The Quantum Rauch-Tung-Striebel Smoothed State

Smoothing is a technique that estimates the state of a system using measurement information both prior and posterior to the estimation time. Two notable examples of this technique are the Rauch-Tung-Striebel and Mayne-Fraser-Potter smoothing techniques for linear Gaussian systems, both resulting in the optimal smoothed estimate of the state. However, when considering a quantum system, classical smoothing techniques can result in an estimate that is not a valid quantum state. Consequently, a different smoothing theory was developed explicitly for quantum systems. This theory has since been applied to the special case of linear Gaussian quantum (LGQ) systems, where, in deriving the LGQ state smoothing equations, the Mayne-Fraser-Potter technique was utilised. As a result, the final equations describing the smoothed state are closely related to the classical Mayne-Fraser-Potter smoothing equations. In this paper, I derive the equivalent Rauch-Tung-Striebel form of the quantum state smoothing equations, which further simplify the calculation for the smoothed quantum state in LGQ systems. Additionally, the new form of the LGQ smoothing equations bring to light a property of the smoothed quantum state that was hidden in the Mayne-Fraser-Potter form, the non-differentiablilty of the smoothed mean. By identifying the non-differentiable part of the smoothed mean, I was then able to derive a necessary and sufficient condition for the quantum smoothed mean to be differentiable in the steady state regime.

One such technique was developed soon after the Kalman-Bucy filtering theory by Rauch, Tung and Striebel [2,3]. This technique, referred to as smoothing, not only utilises the past measurement information as the Kalman-Bucy filter does, but also uses information gathered after the estimation time τ , i.e. the 'future' measurement record, to provide a more accurate estimate of the state. The Rauch-Tung-Striebel (RTS) smoothing technique, first, uses the Kalman-Bucy filtering technique to estimate the state until the final estimation time T . Once at the final estimation time T , the RTS smoothing equations run back over the estimated state, updating the results based on the future measurement record that has been gathered. This results in a smoothed estimate of the state. The only drawback of this smoothing technique, compared to filtering, is that the smoothed state cannot be obtained in real-time since future information is required.
A similar technique was developed by Mayne [4], Fraser [5] and Potter [6] shortly after RTS smoothing which also utilised a past-future measurement record. Mayne-Fraser-Potter (MFP) smoothing, sometimes referred to as two-filter smoothing for reasons that will become apparent, also utilises Kalman-Bucy filtering to condition on the past measurement record. However, to make use of the future information, they introduced a secondary filter, which I will refer to as a retrofilter, that ran backwards from a final uninformative state conditioning on the the future measurement record. This state is the retrofiltered state. Combining the filtered state and retrofiltered state together results in the MFP smoothed state. It has since been shown [6] that the RTS smoothed state and the MFP smoothed state are, in fact, identical.
Moving to quantum systems, an analogous problem to classical state estimation also exists, where instead of estimating a PDF one wishes to estimate the density matrix ρ. Similar to the classical filtering theory, one can estimate the quantum state based on the past measurement record. As a result, this technique is often referred to as quantum filtering [14][15][16]. Interestingly, if one restricts to linear Gaussian quantum (LGQ) systems, the quantum filtering technique reduces to the Kalman-Bucy filtering theory [17][18][19]. Given this, one might assume that classical smoothing techniques, like the RTS or MFP techniques, can also be applied to quantum state estimation. This is not the case. Applying the classical theory to quantum state estimation can result in an unphysical estimate of the quantum state [20][21][22][23][24]. This is due to the non-commutativity of the operators describing the system and the operators describing the future measurement outcomes in quantum theory, which is usually not present in the classical theory. Due to the failure of the classical theory, Guevara and Wiseman [22] devised a new smoothing theory specifically for quantum systems, arXiv:2010.11027v3 [quant-ph] 13 May 2021 the quantum state smoothing theory.
The quantum state smoothing theory [22,24,25] introduces a secondary measurement record that is unobserved by the observer, say Alice, but is observed by someone else, say Bob. The role of Bob's measurement is to gather information about the system that Alice's measurement may have missed. If Alice had access to Bob's record, hereby referred to as the 'unobserved' record, she could condition the estimate of the state on both her past observed record ← − O and unobserved record ← − U to obtain the true state of the system ρ T = ρ← − O ← − U , a state containing the maximum amount of information about the system given the measurements. Here, the leftward arrow indicates that it is the past record and a bidirectional arrow will indicate a past-future record. However, since Alice does not have access to the unobserved record she cannot compute the true state of the system. The best [26] Alice can do is to estimate the true state based only on the observed record. Now, Alice can obtain a smoothed quantum state by averaging over all possible true states conditioned on her past-future observed record where E A|B {C} denotes the ensemble average of C over A conditioned on B, and may appear without the A subscript when A = C.
Following the conception of the smoothed quantum state, the theory was applied to the special case of LGQ systems [24,27,28]. In order to derive the LGQ smoothed quantum state, the classical MFP smoothing techniques was used. These quantum state smoothing equations, due to being closed-form equations, have been able to identify properties of the smoothed quantum state [24,27,28] that would have been difficult to find in the general case. Furthermore, the smoothed state is far simpler to compute for LGQ systems compared to even a simple system in the general case, allowing for easier verification of these properties. However, while the MFP forms of the quantum state smoothing equations have been very useful, they require the calculation of a retrofiltered state. This can be avoided by instead using the RTS form of the smoothing equations, making the smoothed state even simpler to compute for LGQ systems. Additionally, the RTS form of the quantum state smoothing equations are dynamical equation and can provide insight into properties of the smoothed quantum state that would otherwise be hidden in the MFP form. Case in point, I derive a necessary and sufficient constraint for mean of the smoothed quantum state evolve smoothly in the steady state regime. This paper is structured as follows. First, in Sec. II, I will briefly review classical and quantum state smoothing for linear Gaussian systems. Next, in Sec. III will derive the RTS form of the smoothed quantum state. Finally, in Sec. IV, I will discuss the new form of the LGQ state smoothing equations, showing that, under the same condition that makes the classically smoothed mean a continuous function, the path of the smoothed mean is necessarily non-differentiable. Furthermore, I identify a necessary and sufficient condition for the mean of the smoothed quantum state to be differentiable in the steady state regime.

A. Classical
Consider a classical dynamical system. The state of the system is given by a probability density function ℘(x), wherex = (x 1 ,x 2 , ...,x M ) is a vector of M parameters that are required to completely characterize the system and denotes the transpose. Note, for clarity, the wedge accent will be used to denote a dummy variable to differentiate it from the corresponding random variable. For a LG system, the classical state will be a Gaussian distribution, i.e., ℘(x) = g(x; E{x}, V ), completely described by its mean E{x} and covariance matrix V = E{xx } − E{x}E{x }. To guarantee that the state remains Gaussian throughout the evolution of the system, provided that at the initial time t 0 the state is Gaussian with mean E{x(t 0 )} = x 0 and covariance V (t 0 ) = V 0 , it is necessary that the following two constraints are satisfied [7][8][9][10][11][12]19]. Firstly, the evolution of x must be described by a linear Langevin equation Here A (the drift matrix) and E are constant matrices and the process noise dv p is a vector of independent Weiner increments satisfying where I k is the k × k identity matrix. Secondly, any measurement current y used to refine our estimate of the state must also be linear, i.e., where the measurement matrix C is a constant matrix and the measurement noise dv m is a vector of independent Wiener increments satisfying similar conditions to Eq. (3). Usually, it is also assumed that the process noise and measurement noise are uncorrelated [1][2][3][4][5][6][7][8][9][10][11][12], i.e., dv p (dv m ) = 0, however, this assumption is not always true, i.e., there might be measurement backaction. For the sake of generality I will consider the case where the two noises may be correlated, described by the crosscorrelation matrix [19,[29][30][31][32] Γ dt = Edv p (dv m ) .
If all of the above criteria are satisfied, one can calculate the filtered estimate of the state x F , V F ) by conditioning the state on the past measurement record ← − O. The filtered mean are given by the Kalman-Bucy filtering equations [1,19,[29][30][31][32] with initial conditions x F (t 0 ) = x 0 and V F (t 0 ) = V 0 .
Here D = EE is the diffusion matrix, is the optimal Kalman gain, where the minus version will appear shortly, and the vector of innovations is defined as dw c = ydt−Cx c dt for a given conditioning c ∈ {F, R, S}, representing filtering, retrofiltering and smoothing, respectively, and satisfies similar conditions to Eq. (3). One can also calculate the smoothed estimate of the state can be calculated in two ways. The first method is a maximum likelihood argument and results in the RTS smoothing equations [2,3,32], with the final conditions x S (T ) = x F (T ) and V S (T ) = V F (T ). Here,Ã = A − Γ C,D = D − Γ Γ. The second method arises from a Bayesian argument, where one first introduces a retrofiltered state that runs backwards in time from the final estimation time. This retrofiltered state will also be Gaussian with mean with V −1 R (T ) = 0 describing an uninformative state. Combining the filtered and retrofiltered state gives the smoothed state, with the mean and covariance described by the MFP smoothing equations [4][5][6] x It has since been shown [6] that the RTS and MFP forms of the smoothed mean and covariance are identical and is easily verified by differentiating Eqs. (13)- (14) with respect to time.

B. Quantum
To begin, let us consider an open quantum system whose density matrix ρ, assuming a Markovian system, evolves according to the Lindblad master equation [19] dρ with initial condition ρ(t 0 ) = ρ 0 . Here H is the systems Hamiltonian describing unitary evolution,ĉ = (ĉ 1 ,ĉ 2 , ...,ĉ K ) is the vector of Lindblad operators describing the interaction between the system and environment, [A, B] = AB − BA is the commutator and the superoperator with {A, B} = AB + BA being the anticommutator. For this quantum system to be analogous to a classical LG system, we require that the systems observables have an unbounded spectrum. Thus, we will assume that the quantum system that can be described by N bosonic modes with each mode described by a positionq k and momentump k , which satisfy the commutation relation [q k ,p ] = i δ k, . We can then construct a 2N vector x = (q 1 ,p 1 , ...,q N ,p N ) describing all the modes of the bosonic system. For the system to be classified as a LGQ system [17][18][19], the Wigner function, a quasiprobability distribution defined as must initially be Gaussian and remain Gaussian, where the latter can be guaranteed when the Hamiltonian is quadratic and the vector of Lindblad operators is linear inx, i.e., H =x Gx/2 andĉ = Bx, respectively. Note, since we are using the Wigner function to characterize the quantum state, the Hamiltonian must be symmetrically ordered inq k andp k , forcing G to be a symmetric matrix. As the Wigner function is Gaussian, i.e., W (x) = g(x; x , V ), all that is required to know the state is the mean x and covariance matrix, defined sym- Using the Lindbald master equation, we can compute the dynamic equations for the unconditioned mean and covariance, with initial conditions x (t 0 ) = x 0 and V (t 0 ) = V 0 . Here with Σ k = −i[x k ,x ]/ being a symplectic matrix. Additionally, since the position and momentum operators do not commute, it is necessary that the covariance matrix obeys the Schrödinger-Heisenberg uncertainty relation Without any measurement information, this would be the most accurate estimate of the state possible. If one wishes to obtain a better estimate on the quantum state, it is necessary to gather more information about the system by measuring the environment. In the event of a continuous monitoring, we can condition the evolution of the state on the past measurement results and obtain the, so called, quantum filtering equation [19,33] with ρ F (t 0 ) = ρ 0 . For reasons that will become apparent, I have restricted the measurements to diffusive type measurements, like homodyne or heterodyne schemes, with the matrix M (assumed time independent for simplicity) characterizes the particular unravelling. Here the filtered innovation is a vector of independent Wiener increments defined by dw F = ydt − M †ĉ +ĉ † M F dt, satisfying conditions similar to Eq. (3), with the conditioned expectation value defined as • c = Tr[•ρ c ] with c ∈ {F, T, S}, representing the filtered, true and smoothed states, respectively, and ydt being the measurement current, and the superoperator where the Einstein summation convention is being used over repeated indices. To ensure that the resulting filtered state is a valid quantum state, it is necessary and sufficient [33] that the matrix M satisfies M M † = diag(η 1 , η 2 , ..., η 2N ), where 0 ≤ η k ≤ 1 ∀ k and can be interpreted as the fraction of the channelĉ k that has been observed. Note, it must be the case that at least one η k > 0, otherwise no measurement has been performed and both filtering and smoothing are redundant.
When considering an LGQ system, in order to keep the filtered Wigner function Gaussian throughout the entire evolution we require the measurement current to be linear inx (which is the case for diffusive type measurements), i.e., where the measurement matrix C = 2 , and the measurement noise, for simplicity, is assumed to be white, i.e., satisfies the properties in Eq. (3). As was the case for the unconditioned state, since the filtered state has been restricted to have a Gaussian Wigner function, W F (x) = g(x; x F , V F ), we only need information about the mean and covariance matrix to specify the state. Using Eq. (22), on can obtain the equations for the filtered mean and covariance matrix [17][18][19].
. Note, since the filtered state ρ F is a valid quantum state, it is the case that V F satisfies the Schrödinger-Heisenberg uncertainty relation.
If one wishes to obtain an even more accurate estimate of the state, it is possible that we can condition the estimate of the state on not only the past measurement information but also the future measurement record. This leads us to the quantum state smoothing theory of Guevara and Wiseman [22].

III. DERIVING THE QUANTUM RAUCH-TUNG-STRIEBEL SMOOTHED STATE
In order to calculate a smoothed quantum state, one needs to first introduce the true state ρ T . As mentioned in the introduction, the true state is an estimate conditioned on two independent measurement records, the past record observed by Alice and the past record observed by Bob. This situation is merely an extension of the filtering scenario to multiple independent measurement records, and thus it is simple to see that the resulting stochastic master equation will be where, assuming Alice and Bob have the same initial information about the quantum state, the initial condition is ρ T (t 0 ) = ρ 0 . Note, this may not always be the case and is trivial to adapt to the general case, however this assumption seems like a scenario that would occur frequently and is worth considering. As before, we are restricting to the case where both Alice and Bob measure there respective fractions of the measurement channels using a diffusive-type measurements. We have introduced the (r)ecord subscript to distinguish between the record (o)bserved by Alice and the record (u)nobserved by Alice (Bob's record), where the matrices M o and M u describe how Alice and Bob have unravelled the the system, respectively. The observed and unobserved innovations are defined as dw r = y r dt − M † rĉ +ĉ † M r T dt where we have assumed both records are uncorrelated, i.e., dw o dw u = 0 and y r dt is corresponding the measurement current for Alice and Bob.
As was the case for the filtered state, to ensure that the true state is a valid quantum case it is necessary and sufficient that both M o and M u must satisfy M r M † r = diag(η r,1 , η r,2 , ..., η r,M ) with 0 ≤ η r,k ≤ 1 and [24,27,28], that is, together Alice's and Bob's measurements constitute a perfect measurement of the system and the resulting true state will be pure, it not a necessary assumption and what follows will hold generally. As was the case with filtering, it must be that at least one η o,k > 0 and η u,k > 0.
For an LGQ system, to ensure that the true state remains Gaussian, as was the case for filtering, we require that both Alice's and Bob's measurements are linear, where, for simplicity, I have defined the currents in terms of the true mean x T and the innovation, where, in this form, it should be clear that observed innovation Given this restriction, the Wigner function for the true state will be Gaus- with initial conditions x T (t 0 ) = x 0 and V T (t 0 ) = V 0 .
Here, K ± r [V ] = V C r ± Γ r , where C r and Γ r are defined in in the same way as before, with the appropriate M r used in both cases. Now that the true state has been calculated, we can begin to derive the smoothed quantum state for LGQ systems. To begin, as we are interested in the Wigner function of the smoothed state W S (x) in the LGQ setting, we can apply Eq. (17) to both sides of Eq. (1), where, by the linearity of the trace, we obtain As the Wigner function of the true state is restricted to a Gaussian, we know that it only depends on the mean x T and covariance matrix V T , the latter of which is deterministic with the former depending explicitly on both ← − O and ← − U. Hence, averaging over ← − U with a fixed ← → O will be equivalent to averaging over the true mean x T for a fixed observed record. To make the notation simpler for the derivation to come, we take • x = x T , where the ellipse accent will be refereed to as a 'halo' and will denote intermediary variables between the true state and the filtered/smoothed state. Making these changes to Eq. (31), we get where the integral measure dµ( will be a Gaussian distribution. To see why this is the case, we rewrite Eq. (29) as whereĒdv p = r∈{o,u} K + r [V T ]dw r , with the observed measurement current where it is clear that x is described by a linear Langevin equation, as in Eq. (2), and by conditioning on a linear measurement current the resulting PDF will be Gaussian.
As ℘( is a classical object, we can simply apply classical smoothing theory in order to compute this PDF, and since this PDF is Gaussian, ℘( we only need to determine the mean with final conditions Additionally, I have introduced both the haloed filtered mean While the haloed filtered mean and covariance do satisfy there own differential equations, it has been shown [27] that thus the specific equations are not important for computing the haloed smoothed mean and covariance matrices. However, for simplicity, the haloed covariance matrix will be used more often for simplicity. It should also be emphasized that • V F is the covariance of a classical state and thus is not required to satisfy the Schrödinger-Heisenberg uncertainty relation, unlike V F and V T . Finally, we have all the necessary information to compute the smoothed quantum state for LGQ systems. Returning to Eq. (32), we can substitute in the Gaussian where, using the fact that convolving two Gaussian functions will result in another Gaussian, I have replaced the Wigner function of the smoothed state with its Gaussian with mean x S and covariance matrix V S . Using the properties of such a convolution, we find that dt + dVT dt , we obtain the RTS form of the quantum state smoothing equations, with the final condition V S (T ) =

IV. DIFFERENTIABILITY OF THE QUANTUM SMOOTHED MEAN
In addition to being simpler to compute, as one does not need to calculate the haloed retrofiltered estimates [27,28], the RTS forms of the smoothed mean and covariance, Eqs. (38)-(39), are dynamical equations which can provide insight into how the smoothed state evolves. This is hidden in the MFP forms. In particular, we can see the non-differentiability of the smoothed mean. This may not be terribly surprising as even in the classical case the smoothed mean was non-differentiable. However, there is a slight, perhaps predictable difference between the innovation terms in the classical and quantum cases. Specifically, we see that, in the quantum case, there is a dependence on the true covariance V T , which is reasonable as this is the minimum uncertainty in the mean for the quantum system given Alice's and Bob's measurements. Thus it is reasonable to expect that Alice's measurement would change her estimate of the mean by at least an amount proportional to V T . In contrast, the innovation in the classical case only depends on the cross-correlation matrix Γ since the true covariance is zero, corresponding to a delta function PDF, where the Schrödinger-Heisenberg uncertainty relation prevents V T → 0 in quantum systems. Furthermore, because of the uncertainty relation, one might expect that the quantum smoothed mean will always be non-differentiable. This is not the case.
In the classical case, all that is required for the smoothed mean x S to be differentiable is that the crosscorrelation matrix Γ = 0, meaning that Alice's measurement is uncorrelated with the noise affecting the system. However, this condition is not sufficient in general for the quantum case. To find the condition for differentiability in the quantum case, we will assume that the mean is differentiable, that is, d x S ∝ dt, over the interval [τ 1 , τ 2 ]. It is easy to see by looking at Eq. (38), the only term that is preventing the mean from being differentiable is the innovation term K + o [V T ]dw S . Thus, in order for the mean to be differentiable, this innovation term must either be proportional to dt or vanish over the interval. It is impossible for the former to be true as for the innovation term to be proportional to dt, since dw S dw S = I 2N dt, it must be the case that K + o [V T ] = Rdw S , where R is an arbitrary matrix, which cannot be the case as V T , C o and Γ o are all independent of the observed measurement at the estimation time. The latter, on the other hand, is not impossible and occurs when K + o [V T ] = 0. Since we are considering a fixed measurement scheme, i.e., C o and Γ o are time-independent, it is impossible for this condition to be satisfied at all times. Thus, to have a differentiable smoothed mean over a non-infinitesimal time interval, I will only consider time intervals in the steady state regime, i.e., τ 1 ≥ τ ss , where τ ss is the time taken for the true covariance to reach steady state.
Under this condition, the steady-state of the true covariance V ss T satisfies At this point we can see that the steady-state covariance of the true state is identical to the steady-state of a single record filtered state, like Eq. (26), however, rather than using Alice's past record, it is Bob's past record that is used. For comparison, Bob's filtered state is W← − where dw← − U = y u dt − C u x ← − U dt. Thus, we have that if the mean of the smoothed quantum state is differentiable in the steady-state regime then V ss T = V ss ← − U . Importantly, the converse of this is also true, that is, then the mean of the smoothed quantum state is differentiable. This is simple to see since the steady-state of the true covariance in general satisfies and the steady state of Bob's filtered covariance satisfies Since V ss T = V ss However, it is not a necessary condition. This differentiability condition contains useful information about the type of systems and measurements that could exhibit a differentiable smoothed mean. Specifically, we see from V ss T = V ss ← − U that Alice's measurement cannot reduce the uncertainty of the true state beyond what Bob's measurement already has. This immediately means that the quantum system must have more than one Lindblad operator, otherwise the portion of the channel that Alice monitors will always reduce the uncertainty by a fraction more when combine with Bob's measurement.
As an example, I will consider a single mode (N = 1) open quantum system described by the master equation, Note, while this is a toy system to illustrate the differentiability condition for quantum state smoothing, in principle it could be constructed using linear optics. In fact, this system is similar to an optical parametric oscillator [18,19], where the only difference is the additional Lindblad operatorĉ 2 =q, as mentioned before is required.
In this system, since the variance in thep quadrature is bounded by the squeezing Hamiltonian, we can expect that, when κ is large enough, the amount of information about the position quadrature gained by monitoringĉ 2 would reduce the uncertainty inq by enough so that the state is pure. Thus, for this system, if Bob where to perfectly monitorĉ 2 (the κ-channel), with Alice perfectly monitoringĉ 1 (the γ-channel), then the steady state of the true state will be pure, as both Alice and Bob have performed a perfect monitoring, and the steady state of Bob's filtered state will also be pure with the same covariance, satisfying the differentiability condition.
To make this more formal, time will be measured in units of χ −1 and I will consider the case γ = χ. For this system, the drift and diffusion matrices are A = diag(0, −2) and D = ·diag (1, 1+g), where g = κ/χ. I will assume that Alice and Bob both perform homodyne measurements. For homodyne measurements, the matrix M r = diag( √ η γ,r exp[iθ γ,r ], √ η κ,r exp[iθ κ,r ]), with the resulting measurement matrix and cross-correlation matrix Here, η n,r and θ n,r is the measurement efficiency and homodyne phase, respectively, for each channel n ∈ {γ, κ} and observer r ∈ {o, u}.
To demonstrate the smooth evolution of the quantum smoothed mean, Bob must measure the position quadrature of the κ-channel perfectly, i.e. η γ,o = 1 and θ γ,o = π/8, η κ,u = 1 and θ κ,u = 0. Crucially, g is chosen to be unity as this is the point where the steady state solution for Bob's filtered covariance is pure and is equal the true state. Alice, on the other hand, perfectly monitors the γ channel, i.e., η κ,u = 1, with some homodyne phase θ κ,u = π/8. Note, the particular homodyne phase for Alice does not matter in the slightest for the theory and was just chosen for the simulations. As the top graphs of Fig. 1 show, after a sufficient time (t ≈ 0.8) for the true covariance to reach its steady state, the quantum smoothed mean (red line) begins to evolve smoothly, as one would expect from a differentiable function. Whereas, the other three estimates, i.e. the true, the filtered and the classically smoothed means, all evolve stochastically over the entire time interval. The classically smoothed mean was computed using Eqs. (9)-(10), where the measurement matrix C and the cross-correlation matrix Γ have been replaced with Alice's measurement matrix C o and cross-correlation matrix Γ o . Now, if we reverse the channels that Alice and Bob measure, i.e., η γ,u = 1, θ γ,u = π/8 and η κ,o = 1, θ κ,o = 0, we would expect the smoothed mean to be non-differentiable throughout the entire evolution, like the filtered and true state. This is clearly the case, as seen in the bottom graphs of Fig. 1. Note, for this case g = 1 so that Alice's filtered and smoothed state do not reduce to the true state when the system reaches steady state.
The fact that the quantum smoothed mean evolves stochastically might bring into question whether this technique should be referred to as smoothing. While it is idiosyncratic that a 'smoothing' technique does not provide a smooth estimate, I reiterate that the classically smoothed mean suffers from the same issue when Γ o = 0, as seen in Fig. 1. I also remind the reader that the smoothing technique refers to using future measurement information as well as the past information to obtain an estimate [7][8][9][10][11][12][13] and not necessarily obtaining a smooth estimate. Thus, I believe there is no issue in referring to this quantum state estimation technique as smoothing.

V. CONCLUSION
In this paper I have derived the Rauch-Tung-Striebel forms of the quantum state smoothing equations for LGQ systems. These new forms not only make it easier to compute the smoothed quantum state but also provide insight into the dynamics of the state. From these equations, I have derived a necessary and sufficient condition for the quantum smoothed mean to be differentiable in the steady state limit. These equations could prove very useful in the future in identifying more properties of the smoothed state. A good example in the existing literature could be to provide a explanation or even an analytic solution for the optimal measurement strategies for Alice and Bob presented in Ref. [28]. Additionally, there may even be uses for this smoothing technique outside of quantum mechanics. This result would hold for any classical linear Gaussian system with a minimum bound on the covariance, i.e. a system with an uncertainty relation. Lastly, it would be interesting to see if the sufficient condition V T (t) = V← − U (t) for a differentiable quantum smoothed mean is sufficient outside the LGQ regime.

ACKNOWLEDGMENTS
I would like to thank Howard M. Wiseman for his invaluable comments and the example that illustrates the differentiability of the quantum smoothed mean. I acknowledge the traditional owners of the land on which this work was undertaken at Griffith University, the Yuggera people. This research is supported by an Australian Government Research Training Program (RTP) Scholarship.

Appendix A: Non-invertible haloed filtered covariance
In order to show how to treat the smoothed quantum state when does not exist, I will go back to the more general definitions and consider the definition of the filtered state in terms of the true state, that is, 22]. Following similar steps to the derivation of the smoothed quantum state in Sec. III, arriving at In particular, I will look at the probability distribution ℘( x F . Now, since Λ is a diagonal matrix the Gaussian PDF can be factorized as g( • z F,k , λ k ) and we can consider the scenario where • V F has at least one zero eigenvalue.
Let us assume, without loss of generality, that λ k > 0 ∀k ≤ s and the remaining 2N − s eigenvalues are zero. Using the fact that lim σ→0 g(x; a, σ 2 ) = δ(x − a), the Gaussian PDF becomes Computing the integral in Eq. (A2), we find that the transformed filtered mean and covariance are Thus, when • V F has at least one zero eigenvalue, the corresponding components of the mean and covariance matrix are equal the same components of the transformed true mean and covariance.
Moving on to quantum state smoothing, beginning with Eq. (37), if we perform the same change of basis as we did for the filtered state, we obtain g(ž; ẑ S , P V S P ) = dµ( where ẑ S = P x S and • z S = P • x S . However, unlike in the filtering case, P • V S P is not necessarily a diagonal matrix and thus the Gaussian cannot be completely factorized. However, we can still expect the Gaussian to factorize as g( because conditioning the estimate of ẑ T on more information cannot make those components of the probability distribution any more certain than a delta function. Here We see that, like the filtered state, the components of the smoothed mean that have an eigenvalue of zero for the • V F (t) are equal corresponding components of the true mean and similarly for the smoothed covariance matrix. The remaining components of the mean and covariance are computed using the remaining elements of the filtered mean and covariance. This now gives use a method to compute the smoothed quantum state when does not exist at a particular time t. I will comment that in the event that this occurs, the MFP form in Ref. [27] will be more useful because when the inverse does not exist at time t, it will only affect the smoothed mean and covariance at t which can easily be correct.
As an aside, this analysis highlights an interesting property of the smoothed quantum state. In the event that all the eigenvalues of the haloed filtered covariance are zero, using Eqs. (A8)-(A9) the smoothed mean and covariance will be equal to the true mean and true covariance, respectively. While this in itself is not particularly interesting, it becomes interesting when we consider the initial conditions for the smoothed state. As we have assumed throughout this paper we have • V F (t 0 ) = 0. Thus, we have that initially the smoothed quantum state must have mean x S (t 0 ) = x T (t 0 ) = x 0 and covariance V S (t 0 ) = V T (t 0 ) = V 0 , meaning that, in this case, the smoothed quantum state will always start in the same state as both the filtered and true quantum state. Moreover, this fact holds irrespective of whether the true state is pure. As pointed out earlier, we can see this occur in Fig. 1, where we also see that the classically smoothed state does not coincide with the filtered and true means initially. This is because, in the classical case, it is always assumed that the classical true state (the state of maximal knowledge) is a delta function causing the condition to be violated. Note, this does not mean that the quantum smoothed mean can be computed forward in time, just that it is constrained at both t 0 and T .
While the MFP form of the quantum state smoothing equations might be more useful in general, there is a special case when P is time independent over an interval [τ 1 , τ 2 ] with λ k = 0 for k > s. In this case, the Moore-Penrose pseudo-inverse can be used in Eqs. (38)-(39) instead of the usual matrix inverse. Over this interval, following the reasoning prior to this, ẑ S and • V S will be of the forms Eq. (A8) and (A9), respectively, over the interval. I will show that taking the pseudo-inverse causes the mean and haloed covariance matrix (as this is simpler to show analytically) allows the relevant components to evolve as the corresponding components of the true state would, while the remaining components evolve in a similar manner to a system where the inverse exists.
Beginning with the mean, the stochastic differential equation for the transformed mean is d ẑ S =Ā * ẑ S dt+D * Λ + ( ẑ S − ẑ F )dt+P K + o [V T ]y o dt , (A10) where I have used P P = I 2N , an asterisked matrix denotes a transformation by P , i.e. F * = P F P and Λ + is the pseudo-inverse of Λ. For a diagonal matrix, the pseudo-inverse is simple to compute by inverting the non-zero elements and leaving the remaining elements unchanged. Looking at the kth component of the mean we have where for comparison, the evolution of the kth component of the transformed true mean is When k > s, since the evolution of the smoothed mean must be the same as the evolution of the true mean over the time interval, as ẑ k S (τ 2 ) = ẑ k T (τ 2 ), it must be the case that the matrices must have the following block forms:Ā * = Ā * 00Ā * 01 0Ā *

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,D * = D * 00 0 0 0 , where the blocks are divided so that the diagonal matrices have dimensions s × s and (2N − s) × (2N − s). We can understand why this must be the case because ifĀ * was of another form the first term in Eq. (A11) would cause the evolution of the kth component to be influenced by components other than the true mean and hence would cause the kth components to deviate from the true mean. A similar reasoning follows for the form ofD * to eliminate the second term. Note, since the lower half ofD * is zero, we see, usingD = K + u [V T ]K + u [V T ] , that (K + u [V T ]) k, = 0 for all when k > s. This will eliminate the final term in Eq. (A12) and the kth components will evolve identically.
All that remains is to show thatĀ * andD * are of the forms in Eq. (A13). Consider the differential equation for the haloed filtered covariance [27,28], and for the k, th element we have where Einstein's summation convention is not being used. Looking at k, > s and remembering that the zero eigenvalues do not change over the interval, we havē D * k, = 0. By looking at the k, kth element ofD * with k > s, we haveD * = (P K + u [V T ]) 2 k, = 0 and thus (P K + u [V T ]) k, = 0 for all , showing thatD * is of the form in Eq. (A13). Now, returning to Eq. (A16), if we consider the case where k > s and < s + 1, we have λ Ā k, +D * k, = 0. As we have just shown, in this regimē D * k, = 0, and we know λ = 0, resulting inĀ * k, = 0. ThusĀ * is also of the form in Eq. (A13) and completes the proof that the components of the transformed smoothed mean with k > s are equal to the true mean during the interval. Moving onto the haloed covariance matrix, consider the transformed differential equation for the haloed smoothed is It is easy to show using the Eq. (A13) that, given the haloed covariance is of the form as is the case at τ 2 , then the covariance will remain in that form.