
Tianjin University, Tianjin 300072, China

###### Abstract

In this work

we perform real time simulations for probing the non-Abelian fusion of a pair
of Majorana zero modes (MZMs). The nontrivial fusion outcomes can be either a
vacuum, or an unpaired fermion, which reflect the underlying non-Abelian
statistics. The two possible outcomes can cause different charge variations in
the nearby probing quantum dot (QD), while the charge occupation in the dot is
detected by a quantum point contact. In particular, we find that gradual
fusion and gradual coupling of the MZMs to the QD (in nearly adiabatic
switching-on limit) provide a simpler detection scheme than sudden coupling
after fusion to infer the coexistence of two fusion outcomes, by measuring the
occupation probability of the QD. For the scheme of sudden coupling (after
fusion), we propose and analyze continuous weak measurement for the quantum
oscillations of the QD occupancy. From the power spectrum of the measurement
currents, one can identify the characteristic frequencies and infer thus the
coexistence of the fusion outcomes.

Introduction

. — The nonlocal nature of the Majorana zero modes (MZMs) and non-Abelian
statistics obeyed provide an elegant paradigm of topological quantum
computation Kit01 ; Kit03 ; Sar08 ; Ter15 ; Sar15 ; Opp20a . In the past
decade, after great efforts, considerable progress has been achieved for
realizing the MZMs in various experimental platforms. Yet, the main
experimental evidences are largely associated with the zero-bias conductance
peaks (see the recent review GHJ22  and references therein), which cannot
ultimately confirm the realization of MZMs (even a stable quantized
conductance cannot also). Therefore, an essential milestone step is to
identify the MZMs by probing the underlying non-Abelian statistics, via either
braiding or fusion experiments.

Braiding MZMs in real space can result in quantum state evolution in the
manifold of highly degenerate ground states Fish11 ; Opp12 ; Roy19 ; Han20 ,
while fusing the MZMs can yield outcomes of either a vacuum, or an unpaired
fermion (resulting in an extra charge) Ali16 ; BNK20 ; NC22 ; Leij22 ; Zut20 ;
Sau23 . The latter is owing to the fact that the MZMs essentially realize
“Ising” non-Abelian anyons, which obey a particularly simple fusion rule as
Ali16 ; BNK20

$\displaystyle\gamma\times\gamma=I+\psi\,.$ (1)

This means that a pair of MZMs can either annihilate or combine into a fermion
$\psi$. These two “fusion channels” correspond to the regular fermion being
empty or filled. The presence of multiple fusion channels is essentially
related to non-Abelian statistics (actually is commonly used to define non-
Abelian anyons). More specifically, there exist two types of fusion design
Ali16 ; BNK20 . The “trivial” fusion corresponds to the fused pair of MZMs
with a defined parity within the same pair. In this case, the outcome is
deterministic; it leads to unchanged parity with no extra charge. Of more
interest is the case of “nontrivial” fusion, where the fused pair of MZMs are
from different pairs with parities (e.g. even parity) being defined in
advance. In this case, the fusion yields probabilistic outcomes as shown
above.

While directly probing non-Abelian statistics of MZMs is a milestone towards
topological quantum computation, probing fusion should be simpler than
demonstrating braiding Ali16 ; BNK20 . However, so far there is not yet report
of nontrivial fusion experiment NC22 . The basic idea of probing fusion is
bringing a pair of MZMs together to remove energy degeneracy between the two
possible outcomes of fusion (i.e., $I$ and $\psi$), owing to overlap of the
two MZMs. Then, a measurement to distinguish the fermion parity can reveal the
stochastic result being $I$ or $\psi$, with equal probability. This type of
nontrivial fusion demonstration is actually equivalent to demonstrating the
underlying non-Abelian statistics Ali16 ; BNK20 .

Figure 1:  Schematic setup of probing the non-Abelian fusion of a pair of
MZMs, say, $\gamma_{2}$ and $\gamma_{3}$, from different Majorana pairs with
parities (e.g. even parity) being defined in advance. Based on the fusion rule
$\gamma\times\gamma=I+\psi$, the fused MZMs would yield probabilistic outcomes
of vacuum $I$ and a regular fermion $\psi$. A quantum dot is introduced to
couple to the fusing MZMs for probing the fusion outcomes, and a nearby point-
contact detector is introduced to detect the charge occupation of the quantum
dot.

In practice, demonstrating nontrivial fusion of MZMs should require
preparation of initial pair states of MZMs with definite fermion parities and
nonadiabatic moving when bringing the MZMs together to fuse. In this work,
along the line proposed in Ref. NC22  (as schematically shown here in Fig. 1),
we consider fusing a pair of MZMs from two topological superconducting (TSC)
wires (each wire accommodating two MZMs at the ends). This model setup can
correspond to the platform of mini-gate controlled planar Josephson junctions
RenN19 ; RenP19 . The two TSC segments can be formed from a single junction
wire, by making them separated by a topologically trivial segment, via gating
technology. A quantum dot is introduced to couple to the central part of the
coupled wires, for use of probing the fusion outcomes when bringing the MZMs
together to fuse at the central part. Moreover, a nearby point-contact (PC)
detector is introduced to detect the charge occupation of the quantum dot. All
the ingredients in this proposal are within the reach of nowadays state-of-
the-art experiments NC22 . For fusion experiments based on this proposal, one
may encounter some practical complexities, such as the interplay of charge
fluctuations in the quantum dot caused by the two fusion outcomes, which is
relevant to the control of the dot energy level and its coupling strengths to
the MZMs, and the effect on the dot occupation pattern caused by the speed of
fusion and coupling of the MZMs to the quantum dot. Detection schemes
accounting for these issues will be analyzed in this work, and are expected to
be useful for future experiments.

Setup and Basic Consideration

.— The setup proposed in Ref. NC22  can be modeled as Fig. 1, where the two
TSC quantum wires are formed by interrupting a single TSC wire at the center
via mini-gate-voltage control. For each TSC wire, a pair of MZMs are emergent
at the ends, i.e., ($\gamma_{1},\gamma_{2}$) in the left wire and
($\gamma_{3},\gamma_{4}$) in the right wire. The coupling between the central
modes $\gamma_{2}$ and $\gamma_{3}$ is described by
$H^{\prime}_{M}=i\epsilon_{M}\gamma_{2}\gamma_{3}$, with the coupling energy
$\epsilon_{M}$ changeable when $\gamma_{2}$ and $\gamma_{3}$ are separated
away in space by mini-gate-voltage control.

Most naturally, one can combine ($\gamma_{1},\gamma_{2}$) as a regular fermion
$f_{12}$ with occupation $n_{12}=0$ or 1, and ($\gamma_{3},\gamma_{4}$) as
another regular fermion $f_{34}$ with occupation $n_{34}=0$ or 1. For fusion
experiment, one can prepare the specific initial state $|0_{12}0_{34}\rangle$
as proposed in Ref. NC22 . That is, by means of mini-gate-voltage control,
move $\gamma_{2}$ and $\gamma_{3}$ to the ends of the two wires, close to
$\gamma_{1}$ and $\gamma_{4}$, respectively; then, empty the possible
occupations of the regular fermions $f_{12}$ and $f_{34}$ by introducing
tunnel-coupled side quantum dots and modulating the dot energies (while the
quantum dots are also tunnel-coupled to outside reservoirs). Starting with
$|0_{12}0_{34}\rangle$, consider simultaneously moving $\gamma_{2}$ and
$\gamma_{3}$ from the two terminal sides back to the central part to fuse (to
couple each other such that $\epsilon_{M}\neq 0$), as shown in Fig. 1. For the
final state, in the representation of $n_{12}$ and $n_{34}$ occupations, it is
still $|0_{12}0_{34}\rangle$. However, in the representation of $n_{23}$ and
$n_{14}$, i.e., the occupations of the regular fermions $f_{23}$ and $f_{14}$
associated with the Majorana pairs ($\gamma_{2},\gamma_{3}$) and
($\gamma_{1},\gamma_{4}$), we can reexpress this state as (for derivation of
this transformation rule, or the so-called fusion rule, see Appendix A)

$\displaystyle|0_{12}0_{34}\rangle=\frac{1}{\sqrt{2}}(|0_{23}0_{14}\rangle+i|1_{23}1_{14}\rangle)\,.$
(2)

We find that, in the new representation, the occupation of the $f_{23}$
fermion can be empty or occupied, i.e., $|0_{23}\rangle$ or $|1_{23}\rangle$.
This is nothing but the two possible outcomes $I$ and $\psi$ of the nontrivial
fusion of Ising anyons, as shown by Eq. (1).

The fusion coupling between the Majorana modes $\gamma_{2}$ and $\gamma_{3}$
would lift the energy degeneracy of the states $|0_{23}\rangle$ and
$|1_{23}\rangle$, thus allowing to identify the fusion outcomes $I$ and
$\psi$. Following Ref. NC22 , we consider to introduce a nearby quantum dot
(QD) to couple to the central segment of the quantum wire, as shown in Fig. 1,
where the Majorana modes $\gamma_{2}$ and $\gamma_{3}$ are located. The QD is
assumed to have a single relevant energy level, described by
$H_{D}=\epsilon_{D}d^{\dagger}d$. We thus expect different charge occupation
patterns of the QD, for the different fusion outcomes $I$ and $\psi$. In this
context, it would be more convenient to describe the coupling between
$(\gamma_{2},\gamma_{3})$ and the QD using the regular fermion $f_{23}$
picture, as follows

$\displaystyle
H^{\prime}_{DF}=(\lambda_{N}d^{\dagger}f_{23}+\lambda_{A}d^{\dagger}f_{23}^{\dagger})+{\rm
h.c.}\,.$ (3)

Here we have used the definition $f_{23}=(\gamma_{2}+i\gamma_{3})/2$.
Physically, the first term describes the usual normal tunneling process and
the second term describes the Andreev process owing to Cooper pair splitting
and recombination. The coupling amplitudes are associated with the couplings
of $\gamma_{2}$ and $\gamma_{3}$ to the QD, say, $\lambda_{2}$ and
$\lambda_{3}$ as shown in Fig. 1, as $\lambda_{N,A}=\lambda_{2}\pm
i\lambda_{3}$.

Also following the proposal of Ref. NC22 , the charge fluctuation in the
quantum dot (occupied or unoccupied) is measured by a point-contact (PC)
detector, as schematically shown in Fig. 1. PC detectors with sensitivity at
single electron level have been experimentally demonstrated and broadly
applied in practice Ens06 ; Fuj06 ; Guo13 ; Guo15 . Actually, the measurement
dynamics of a charge qubit by a PC detector has been a long standing
theoretical problem and has attracted intensive interest in the community of
quantum and mesoscopic physics Kor01 ; Mil01 ; Li05 ; Opp20 . In this work, we
will perform real-time simulations for probing the non-Abelian fusion of a
pair of MZMs. In particular, within the scheme of continuous quantum weak
measurement, we will carry out results of individual quantum trajectories and
power spectrum of the measurement currents. The characteristic frequencies in
the power spectrum indicate the quantum oscillations of charge transfer
associated with the two outcomes of fusion. The coexistence of two
characteristic frequencies should be a promising evidence for the non-Abelian
fusion of MZMs.

QD occupations caused by the two fusion outcomes

.— Let us consider first the detection scheme of switching on the coupling of
the QD to the Majorana modes $\gamma_{2}$ and $\gamma_{3}$, after their fusion
from the deterministic initial state $|0_{12}0_{34}\rangle$. This can be
realized by initially setting the QD energy level much higher than the final
coupling energy $\epsilon_{M}$ between $\gamma_{2}$ and $\gamma_{3}$. Then,
during the moving and fusion process, the QD level is effectively decoupled
with $\gamma_{2}$ and $\gamma_{3}$, owing to the large mismatch of energies.
After fusion, switch on the coupling by lowering the QD energy level such that
$\epsilon_{D}=\epsilon_{M}$.

For this scheme, the time dependent charge occupation in the dot is shown in
Fig. 2 (the result of the green curve). To understand this result, we should
notice the coexistence of two channels of charge transfer oscillations between
the QD and the MZMs $\gamma_{2}$ and $\gamma_{3}$. One channel is governed by
normal tunneling between the states $|1_{23}0_{d}\rangle$ and
$|0_{23}1_{d}\rangle$, resulting in a quantum oscillation state as
$\alpha_{N}(t)|1_{23}0_{d}\rangle+\beta_{N}(t)|0_{23}1_{d}\rangle$, with the
dot occupation probability $p_{d}^{(N)}(t)=|\beta_{N}(t)|^{2}$ plotted in Fig.
2 by the full Rabi-type oscillating red curve. The other channel is governed
by the Andreev process between $|0_{23}0_{d}\rangle$ and
$|1_{23}1_{d}\rangle$, resulting in a quantum oscillation given by
$\alpha_{A}(t)|0_{23}0_{d}\rangle+\beta_{A}(t)|1_{23}1_{d}\rangle$, with the
dot occupation probability $p_{d}^{(A)}(t)=|\beta_{A}(t)|^{2}$ plotted in Fig.
2 by the smaller amplitude blue curve. These two channels are independent to
each other. Thus the electron occupation in the dot is simply an equal
probability weighted sum, based on Eq. (2), as

$\displaystyle
p_{d}(t)=\left[|\beta_{N}(t)|^{2}+|\beta_{A}(t)|^{2}\right]/2\,.$ (4)

Actually, the quantum oscillations in the two channels have simple analytic
solutions, with dot occupation probabilities given by

$\displaystyle|\beta_{N,A}(t)|^{2}=\left(|\lambda_{N,A}|^{2}/\Omega_{N,A}^{2}\right)\sin^{2}\left(\Omega_{N,A}t\right)\,,$
(5)

where $\Omega_{N,A}=\sqrt{\Delta_{N,A}^{2}+|\lambda_{N,A}|^{2}}$ and
$\Delta_{N,A}=|\epsilon_{D}\mp\epsilon_{M}|/2$. We then understand that, when
$\epsilon_{D}\simeq\epsilon_{M}>>|\lambda_{N,A}|$, the quantum oscillation
associated with the fusion outcome $\psi$ is dominant, while the Andreev
process following the $I$ outcome is largely suppressed. However, viewing that
the coupling energy $\epsilon_{M}$ between $\gamma_{2}$ and $\gamma_{3}$ is
small (might be comparable with $\lambda_{2}$ and $\lambda_{3}$ in practice,
see Fig. 1), one may encounter the complexity that both channels coexist
during detection of the charge variations in the quantum dot, having thus the
result as shown in Fig. 2 by the green curve.

Figure 2:  Dot occupation probability $p_{d}$ (green curve) associated with
the detection scheme of sudden coupling of the MZMs after fusion to the
probing QD. According to the fusion rule, Eq. (2), $p_{d}$ is half of the sum
of $p_{d}^{(N)}$ (red curve, labeled by $\psi$) and $p_{d}^{(A)}$ (blue curve,
labeled by $I$). The meaning of $p_{d}^{(N)}$ and $p_{d}^{(A)}$ is referred to
the main text. Parameters are assumed as $\lambda_{2}=\lambda_{3}=\lambda=1$,
and $\epsilon_{D}=\epsilon_{M}=1.5$. In this work, we use the arbitrary system
of units, taking $\lambda$ as the unit of energy and $\lambda^{-1}$ as the
unit of time.

Next, let us consider an alternative detection scheme of gradually coupling
the QD with the Majorana modes $\gamma_{2}$ and $\gamma_{3}$. The initial
state preparation and moving of the Majorana modes $\gamma_{2}$ and
$\gamma_{3}$ are the same as above (the first scheme). However, we consider
now initially setting the QD level in resonance with the final Majorana
coupling energy, i.e., $\epsilon_{D}=\epsilon_{M}$. Then, in this scheme,
modulation of the QD energy level after Majorana fusion is not needed. When
$\gamma_{2}$ and $\gamma_{3}$ are somehow slowly brought to close to each
other, they also couple to the QD gradually. We may model the gradual coupling
as follows

$\displaystyle\lambda_{2}=\lambda_{3}=\lambda\left[\frac{t}{\tau}\Theta(1-\frac{t}{\tau})+\Theta(\frac{t}{\tau}-1)\right]\,,$
$\displaystyle\epsilon_{23}=\epsilon_{M}\left[\frac{t}{\tau}\Theta(1-\frac{t}{\tau})+\Theta(\frac{t}{\tau}-1)\right]\,,$
(6)

where $\Theta(\cdots)$ is the step function. In this simple model, $\tau$ is
introduced to characterize the speed of moving $\gamma_{2}$ and $\gamma_{3}$.
Here we assume that the coupling of $\gamma_{2}$ and $\gamma_{3}$ to the QD
(nonzero $\lambda_{2}$ and $\lambda_{3}$) and the coupling between them
(nonzero $\epsilon_{23}$) are started at same time. We also use it as the
initial time for latter state evolution, which is associated with the fusion
and detection. Before $\gamma_{2}$, $\gamma_{3}$ and the QD start to couple to
each other, the moving of $\gamma_{2}$ and $\gamma_{3}$ can be performed at
different speed (faster speed). However, the moving should satisfy the
adiabatic condition determined by the energy gap of the TSC wire, which is
much larger than the coupling energies $\epsilon_{M}$, $\lambda_{2}$ and
$\lambda_{3}$, and the dot energy $\epsilon_{D}$. We may remark that the
modeling, in terms of Eq. (Probing the non-Abelian fusion of a pair of
Majorana zero modes), might not be very accurate, but it captures the
underlying physics and can predict valid behavior of electron occupation in
the quantum dot, in comparison with more accurate simulation based on the more
realistic lattice model.

Figure 3:  Dot occupation probability $p_{d}$ associated with the detection
scheme of gradual fusion and coupling to the probing QD, as modeled by Eq.
(Probing the non-Abelian fusion of a pair of Majorana zero modes). Results of
different coupling speeds are shown for $\tau=1$ (green curve), 10 (blue
curve), and 80 (red curve), respectively. Prominent feature is that the
quantum oscillations tend to disappear when decreasing the speed of coupling,
i.e., when approaching the adiabatic limit of switching on the coupling.
Parameters are the same as assumed in Fig. 2.

In Fig. 3 we show results of different coupling speeds, which are
characterized by the parameter $\tau$. For fast coupling, the result (green
curve) is similar to that shown in Fig. 2. However, if decreasing the speed of
coupling, the charge occupation pattern in the quantum dot becomes different.
The most prominent feature is that the quantum oscillations tend to disappear
(see the blue and red curves in Fig. 2). This can be understood as follows. In
this scheme of fusion and detection, there exist also two charge transfer
channels associated with, respectively, the fusion outcomes $I$ and $\psi$.
However, in the limit of adiabatically switching on the coupling, in each
channel the state will largely follow an instantaneous eigenstate. For
instance, for the $\psi$-related dominant channel (governed by the normal
tunneling between the states $|1_{23}0_{d}\rangle$ and $|0_{23}1_{d}\rangle$),
the state can be expressed also as
$\alpha_{N}(t)|1_{23}0_{d}\rangle+\beta_{N}(t)|0_{23}1_{d}\rangle$. Yet, in
the adiabatic limit, the superposition coefficients in the instantaneous
eigenstate do not reveal the feature of Rabi-type quantum oscillations.
Actually, we can easily obtain $|\beta_{N}(t\geq\tau)|^{2}=p^{(N)}_{d}=1/2$.
Similarly, for the outcome-$I$-related channel, the instantaneous eigenstate
can be expressed as
$\alpha_{A}(t)|0_{23}0_{d}\rangle+\beta_{A}(t)|1_{23}1_{d}\rangle$. In the
adiabatic limit, $|\beta_{A}(t)|^{2}$ does not oscillate with time and, when
$t\geq\tau$, we obtain

$\displaystyle
p^{(A)}_{d}=|\beta_{A}|^{2}=\frac{(\Omega_{A}-\epsilon_{M})^{2}}{(\Omega_{A}-\epsilon_{M})^{2}+2|\lambda|^{2}}\,.$
(7)

Here, $\Omega_{A}=\sqrt{\epsilon^{2}_{M}+2|\lambda|^{2}}$, under the
conditions $\epsilon_{D}=\epsilon_{M}$ and $\lambda_{2}=\lambda_{3}=\lambda$.
Based on the fusion rule Eq. (2), we expect the overall occupation probability
of an electron in the QD to be $p_{d}=(p^{(N)}_{d}+p^{(A)}_{d})/2$. Indeed,
this is the asymptotic result observed in Fig. 3, in the adiabatic limit.

The QD occupations predicted in Figs. 2 and 3 can be measured by using a
charge-sensitive PC-detector as shown in Fig. 1. The standard method is
performing the so-called single shot projective measurement to infer the QD
being occupied or not by an electron. After a large number of ensemble
measurements, the occupation probability can be obtained. However, the result
in Fig. 2 (the overall pattern plotted by the green curve) does not very
directly reveal the coexistence of the two fusion outcomes. Also, measuring
this oscillation pattern and the subsequent analyzing will be more complicated
than handling the result from the adiabatic coupling as shown in Fig. 3. That
is, importantly, measuring the final single constant occupation probability is
much simpler than measuring the oscillation pattern in Fig. 2, while the
result can more directly inform us the coexistence of the two fusion outcomes,
by using the formula $p_{d}=(p^{(N)}_{d}+p^{(A)}_{d})/2$ and the result
$p^{(N)}_{d}=0.5$. Therefore, the second detection scheme proposed above is
expected to be useful in practice, by adiabatically switching on the probe
coupling.

Continuous weak measurements.— Besides the usual strong (projective)
measurements, as discussed above for measuring the QD occupation probability,
continuous quantum weak measurement is an interesting and different type of
choice, suitable in particular for measuring quantum oscillations. For
instance, the problem of continuous weak measurement of charge qubit
oscillations by a PC detector has attracted strong interest for intensive
studies Kor01 ; Mil01 ; Li05 . In the following, we consider continuous weak
measurement for the quantum oscillations displayed in Fig. 2 (by the green
curve). Specifically, for the setup shown in Fig. 1, the PC detector is
switched on (applied bias voltage) from the beginning of state evolution after
fusion, owing to tunnel-coupling with the QD. The noisy output current in the
PC detector can be expressed as Kor01 ; Mil01 ; Opp20

$\displaystyle I_{c}(t)=n_{d,c}(t)+\frac{1}{\sqrt{4\kappa}}\,\xi(t)\,.$ (8)

This (rescaled) expression of current is valid up to a constant factor (with
current dimension). Then, the first term is simply the quantum average
occupation of an electron in the quantum dot, i.e., $n_{d,c}(t)={\rm
Tr}[\hat{n}_{d}\rho_{c}(t)]$, with $\rho_{c}(t)$ the PC-current-conditioned
state of the measured system. The second term describes the deviation of the
real noisy current from the quantum average occupation $n_{d,c}(t)$, owing to
classical events of random tunneling of electrons in the PC detector. The rate
parameter $\kappa$ characterizes the measurement strength and $\xi(t)$ is the
Gaussian white noise. For completeness, we also present here the quantum
trajectory (QT) equation for the conditional state as Mil01

$\displaystyle\dot{\rho}_{c}=\mathcal{L}\rho_{c}+\sqrt{\kappa}\mathcal{H}[\hat{n}_{d}]\rho_{c}\xi(t)\,.$
(9)

The first deterministic part is given by
$\mathcal{L}\rho_{c}=-i\left[H,\rho_{c}\right]+\kappa\mathcal{D}[\hat{n}_{d}]\rho_{c}$,
with the Lindblad superoperator defined as
$\mathcal{D}[x]\rho_{c}=x\rho_{c}x^{\dagger}-\frac{1}{2}\\{x^{\dagger}x,\rho_{c}\\}$.
The second noisy term stems from measurement backaction owing to information
gain in the single realization of continuous weak measurement, while the
superoperator is defined as
$\mathcal{H}[x]\rho_{c}=x\rho_{c}+\rho_{c}x^{\dagger}-\mathrm{Tr}\left[(x+x^{\dagger})\rho_{c}\right]\rho_{c}$.

Figure 4:  Quantum trajectories of continuous weak measurement, associated
with the detection scheme of sudden coupling of the MZMs after fusion to the
probing QD. The measurement-current-conditioned occupation $n_{d,c}(t)$ and
the current $\bar{I}_{c}(t)$ (low-pass-filtered with a time window $T=1$) are
shown for measurement strengths $\kappa=0.8$ in (a), 0.2 in (b), and 0.05 in
(c), respectively. Other parameters are the same as in Fig. 2.

Jointly simulating the evolution of Eqs. (8) and (9) we can obtain
$\rho_{c}(t)$, $n_{d,c}(t)$, and $I_{c}(t)$. From Eq. (8), we understand that
the measurement current does encode the information of the QD occupation.
However, owning to measurement backaction, the measurement-current-conditioned
occupation $n_{d,c}(t)$ is different from the occupation probability
$p_{d}(t)$ shown in Fig. 2 (in the absence of measurement). For considerably
weak measurement, $n_{d,c}(t)$ should be quite close to $p_{d}(t)$, yet the
noisy term in Eq. (8) will hide the informational term, thus preventing us
inferring the dot occupation. In contrast, if increasing the measurement
strength, the noisy term will decrease, but the measurement backaction will
make $n_{d,c}(t)$ more seriously deviate from the original result $p_{d}(t)$.
In Fig. 4, we show the results of $n_{d,c}(t)$ and $I_{c}(t)$, for a couple of
measurement strengths. In addition to properly choosing the measurement
strengths, we also applied the so-called low-pass-filtering technique. That
is, we averaged the current over a sliding time window of duration $T$,
$\bar{I}_{c}(t)=\frac{1}{T}\int_{t-T/2}^{t+T/2}dI_{c}(\tau)$, which gives a
smoothed current for better reflecting the dot occupation. However, even after
making these efforts, we find that from the noisy output current $I_{c}(t)$,
it is hard to track the quantum oscillations of the QD occupation shown in
Fig. 2.

Actually, in continuous weak measurements, a useful technique is extracting
information from the power spectrum of the measurement currents Kor01 ; Mil01
; Li05 . The steady-state current correlation function is obtained through the
ensemble average $S_{I}(\tau)={\rm E}[I_{c}(t+\tau)I_{c}(t)]-{\rm
E}[I_{c}(t+\tau)]{\rm E}[I_{c}(t)]$, at long time limit (large $t$ limit for
achieving steady state). From the power spectrum,
$S_{I}(\omega)=2\int^{\infty}_{-\infty}S_{I}(\tau)e^{i\omega\tau}$, one can
identify the characteristic frequencies and infer thus quantum coherent
oscillations inside the system under measurement. For the problem under study,
the goal is to identify the quantum oscillations shown in Fig. 2, which are
associated with the two fusion outcomes. Based on the result of Eq. (8), it
can be proved Kor01 ; Mil01  that

$\displaystyle S_{I}(\tau)$ $\displaystyle=$ $\displaystyle
S_{d}(\tau)+\frac{1}{4\kappa}\delta(\tau)-\frac{1}{4}\,.$ (10)

In this result, the correlation function of the dot occupation is given by
$S_{d}(\tau)={\rm E}[n_{d,c}(t+\tau)n_{d,c}(t)]={\rm
Tr}[{\hat{n}_{d}}e^{\mathcal{L}|\tau|}(\hat{n}_{d}\rho_{st})]$, with
$\rho_{st}$ the reduced density matrix of steady state. Then, we know the
structure of the current power spectrum as
$S_{I}(\omega)=S_{0}+S_{d}(\omega)$, with $S_{0}$ the frequency-free
background noise and $S_{d}(\omega)$ the information-contained part. Based on
the master equation $\dot{\rho}={\cal L}\rho$, which is the ensemble average
of Eq. (9), using the so-called quantum regression theorem we obtain

$\displaystyle S_{d}^{(j)}(\omega)$ (11) $\displaystyle=$
$\displaystyle\frac{2\kappa|\lambda_{j}|^{2}(16\Delta_{j}^{2}+\kappa^{2}+4\omega^{2})}{\omega^{2}(16\Omega_{j}^{2}+\kappa^{2}-4\omega^{2})^{2}+16\kappa^{2}(2|\lambda_{j}|^{2}-\omega^{2})^{2}}.$

Here we use $j=N,A$ to denote the two charge transfer channels, say, the
Andreev process and normal tunneling, with coupling amplitudes
$\lambda_{N,A}=\lambda_{2}\pm i\lambda_{3}$. Since the two channels are
independent, we obtained the above results independently for each process. The
overall spectrum $S_{d}(\omega)$ is the weight-averaged sum of
$S_{d}^{(N)}(\omega)$ and $S_{d}^{(A)}(\omega)$, i.e.,
$S_{d}(\omega)=[S_{d}^{(N)}(\omega)+S_{d}^{(A)}(\omega)]/2$, owing to the
fusion rule of Eq. (2). Moreover, under the condition of weak-coupling
measurement, we can further approximate the result as

$\displaystyle
S_{d}^{(j)}(\omega)\simeq\frac{\Delta_{j}^{2}}{4\Omega_{j}^{2}}\frac{\kappa
R_{j}/2}{\omega^{2}+(\kappa R_{j}/2)^{2}}$
$\displaystyle~{}+\frac{|\lambda_{j}|^{2}}{8\Omega_{j}^{2}}\frac{\frac{\kappa}{2}(1-\frac{R_{j}}{2})}{(\omega-2\Omega_{j})^{2}+[\frac{\kappa}{2}(1-\frac{R_{j}}{2})]^{2}}\,.$
(12)

Here we have introduced $R_{j}\equiv|\lambda_{j}|^{2}/\Omega_{j}^{2}$. From
this standard Lorentzian form, one can extract the characteristic frequencies
$\Omega_{N}$ and $\Omega_{A}$, which reflect in essence the quantum
oscillations given by Eq. (5).

Figure 5:  Characteristic frequency spectrum, $S_{d}(\omega)$, the Fourier
transform of the QD-occupation correlation function, from the output currents
of continuous weak measurement associated with the detection scheme of sudden
coupling of the MZMs after fusion to the probing QD. Theoretically, owing to
the fusion rule Eq. (2),
$S_{d}(\omega)=[S_{d}^{(N)}(\omega)+S_{d}^{(A)}(\omega)]/2$, where
$S_{d}^{(N)}(\omega)$ and $S_{d}^{(A)}(\omega)$ are the spectrums related to
the fusion outcomes $\psi$ and $I$. Exact results of Eq. (11) are displayed by
the solid-red and solid-blue curves, while approximate results of the
Lorentzian form Eq. (Probing the non-Abelian fusion of a pair of Majorana zero
modes) are plotted by the dashed-yellow and dashed-blue curves.
$S_{d}(\omega)$ is from numerically solving the full master equation, which
includes the two fusion outcomes caused charge transfer channels. Satisfactory
agreement between all these results is demonstrated. The coexistence of two
Lorentzian peaks (at $2\Omega_{N}$ and $2\Omega_{A}$) in $S_{d}(\omega)$
indicates the appearance of the two fusion outcomes, predicted by Eq. (1).
Measurement strength $\kappa=0.4$ is assumed, while other parameters are the
same as in Fig. 2.

In Fig. 5, we plot the result of $2S_{d}(\omega)$ from numerically solving the
full master equation, which includes the two charger transfer channels. We
find that it is indeed the sum of $S_{d}^{N}(\omega)$ and $S_{d}^{A}(\omega)$,
while in this plot, for the purpose of self-consistence verification, we use
their analytic solutions given by Eq. (11). We also compare the results with
the approximate Lorentzian form solutions and find satisfactory agreement.
Very importantly, the coexistence of two Lorentzian peaks (at $2\Omega_{N}$
and $2\Omega_{A}$) in $S_{d}(\omega)$ simply indicates the appearance of the
two fusion outcomes, predicted by Eq. (1).

We may remark that, within the scheme of continuous quantum weak measurement,
from its output current power spectrum $S_{I}(\omega)$ to infer the intrinsic
quantum oscillations is a very useful scheme, which is much simpler than the
ensemble single shot projective measurements of the dot occupation, in order
to obtain the probabilities as shown in Figs. 2 and 3. This type of technique
has been analyzed in detail in the context of charge-qubit measurements Kor01
; Mil01 ; Li05 . The present proposal is an extension along this line,
hopefully to be employed to identify the non-Abelian fusion of MZMs, through
the two different characteristic frequencies of quantum oscillations, which
are associated with the two fusion outcomes.

Summary

.— We have analyzed two schemes of detecting the nontrivial fusion of a pair
of MZMs. The two possible stochastic fusion outcomes reflect the non-Abalian
statistics nature of the MZMs, whose experimental demonstration will be a
milestone for ultimately identifying the MZMs and paving the way to
topological quantum computation. One scheme, the most natural choice, is to
switch on sudden coupling of the fused MZMs to the probing QD, with the
subsequent QD oscillating occupation being monitored by a PC detection in
terms of continuous weak measurement. From the power spectrum of the
measurement currents, one can identify two characteristic frequencies of
quantum oscillations and infer thus the two fusion outcomes of the pair of
MZMs. The other scheme is to switch on, almost adiabatically, gradual fusion
coupling between the MZMs and their coupling to the probing QD. This type of
detection scheme will result in the QD occupation not oscillating with time,
thus allowing a simpler way to measure the single value of the QD occupation
probability and using it to infer the two outcomes of nontrivial fusion. We
expect that both detection schemes analyzed in this work can be useful for
future fusion experiments.

Acknowledgements.

— This work was supported by the NNSF of China (Grants No. 11974011 and No.
11904261).

##  Appendix A Derivation of the Fusion Rule

In this Appendix let us prove the following transformation rule (fusion rule):

$\displaystyle|0_{12}0_{34}\rangle=\frac{1}{\sqrt{2}}(|0_{23}0_{14}\rangle+i|1_{23}1_{14}\rangle)\,.$

Under the constraint of fermion parity, in general, we may first construct the
transformation ansatz as
$|0_{12}0_{34}\rangle=a|0_{23}0_{14}\rangle+b|1_{23}1_{14}\rangle)$. Then, we
express the operator $f_{12}=(\gamma_{1}+i\gamma_{2})/2$ in terms of the
regular fermion operators $f_{14}$ and $f_{23}$ as

$\displaystyle
f_{12}=\frac{1}{2}\left(f_{14}+f^{\dagger}_{14}+if_{23}+if^{\dagger}_{23}\right)\,.$
(13)

This result is obtained by simply associating the Majorana fermions
$\gamma_{1}$ and $\gamma_{4}$ with the regular fermion $f_{14}$, and
$\gamma_{2}$ and $\gamma_{3}$ with $f_{23}$, respectively. Thus we have
$\gamma_{1}=f_{14}+f^{\dagger}_{14}$ and $\gamma_{2}=f_{23}+f^{\dagger}_{23}$.
Further, acting the annihilation operator $f_{12}$ on both sides of the ansatz
equation, we have

$\displaystyle 0=a(|0_{23}1_{14}\rangle+i|1_{23}0_{14}\rangle)$
$\displaystyle~{}~{}~{}+b(i|0_{23}1_{14}\rangle-|1_{23}0_{14}\rangle)\,.$ (14)

During the algebra, one should notice the difference of a minus sign between
$f_{23}|1_{23}1_{14}\rangle=|0_{23}1_{14}\rangle$ and
$f_{14}|1_{23}1_{14}\rangle=-|1_{23}0_{14}\rangle$. From this result, we
obtain $ia-b=0$ and $a=1/\sqrt{2}$, which finishes the proof of the above
formula of transformation. Applying the same method outlined above, one can
carry out all the transformation formulas between the two sets of basis
states, $|n_{12}n_{34}\rangle$ and $|n_{23}n_{14}\rangle$.

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numeric

# When exposure affects subgroup membership: Framing relevant causal questions
in perinatal epidemiology and beyond

Shalika Gupta1 Laura B. Balzer2 Moses R. Kamya3 Diane V. Havlir4 Maya L.
Petersen1,2,5

(1Division of Epidemiology, University of California, Berkeley
2Division of Biostatistics, University of California, Berkeley
3Department of Medicine, Makerere University College of Health Sciences
4Division of HIV, Infectious Diseases, and Global Medicine, Department of
Medicine, University of California, San Francisco
5UCSF-UC Berkeley Program in Computational Precision Health
)

###### Abstract

Perinatal epidemiology often aims to evaluate exposures on infant outcomes.
When the exposure affects the composition of people who give birth to live
infants (e.g., by affecting fertility, behavior, or birth outcomes), this
“live birth process” mediates the exposure effect on infant outcomes. Causal
estimands previously proposed for this setting include the total exposure
effect on composite birth and infant outcomes, controlled direct effects
(e.g., enforcing birth), and principal stratum direct effects. Using perinatal
HIV transmission in the SEARCH Study as a motivating example, we present two
alternative causal estimands: 1) conditional total effects; and 2) conditional
stochastic direct effects, formulated under a hypothetical intervention to
draw mediator values from some distribution (possibly conditional on
covariates). The proposed conditional total effect includes impacts of an
intervention that operate by changing the types of people who have a live
birth and the timing of births. The proposed conditional stochastic direct
effects isolate the effect of an exposure on infant outcomes excluding any
impacts through this live birth process. In SEARCH, this approach quantifies
the impact of a universal testing and treatment intervention on infant HIV-
free survival absent any effect of the intervention on the live birth process,
within a clearly defined target population of women of reproductive age with
HIV at study baseline. Our approach has implications for the evaluation of
intervention effects in perinatal epidemiology broadly, and whenever causal
effects within a subgroup are of interest and exposure affects membership in
the subgroup.

##### Keywords:

Epidemiologic methods, pregnancy, HIV infections, causal inference, mediation,
perinatal, direct effects

##  1 Introduction

Perinatal epidemiology often evaluates the effects of exposures on infant
outcomes when the exposure precedes or occurs during pregnancy and may affect
the probability of live birth (Figure 1). This poses a profound challenge to
specifying relevant and interpretable causal effects of exposures on infant
outcomes. Among individuals at risk of pregnancy, infant outcomes are only
defined once a live birth occurs. Importantly, the timing of birth and the
risk of adverse infant outcomes due to baseline characteristics of the
individuals who give birth may differ by exposure status, even when the
exposure is randomized. This fundamental challenge, although particularly
acute in perinatal epidemiology, occurs whenever causal effects are of
interest within a subset of the target population, and membership in that
subset is affected by the exposure.[29]

In perinatal epidemiology, Chiu et al.[30] proposed defining total effects
based on a composite endpoint of live birth and infant outcomes. This
definition implicitly assumes that all individuals in the population are
pregnancy seeking (or all are not pregnancy seeking), which may be applicable
in some settings, but not all.[30] It further includes any exposure effects on
the live birth process, which may or may not be of interest. A range of
mediation-based causal estimands have been proposed in perinatal epidemiology
to disentangle these effects, including controlled direct, separable, and
principal stratum direct effects.[30, 31, 32] While useful in some settings,
each has limitations.[31, 33]

ptExposurePregnancyInfant outcome Figure 1: Directed Acyclic Graph (DAG) with
pregnancy as a mediator of the effect of an exposure on infant outcomes. For
illustration, we focus on the setting of a randomized trial and omitting
unmeasured factors. ptConfounder at $t=1$Confounder at $t=2$Pregnancy at
$t=1$ExposurePregnancy at $t=2$Infant outcome Figure 2: DAG with pregnancy as
a mediator of a randomized exposure on infant outcomes including time-varying
confounders at two time points (and omitting unmeasured factors).

In this paper, we first discuss an alternative definition of total effects in
perinatal studies, using a composite outcome as in Chiu et al.,[30] but
instead focusing on a conditional total effect (CTE) among counterfactual live
births. The proposed conditional total effect includes by definition any
impact of an intervention that operates by changing the types of people who
give birth and timing of live births. Such differences are not a failure of
causal identification; instead, these differences are a result of the paths by
which an exposure can affect infant outcomes in settings where group
membership may be affected by treatment.

Next, in order to isolate the exposure effect on infant outcomes not mediated
by the birth process, we present an alternative class of conditional mediation
causal parameters that are identifiable, interpretable, and which answer
distinct questions in perinatal epidemiology from those previously proposed.
Following the causal roadmap,[34] we consider a longitudinal data structure
with time-varying covariates and a time-varying mediator (live birth process)
(Figure 2). We present conditional stochastic direct effect (CSDE) parameters
formulated using stochastic interventions on the live birth process, which
allows us to isolate the exposure effect on infant outcomes not due to any
effects on the types of people who give birth and timing of live births (i.e.,
within the same counterfactual population at-risk), and discuss identification
assumptions for these parameters. The SEARCH Study provides a detailed
illustration. Our approach is applicable to a wide range of perinatal
epidemiologic research, as well as more generally to the broad class of
studies in which the causal effect of interest is defined among a subset of
the population, and membership in this subset may be affected by the exposure.

###  1.1 The SEARCH Study

The SEARCH Study was a cluster randomized trial that evaluated the
effectiveness of universal antiretroviral treatment (ART) and person-centered
HIV care on population health in rural Kenya and Uganda.[35, 36] The study
tested 90% of adults for HIV at baseline[37] and ascertained vital status and
HIV status after three years for infants born during the study. In pre-
specified analyses, the SEARCH intervention reduced perinatal HIV transmission
and increased infant HIV-free survival among women of reproductive age living
with HIV.[38] However, the extent to which this total effect may have operated
through intervention-induced changes to the counterfactual population of
infants at-risk, due to timing of births (in the context of time-varying
transmission risk across the pre and postnatal risk periods) and/or the types
of people who gave birth (e.g., through changes to maternal survival,
fertility, or behavior) remains unclear. CSDEs provide an approach to
answering this question.

##  2 Previous Mediation-Based Approaches

A number of mediation-based perinatal epidemiology parameters have been
previously proposed. In particular, prior authors[30, 31] have sought to
exclude from the total effect of the exposure on infant outcomes any impact of
the exposure on the live birth process (the mediator) through: 1) hypothetical
static interventions on the live birth process (i.e., controlled direct
effects); 2) restriction to the group of individuals who would have had a live
birth irrespective of exposure (i.e., principal stratum direct effects); and,
3) partitioning of the exposure into distinct components that operate in
different ways on the live birth process and infant outcomes (i.e., separable
direct effects).

Each of these approaches has limitations. First, a controlled direct effect
evaluates the exposure effect on infant outcomes if all individuals in the
target population were to give birth. This type of hypothetical intervention
may be both hard to interpret and poorly supported by the data, particularly
if the probability of live birth is low and/or many are not seeking pregnancy.
A controlled direct effect quantifies the impact of the SEARCH intervention on
infant HIV-free survival if all reproductive age women living with HIV at
study baseline were to have a birth during the follow-up period, a
counterfactual scenario that is neither realistic nor informative (Figure
LABEL:dag:interventionsLABEL:dag:cde).

Second, identification of principal stratum direct effects requires strong
assumptions, such as an assumption that no ‘defiers’ exist. In a perinatal
setting with live birth as a mediator, this corresponds to assuming that there
are no individuals who would have a live birth if unexposed, but not if
exposed. Moreover, if exposure affects the timing of births within the
principal stratum, this direct effect parameter would still include any impact
that operates through time at risk. For example, in SEARCH this would include
any effect due to changes in timing of birth during follow-up and
corresponding shifts in the distribution of infant exposure to pre versus and
postnatal transmission risk.

Third, separable direct effect parameters are predicated on the decomposition
of the exposure into two components, one that acts directly on the outcome
without affecting the mediator (live birth process) and a second that operates
only on the (infant) outcome through the mediator. Often, this decomposition,
which requires specifying mechanisms by which exposure affects infant survival
but not maternal survival, may not be feasible. In the SEARCH trial, where the
outcome of interest involves infant HIV-free survival and the mediator
includes maternal survival, we cannot disentangle the portion of the
intervention that impacts only the outcome without affecting the mediator.

To answer many questions in perinatal epidemiology, alternative approaches are
thus needed that account for the live birth process while addressing
interpretable causal questions that are identified under plausible
assumptions.

##  3 Conditional Stochastic Direct Effects in Perinatal Epidemiology

We propose an alternative approach to defining causal parameters in perinatal
epidemiology, using direct effects based on stochastic interventions on the
mediating variables. Stochastic interventions consider a hypothetical
intervention on the mediators where their values are randomly drawn from a
distribution (possibly conditional on covariates) rather than
deterministically set to a particular value.

Direct effect parameters formulated using stochastic interventions on
mediating variables [39, 40, 41, 42, 43, 44] have several advantages,
including that they define more realistic causal parameters (by allowing for
variation in mediator values), reduce positivity concerns, naturally
generalize to longitudinal settings, and remain identifiable in the common
setting of post-treatment confounding of the mediator-outcome relationship. In
perinatal settings, stochastic direct effects are particularly appealing
because they can be formulated to exclude from the total effect any
intervention impact that affects infant risk that is mediated by the live
birth process. In the remainder of the paper, we discuss the use of
conditional stochastic direct effects to frame and identify causal questions
in perinatal epidemiology, using the SEARCH Study as a running example.

###  3.1 Structural Causal Model

In many perinatal applications, the observed data consist of baseline
covariates, an indicator of exposure, time-varying covariates and mediators,
and infant outcome (e.g., HIV-free survival in SEARCH). A longitudinal
structural causal model (SCM) can be used to formalize knowledge about the
data-generating process.[45] For simplicity, we focus on an individual-level
SCM (noting that this framework generalizes to hierarchical settings[46, 47]).
We use $\bar{V}_{t}$ to denote $(V_{1},\dots,V_{t})$, which represents the
values of the variable $V$ from time $1$ through time $t$. $L_{0}$ represents
baseline covariates (e.g., maternal age, prior ART use, and community HIV
prevalence). $A$ denotes exposure (e.g., if the individual was randomized to
receive the intervention). $L_{t}$ is the set of time-varying covariates
(e.g., maternal ART status and infant vital status). $Z_{t}$ represents the
mediator(s) of interest, which in SEARCH consist of $Z1_{t}$ (maternal
mortality by time $t$) and $Z2_{t}$ (live birth status at time $t$). Let
$Y_{\tau}$ denote the final outcome, where $\tau$ represents the final time
point. In SEARCH, $Y_{\tau}$ denotes an indicator of an infant being born and
surviving HIV-free until the end of follow-up, a composite of $Y1_{\tau}$ (an
indicator of an infant being born and remaining alive) and $Y2_{\tau}$ (an
indicator of an infant remaining HIV-uninfected) by the end of follow-up
$\tau$. The exogenous nodes, representing unmeasured variables, are:
$U=(U_{L_{0}},U_{A},U_{L_{t}},U_{Z_{t}},U_{Y_{\tau}})$ for $t=1,\dots,\tau$.
The endogenous nodes are: $X=(L_{0},A,\bar{L}_{t},\bar{Z}_{t},Y_{\tau})$ for
$t=1,\dots,\tau$. In randomized trials such as SEARCH, the exposure assignment
mechanism is known. Less is known about the time-varying covariates, mediator
and outcome processes, which, in particular, may have been affected by
exposure and in turn may have affected subsequent covariates, mediators, and
outcomes (Appendix A). Figure LABEL:dag:interventionsLABEL:dag:orig
illustrates (using fewer timepoints) the mechanisms of interest by which the
SEARCH intervention may have impacted infant HIV-free survival, which include
the intervention’s effects through changes to maternal ART initiation,
maternal mortality, and live birth.

###  3.2 Causal Estimands

The relevance of conditional stochastic direct effect parameters for perinatal
epidemiology, as well as the applicability of the CSDE approach for a much
broader class of conditional causal effects, is clarified by first revisiting
the definition of total effects in these settings.

####  3.2.1 Conditional Total Effects

Epidemiology commonly studies total effects, defined using marginal summaries
of the distribution of counterfactual outcomes. However, in many settings,
including perinatal epidemiology, conditional effects defined using summaries
of the distribution of counterfactual outcomes among a counterfactual subset
of the population are often of greater interest. To formulate such conditional
total effect parameters, as in Chiu et al.,[30] we use a composite outcome of
live birth and infant outcomes. However, in contrast to Chiu et al.,[30] who
define the total effect as the effect of the exposure on the probability of
the composite event of live birth and infant outcomes, we instead define a
conditional total effect (Table 1):

$\displaystyle P[Y1_{\tau}(a=1)=1,Y2_{\tau}(a=1)=1|Z2_{\tau}(a=1)=1]$
$\displaystyle\quad\quad\quad\quad-P[Y1_{\tau}(a=0)=1,Y2_{\tau}(a=0)=1|Z2_{\tau}(a=0)=1]$
$\displaystyle=\frac{P[Y_{\tau}(a=1)=1]}{P[Z2_{\tau}(a=1)=1]}-\frac{P[Y_{\tau}(a=0)=1]}{P[Z2_{\tau}(a=0)=1]}$

In the SEARCH Study, this conditional total effect evaluates the effect of the
exposure on the outcome of live birth and infant HIV-free survival
($Y1_{\tau}(a)=1,Y2_{\tau}(a)=1$), but only among counterfactual live births
($Z2_{\tau}(a)=1$). By the definition of conditional probabilities, the
conditional causal effect can alternatively be expressed by contrasting the
joint probability of the composite counterfactual outcome of live birth and
infant outcomes divided by the probability of counterfactual live birth for
each exposure of interest.

Conditioning on the probability of counterfactual live birth will exclude from
the traditional total effect parameter any effect of the intervention on
infant outcomes that operates through changes to the marginal probability of
live birth by the end of follow-up; however, this conditional total effect
parameter does not ensure equivalence of other aspects of the live birth
process under differing exposures, and thus will still include any effects due
to exposure-induced changes to the the types of people who have a live birth
and the timing of births (i.e., composition of the counterfactual population
at-risk). For example, in SEARCH, the conditional total effect of the
intervention on infant HIV-free survival would include both any direct
reduction in vertical transmission (e.g., by improving HIV viral suppression)
and any increase in transmission due to facilitating live birth among persons
at higher risk of transmission (e.g., resulting from barriers to care) or
changing the time of birth (e.g., such that a higher proportion of follow-up
time is prenatal). Such conditional total effects are often of real-world
interest in that they reflect the total change in infant outcomes due to
multiple mechanisms expected were an intervention to be implemented.
Nonetheless, in many cases it remains of interest to isolate intervention
effects on outcomes among the same types of people with the same timing of
births.

####  3.2.2 Conditional Stochastic Direct Effect

Conditional stochastic direct effect parameters allow us to quantify the
effect of an exposure on infant outcome absent any effect of the intervention
on the live birth process, or in other words, the exposure effect in
populations with the same types of people (Table 1). We build on our
definition of a conditional total effect to define such a conditional
stochastic direct effect. In contrast to a controlled direct effect parameter,
which would compare hypothetical interventions in which exposure is varied but
all persons in the target population survive to give birth (Figure
LABEL:dag:interventionsLABEL:dag:cde), stochastic direct effect parameters,
[39, 40, 41] compare hypothetical interventions in which exposure is varied
but mediator values $Z_{\tau}$ are drawn from a distribution that is held
constant, denoted here (for ease of of notation) as simply $\bar{G}$. Given
some choice of stochastic mediator distribution $\bar{G}$, we use this
approach to define the following conditional stochastic direct effect
parameter:

$\displaystyle
P[Y1_{\tau}(a=1,\bar{G})=1,Y2_{\tau}(a=1,\bar{G})=1|Z2_{\tau}(\bar{G})=1]$
$\displaystyle\quad\quad\quad\quad-P[Y1_{\tau}(a=0,\bar{G})=1,Y2_{\tau}(a=0,\bar{G})=1|Z2_{\tau}(\bar{G})=1]$
$\displaystyle=\frac{P[Y_{\tau}(a=1,\bar{G})=1]}{P(Z2_{\tau}(\bar{G})=1)}-\frac{P[Y_{\tau}(a=0,\bar{G})=1]}{P(Z2_{\tau}(\bar{G})=1)}$

As with the CTE, this CSDE is defined among infants born to persons in the
target population under each exposure level of interest; however, in contrast
to the conditional total effect, the hypothetical intervention to draw
maternal survival and live birth from some fixed distribution $\bar{G}$
ensures that the exposure effect is evaluated among the same counterfactual
population. For example, in SEARCH, the CSDE corresponds to the difference in
counterfactual probability of HIV-free survival among infants born to persons
with HIV under intervention or control, had maternal mortality and live birth
over time, $\overline{Z1}_{\tau}$ and $\overline{Z2}_{\tau}$, respectively,
been drawn from the same underlying distribution.

In contrast to the principal stratum direct effect, which evaluates the impact
of the SEARCH intervention in the group who would give birth regardless of
treatment assignment, the stochastic direct effect parameter can evaluate the
direct effect of an exposure among different groups of people, depending on
the choice of the distribution $\bar{G}$ from which mediator values are drawn.
A variety of CSDEs can be formulated using different choices of stochastic
interventions on the mediator variables to help understand how the
intervention effect varies by the timing of births and composition of people
who give birth.

####  3.2.3 Choice of Mediator Distribution: Interpretation and Identification

In specifying a distribution $\bar{G}$ from which the mediator value(s) are
drawn, there are several options, with implications for both parameter
identification (or ability to rewrite the causal parameter as a parameter of
the observed data distribution and thus estimate it) and interpretation. One
approach[48, 41, 39] is to define the (potentially longitudinal) mediator
distribution as the unknown counterfactual distribution in the absence of
exposure, possibly conditional on baseline covariates (also referred to as a
marginal mediator or fixed stochastic direct effect in the context of non-
conditional effect [43, 49] (Appendix B)). In perinatal settings, such a
natural direct effect formulated conditional on baseline covariates holds
constant the types and timing of births (Figure
LABEL:dag:interventionsLABEL:dag:sde) across exposure conditions; however, in
the presence of time-varying confounding, it will not decompose the original
total effect. [44, 43] Identification of this parameter requires no unmeasured
confounding of the 1) exposure-outcome, 2) exposure-mediator, and 3) mediator-
outcome relationships at all time points (but avoids a stronger "cross-world"
assumption required by earlier definitions[50, 51]). Identification requires a
positive probability of 1) the values the mediator takes under the control
condition given each exposure of interest within all values of the mediator-
outcome confounders (baseline and time-varying covariates) at each time $t$
and 2) of each possible exposure within all values of baseline covariates.

Another choice for the stochastic intervention on the mediator(s) is to draw
mediator values from the counterfactual control distribution conditional both
on baseline covariates and time-varying history, as proposed by Zheng and van
der Laan.[44] While this conditional mediator formulation of the natural
direct effect is appealing because it provides a decomposition of the average
treatment effect,[44] in perinatal settings, this type of parameter does not
ensure that the timing of births and types of people who give birth under
different exposures will be the same (i.e., does not ensure equivalent
counterfactual populations at-risk across differing exposures). In perinatal
settings (Figure LABEL:dag:interventionsLABEL:dag:nde), this corresponds to a
hypothetical intervention where live birth is a function of both the baseline
covariates and the time-varying past (e.g., maternal time-to-ART start).
Because the time-varying past is affected by the static intervention on $A$,
the types of people who have a live birth under an intervention to assign
everyone to receive the intervention may not be the same as those who would
have a live birth if everyone were to be assigned to the control condition.
This parameter also requires stronger assumptions for identifiability than the
natural direct effect using marginal mediator distributions (i.e., strong
sequential randomization and its associated positivity assumptions).[44]

Alternatively, conditional stochastic direct effect parameters can be
formulated such that mediator values are drawn from some known distribution,
conditional only on baseline covariates (Figure
LABEL:dag:interventionsLABEL:dag:sde), or from a corresponding distribution
that is estimated from the observed data (i.e., a data-adaptive parameter as
proposed by Rudolph et al.[52, 49]). Data-dependent choices for the stochastic
intervention on the mediator include the estimated mediator distribution in
the intervention or comparator conditions or an intervention that is well-
supported in the study sample. In SEARCH, for example, we could formulate a
data-adaptive conditional stochastic direct effect parameter which compared
infant HIV-free survival among live births if everyone received the
intervention versus the control condition and if maternal mortality and time-
to-live birth were drawn from their observed distribution in the intervention
arm conditional on baseline covariates, such as maternal age, community-
specific HIV prevalence, and baseline ART initiation status.

Such stochastic direct effect parameters, in which the distribution from which
the mediators is drawn is known or estimated, require fewer assumptions for
identifiability than mediated effects formulated using either the marginal or
conditional mediator approaches described previously. More specifically,
identification requires no unmeasured confounding of the 1) exposure-outcome
and 2) mediator-outcome relationships at all time points. In addition,
identification requires positive probabilities of 1) each exposure of interest
given baseline covariates and 2) of the values the mediator can take under its
specified known distribution given each exposure within all past values of
mediator-outcome confounders (baseline and time-varying factors) at each
timepoint. If the observed mediator distribution under one of the exposure
values is selected as the known distribution from which mediator values are
drawn, then the positivity assumption will be satisfied for that value of the
exposure variable.

The assumption of no unmeasured confounding of the exposure and outcome will
be satisfied by design in a randomized controlled trial such as the SEARCH
Study. Randomization of the exposure variable is not sufficient to satisfy the
second assumption of no unmeasured confounding of the mediator-outcome
relationship(s). In the SEARCH trial, the second assumption will be satisfied
if we believe that all confounders of the live birth process (maternal
mortality and live birth at time $t$) and outcome (infant HIV-free survival)
at each time point were measured. In SEARCH, a potential set of such variables
includes maternal ART initiation through time $t$, mediator history (maternal
mortality and live birth through time $t-1$), and baseline covariates.
Measurement of additional time-varying covariates, such as maternal health
(e.g., CD4 cell count and viral load) over time, makes it more likely that
this assumption will be satisfied. In general, measuring a rich set of
baseline and time-varying covariates will improve the plausibility of both of
these assumptions on confounding.

The conditional total effect will be identified if there is 1) no unmeasured
exposure-outcome confounding and 2) a positive probability of each exposure
within all baseline covariate values. Because the outcome of interest is
defined as the composite event of the mediator live birth and infant outcomes,
in SEARCH, this randomization assumption more specifically corresponds to
assumptions of no unmeasured confounding of the exposure-live birth
($A-Z2_{\tau}$), exposure-infant HIV status ($A-Y1_{\tau}$), and exposure-
infant vital status ($A-Y2_{\tau}$) relationships. Unlike the CTE, the CSDE
does not require an assumption of no unmeasured confounding of the exposure-
live birth relationship because the mediator, the live birth process
($\bar{Z}_{\tau}$), is intervened upon in the formulation of the CSDE
parameter.[53] As a result, in observational settings, it is possible that the
CSDE may be identified even when the CTE is not, due to unmeasured exposure-
mediator confounding. (In the setting of a randomized trial, randomization
assumptions involving the exposure will be satisfied by design.)

Table 1: Definition and interpretations of causal estimands proposed for the perinatal epidemiologic setting. Parameter |  Interpretation (General) |  Interpretation (SEARCH) |  Counterfactuals (SEARCH) |  Hypothetical Intervention
---|---|---|---|---
Total effect parameter |  |  |  |
$\begin{aligned} &\frac{P[Y_{\tau}(a=1)=1]}{P[Z2_{\tau}(a=1)=1]}-\frac{P[Y_{\tau}(a=0)=1]}{P[Z2_{\tau}(a=0)=1]}\\\ &=P[Y1_{\tau}(a=1)=0,Y2_{\tau}(a=1)=0|Z2_{\tau}(a=1)=1]\\\ &-P[Y1_{\tau}(a=0)=0,Y2_{\tau}(a=0)=0|Z2_{\tau}(a=0)=1]\end{aligned}$ |  • The total effect of an intervention on infant health outcomes among live births • Includes: any effect of the intervention operating through changes to the timing and types of people who give birth • Excludes: any effect of the intervention on infant health outcomes through changes to the marginal probability of live birth by the end of the follow-up period |  • The total effect of the SEARCH intervention on infant HIV-free survival among live births |  • $Y_{\tau}(a)$: counterfactual outcome of live birth and infant HIV-free survival at time $\tau$ when $A=a$ • $Y1_{\tau}(a)$: counterfactual infant vital status at time $\tau$ when $A=a$ • $Y2_{\tau}(a)$: counterfactual infant HIV status at time $\tau$ when $A=a$ • $Z2_{\tau}(a)$: counterfactual live birth status by time $\tau$ when $A=a$ |  • Compare intervening to set $a=1$ (i.e., receive the SEARCH intervention) for everyone to setting $a=0$ (i.e., receive country standard of care) for everyone
Stochastic direct effect parameters |  |  |  |
$\begin{aligned}
&\frac{P[Y_{\tau}(a=1,\bar{G})=1]}{P(Z2_{\tau}(\bar{G})=1)}-\frac{P[Y_{\tau}(a=0,\bar{G})=1]}{P(Z2_{\tau}(\bar{G})=1)}\\\
&=P[Y1_{\tau}(a=1,\bar{G})=0,Y2_{\tau}(a=1,\bar{G})=0|Z2_{\tau}(\bar{G})=1]\\\
&-P[Y1_{\tau}(a=0,\bar{G})=0,Y2_{\tau}(a=0,\bar{G})=0|Z2_{\tau}(\bar{G})=1]\end{aligned}$
where $\bar{G}\equiv\bar{G}_{\tau}$ and $\bar{G}_{t}\big{(}\bar{z}_{t}|L_{0}\big{)}\equiv P\big{(}\bar{Z}_{t}=\bar{z}_{t}|L_{0}\big{)}$ is a known distribution (conditional on baseline covariates)aaaIn the specific case where the known distribution of the mediators conditional on baseline covariates is equivalent to the true marginal mediator distribution conditional on baseline covariates in the control arm, this stochastic direct effect parameter will be equal to the natural direct effect parameter formulated using marginal mediator distributions (and will provide a decomposition of a type of total effect). |  • A direct effect of an intervention on infant health outcomes where the live birth process (a person’s probability of giving birth at each time point) is drawn from the same known distribution conditional on baseline covariates • Includes: any effect of the intervention on the outcome that operates through mechanisms other than the live birth process • Excludes: any effect of the intervention on infant health outcomes through changes to the mediator: timing/occurrence of live births and to the types of people who give birth |  • The direct effect of the SEARCH intervention on infant HIV-free survival if the live birth process was drawn from some known distribution conditional on the baseline covariates of a woman in the target population |  • $Y_{\tau}(a,\bar{G})$: counterfactual outcome of live birth and infant HIV-free survival at $t=\tau$ under an intervention to set $A=a$ and draw maternal survival and time-to-live birth from a known distribution that is conditional on observed baseline covariates • $Y1_{\tau}(a^{\prime},\bar{G})$: counterfactual infant vital status at $t=\tau$ under an intervention to set $A=a$ and draw maternal survival and time-to-live birth from the known mediator distribution • $Y2_{\tau}(a^{\prime},\bar{G})$: counterfactual infant HIV status at $t=\tau$ under an intervention to set $A=a$ and draw maternal survival and time-to-live birth from the known mediator distribution • $Z2_{\tau}(\bar{G})$: counterfactual live birth status at $t=\tau$ under the known stochastic intervention on the mediators |  • Compare intervening to set $a=1$ to setting $a=0$ for all persons in the target population, while drawing the live birth process from the same known distribution conditional on baseline covariates
|  |  |  |

##  4 Discussion

A common challenge of causal effect estimation in perinatal epidemiology is
that outcomes are only measured among infants born alive, but exposures
typically predate birth and may affect the live birth process, resulting in
different types of people who have live births. While particularly notable in
the perinatal epidemiology literature, this challenge also occurs ubiquitously
across settings in where an exposure may affect the composition of the group
that is at risk for the study outcome.[54] We add to existing literature
addressing this challenge by first explicitly defining and interpreting
conditional total effects in this setting, highlighting two components of the
total effect, one due to due to exposure effects among counterfactual group
members, and a second due to exposure effects on counterfactual group
membership.

We then suggest the use of a class of conditional stochastic direct effects to
disentangle these two exposure pathways and discuss setting-specific
advantages over other types of direct effect parameters described in the
perinatal literature. Such conditional direct effect parameters are formulated
using stochastic interventions on the mediator process conditional on baseline
covariates. The choice of stochastic intervention on the mediator(s) requires
careful consideration as it determines the subgroup in which the intervention
is assessed. In perinatal settings, the choice of stochastic intervention will
determine the timing of births and composition of people who have live births,
and will thus affect the interpretation of the stochastic direct effect
parameter. This flexibility also represents an advantage of this approach;
applying different choices of stochastic interventions on the mediating
variables can reveal how the direct effect of the intervention is mediated by
the live birth process that occurs in perinatal studies, as well as providing
a straightforward approach to transporting (under assumptions) perinatal
exposure effect estimates to new settings with differing live birth processes,
or more generally, of transporting stochastic direct effects among selected
subgroups to new settings in which the group selection process my differ.

Approaches for the estimation of stochastic direct effect parameters have been
developed by van der Laan and Petersen,[41] Rudolph et al.,[49] Zheng and van
der Laan,[44] VanderWeele and Tchetgen Tchetgen,[43] and Diaz et al.,[55]
among others, and include a range of inverse probability weighted and
semiparametric efficient multiply robust estimators. However, to date, while
stochastic mediation effects have been applied in perinatal studies
previously,[42, 56] to the best of our knowledge, they have not been used to
formulate conditional causal effect parameters that account for exposure-
induced changes to the live birth process in these types of settings, nor more
generally to account for exposure-induced changes in subgroup membership when
conditional causal effects are of interest. This paper aims to provide a
bridge to the wider use of these methods in settings where they are perhaps
uniquely positioned to address the underlying question of interest.

## Acknowledgements

We thank the SEARCH Collaboration, Patrick Bradshaw, the Ministry of Health of
Uganda and the Ministry of Health of Kenya, our research teams and
administrative teams in San Francisco, Uganda, and Kenya, collaborators and
advisory boards, and especially all the communities and participants involved
in the SEARCH trial.

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## References

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  * [30] Yu-Han Chiu et al.  “The Effect of Prenatal Treatments on Offspring Events in the Presence of Competing Events An Application to a Randomized Trial of Fertility Therapies”  In _Epidemiology_ 31.5, 2020, pp. 636–643  DOI: 10.1097/EDE.0000000000001222
  * [31] Jessica G. Young and Mats J. Stensrud  “Identified Versus Interesting Causal Effects in Fertility Trials and Other Settings With Competing or Truncation Events”  In _Epidemiology_ 32.4, 2021, pp. 569–572  DOI: 10.1097/EDE.0000000000001357
  * [32] Mats J. Stensrud et al.  “Conditional Separable Effects”  In _Journal of the American Statistical Association_ 118.544 Taylor & Francis, 2023, pp. 2671–2683  DOI: 10.1080/01621459.2022.2071276
  * [33] Jonathan M. Snowden, Kelly M. Reavis and Michelle C. Odden  “Conceiving of Questions Before Delivering Analyses Relevant Question Formulation in Reproductive and Perinatal Epidemiology”  In _Epidemiology_ 31.5, 2020, pp. 644–648  DOI: 10.1097/EDE.0000000000001223
  * [34] Maya L. Petersen and Mark J. van der Laan  “Causal Models and Learning from Data: Integrating Causal Modeling and Statistical Estimation”  In _Epidemiology_ 25.3, 2014, pp. 418–426  DOI: 10.1097/EDE.0000000000000078
  * [35] Diane V. Havlir et al.  “HIV Testing and Treatment with the Use of a Community Health Approach in Rural Africa”  In _New England Journal of Medicine_ 381.3, 2019, pp. 219–229  DOI: 10.1056/NEJMoa1809866
  * [36] James Ayieko et al.  “A Patient-Centered Multicomponent Strategy for Accelerated Linkage to Care Following Community-Wide HIV Testing in Rural Uganda and Kenya”  In _Journal of Acquired Immune Deficiency Syndromes (1999)_ 80.4, 2019, pp. 414–422  DOI: 10.1097/QAI.0000000000001939
  * [37] Gabriel Chamie et al.  “A Hybrid Mobile Approach for Population-Wide HIV Testing in Rural East Africa: An Observational Study”  In _The lancet. HIV_ 3.3, 2016, pp. e111–119  DOI: 10.1016/S2352-3018(15)00251-9
  * [38] Shalika Gupta et al.  “Population-Level HIV-free Infant Survival in the SEARCH Trial. Conference on Retroviruses and Opportunistic Infections (CROI)”, 2020
  * [39] Vanessa Didelez, Philip Dawid and Sara Geneletti  “Direct and Indirect Effects of Sequential Treatments”  In _Proceedings of the 22nd Conference on Uncertainty in Artificial Intelligence_ Cambridge, MA, USA: AUAI Press, 2006, pp. 138–146
  * [40] Sara Geneletti  “Identifying Direct and Indirect Effects in a Non-Counterfactual Framework”  In _Journal of the Royal Statistical Society Series B: Statistical Methodology_ 69.2, 2007, pp. 199–215  DOI: 10.1111/j.1467-9868.2007.00584.x
  * [41] Mark J. van der Laan and Maya L. Petersen  “Direct Effect Models”  In _The International Journal of Biostatistics_ 4.1 Germany: bepress, 2008, pp. Article 23  DOI: 10.2202/1557-4679.1064
  * [42] Tyler J. VanderWeele, Stijn Vansteelandt and James M. Robins  “Effect Decomposition in the Presence of an Exposure-Induced Mediator-Outcome Confounder”  In _Epidemiology_ 25.2, 2014, pp. 300–306  DOI: 10.1097/EDE.0000000000000034
  * [43] Tyler J. VanderWeele and Eric J. Tchetgen Tchetgen  “Mediation Analysis with Time Varying Exposures and Mediators”  In _Journal of the Royal Statistical Society Series B: Statistical Methodology_ 79.3, 2017, pp. 917–938  DOI: 10.1111/rssb.12194
  * [44] Wenjing Zheng and Mark van der Laan  “Longitudinal Mediation Analysis with Time-varying Mediators and Exposures, with Application to Survival Outcomes”  In _Journal of Causal Inference_ 5.2, 2017  DOI: 10.1515/jci-2016-0006
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  * [46] Laura B Balzer, Wenjing Zheng, Mark J van der Laan and Maya L Petersen  “A New Approach to Hierarchical Data Analysis: Targeted Maximum Likelihood Estimation for the Causal Effect of a Cluster-Level Exposure”  In _Statistical Methods in Medical Research_ 28.6 SAGE Publications Ltd STM, 2019, pp. 1761–1780  DOI: 10.1177/0962280218774936
  * [47] Alejandra Benitez et al.  “Defining and Estimating Effects in Cluster Randomized Trials: A Methods Comparison”  In _Statistics in Medicine_ 42.19, 2023, pp. 3443–3466  DOI: 10.1002/sim.9813
  * [48] Mark van der Laan and Maya Petersen  “Direct Effect Models”  In _U.C. Berkeley Division of Biostatistics Working Paper Series_ , 2005
  * [49] Kara E. Rudolph, Oleg Sofrygin, Wenjing Zheng and Mark J. van der Laan  “Robust and Flexible Estimation of Stochastic Mediation Effects: A Proposed Method and Example in a Randomized Trial Setting”  In _Epidemiologic methods_ 7.1, 2017, pp. 20170007  DOI: 10.1515/em-2017-0007
  * [50] James M. Robins and Sander Greenland  “Identifiability and Exchangeability for Direct and Indirect Effects”  In _Epidemiology_ 3.2 Williams & Wilkins and Epidemiology Resources Inc., 1992, pp. 143–155
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  * [55] Iván Díaz, Nima S. Hejazi, Kara E. Rudolph and Mark J. Van der Laan  “Nonparametric Efficient Causal Mediation with Intermediate Confounders”  In _Biometrika_ 108.3, 2021, pp. 627–641  DOI: 10.1093/biomet/asaa085
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##  Appendix A Structural Causal Model

A longitudinal structural causal model is used to represent the data
generating process of perinatal HIV transmission in the target population of
women living with HIV who are at risk for pregnancy over the follow-up period.
$L_{0}$ denotes the baseline covariates and $A$ the intervention node. $L_{t}$
represents time-varying covariates (which may be confounders of the mediator-
outcome relationship(s)) at each time $t$. $Z_{t}$ denotes the set of
mediators, $Z1_{t}$ and $Z2_{t}$, at each time point. We use two mediators to
capture two mechanisms by which the intervention may affect the timing of
births and the types of people who give birth: 1) by the effect of the
intervention on maternal morality ($Z1_{t}$) and 2) by the effect of the
intervention on the timing of live birth ($Z2_{t}$). Both $Z1_{t}$ and
$Z2_{t}$ are time-to-event variables that each take a value of 1
deterministically at time $t+1$ if they are equal to 1 at time $t$. The final
outcome of interest is the composite variable $Y_{\tau}$, which takes a value
of 1 if, in terms of our motivating example, a live birth occurs and the
infant survives and is HIV-free at the end of follow-up and a value of 0
otherwise. Infant survival is represented by the node $Y1_{t}$; its value is
missing if a live birth has not yet occurred by time $t$. $Y2_{\tau}$ denotes
whether a live infant was HIV-free at the end of follow-up. Because infection
status was only assessed at the end of the study, it is not a time-varying
covariate, and its value will be missing in the absence of a live birth or in
the event of infant mortality prior to the end of the follow-up period.

$\displaystyle L_{0}$ $\displaystyle=f_{L_{0}}(U_{L_{0}})$ $\displaystyle A$
$\displaystyle=f_{A}(U_{A})$ $\displaystyle L_{t}$
$\displaystyle=f_{L_{t}}(A,\bar{L}_{t-1},\bar{Z}_{t-1},U_{L_{t}})$
$\displaystyle Z_{t}$ $\displaystyle=\\{Z1_{t},Z2_{t}\\}$ $\displaystyle
Z1_{t}$ $\displaystyle=\begin{cases}1,&\text{ if }Z1_{t-1}=1\\\
f_{Z1_{t}}(A,\bar{L}_{t},\bar{Z}_{t-1},U_{Z_{t}}),&\text{ otherwise }\\\
\end{cases}$ $\displaystyle Z2_{t}$ $\displaystyle=\begin{cases}1,&\text{ if
}Z2_{t-1}=1\\\ f_{Z2_{t}}(A,\bar{L}_{t},\bar{Z}_{t-1},U_{Z_{t}}),&\text{
otherwise }\\\ \end{cases}$ $\displaystyle Y1_{t}$ $\displaystyle\in L_{t+1}$
$\displaystyle Y1_{t}$ $\displaystyle=\begin{cases}NA,&\text{ if }Z2_{t}=0\\\
0,&\text{ if }Y1_{t-1}=0\\\
f_{Y1_{t}}(A,\bar{L}_{t},\bar{Z}_{t},U_{Y1_{t}}),&\text{ otherwise }\\\
\end{cases}$ $\displaystyle Y2_{\tau}$ $\displaystyle=\begin{cases}NA,&\text{
if }Z2_{\tau}=0\text{ or }Y1_{\tau}=0\\\
f_{Y2_{\tau}}(A,\bar{L}_{\tau},\bar{Z}_{\tau},U_{Y2_{\tau}}),&\text{ otherwise
}\\\ \end{cases}$ $\displaystyle Y_{\tau}$
$\displaystyle=\begin{cases}1,&\text{ if }Z2_{\tau}=1,Y1_{\tau}=1,\text{ and
}Y2_{\tau}=1\\\ 0,&\text{ if }Z2_{\tau}=0\text{ or }(Z2_{\tau}=1\text{ and
}(Y1_{\tau}=0\text{ or }Y2_{\tau}=0))\\\ \end{cases}$

##  Appendix B Summary of Natural Direct Effect Parameters

Table 2: Definition and interpretations of alternative target natural direct effect parameters. Parameter |  Interpretation (General) |  Interpretation (SEARCH) |  Counterfactuals (SEARCH) |  Hypothetical Intervention
---|---|---|---|---
Marginal mediator distributionsbbbDenominator is equal to $P[Z2_{\tau}(a=0)=1]$ |  |  |  |
$\begin{aligned}
&\frac{P[Y_{\tau}(a=1,\bar{G}^{a=0})=1]}{P(Z2_{\tau}(\bar{G}^{a=0})=1)}-\frac{P[Y_{\tau}(a=0,\bar{G}^{a=0})=1]}{P(Z2_{\tau}(\bar{G}^{a=0})=1)}\\\
&=P[Y1_{\tau}(a=1,\bar{G}^{a=0})=0,Y2_{\tau}(a=1,\bar{G}^{a=0})=0|Z2_{\tau}(\bar{G}^{a=0})=1]\\\
&=P[Y1_{\tau}(a=0,\bar{G}^{a=0})=0,Y2_{\tau}(a=0,\bar{G}^{a=0})=0|Z2_{\tau}(\bar{G}^{a=0})=1]\end{aligned}$
where $\bar{G}^{a=0}\equiv(G_{1}^{a=0},\dots,G_{\tau}^{a=0})$ and
${G}_{t}^{a=0}\big{(}z_{t}|l_{0}\big{)}\equiv P\big{(}Z_{t}(a=0)=z_{t}|L_{0}=l_{0}\big{)}$ |  • A natural direct effect of an intervention on infant health outcomes where the live birth process is drawn from its true distribution of live births conditional on baseline covariates under the control condition • Includes: any effect of the intervention on the outcome that operates through mechanisms other than the live birth process • Excludes: any effect of the intervention on infant health outcomes through changes to the timing/occurrence of live births and to the types of people who give birth |  • The natural direct effect of the SEARCH intervention on infant health outcomes where the live birth process is drawn from the true distribution of live births conditional on baseline covariates under the control condition |  • $Y_{\tau}(a,\bar{G}^{a^{\prime}})$ represents counterfactual live birth and infant HIV-free survival at $t=\tau$ under an intervention to set $A=a$ and draw the mediators from their (unknown) distribution under $A=a^{\prime}$ conditional on baseline covariates • $Y1_{\tau}(a,\bar{G}^{a^{\prime}})$ represents counterfactual infant vital status at $t=\tau$ under an intervention to set $A=a^{\prime}$ and draw maternal survival and time-to-live birth from their distribution under $A=a^{\prime}$ conditional on baseline covariates • $Y2_{\tau}(a,\bar{G}^{a^{\prime}})$ represents counterfactual infant HIV status at $t=\tau$ under an intervention to set $A=a^{\prime}$ and draw maternal survival and time-to-live birth from their distribution under $A=a^{\prime}$ conditional on baseline covariates • $Z2_{\tau}(\bar{G}^{a^{\prime}})$ is the counterfactual live birth status at $t=\tau$ under the stochastic intervention |  • Compare intervening to set $A=1$ for everyone to setting $A=0$ for everyone and drawing the live birth process from the true, unknown live birth distribution conditional on baseline covariates under the control exposure
Conditional mediator distributions |  |  |  |
$\begin{aligned}
&\frac{P[Y_{\tau}(a=1,\bar{\Gamma}^{a=0})=1]}{P(Z2_{\tau}(\bar{\Gamma}^{a=0})=1)}-\frac{P[Y_{\tau}(a=0)=1]}{P(Z2_{\tau}(a=0)=1)}\\\
&=P[Y1_{\tau}(a=1,\bar{\Gamma}^{a=0})=0,Y2_{\tau}(a=1,\bar{\Gamma}^{a=0})=0|Z2_{\tau}(\bar{\Gamma}^{a=0})=1]\\\
&-P[Y1_{\tau}(a=0)=0,Y2_{\tau}(a=0)=0|Z2_{\tau}(a=0)=1]\end{aligned}$
where $\bar{\Gamma}^{a=0}\equiv(\Gamma_{1}^{a=0},\dots,\Gamma_{\tau}^{a=0})$
and
${\Gamma}_{t}^{a=0}\big{(}z_{t}|\bar{l}_{t},\bar{z}_{t-1}\big{)}\equiv P\big{(}Z_{t}(a=0)=z_{t}|\bar{L}_{t}(a=0)=\bar{l}_{t},\bar{Z}_{t-1}(a=0)=\bar{z}_{t-1}\big{)}$ |  • A natural direct effect of an intervention on infant health outcomes where the live birth process is drawn from its true distribution of live births conditional on baseline and time-varying history under the control condition • Includes: any effect of the intervention on the outcome through changes to time-varying covariates which may in turn impact the timing/occurrence of live births and the types of people who give birth • Excludes: any effect of the intervention on infant health outcomes that operating only through the survival of those at risk for pregnancy and/or live birth |  • The natural direct effect of the SEARCH intervention on infant HIV-free survival if the live birth process was drawn from its true, unknown distribution conditional on baseline and time-varying covariate history under the control condition |  • $Y_{\tau}(a,\bar{\Gamma}^{a^{\prime}})$ is counterfactual live birth and infant HIV-free survival at $t=\tau$ under an intervention to set $A=a$ and draw maternal survival and time-to-live birth from their distribution under $A=a^{\prime}$ conditional on observed baseline and time-varying covariate history • $Y_{\tau}(a)$ is counterfactual live birth and infant HIV-free survival at $t=\tau$ under an intervention to set $A=a$ • $Y1_{\tau}(a,\bar{\Gamma}^{a^{\prime}})$ is counterfactual infant vital status at $t=\tau$ under an intervention to set $A=a$ and draw maternal survival and time-to-live birth from their distribution under $A=a^{\prime}$ conditional on observed baseline and time-varying covariate history • $Y1_{\tau}(a)$ is counterfactual infant vital status at $t=\tau$ under an intervention to set $A=a$ • $Y2_{\tau}(a,\bar{\Gamma}^{a^{\prime}})$ is counterfactual infant HIV status at $t=\tau$ under an intervention to set $A=a$ and draw maternal survival and time-to-live birth from their distribution under $A=a^{\prime}$ conditional on observed baseline and time-varying covariate history • $Y2_{\tau}(a)$ is counterfactual infant HIV status at $t=\tau$ under an intervention to set $A=a$ • $Z2_{\tau}(a,\bar{\Gamma}^{a^{\prime}})$ is the counterfactual live birth status at $t=\tau$ under this stochastic intervention conditional on baseline and time-varying covariate history • $Z2_{\tau}(a)$ is the counterfactual live birth at $t=\tau$ under an intervention to set $A=a$ |  • Compare intervening to set $A=1$ for everyone to setting $A=0$ for everyone while drawing the live birth process from its true, unknown distribution conditional on baseline and time-varying history