From the map Eq.~\eqref{eq-updatetrue}, we can derive a stochastic master equation (SME) for the unnormalized state, which averages to the unconditioned master equation. By referring to Eq.~\eqref{eq-masterunnorm} and the discussion surrounded, we should treat the measurement result $u_t$ as having a zero-mean Gaussian statistics in order to generate a correct statistics for the unnormalized conditioned states. Therefore, we can derive the SME in the It\^o interpretation~[148, 149] by taking $lim_→ u_t^2 ^2 = $, which gives
\begin{align}\label{eq-untruequbit}
\dd \trho\god = & - i\, \dt\, [\hat H, \trho\god] + \dt\, {\cal D}[{\hat c}_{\rm u}]\trho\god + u_t \dt \, {\bar{\cal H}}[{\hat c}_{\rm u}]\trho\god -  \dt\, {\bar{ \cal H}}\left[\tfrac{1}{2} {\hat c}_{\rm o}^{\dagger}{\hat c}_{\rm o} \right]\trho\god.
\end{align}
Here we are omitting the $t$-dependence argument for the state whenever possible. Moreover, since the initial state $ρ(t=0)$ is a pure state, the true state at any time is always pure, and Eq.~\eqref{eq-untruequbit} could be replaced with a stochastic Schr\"odinger equation (SSE) [124, 18] or, indeed, something even simpler as we will see below in an equation for the qubit's angle $θ$.

%We also use the fact that, for the fluorescence channel, $\hat c_{\rm u} \propto \hat\sigma_{-}$, we have $\hat c_{\rm u}^2 = ( \hat c_{\rm u}^\dagger)^2 = 0$, making the term $\bar{\cal H}[\hat c^2_{\rm u}]$ in Eq.~\eqref{eq-updatetrue} vanish.

For completeness, we also give the normalized version of the SME in Eq.~\eqref{eq-untruequbit}:
\begin{align}\label{eq-truequbit}
\dd \rho\god =& - i\, \dt\, [\hat H, \rho\god] + \dt\, {\cal D}[{\hat c}_{\rm u}]\rho\god  -  \dt\, {\cal H}\left[\tfrac{1}{2} {\hat c}_{\rm o}^{\dagger}{\hat c}_{\rm o} \right]\rho\god + \dt\{ u_t - {\rm Tr}[ (\hat c_{\rm u} + \hat c_{\rm u}^\dagger) \rho\god ] \} \, {\cal H}[{\hat c}_{\rm u}]\rho\god.
\end{align}
This is the usual It\^o SME for a normalized state conditioned on both Alice's no-jump and Bob's diffusive records, where $ H[ĉ] ∙= ĉ ∙+ ∙ĉ^† - Tr(  ĉ ∙+ ∙ĉ^† )∙$. Note that the statistics of the unknown record $u$ in this case is governed by its actual conditional PDF $℘_(u)$. The PDF can be approximated as a Gaussian function with a mean given by $_| = Tr[ (ĉ_u + ĉ_u^†)ρ]$; see Eq.~\eqref{eq-app-probupast} in Appendix~\ref{sec-app-pdf}. Therefore, the SME for the normalized state in
Eq.~\eqref{eq-truequbit} is usually seen written in terms of an innovation, $w(t) ≡u_t - Tr[ (ĉ_u + ĉ_u^†) ρ]$, which has the same statistics as the Wiener increment, \ie, $w= 0$ and $w^2 = $. %The second and third terms represent the diffusive ($y$-homodyne) dynamics, and the last term describes the backaction from no-jump observed record. We defined the Lindblad operators ${\hat c}\subp = \sqrt{\Gamma} \,{\hat \sigma}_- e^{-i \Phi} $ and ${\hat c}\subn = \sqrt{\gamma} \,{\hat \sigma}_-  $ for the diffusive and jump processes, with the coupling rate $\Gamma$ and $\gamma$, respectively. The diffusive measurement record and the Wiener increment $\dd W$ is related by $\dd J_{\Phi,t}= {\rm Tr}\{({\hat c}\subp + {\hat c}\subp^\dagger)\rho\god(t)\} \dt+ \dd W(t)$. We note that the possible true state $\rho\god$ at any time is always pure, if the initial state $\rho\god(t=0) = \rho\god(t_j)$ is a pure state.

Alice, with only the record of observed jumps, can calculate her filtered state trajectory $ρ$ conditioned on her past observed record following Eq.~\eqref{eq-filstate}. This can also be obtained by averaging over all possible unknown records from Eq.~\eqref{eq-truequbit}, which is equivalent to averaging the equation over the innovation $w$, yielding
\begin{align}\label{eq-filterqubit}
\dd \rho\fil = \!- i\, \dt\, [{\hat H}, \rho\fil] + \dt\, {\cal D}[{\hat c}_{\rm u}]\rho\fil -  \dt\, {\cal H}\left[\tfrac{1}{2} {\hat c}_{\rm o}^{\dagger}{\hat c}_{\rm o} \right]\rho\fil,
\end{align}
which is Eq.~\eqref{eq-truequbit} with its last term removed.
Contrary to the true state dynamics, this filtered state does not have unit purity, even though the initial state is pure. However, for a better estimate of the true state, the observer can use the entire observed record (both past-future information) in the estimation, following the estimators based on the cost functions presented in Section~\ref{sec-sevenQSE}.

Since, the true qubit state in Eq.~\eqref{eq-truequbit} is limited to the $y$--$z$ great circle of the Bloch sphere, other states, such as the unconditioned state Eq.~\eqref{eq-master}  or the filtered state Eq.~\eqref{eq-filterqubit}, should always be on the $y$--$z$ plane. Therefore, we can reparametrize the qubit states with two parameters, $θ$ and $R$, where $z = R cosθ$ and $y= R sinθ$ are the two coordinates of the Bloch vector (for the pure true state, $R = 1$). The excited state is $θ= 0$ and the ground state is $θ= π$. The normalized and unnormalized true states at any time $t$ are given by $ρ(t) = (1/2)( Î + sinθ_t σ̂_y + cosθ_t σ̂_z)$ and $(t)= (_t/2)( Î + sinθ_t σ̂_y + cosθ_t σ̂_z)$, respectively, where $_t ≡Tr[(t)]$ takes care of the state's norm. We then obtain the state update Eq.~\eqref{eq-updatetrue} for these new variables,
\begin{subequations}\label{eq-updatetheta}
\begin{align}
\theta_{t+\ddt} = &\,\, \theta_t - \ddt\, \Omega + \ddt \tfrac{\gamma}{2} \sin\theta_t + (\ddt-u_t^2\ddt^2)\tfrac{\Gamma}{2}\sin\theta_t - u_t^2 \ddt^2\tfrac{\Gamma}{2}  \cos\theta_t \sin\theta_t - u_t \ddt \sqrt{\Gamma}(\cos\theta_t+1), \\
\lambda_{t+\ddt} =& \,\, \lambda_t - \ddt \tfrac{\gamma}{2} ( \cos\theta_t + 1 ) \lambda - u_t \ddt \sqrt{\Gamma} \sin\theta_t \lambda_t  - (\ddt- u_t^2\ddt^2)\tfrac{\Gamma}{2} \lambda_t - (\ddt- u_t^2\ddt^2)\tfrac{\Gamma}{2} \cos\theta_t \lambda_t ,
\end{align}
\end{subequations}
with terms up to the order $u^2 ^2$. These equations will be used in deriving two types of differential equations in the next subsections, as mentioned above.

In the following, we present the calculation methods for the eight estimators, and the numerical simulation results, in three subsections. Each subsection contains the estimators that are calculated with similar mathematical techniques, which are the Fokker-Planck equation, the CDJ most-likely path technique, and the local optimal records. We finish the section with the calculation of the expected cost functions and their averages for all possible jump times, where possible.

%The first subsection involves the calculation of the first three estimators, for which their cost functions are all defined in the state's \red state  matrix space. We proposed a technique using a \emph{quasi} Fokker-Planck equation to solve for a probability density function for true states parametrized in a qubit's angle variable. For the second subsection,..  Because there is a mapping between a Langevin equation to the Fokker-Planck equation, we will use this technique to simplify our problem, and not having to simulate trajectories as in [REF]. We note that the calculations are only for the qubit's evolution between any two jumps, since the jump records reset the qubit to the ground state making the evolution between jumps independent from each other.

\subsection{State estimators with Fokker-Planck equation}\label{sec-fpe}
The first three cost functions, $Q_1$--$Q_3$, are defined in the space of unknown true states at any time, from Eq.~\eqref{eq-qss-alter}, Eq.~\eqref{eq-th-q-nf} and Eq.~\eqref{eq-th-q-ne}. The three estimators are, respectively, the mean, the max-eigenstate of the mean, and the mode, of the PDF $℘_(ψ̂)$ of the true state conditioned on the past-future observed record. One could use the stochastic simulation to find the PDF of the true state. However, in this subsection, the true state can be parametrized by the one-dimensional variable, $θ$. This allows us to solve the PDF $℘_(θ)$ via a semi-analytical method similar to solving a Fokker-Planck equation.

The dynamical equations for the true state are shown in Eq.~\eqref{eq-untruequbit} (unnormalized) and Eq.~\eqref{eq-truequbit} (normalized version) and one might be tempted to think that the equation for the normalized true state could be enough. However, as mentioned in the quantum state smoothing formalism, Section~\ref{sec-qss}, the correct statistics of true states or unknown records, conditioned on the fixed observed record, cannot be obtained from the normalized state and should be derived from the unnormalized state equations and their norms. Therefore, let us refer back to the PDF of the unobserved record written in terms of the unnormalized state in Eq.~\eqref{eq-bothoprob1}. From this, we can derive a general form of the PDF $℘_(ψ̂)$ of the true state:
\begin{align}\label{eq-findpdf}
\wp_{\pasts{\bo}}(\hat\psi) \propto & \int \!\! \dd\mu(\,\past{u}\,)  \delta[\hat\psi - \rho_{\pasts{\bo}, \pasts{u}}] \wp\ost(\,\past{u}\,) {\rm Tr}\left( {\trho}_{\pasts{\bo}, \pasts{u}} \right), \nonumber \\
= & \int \!\! \dd\mu(\,\past{u}\,) \!\! \int\!\! \dd \lambda\, \delta[\lambda - {\rm Tr}(\trho_{\pasts{\bo}, \pasts{u}})] \,\delta[\hat\psi - \rho_{\pasts{\bo}, \pasts{u}}] \, \wp\ost(\,\past{u}\,) \lambda.
\end{align}
In the second line, we added the $δ$-function mapping the state norm to the variable $λ$. This is so that we can separately define an ostensible PDF,
\begin{align}
Q_{\pasts{\bo}}(\lambda, \hat\psi) =  \int \!\! \dd\mu&(\,\past{u}\,)\,  \delta[\lambda - {\rm Tr}(\trho_{\pasts{\bo}, \pasts{u}})]\,  \delta[\hat\psi - \rho_{\pasts{\bo}, \pasts{u}}]\, \wp\ost(\,\past{u}\,),
\end{align}
which has an advantage that it can be solved for using the dynamical equation for the unnormalized state and its norm, for example, as in Eq.~\eqref{eq-updatetheta}.

It is important to note that the ostensible PDF $Q_(λ, ψ̂)$ is not the actual joint PDF of $λ$ and $ψ̂$. However, the actual PDF of the true state can be obtained via
\begin{align}\label{eq-quasi-int}
\wp_{\pasts{\bo}}(\hat\psi) \propto \tilde\wp_{\pasts{\bo}}(\hat\psi) =  \int \!\! \dd \lambda \, Q_{\pasts{\bo}}(\lambda, \hat\psi) \lambda,
\end{align}
where the tilde indicates an unnormalized PDF, \ie, $℘_(ψ̂) = ℘̃_(ψ̂) / ∫μ_H(ψ̂) ℘_(ψ̂)$. Note the multiplication by $λ$ in the integrand in Eq.~\eqref{eq-quasi-int}. The past-future conditional PDF $℘_(ψ̂)$ is then given by the Bayesian rule in Eq.~\eqref{eq-bayes}. For the particular example of the qubit pure state parametrized by the Bloch angle $θ$, we can use $μ_H(ψ̂) = θ$ as the Haar measure and replace
\begin{align}
\hat\psi = \hat S(\theta) = (1/2)( \hat I + \sin\theta \hat \sigma_y + \cos\theta \hat \sigma_z),
\end{align}
where the PDF of the pure state becomes $℘_(θ)$.

In order to solve for the ostensible PDF $Q_(λ, θ)$, we start with a Langevin-type equation for the unnormalized state from Eqs.~\eqref{eq-updatetheta}. As we mentioned in the previous subsection, the statistics of $u$ are governed by the ostensible PDF $℘(u)$, appropriate to the unnormalized state. Therefore, we can take $v = lim_→ u_t$ to have the same statistics as the Wiener increment. This then leads to the It\^o rule, $ lim_→ u_t^2^2 = $, which makes the terms with $(- u_t^2^2)$ in Eqs.~\eqref{eq-updatetheta} disappear. We obtain a two-dimensional Langevin equation for $θ$ and $$,
\begin{subequations}
\begin{align}\label{eq-langevin}
\dd \theta = & \, A_{\theta}(\theta) \dt  + B_{\theta}(\theta) \dd v,\\
\dd \qp  = & A_\qp(\qp, \theta) \dt + B_\qp(\qp,\theta) \dd v,
\end{align}
\end{subequations}
where we have defined
\begin{subequations}
\begin{gather}
A_\theta(\theta) = - \Omega  + \tfrac{\gamma}{2}  \sin\theta - \tfrac{\Gamma}{2}  \cos\theta \sin\theta, \quad B_\theta(\theta) = - \sqrt{\Gamma} (\cos\theta + 1),\\
A_\qp(\qp,\theta) = - \tfrac{\gamma}{2}   (\cos\theta+1)\, \qp,  \quad B_\qp(\qp, \theta) = -  \sqrt{\Gamma} \sin\theta \, \qp.
\end{gather}
\end{subequations}
From the Langevin equation, we can write a corresponding Fokker-Planck equation for $Q_(,θ)$, describing the diffusion in the two-dimensional space $(θ,)$,
\begin{align}\label{eq-FPE}
\partial_t Q_{\pasts{\bo}}&(\qp, \theta) = - \sum_{j} \partial_j [ A_j(\qp,\theta) \, Q_{\pasts{\bo}}(\qp,\theta )] +  \sum_{i,j} \tfrac{1}{2}\partial_i\partial_j \left[ B_i (\qp,\theta) B_j (\qp,\theta) \, Q_{\pasts{\bo}}(\qp,\theta) \right],
\end{align}
where the summations are over the coordinate labels, $i, j ∈{ , θ}$. %which can be easily normalized by dividing it with a norm ${\cal N}(t) = \int \! \dd \theta\, \twp_{\past\bo}(\theta,t)$.

The unnormalized true PDF in the $θ$ variable is defined in Eq.~\eqref{eq-quasi-int}.
Substituting this into Eq.~\eqref{eq-FPE}, we obtain a partial differential equation for the unnormalized true PDF,
\begin{align}\label{eq-PDEsol}
\partial_t \twp_{\pasts{\bo}}(\theta) = &- \tfrac{\gamma}{2}(1+\cos\theta)\twp_{\pasts{\bo}}(\theta)  + \partial_\theta \left\{  \left[ \sqrt{\Gamma}\sin\theta \, B_\theta(\theta)- A_\theta(\theta) \right] \twp_{\pasts{\bo}}(\theta)\right\} +   \tfrac{1}{2} \partial^2_\theta\left[ B_\theta^2(\theta) \twp_{\pasts{\bo}}(\theta) \right].
\end{align}
This can be solved numerically with an appropriate initial condition and a periodic boundary condition for the variable $θ$. Thus we can finally obtain the normalized past-only and past-future conditional PDFs of the true state,
\begin{align}
\wp_{\pasts{\bo}}(\theta)  = &  \frac{\twp_{\pasts{\bo}}(\theta)}{\int\!  \dd \theta  \, \twp_{\pasts{\bo}}(\theta)},\\
\wp_{\bothps{\bo}}(\theta) =  & \frac{\twp_{\pasts{\bo}}(\theta) {\rm Tr} \big [ \hat E_{\futps{\bo}}\, \hat S(\theta) \big ] }{\int\!  \dd \theta\, \twp_{\pasts{\bo}}(\theta) {\rm Tr}\big[\hat E_{\futps{\bo}} \,\hat S(\theta) \big ]}.
\end{align}
We show in Appendix~\ref{sec-app-retroeff} how the retrofiltered matrix $Ê_$ can be simply calculated from the no-observed-jump evolution. These PDFs above are then used to compute the smoothed state (the estimator for $Q_1$):
\begin{align}
\rho_{\textbf{Q}_1} = &\, \rho\sm =   \int\!\! \dd \theta \, \hat S(\theta) \,\wp_{\bothps{\bo}}(\theta).
\end{align}
Moreover, following Eqs.~\eqref{eq-th-q-nf} and \eqref{eq-th-q-ne}, the other two estimators in this subsection, $ρ_Q_2$ and $ρ_Q_3$, are found as
\begin{align}
\rho_{\textbf{Q}_2} & =  \hat{\psi}\sm^{\rm max} = \hat S(\theta_{\textbf{Q}_2}),\\
\rho_{\textbf{Q}_3} & = \hat S(\theta_{\textbf{Q}_3}).
\end{align}
The first line corresponds to the max-eigenvalue eigenvector of the smoothed state, \ie, $ ρ|ψ^max = λ^max|ψ^max $, which leads to $θ_Q_2 = Tr[(σ̂_z + i σ̂_y)ρ]$. The second line is the most-likely state obtained from the most-likely angle, $θ_Q_3 = _θ ℘_(θ)$.

\begin{figure}[t]
\centering
\includegraphics[width=16.6cm]{Fig5-trajs.pdf}
\caption{Dynamics of the eight estimators (seven estimators of the true states, $\rho_{\textbf{Q}_1}$\,--\,$\rho_{\textbf{Q}_7}$, and the SWV state $\varrho_{\textbf{Q}_8}$) for the example of a resonantly driven and damped qubit. The estimators at any time $t$ are conditioned on the past-future observed measurement record, which is a string of photon detection events. The time interval shown, $[0, T)$, is the block of time between two such detections (jumps),
not including the state after the second jump (which is the same as its initial state, except for the case $\varrho_{\textbf{Q}_8}$).   The eight estimators are shown in: (a) $y$-coordinate, (b) $z$-coordinate, (c) qubit's angle $\theta$ around the $x =0$ plane, and (d) qubit's radius $R$. The usual filtered (past-only) quantum trajectory $\rho\fil$ is also shown in blue curves for comparison. The parameters used in the numerical simulation are: $T = 4 T_\gamma$ and $\Omega = 2 T_\gamma^{-1}$, in the units of the system's total decay time  $T_\gamma$.  The parameters are chosen such that we can see an interesting feature: the discontinuity in the most-likely state $\rho_{\textbf{Q}_3}$ right after the time $t = 2.09 T_\gamma$.}
\label{fig-trajs}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics[width=11cm]{Fig6-blochprob.pdf}
\caption{The Bloch great circles (dotted black) and the PDFs of (pure) true states, $\wp_{{\protect \pasts{\bo}}}(\theta)$ (solid blue curves) and $\wp_{{\protect \bothps{\bo}}}(\theta)$ (solid red curves), at any qubit's angle $\theta \in (0,2\pi]$, for two different times: (a) $t = 2.09 T_\gamma$ and (b) $t = 2.15 T_\gamma$. The `height' of the probability density is measured in the radial direction. Colored dots show some qubit's state estimators ($\rho\fil$, $\rho\sm$, $\rho_{\textbf{Q}_2}$, $\rho_{\textbf{Q}_3}$ and $\rho_{\textbf{Q}_5}$) for both times, using the same color scheme as in Figure~\ref{fig-trajs}. The time of the final jump, and other parameters, are the same as in Figure~\ref{fig-trajs}. Note the interesting large jump that happens for the most-likely state $\rho_{\textbf{Q}_3}$ (magenta) between the two times. The orange dashed lines are to show that the estimator $\rho_{\textbf{Q}_2}$ (cyan) always has exactly the same Bloch angle as the smoothed state (red).}
\label{fig-blochprob}
\end{figure}

We show in Figure~\ref{fig-trajs} the numerical results of qubit state estimators between two jumps, compared with the regular filtered quantum trajectory $ρ$ obtained from Eq.~\eqref{eq-filterqubit} (blue curves). We solve the partial differential equation in Eq.~\eqref{eq-PDEsol} (using Mathematica's `NDSolve' package) with a delta-function initial state replaced with an approximated narrow-width Gaussian function $℘_t=0 (θ) = (2πσ^2)^-1/2exp[ - (θ-π)^2/2σ^2]$, for the initial ground state $θ_0 = π$. We chose $σ= 0.01$, which gives the value at its peak to be $℘_t=0(π) ≈39.89$ instead of infinity.

The first three estimators, $ρ_Q_1 = ρ$, $ρ_Q_2$, and $ρ_Q_3$, are shown in red, dashed light blue, and dot-dashed magenta curves, respectively. The smoothed state $ρ$ is not necessarily pure, while the other two estimators are pure states with unit radius; see Figure~\ref{fig-trajs}(d). At the final time, $T=4 T_γ$, one might expect that $ρ(T) = ρ(T)$ since the conditioning on the past record should be the same as the conditioning on the whole record at the final time. However, this is not true for our example because the last jump is not included in the past record $$. One could include the last jump by considering the state estimators with one more time step, but all estimators would simply jump to the ground state at the new final time. Also note that, in Figure~\ref{fig-trajs}(c), the estimator $ρ_Q_2$ always has exactly the same Bloch angle as the smoothed state  (also shown as orange dashed lines in Figure~\ref{fig-blochprob}). This is because $ρ_Q_2$, in Eq.~\eqref{eq-th-q-nf}, is simply the purest state possible with the same Bloch angle as the smoothed state.

We have chosen the parameter regime (detail in Figure~\ref{fig-trajs}'s caption) such that we can see an interesting feature: the discontinuity in the most-likely state $ρ_Q_3$ at $t = 2.09 T_γ$. This occurs because the most-likely state follows the highest point of the PDF of true states Eq.~\eqref{eq-bayes} at any local time. The PDF can have multiple competing peaks, which results in most-likely states jumping between local maxima as the PDF changes in time. We show in Figure~\ref{fig-blochprob} how the PDFs and the jumping states (magenta dots) look, right before the jump time, $t = 2.09 T_γ$, and not long after the jump, $t = 2.15 T_γ$. Also, in Figure~\ref{fig-blochprob}(a)-(b), we can see that the smoothed state and $ρ_Q_2$ have the same angle on the Bloch sphere at both times.

%\label{eq-maptheta} \bm{{\cal E}}(\theta\ik, u\ik, \bo\ik = 0) = &\,\, \theta\ik -\dt \, \Omega + \dt \tfrac{\Gamma + \gamma}{2}  \sin\theta\ik \nonumber \\
%& - u\ik \dt  \sqrt{\Gamma}(\cos\theta\ik+1) ,

\subsection{State estimators with CDJ optimization}\label{sec-mlp}
For the estimators of $Q_4$ and $Q_5$, we need to optimize the PDFs of the unknown measurement record conditioned on the observed record. We follow the optimization approach presented in Section~\ref{sec-sevenQSE}. As in the previous subsection, we are interested in the block of qubit's dynamics between any two jumps, where the observed record is given by $ = { _0 , _, ..., _T- ,_T }= { 0,0,..., 0,1}$ and the unknown record $u$ is from the $y$-homodyne measurement.

Following the CDJ optimization, we need to solve the difference equations in Eqs.~\eqref{eq-diffeq1} and Eqs.~\eqref{eq-diffeq2}, for the estimators $ρ_Q_4$ and $ρ_Q_5$, respectively. These difference equations have to be derived from the state update for normalized states, given by the $θ$ equation in Eqs.~\eqref{eq-updatetheta}, and the PDFs of time-local unknown measurement results. For the state update, since a solution of the difference equations gives smooth optimal unknown records and their corresponding smooth state paths, we need to treat the variable $u_t$ as zeroth order in $$, instead of $O(^-1/2)$ as in the Fokker-Planck approach. Therefore, the state mapping $θ_t+ = E(θ, u, =0)$ is given by
\begin{align}
\theta_{t+\ddt} = &\, \theta_t - \ddt\, \Omega + \ddt \tfrac{\gamma+\Gamma}{2} \sin\theta_t - u_t \ddt \sqrt{\Gamma}(\cos\theta_t+1),  \\
\label{eq-lambdt}\lambda_{t+\ddt} =& \, \left[1 - \ddt \tfrac{\gamma+\Gamma}{2} ( \cos\theta_t + 1 ) - u_t \ddt \sqrt{\Gamma} \sin\theta_t \right ] \lambda_t.
\end{align}
This is the update equation for the true state Eqs.~\eqref{eq-updatetheta} keeping up to first order in $$. For the PDF of the unknown result, we can use the equation of the state norm,
\begin{align}
\wp(u\ik, \bo\ik = 0 | \theta\ik) =&\, \, \wp\ost(u_t) {\rm Tr}[{\cal N}_0 {\cal M}_{u\ik} \hat S(\theta\ik) ] \nonumber \\
= & \,\, \wp\ost(u_t)\lambda_{t+\ddt}(\theta\ik, u\ik,\lambda_t)/\lambda_t,
\end{align}
with the form of $λ_t+$ given in Eq.~\eqref{eq-lambdt}. This leads to the logarithm of the probability to first order in $$,
%\wp(u\ik, \bo\ik = 0 | \theta\ik) \approx & \exp\left\{ - \frac{\dt}{2} \left(u\ik + \sqrt{\Gamma}\sin\theta\ik \right)^2 \right\} \nonumber \\
%& \times \exp\left\{- \frac{\dt}{2}\gamma\,(1+\cos\theta\ik) \right\},
%where the measurement operation is defined with a measurement operator ${\hat M}_{u\ik,O\ik=0} = (\dt/2\pi)^{1/4}\exp(u_t^2 \dt/4)\{\hat{1} - \tfrac{1}{2}({\hat c}\subn^\dagger {\hat c}\subn + {\hat c}\subn^2 + {\hat c}\subp^\dagger {\hat c}\subp + {\hat c}\subp^2) \dt +{\hat c}\subp u_t \dt + {\cal O}(\dt^2)\}$~\cite{Wong2020} (see Appendix~\ref{sec-app-ppdf} for derivations).
\begin{align}
\ln \wp(u\ik, & \bo\ik = 0 | \theta\ik) = \tfrac{1}{2} \ln (\ddt/2\pi) -  \ddt \left[ \tfrac{1}{2} u_t^2 +  \tfrac{\gamma+\Gamma}{2} ( \cos\theta_t + 1 ) + u_t  \sqrt{\Gamma} \sin\theta_t \right].
\end{align}
Putting all the components together, we can construct the difference equations as in Eqs.~\eqref{eq-diffeq1} or \eqref{eq-diffeq2}.
Taking the continuum limit $→$, we obtain a set of ordinary differential equations (ODEs) for $θ_t$, its conjugate $p_t$, and an optimal record $u_t$,
\begin{subequations}\label{eq-diffeqcont}
\begin{align}
\partial_t \theta_t  = & - \Omega  + \tfrac{\Gamma+\gamma}{2} \sin\theta_t -  \sqrt{\Gamma}(\cos\theta_t+1) u_t, \label{eq-thetaeq}\\
\partial_t p_t = &  -  \tfrac{\Gamma+\gamma}{2}\,p_t\, \cos\theta_t - \sqrt{\Gamma} \, p_t\, \sin\theta_t\, u_t
+ \sqrt{\Gamma}\cos\theta_t\, u_t - \tfrac{\Gamma + \gamma}{2}  \sin\theta_t ,\\
u_t = & -  \sqrt{\Gamma}\left( p_t + p_t \cos\theta_t + \sin\theta_t \right).
\end{align}
\end{subequations}
The initial condition for these ODEs is the qubit's initial state $θ_0 = π$ (ground state). However, the final condition for the ODEs will depend on the problem of interest; that is, whether Eqs.~\eqref{eq-diffeqcont} will be used for $Q_4$ or $Q_5$.

Let us first consider the simpler case, $Q_5$, where the optimal unknown record and its corresponding state estimator $ρ_Q_3$ are calculated from the ODEs just once with the final boundary condition at time $T$. Following Eqs.~\eqref{eq-diffeq2}, the final condition is in the third line. Using the final observed jump record, we get $Ê_T = ĉ_o^† ĉ_o = γ|ee |$ and $℘(_T = 1 | θ_T) = Tr[ ĉ_o^† ĉ_o Ŝ(θ_T) ] = γ2(1+cosθ_T)$. This leads to the final condition for the conjugate variable,
\begin{align}\label{eq-finalbc2}
p_T =  \,  - \frac{\sin\theta_T}{1+\cos\theta_T},
\end{align}
where we have approximated $p_T = lim_→ p_T-δt$ for the time-continuum limit. Solving Eq.~\eqref{eq-diffeqcont} with the final condition Eq.~\eqref{eq-finalbc2}, we obtain the optimal $u = { u_t : t ∈[0, T)}$ and its state path ${ θ_t : t ∈[0, T]}$, where the latter is exactly the qubit state estimator $ρ_Q_5$.

For the estimator of $Q_4$, the calculation is more involved, since we need to solve the ODEs for every intermediate time $t=τ$ during the time period of interest $τ∈(0, T]$. Following Eqs.~\eqref{eq-diffeq1}, we numerically solve the ODEs, Eqs.~\eqref{eq-diffeqcont}, for discrete values of $τ∈{ , 2, ..., T}$, with the final conditions,
\begin{align}
p_\tau = & \,\,\frac{\partial}{\partial \theta_\tau} \ln {\rm Tr} [ {\hat E}_{\futps{\bo}}\, \hat S(\theta_\tau) ]  \nonumber \\
= &\, \frac{  - \zeta \sin\theta_\tau +  \beta \cos\theta_\tau}{\alpha + \zeta \cos\theta_\tau +  \beta \sin\theta_\tau},
\end{align}
for any time $τ$. Here we have defined the retrofiltered matrix's elements as
\begin{align}
{\hat E}_{\futps{\bo}}=  \alpha\, \hat 1 + \beta \, \hat\sigma_y + \zeta \, \hat\sigma_z,
\end{align}
where all elements are functions of $τ$ (see Appendix~\ref{sec-app-retroeff}). In contrast to the previous $ρ_Q_5$, the state estimator $ρ_Q_4$ is a series of $θ_τ$, each of which is a `final' qubit's state via the solution of the ODEs, Eqs.~\eqref{eq-diffeqcont}, for each value of $τ∈{ , 2, ..., T}$. As noted before, the qubit state estimator $ρ_Q_4$ at the final time $τ= T$ is exactly the same as $ρ_Q_5$.

We solve for $ρ_Q_4$ and $ρ_Q_5$ using the numerical method and Python codes developed by Lewalle \emph{et~al.}~[134, 129]. Since the ODEs are non-linear, there can be multiple solutions, which serve as multiple optimal paths (in a similar way that a PDF could have multiple peaks). We therefore solve for all candidate most probable solutions and pick the one which does maximize the joint probability density of the path, \eg calculated from Eqs.~\eqref{eq-mlp1-prob} or \eqref{eq-probmlp2}. The solutions of $ρ_Q_4$ and $ρ_Q_5$ are shown as solid orange and dotted brown curves, respectively, in Figure~\ref{fig-trajs}. They are similar in shape, have unit purities, and the same final states $ρ_Q_4(T) = ρ_Q_5(T)$ as expected, but they are not exactly the same. In Figure~\ref{fig-blochprob}, only $ρ_Q_5$ is shown as the brown dots. The corresponding optimal record for $ρ_Q_5$ will be discussed more in Section~\ref{sec-eightestqubit}.

\subsection{State estimators using local optimal records}
For the last three estimators of $Q_6$--$Q_8$, their cost functions are defined with the time-local unknown records. We have assumed that the unknown record was from a continuous diffusive measurement ($y$-homodyne measurement for the qubit example). Therefore, the measurement can be thought of as a series of infinitely weak measurements, each with an infinitesimal duration $$. As mentioned in Section~\ref{sec-sevenQSEdiff}, the maximum value of the PDF in Eq.~\eqref{eq-probut} is the same as its mean value given by the weak value in Eq.~\eqref{eq-wv-gen2}. Thus we can calculate the state estimators $ρ_Q_6 = ρ_Q_7$ from the state update Eqs.~\eqref{eq-updatetheta} using the weak value record in place of $u$. For the last estimator $ϱ_Q_8$, we use Eq.~\eqref{eq-wvsO} to compute the SWV state. It is important to note that since, in our case, the Lindblad operator for the unknown measurement record $ĉ_u$ is not Hermitian, the SWV state is not associated with the weak value defined in Eq.~\eqref{eq-wv-gen}, \ie, $_τ_ Tr [ (ĉ_u + ĉ^†_u) ϱ_Q_8  ] $, given the definition of the mean in Eq.~\eqref{eq-wv-gen2}. This would be true if $ĉ_u$ were Hermitian, when Eqs.~\eqref{eq-wv-gen} and \eqref{eq-wv} coincide.

We show the numerical results for the last three estimators in Figure~\ref{fig-trajs} as dashed green (for $ρ_Q_6 = ρ_Q_7$) and solid dark green (for the SWV state $ϱ_Q_8$) curves. As expected, we see in Figure~\ref{fig-trajs}(d) that the former estimators are pure quantum states with unit Bloch radius, while the the SWV `state' is not even constrained within the valid quantum state space as it can have a Bloch radius larger than one.

\begin{figure}
\centering
\includegraphics[width=16.6cm]{Fig7-cost123.pdf}
\caption{Expected costs for ${\textbf{Q}_1}$, ${\textbf{Q}_2}$, and ${\textbf{Q}_8}$ (in panels (a)-(c), respectively) are numerically evaluated for all eight estimators. The first two expected costs are defined in the true state space: the trace square deviation (${\cal C}_1$) and the negative fidelity (${\cal C}_2$) from the true state. The optimal estimators for the two costs are the smoothed quantum state, $\rho_{\textbf{Q}_1}$, and the max-eigenvalue eigenvector of the smoothed state, $\rho_{\textbf{Q}_2}$. The last expected cost, ${\cal C}_8$, in panel (c), is defined in Eq.~\eqref{eq-cq8}, for which the SWV state, $\varrho_{\textbf{Q}_8}$, is the optimal estimator.}
\label{fig-cost1}
\end{figure}

%For the last, the expected cost is the negative probability density, $-\wp_{{\protect \bothp\bo}}(\theta,t)$, plotted on a log scale for the absolute value. This is numerically solved using a narrow-width Gaussian function as its initial state, so that $-\wp_{{\protect \bothp\bo}}(\theta = \pi,t=0) \approx -39.89$ \hmw{The plot seems to show it starting at $-50$. Also here's an idea to consider: just truncate the plot at $-5$ because they are all practically identical there, so that we can see the separation better. Q1 should not be in the legend.} instead of minus infinity at the initial time. The results show that the associated estimator for each of the costs gives the smallest values, as expected.

\subsection{Expected costs for qubit estimators}\label{sec-eightestqubit}

From the eight estimators presented in Figure~\ref{fig-trajs}, we can also show that they do optimize their corresponding expected cost functions compared to all the other estimators. We do this by evaluating, where possible, the expected costs (or equivalent versions theoreof) discussed in Section~\ref{sec-expcost} for the eight qubit estimators. We first numerically calculate the local-state expected costs that can be applied to all eight estimators. The expected costs are $C_1$, $C_2$, and $C_8$ and the results are shown in Figure~\ref{fig-cost1}(a), (b), and (c), respectively. We see that the estimators $ρ_Q_1$, $ρ_Q_2$, and $ϱ_Q_8$ give the minimum values for the expected costs $C_1$, $C_2$, and $C_8$, respectively.

For $C_3$, the expected cost (negative equality with the true state) can only be applied to pure-state estimators. This is because we assume the true state is pure and thus the likelihood for a mixed state to be the true state is zero. The results are shown in Figure~\ref{fig-cost2}(a), where the expected cost $C_3$ is evaluated for the pure-state estimators: $ρ_Q_2$\,--$ρ_Q_7$.
The most-likely state $ρ_Q_3$ gives the lowest value for the negative probability density at all time. As we have discussed in Section~\ref{sec-expcost}, the expected cost $C_4$ is excluded from our analysis because it is extremely difficult to calculate.
%\hmw{We should be clear: that means that the cost for the others is higher. They are not possible true states, so the probability is zero, which would be infinitely high on that log-scale plot. We should say that.}
%\hmw{Even though it is out of order, I think Q8 should go here. It is, after all, a local-state expected cost. If you want to avoid jumping backwards and forwards you could even start with Q8. After all, the plot looks closest to Q1, which is not surprising since classically (or for filtering only) they are the same estimator. So Fig. 4 would have 3 sublots, and Fig. 5 would have 3 also, including Q3, or 4, including a narrow rectangle for Q5. That way it is (almost) the same set of estimators being plotted in each subfigure for 4 and for 5. And so rewrite. But actually, how come 4(c) has a legend that says all 8 of the states are plotted there, when it should only be 5 of them. In fact, the red curve should definitely not be there.}

\begin{figure}
\centering
\includegraphics[width=16.6cm]{Fig8-cost567.pdf}
\caption{Expected costs for ${\textbf{Q}_3}$, ${\textbf{Q}_6}$ and ${\textbf{Q}_7}$ are numerically evaluated for relevant optimal and sub-optimal state estimators (pure-state estimators only). We show in (a) the expected costs ${\cal C}_3$ which can be applied to all six pure-state estimators: $\rho_{\textbf{Q}_2}$\,--$\rho_{\textbf{Q}_7}$, showing that the most-likely state, ${\rho}_{\textbf{Q}_3}$ is optimal. In panel (b), we show the associated (estimated) unknown records solved for $\rho_{\textbf{Q}_2}$, $\rho_{\textbf{Q}_4}$\,--$\rho_{\textbf{Q}_7}$, excluding $\rho_{\textbf{Q}_3}$ because it has a discontinuity in the state evolution (see texts). The modified expected costs ${\cal C}_6 = {\cal C}_7$ in Eqs.~\eqref{eq-cq6} and \eqref{eq-cq7} are shown in panel (c).}
\label{fig-cost2}
\end{figure}

For the expected costs defined in the unknown record space, $C_5$--$C_7$,
the relevant estimators have to also be pure states, because we need to solve for corresponding unknown records that reproduce them as possible evolutions of the pure true state.  The calculation for the unknown records can be done in different ways, but it is not guarantee that any pure-state evolution will have a physical corresponding unknown record. For $ρ_Q_2$, $ρ_Q_3$, and $ρ_Q_4$, their records can be calculated by reversing the dynamics in Eq.~\eqref{eq-thetaeq}. For $ρ_Q_5$ and $ρ_Q_6$ (or $ρ_Q_7$), their associated unknown records can be determined in the process of calculating the state estimators in Eq.~\eqref{eq-diffeq2} and \eqref{eq-wv-gen2}. However, for our driven-qubit example, the state estimator $ρ_Q_3$ exhibits the discontinuity (as shown in Figure~\ref{fig-trajs}) at $T = 2.09 T_γ$, which leads to an unphysical infinite measurement result at that particular time. Therefore, we only show the estimated unknown records for the five pure-state estimators in Figure~\ref{fig-cost2}(b), excluding the estimator $ρ_Q_3$.

Using the estimated unknown records, we can calculate the expected costs $C_5$--$C_7$. The expected cost, $C_5$, is the log ratio of the probability densities of the entire unknown record, as shown in Eq.~\eqref{eq-cq5}. This quantity is non-local in time and therefore is a single number associated with each of the state estimators: $1.8×10^16$ for $ρ_Q_2$, $8.7×10^10$ for $ρ_Q_4$, $0$ for $ρ_Q_5$, and $3.3×10^1$ for $ρ_Q_6$ and $ρ_Q_7$. The expected costs $C_6$ and $C_7$ in Eqs.~\eqref{eq-cq6} and \eqref{eq-cq7} are exactly the same and are presented in Figure~\ref{fig-cost2}(b), showing that the estimators $ρ_Q_6=ρ_Q_7$ give the minimum value.

%Lastly, as
The expected cost functions we have presented so far are dependent on the duration of time $T$ between two jumps.
However, in some cases,
we can also calculate an average of the expected cost over all possible jump times, $T$. This can be done by defining a jump-time-average expected cost as
\begin{align}\label{eq-avetimecost}
\bar{\cal C}_\kappa (\bullet)  \equiv & \int^\infty_0 \!\!\! \dd T \, \wp_{\rm wait}(T) \,  {T}^{-1} \!\! \int_0^T\!\!\! \dt \, \, {\cal C}_\kappa (\bullet_t),
\end{align}
where the first time-integral is weighted by the waiting-time PDF, %probability  for the wait time (for jumps), $P_{\rm wait}(T)$,
normalized as $∫^∞_0  T   ℘_wait(T) = 1$. The second time-integral, $T^-1∫_0^T   ∙$, is to average over the local time, $t$, which is applied to the expected cost that is $t$-dependent.  The waiting-time PDF can be calculated from,
\begin{align}\label{eq-jumptimeavg}
\wp_{\rm wait}(T) & = {\rm Tr}\left[ \hat{c}_{\rm o}^\dagger \hat{c}_{\rm o}\, \rho_{\pasts{\bo}}(T) \right] {\rm Tr}\left[ {\trho}_{\pasts{\bo}}(T)\right]  \nonumber \\
& = {\rm Tr}\left[ \hat{c}_{\rm o}^\dagger \hat{c}_{\rm o}\, \trho_{\pasts{\bo}}(T) \right],
\end{align}
where the unnormalized filtered state is given by
\begin{align}
\trho_{\pasts{\bo}}(T) =  {\cal M}_{O_{T-\ddt}} e^{\ddt{\cal L}_{\rm u}}\cdots \,  {\cal M}_{O_{0}} e^{\ddt{\cal L}_{\rm u}}\rho_0.
\end{align}
The first line in Eq.~\eqref{eq-jumptimeavg} is a product of the probability of an observed jump occurred at time $T$ (given that the system's state is $ρ(T)$) multiplied by the probability of having no jumps before that (for the length of time $T$). The latter is given by the trace of the unnormalized filtered state, which tells us the probability of the observed no-jump record from $t=0$ to $t = T$.

We approximate the integrals in Eq.~\eqref{eq-jumptimeavg} by using numerical summations (details are in Appendix~\ref{sec-app-numer}). The results are shown in Table~\ref{tab-avecost}, where the jump-time-average expected costs are calculated for $Q_1$, $Q_2$, $Q_3$, and $Q_8$, applied to relevant estimators. The minimal values of the average are shown in bold  font.  These results show that the optimal estimators minimize their expected costs,
not just for the value $T=4 T_γ$. We note that we did  not include $Q_5$ in our calculation for the jump-time-average expected cost, because solving the whole-record optimization becomes extremely resource-extensive for large values of $T$. The same reason holds for why we did not apply the calculation to $ρ_Q_4$ and $ρ_Q_5$. For $Q_6$ and $Q_7$, there were problems in reversing the dynamics to get reasonable estimated unknown records, and records could be infinite in some cases.  (Evidence of this divergence can be seen for the estimated record for $ρ_Q_2$ and $ρ_Q_4$ shown in Figure~\ref{fig-cost2}(b).)

\renewcommand{\tabcolsep}{0.2cm}
\begin{table}[t]
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\backslashbox{Cost function}{Estimator} & $\textcolor{colq1}{\rho_{\textbf{Q}_1}}$  & $\textcolor{colq2}{\rho_{\textbf{Q}_2}}$ & $\textcolor{colq3}{\rho_{\textbf{Q}_3}}$ & $\textcolor{colq67}{\rho_{6,7}}$ & $\textcolor{colq8}{\varrho_{\textbf{Q}_8}}$ \\
\hline
{\bf $\la$TrSDf$\rhoU\ra$}: $\textcolor{colq1}{{{\cal C}_1}}$, Eq.~(\ref{eq-cq1}) &  {\bf 0.31} & 0.46 & 0.48 & 0.62 & 0.34 \\
{\bf $\la$nFw$\rhoU\ra$}: $\textcolor{colq2}{{{\cal C}_2}}$, Eq.~(\ref{eq-cq2}) &  -0.69 & {\bf -0.77} & -0.76 & -0.69 & -0.70 \\
{\bf $\la$nEw$\rhoU\ra$}: $\textcolor{colq3}{{{\cal C}_3}}$, Eq.~(\ref{eq-cq3}) &  NA & -2.76 & {\bf -2.85} & -2.69 & NA \\
{\bf SWV}: $\textcolor{colq8}{{{\cal C}_8}}$, Eq.~(\ref{eq-cq8}) &  -0.44 & -0.18 & -0.07 & 0.22 & {\bf -0.51} \\
\hline
\end{tabular}

\caption{Numerical results for the jump-time-average expected costs ${\bar {\cal C}}_\kappa(\rho)$ for $\kappa = 1,2,3,8$ applied to relevant estimators: $\rho_{\textbf{Q}_1}$, $\rho_{\textbf{Q}_2}$, $\rho_{\textbf{Q}_3}$, $\rho_{\textbf{Q}_6}=\rho_{\textbf{Q}_7}$, and $\varrho_{\textbf{Q}_8}$.  `NA' indicates `Not applicable'. The bold numbers are to emphasize the minimum values in each row. Details of the numerical calculation are presented in Appendix~\ref{sec-app-numer}. }
\label{tab-avecost}
\end{table}}

% Discussion
\section{Discussion}\label{sec-discussion}
%Discuss about the generalization to true mixed states and which cost functions would not apply.
% [Discuss: what we did, how can this be useful and impact on the field, and what are the leftover questions to be asked ]

We have presented the cost-optimized quantum state estimation theory, unifying various existing formalisms that use the past-future information for quantum systems. The three main existing theories are the smoothed quantum state (minimizing the trace square deviation cost from true states), the most-likely path (minimizing the negative equality with entire true unknown records), and the smoothed weak-value state (minimizing the square deviation of an expectation value from true weak measurement results). As shown in Figure~\ref{fig-diagramQ}, the theory outlines explicit connections of possible estimators using various choices of cost functions, where the latter can be defined either in the unknown state space or the unknown measurement record space.
%We also used the classical-quantum state correspondence and showed that the classical analogue of such state estimation theory can be defined. Interesting connections were shown in Figure~\ref{fig-diagramC}, where the quantum state smoothing and the smoothed weak-value state both reduce to the classical state smoothing as expected.

In addition to clarifying the connection among existing formalisms, the theory suggested six new quantum state estimators ($ρ_Q_2$\,--$ρ_Q_8$) that could be used in estimating unknown quantum states in different contexts. It is noted that, for problems with unknown quantities, there is no single best guess for an unknown, but there is an optimal guess with respect to a chosen cost function. The theory was then applied to the coherently driven qubit coupled to bosonic baths, showing that all but two of the estimators are distinct, even though they all represent an optimal estimate of the unknown qubit's state. We do not claim to have compiled all possible cost functions in this work; however, we have built the framework that allows other estimators to be introduced.

One obvious generalization of our proposed formalism would be to problems where the true states are \emph{mixed}. Indeed, this is necessary for experimental tests or  practical applications of the formalism, in which Bob is a real party with a real measurement record, not a hypothetical omniscient observer. That is because an actual experiment performed by Bob will not be able to garner all of the information unavailable to Alice. Some information will be irrevocably lost to the environment, causing the state jointly conditioned on Alices's and Bob's measurement records to be mixed.   %only a portion of it.
%since the unknown true states in real experiments are generally mixed.
For most cases, except $Q_2$, the form of the estimators presented here still hold, after replacing the true states, $= $, with pure mixed states. For $Q_2$, the fidelity simplification in Eq.~\eqref{eq-fidel} will need to be modified for mixed states, which will likely result in an optimal estimator different from Eq.~\eqref{eq-th-q-nf}.

The central concern of this paper was conditioning on the past-future information.
However, the idea of different optimal estimates resulting from different cost function can also be applied to  conditioning only on past observation, or even no observation at all. Some estimators can be easily surmised. For example, for $Q_1$, the quantum smoothed state will be replaced by the quantum filtered state (for the past-only conditioning), or by a solution of the Lindblad master equation (for the no observation conditioning). However, for some other estimators, the generalization is quite involved, and this is a topic for future investigation.

\section{Acknowledgments}
We would like to thank Philippe Lewalle and John Steinmetz for providing Python software used in numerically solving the most-likely paths. We acknowledge the traditional owners of the land on which this work was undertaken
at Griffith University, the Yuggera people. This research is supported by the Australian Research Council
Centre of Excellence Program CE170100012 and Mahidol University (Basic Research Fund: fiscal year 2021) Grant Number BRF1-A29/2564.  A.C.~acknowledges the support of the Griffith University
Postdoctoral Fellowship scheme and the Australia Awards Endeavour Scholarships and Fellowships.

%% The Appendices part is started with the command \appendix;
%% appendix sections are then done as normal sections
%% \appendix

%% \section{}
%% \label{}

\appendix

\setcounter{section}{0}% Reset numbering for sections
\renewcommand{\thesection}{\Alph{section}}

\section{ Probability density function for time-local unknown records}\label{sec-app-pdf}
Here we discuss some properties of the PDF of the time-local unknown record, $℘_(u_τ)$, and show that the mean and the mode of both distributions are the same for a weak measurement with a Lindblad operator $c = c_u$. We note that the derivation below is applied to the past-only conditional PDF, $℘_(u_τ)$, by taking $E_ = Î$.

From the definition of the PDF in Eq.~\eqref{eq-probut}, we obtain
\begin{align}
\wp_{\bothps{\bo}}(u_\tau) = & \,  \frac{\Tr{ {\op E}_{\futps{\bo}}\,{\hat M}_{u_\tau} \rho_{\pasts{\bo}}\, {\hat M}_{u_\tau}^\dagger}}{\int \!\!\dd u_t \, \Tr{ {\op E}_{\futps{\bo}}\,{\hat M}_{u_t} \rho_{\pasts{\bo}}\, {\hat M}_{u_t}^\dagger}},
\end{align}
where $M̂_u = (/2π)^1/4exp(u_t^2/4) [  1̂ - 12( ĉ^†ĉ + ĉ^2) +ĉ  u_t + O(^2) ]$ (similar to Eq.~\eqref{eq-moput} in the main text, but without the Hamiltonian and only keeping to first order in $$) is the measurement operator for the unknown record $u$ at any time $t$. We can show that the PDF $℘_(u_τ)$ can be approximated as a Gaussian function, correct up to first order in $$,
\begin{align}\label{eq-app-probuboth}
\wp_{\bothps{\bo}}(u_\tau) \approx & \left(\frac{\ddt}{2\pi}\right)^{1/2} \!\!\!\! \exp\left(-u_\tau^2 \ddt/2\right) \left[   \op{1} + u_\tau \frac{\Tr{\op E_{\futps{\bo}}\,  \op c  \, \rho_{\pasts{\bo}} + \rho_{\pasts{\bo}}\, \op c^\dagger\, \op E_{\futps{\bo}}}}{\Tr{\op E_{\futps{\bo}} \,\rho_{\pasts{\bo}}}}  \ddt +{\cal O}(\ddt^2) \right] .
%\wp_{\past\bo}(u_\tau) \approx \left(\frac{\dt}{2\pi}\right)^{\tfrac{1}{2}} \exp\left\{ - \left(u_\tau + \sqrt{\Gamma}\sin\theta \right)^2\right\},
\end{align}
The PDF definitely has the same mean and mode, \ie,
\begin{align}\label{eq-app-probupast}
\la u_\tau \ra_{\bothps{\bo}} = \argmax_{u_\tau} \wp_{\bothps{\bo}} (u_\tau) =\frac{\Tr{\op E_{\futps{\bo}}\,  \op c  \, \rho_{\pasts{\bo}} + \rho_{\pasts{\bo}}\, \op c^\dagger\, \op E_{\futps{\bo}}}}{\Tr{\op E_{\futps{\bo}} \,\rho_{\pasts\bo}}},
\end{align}
which is exactly the real part of the weak value in Eq.~\eqref{eq-wv-gen2}.
The second moment of this distribution is dominated by $1/$, that is $u_τ^2 _ ≈1/$, therefore the distribution can be approximated as a Gaussian function with variance $1/$. Similarly, the mean and mode for the filtering PDF, $℘_(u_τ)$, are then simply $u_τ_ = _u_τ ℘_ (u_τ) = Tr [ (  c  + c^†) ρ_ ]$.

\section{ Retrodictive effect matrix for the driven qubit example}\label{sec-app-retroeff}
The POVM $Ê_$ can be computed semi-analytically for the coherently-driven qubit example used in the main text. Here, we rewrite the POVM for the partially observed system, Eq.~\eqref{eq-povm} in the main text,
\begin{align}\label{eq-povm-app}
{\hat E}_{\futps{\bo}} = e^{\ddt{\cal L}_{\rm u}^\dagger} {\cal M}_{O_\tau}^\dagger \cdots e^{\ddt{\cal L}_{\rm u}^\dagger} {\cal M}_{O_{T-\ddt}}^\dagger \hat E_T,
\end{align}
which can be used to compute a PDF of the future record given a state $ρ_τ$ at time $τ$,
\begin{align}
\wp(\,\futp{\bo}\,|\rho_\tau) = \Tr{ {\hat E}_{\futps{\bo}}\, \rho_\tau}.
\end{align}
In the main text, we are interested in the observed record $ = { _0 , _, ..., _T- ,_T }= { 0,0,..., 0,1}$ including the final jump at time $T$. Therefore, the final POVM is given by
\begin{align}
{\hat E}_T = {\hat c}_{\rm o}^{\dagger} {\hat c}_{\rm o} \ddt = \gamma \ddt |e\ra \la e |,
\end{align}
and the POVM at other times can be calculated from a dynamical equation
\begin{align}
{\hat E}_{t-\ddt}  = e^{\ddt{\cal L}_{\rm u}^\dagger} {\cal N}^\dagger_0 {\hat E}_t,
\end{align}
where we have replaced all $M^†_O_t$ for $t ∈[τ, T)$ in Eq.~\eqref{eq-povm-app} by the adjoint of the no-jump map $N^†_0$. We can take the continuum limit $→$ and write the dynamical equation in the form, $Ê_t-  = Ê_t -  dÊ$, where $dÊ$ is given by
\begin{align}
- {\rm d}{\hat E} =& + i \dt [\tfrac{\Omega}{2} \hat \sigma_x, {\hat E}_t] + \dt {\cal D}^{\dagger}[{\hat c}_{\rm u}]{\hat E}_t - \dt\, \frac{1}{2} \left( {\hat c}_{\rm o}^{\dagger} {\hat c}_{\rm o} {\hat E}_t + {\hat E}_t{\hat c}_{\rm o}^{\dagger} {\hat c}_{\rm o}\right).
\end{align}
The first two terms and the last term are from expanding the operations $e^L_u^†$ and $N^†_0$ to first order in $$ (or $$), respectively.

Given the form of the Hermitian POVM matrix
\begin{align}
{\hat E}_{\futps{\bo}}=  \left( \begin{matrix} \alpha + \zeta & -i \beta \\ i \beta & \alpha - \zeta\end{matrix} \right),
\end{align}
where  $α$, $β$, and $ζ$  are real numbers as functions of time (one can check that the real part of its diagonal elements has to be zero).  From this we obtain a set of differential equations for the matrix elements,
\begin{align}  %-   (I removed a minus from both sides -- it is the same eqn as before.
-\frac{{\rm d}}{\dt}\left( \begin{matrix} \alpha \\  \beta \\  \zeta \end{matrix} \right) = -\frac{1}{2}\left( \begin{matrix} {\gamma} &  0 & \gamma+2\Gamma\\
0& \gamma+\Gamma &-2\Omega \\
\gamma&  2\Omega & \gamma+2\Gamma\end{matrix} \right)
\left( \begin{matrix} \alpha \\ \beta \\ \zeta \end{matrix} \right),
\end{align}
which can be solved backwards in time (as indicated by the sign of the left hand side)  to get $Ê_$ at time $τ$ from the final condition $Ê_T =  γ|ee |$,   \ie, $α=ζ= γ/2$, $β=0$.

\section{ Numerical calculation for time-average expected costs}\label{sec-app-numer}

The jump-time-average expected cost and the waiting-time PDF are given in Eqs.~\eqref{eq-avetimecost} and \eqref{eq-jumptimeavg}, respectively. However, the time integral with the infinite wait time limit is impractical, given that one needs to calculate the state estimators for any wait time $T$ numerically. We therefore approximate the wait time integral with a finite summation such that an average of an arbitrary function $f(T)$ is given by
\begin{align}
\int_0^\infty \!\!\! \dd T \, \wp_{\rm wait}(T) f(T) & = \int_0^1\!\!\! \dd x \, f( G^{-1}(x)),\\
& = \Delta x \sum_{j = 1}^J  f( G^{-1}(j \Delta x)),
\end{align}
where $G(τ) ≡∫_0^τ T   ℘_wait(T)$ and $Δx = 1/J$. For simplicity, let us denote a set of jump times by ${ T_j }$ where each element $T_j ≡G^-1(j Δx)$, for $j = 1, 2,...,J$, we can write the time-average expected cost Eq.~\eqref{eq-avetimecost} as
\begin{align}
\bar{\cal C}_\kappa(\bullet) \approx  \frac{\sum_{j=1}^J  \Delta t \sum_{k=0}^{K_j}  \,  {\cal C}_{\textbf{Q}_\kappa}(\bullet_k)   }{\sum_{j=1}^J  \sum_{k=0}^{K_j} \Delta t},
\end{align}
where we have also discretized the second time integral in Eq.~\eqref{eq-avetimecost} with a step size $Δt$ and $K_j$ is the time steps given the jump time $T_j = K_j Δt$. The numerical results shown in Table~\ref{tab-avecost} are calculated with $Δx = 0.1$ and $Δt = 0.05$.

Moreover, in order to show that our technique solving the filtered and smoothed states using the Fokker-Planck equation Section~\ref{sec-fpe} are correct, we show in Figure~\ref{fig-compare} the comparison of the states with solutions from simulating all possible true states and averaging them as in Eq.~\eqref{eq-filstate} and \eqref{eq-qssstate}. We use the numerical techniques and C++ programming codes used  in Refs.~[8, 35, 36].

\begin{figure}[h]
\centering
\includegraphics[width=16.6cm]{FigApp-compare.pdf}
\caption{Comparison of the filtered and smoothed states calculated using two different techniques. The dashed curves are from the Fokker-Planck techniques presented in Section~\ref{sec-fpe} and the solid colored are from trajectory simulation in Ref.~[8, 35].}
\label{fig-compare}
\end{figure}
%% If you have bibdatabase file and want bibtex to generate the
%% bibitems, please use

\section{ Glossary of acronyms and abbreviations}
\label{Glossary}

\begin{table}[h]
\centering
\begin{tabular}{|c|c|}
\hline
Abbreviation & Meaning\\
\hline
SD & Square Deviation\\[1mm]
nE & negative Equality\\[1mm]
BME & Bayesian Mean Estimator\\[1mm]
MLE & Maximum Likelihood Estimator\\[1mm]
PDF & Probability Density Function\\[1mm]
$\Sigma$SD & sum Square Deviation\\[1mm]
TrSD & Trace Square Deviation\\[1mm]
TSVF & Two-State Vector Formalism\\[1mm]
POVM & Positive Operator-Valued Measure\\[1mm]
SWV & Smoothed Weak-Value\\[1mm]
QED & Quantum Electrodynamics\\[1mm]
CDJ & Chantasri, Dressel and Jordan\\[1mm]
ODE & Ordinary Differential Equation\\[1mm]
\hline
\end{tabular}
\caption{List of abbreviations (in order of appearance)}
\end{table}

\begin{table}[h]
\centering
\setlength{\extrarowheight}{8pt}
\begin{tabular}{|p{6mm}|p{0.32\textwidth}|c|c|}
\hline
& (Expected) Cost Function & Acronym & Definition\\\hline
${\bf Q}_1$ & {\bf Tr}ace-{\bf S}quare {\bf D}eviation \newline {\bf f}rom the true state & {\bf $\la$TrSDf$\rhoU\ra$} & $\left\la {\rm Tr} \left [  \left(\rho - \rhoOU\right)^2  \right ] \right \ra_{\pasts{\bu} | \bothps{\bo}}$\\\hline
${\bf Q}_2$ & {\bf n}egative {\bf F}idelity \newline {\bf w}ith the true state & {\bf $\la$nFw$\rhoU\ra$} & $\left\la -F\!\left[\rho, \rhoOU \right] \right\ra_{\pasts{\bu} | \bothps{\bo}}$\\\hline
${\bf Q}_3$ &  {\bf n}egative {\bf E}quality \newline {\bf w}ith the true state & {\bf $\la$nEw$\rhoU\ra$} & $\left\la - \delta_{\rm H}\!\left[ \rho - \rhoOU \right] \right\ra_{\pasts{\bu} | \bothps{\bo}}$\\\hline
${\bf Q}_4$ & {\bf n}egative {\bf E}quality \newline {\bf w}ith the past unknown record & {\bf $\la$nEw$\past\bu\ra$} & $\left\la -\delta \left(\past{u} - \past{\bu} \right) \right\ra_{\pasts{\bu} | \bothps{\bo}}$\\\hline
${\bf Q}_5$ & {\bf n}egative {\bf E}quality \newline {\bf w}ith the entire unknown record & {\bf $\la$nEw$\both{\bu}\ra$} & $\left\la - \delta \left(\both{u} - \both{\bu} \right) \right\ra_{\boths{\bu} | \bothps{\bo}}$\\\hline
${\bf Q}_6$ & {\bf n}egative {\bf E}quality \newline {\bf w}ith the unknown record at $\tau$ & {\bf $\la$nEw$\bu_\tau \ra$} & $\left\la - \delta \left(u_\tau - \bu_\tau \right) \right\ra_{\bu_\tau | \bothps \bo}$\\\hline
${\bf Q}_7$ & {\bf S}quare {\bf D}eviation \newline {\bf f}rom the unknown record at $\tau$ & {\bf $\la$SDf$\,\bu_\tau\ra$} & $\left\la \left(u_\tau - \bu_\tau \right)^2 \right\ra_{ \bu_\tau | \bothps{\bo}}$\\\hline
${\bf Q}_8$ & Sum Square Deviation \newline from the weak measurement result ({\bf S}moothed {\bf W}eak-{\bf V}alue state) & {\bf SWV}& $\displaystyle{\sum_{j=1}^{d^2-1}\left\la \left[ {\rm Tr}(\varrho \op{\Lambda}_j) - \Lambda^w_j \right]^2\right\ra_{\Lambda^w_j  | \bothps{\bo}}}$\\
\hline
\end{tabular}
\caption{List of cost functions and definitions of expected cost functions}
\end{table}

\begin{table}
\centering
\begin{tabular}{|c|c|l|}
\hline
Notation & Definition & Description\\
\hline
${\bf x}$ & & The configuration, a vector of $d$ classical variables.
\\[2mm]
$\mathbb{X}$ & & The set of possible configurations ${\bf x}$ for the classical system.
\\[2mm]
$\wp({\bf x})$ & & \makecell{A probability density function of the configuration ${\bf x}$. \\ Also referred to as the classical state.}
\\[4mm]
$\wp_{\rm D}({\bf x})$ & $\wp({\bf x}|{\rm D})$ & \makecell{The probability density function of the configuration ${\bf x}$ \\ conditioned  on $\rm D$. Also referred to as the conditioned \\ (classical) state.}
\\[5mm]
$\langle\bullet\rangle_{A|{\rm D}}$ & $\displaystyle{\int}\dd\mu(A) \wp_{\rm D}(A) \bullet$ & \makecell{The expectation value, averaging over $A$ and conditioned on $\rm D$, \\ where the integration measure $\dd\mu(A)$ will vary depending on $A$.}
\\[4mm]
$\est A_{\rm C}$ & & \makecell{An optimal estimator, given $D$, for cost function \\ with abbreviation ${\rm C}$.}
\\[4mm]
$\mathbb{P}$ & $\{\wp({\bf x}): {\bf x} \in \mathbb{X} \}$ & \makecell{The set of all possible normalized probability density functions \\ of the configuration.}
\\[4mm]
$\rho$ &  & A density operator/quantum state.
\\[2mm]
$\tilde{\rho}$ & & An unnormalized quantum state.
\\[2mm]
$\rho_{\rm D}$ & & A quantum state conditioned on $\rm D$.
\\[2mm]
$\hat{\psi}$ & $\ket{\psi}\bra{\psi}$ & A pure quantum state.
\\[2mm]
$\varrho$ & &  An indefinite Hermitian matrix with unit trace.
\\[2mm]
$\hat{E}$ & & A POVM element/quantum effect.
\\[2mm]
$\hat{E}_{\rm D}$ & & A quantum effect conditioned on D.
\\[2mm]
$\mathbb{H}$ & & The Hilbert space of the quantum system.
\\[2mm]
$\mathfrak{G}({\mathbb{H}})$ & & The set of all valid density operators in the Hilbert space.
\\[2mm]
$\mathfrak{G}'({\mathbb{H}})$ & & The set of unit-trace, Hermitian operators in the Hilbert space.
\\[2mm]
$\dd\mu_{\rm H}(\hat{\psi})$ & & The Haar measure over pure states.
\\[2mm]
$\past{R}$ & $\{R_t: t\in[0, \tau)\}\tp$ & The measurement results $R_t$ {\em prior} to the estimation time
\\[2mm]
$\fut{R}$ & $\{R_t: t\in[\tau, T)\}\tp$ & \makecell{The measurement results $R_t$ including at and {\em posterior} \\ to the estimation time $\tau$, {\em excluding} the result at the final time $T$. }
\\[4.5mm]
$\futp{R}$ & $\{R_t: t\in[\tau, T]\}\tp$ & \makecell{The measurement results $R_t$ including at and {\em posterior} \\ to the estimation time $\tau$, {\em including} the result at the final time $T$. }\\[4.5mm]
$\both{R}$ & $\{R_t: t\in[0, T)\}\tp$ & \makecell{The set of all measurement results $R_t$ both {\em prior} and {\em posterior} \\ to the estimation time $\tau$, {\em excluding} the result at the final time $T$. }\\[4.5mm]
$\bothp{R}$ & $\{R_t: t\in[0, T]\}\tp$ & \makecell{The set of all measurement results $R_t$ both {\em prior} and {\em posterior} \\ to the estimation time $\tau$, {\em including} the result at the final time $T$. } \\[4.5mm]
$ _{\hat f}\langle X^w\rangle_{\hat i}$ & $\displaystyle{\lim_{\epsilon \to 0}\int_{-\infty} ^{\infty}\dd x \,x\,\wp(x|\hat{i},\hat{f}) }$ & \makecell{The weak-value associated with pre- ($\hat{i}$) and post-selecting ($\hat{f}$) \\  on the  outcome of a weak measurement of $\hat{X}$; \\ $\epsilon$ is the measurement strength.} \\[5mm]
\hline
\end{tabular}
\caption{List of notations and descriptions}
\end{table}

\clearpage

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