Quantum State Smoothing for Linear Gaussian Systems
QQuantum State Smoothing for Linear Gaussian Systems
Kiarn T. Laverick, Areeya Chantasri, and Howard M. Wiseman
Centre for Quantum Computation and Communication Technology (Australian Research Council),Centre for Quantum Dynamics, Griffith University, Nathan, Queensland 4111, Australia (Dated: May 17, 2019)Quantum state smoothing is a technique for assigning a valid quantum state to a partially observeddynamical system, using measurement records both prior and posterior to an estimation time. Weshow that the technique is greatly simplified for Linear Gaussian quantum systems, which have widephysical applicability. We derive a closed-form solution for the quantum smoothed state, which ismore pure than the standard filtered state, whilst still being described by a physical quantum state,unlike other proposed quantum smoothing techniques. We apply the theory to an on-thresholdoptical parametric oscillator, exploring optimal conditions for purity recovery by smoothing. Therole of quantum efficiency is elucidated, in both low and high efficiency limits.
Smoothing and filtering are techniques in classical es-timation of dynamical systems to calculate probabilitydensity functions (PDFs) of quantities of interest at sometime t , based on available data from noisy observation ofsuch quantities in time. In filtering, the observed data upto time t is used in the calculation. In smoothing, the ob-served data both before (past) and after (future) t can beused. For dynamical systems where real-time estimationof the unknown parameters is not required, smoothing al-most always gives more accurate estimates than filtering.In the quantum realm, numerous formalisms have beenintroduced which use past and future information [1–7].Many of these ideas have been applied, theoretically andexperimentally, to the estimation of unknown classicalparameters affecting quantum systems [8–14], or of hid-den results of quantum measurements [15–20]. The op-timal improvement obtained by using future informationin these applications comes from using classical Bayesiansmoothing to obtain the PDF of the variables of interest.Despite such applications of smoothing to quantum pa-rameter estimation, a quantum analogue for the classi-cal smoothed state (i.e. the PDF) was still missing. Asquantum operators for a system at time t do not com-mute with operators representing the results of later mea-surements on that system [21], a na¨ıve generalisation ofthe classical smoothing technique would not result in aproper quantum state [4, 5, 7]. As elucidated by Tsang[4] (see also the Supplemental Material of [5]), such aprocedure would result in a “state” that gives the (typi-cally anomalous) weak-value [2] as its expectation valuefor any observable. Thus, we will refer to this type ofsmoothed “state” for a quantum system as the SmoothedWeak-Value (SWV) state. In contrast to this, Guevaraand Wiseman [22] recently proposed a theory of quantumstate smoothing which also generalises classical smooth-ing but which gives a proper smoothed quantum state,i.e., both Hermitian and positive semi-definite.The quantum state smoothing theory of Ref. [22] con-siders an open quantum system coupled to two baths (seeRef. [12] for a similar idea). An observer, Alice, monitorsone bath and thereby obtains an “observed” measure-ment record O . Another observer, Bob (who is hiddenfrom Alice), monitors the remaining bath, unobserved by Alice, and thereby obtains an “unobserved” record U . IfAlice knew ←− U as well as ←− O (the back-arrows indicatingrecords in the past), she would have maximum knowl-edge of the quantum system, i.e., the “true” state ρ ←− O , ←− U at that time. Thus, Alice’s filtered and smoothed statescan be defined in the same form of a conditioned state, ρ C = (cid:88) ←− U ℘ C ( ←− U ) ρ ←− O , ←− U , (1)where the summation is over all possible records unob-served by Alice. For filtering ( ρ C = ρ F ), the PDF ofunobserved records is ℘ C ( ←− U ) = ℘ ( ←− U |←− O ) conditioned onher past record ←− O . For smoothing ( ρ C = ρ S ), one has ℘ C ( ←− U ) = ℘ ( ←− U |←→ O ) conditioned on Alice’s past-futurerecord ←→ O . By construction, Eq. (1) guarantees the posi-tivity of the smoothed quantum state.In this Letter we present the theory of quantum statesmoothing for Linear Gaussian Quantum (LGQ) systems.This can be applied to a large number of physical sys-tems, e.g., multimodal light fields [23, 24], optical andoptomechanical systems [13, 20, 21, 25–35], atomic en-sembles [36–38], and Bose-Einstein condensates [39]. Dueto the nice properties of LGQ systems, we are able to ob-tain closed-form solutions for the smoothed LGQ state.This makes them much easier to study even than the two-level system originally considered in [22], as there is noneed to generate numerically the numerous unobservedrecords appearing in the summation of Eq. (1). LQGsmoothing only requires solving a few additional equa-tions compared to classical smoothing for Linear Gaus-sian (LG) systems. The simplicity of our theory will en-able easy application to numerous physical systems, andalso allows analytical treatment of various measurementefficiency regimes. We give such a treatment here for anoptical parametric oscillator (OPO) on threshold [21, 25].As expected, our smoothed quantum state has higher pu-rity than the usual filtered quantum state, while the SWVstate is often unphysical, with purity larger than one.We begin by reviewing the necessary theoretical back-ground of classical LG systems and LGQ systems. Wethen develop quantum state smoothing for LGQ systems a r X i v : . [ qu a n t - ph ] M a y and obtain analytic results in different limits. Finally, weapply LGQ smoothing to the on-threshold OPO. LG systems and classical smoothing. — Consider a clas-sical dynamical system described by a vector of M pa-rameters x = { x , x , ..., x M } (cid:62) . Here (cid:62) denotes trans-pose. This system is regarded as an LG system if andonly if it satisfies three conditions [21, 40–45]. First, itsevolution can be described by a linear Langevin equationd x = A x d t + E d v p . (2)Here A (the drift matrix) and E are constant matricesand d v p is the process noise, i.e., a vector of independentWiener increments satisfyingE[d v p ] = , d v p (d v p ) (cid:62) = I d t . (3)Here E[ ... ] represents an ensemble average, and I is the M × M identity matrix. Second, knowledge about thesystem is conditioned on a measurement record y that islinear in x , y d t = C x d t + d v m , (4)where C is a constant matrix and the measurement noised v m is a vector of independent Wiener increments satis-fying similar conditions to Eq. (3). It is possible for theprocess noise and the measurement noise to be correlated,e.g., from measurement back-action, which is describedby a nonzero cross-correlation matrix Γ, computed fromΓ (cid:62) d t = E d v p (d v m ) (cid:62) . The third condition is that theinitial state of the system (i.e., the initial PDF of x , de-noted as ℘ ( x ) | t =0 ) is Gaussian; then the linearity condi-tions (first and second) guarantee the conditioned statewill remain Gaussian: ℘ C ( x ) = g ( x ; (cid:104) x (cid:105) C , V C ) , (5)which is fully described by its mean (cid:104) x (cid:105) C and variance(strictly, covariance matrix) V C ≡ (cid:104) xx (cid:62) (cid:105) C − (cid:104) x (cid:105) C (cid:104) x (cid:105) (cid:62) C ,throughout the entire evolution.If the above criteria are met, one can compute a filteredLG state conditioned only on the past record (before theestimation time t ). The filtered mean and variance aregiven by,d (cid:104) x (cid:105) F = A (cid:104) x (cid:105) F d t + K + [ V F ]d w F , (6)d V F d t = AV F + V F A (cid:62) + D − K + [ V F ] K + [ V F ] (cid:62) , (7)where d w F ≡ y d t − C (cid:104) x (cid:105) F d t is a vector of innovations, D = EE (cid:62) is the diffusion matrix, and we have defineda “kick” matrix, a function of V , via K ± [ V ] ≡ V C (cid:62) ± Γ (cid:62) . Initial conditions for these filtering equations are themean and variance of the initial Gaussian state.To solve for a smoothed LG state, one needs to includeconditioning on the future record, which can be obtainedfrom the retrofiltering equations − d (cid:104) x (cid:105) R = − A (cid:104) x (cid:105) R d t + K − [ V R ]d w R , (8) − d V R d t = − AV R − V R A (cid:62) + D − K − [ V R ] K − [ V R ] (cid:62) , (9) where K − [ V ] was defined above and d w R ≡ y d t − C (cid:104) x (cid:105) R d t . As the leading negative signs suggest, theseequations are evolved backward in time, from a final con-dition at t = T . This is typically taken to be an unin-formative PDF. Combining the filtered and retrofilteredsolutions Eqs. (6)–(9), one obtains a smoothed LG stateconditioned on the entire measurement record [40–44], (cid:104) x (cid:105) S = V S ( V − (cid:104) x (cid:105) F + V − (cid:104) x (cid:105) R ) , (10) V S = ( V − + V − ) − . (11) LGQ systems. — For a quantum system analogous tothe classical LG one, the system’s observables requireunbounded spectrums, represented by N bosonic modes.We denote such a system by a vector of M = 2 N observ-able operators ˆ x = (ˆ q , ˆ p , ..., ˆ q N , ˆ p N ) (cid:62) , where ˆ q k and ˆ p k are canonically conjugate position and momentum opera-tors for the k th mode, obeying the commutation relation[ˆ q k , ˆ p l ] = i (cid:126) δ kl . The system is called an LGQ systemif its dynamical and measurement equations are isomor-phic to those of a classical LG system [21, 25, 46–49].For quantum systems there are additional constraintson the system’s dynamics [21]. For example the ini-tial state must satisfy the Schr¨odinger-Heisenberg uncer-tainty relation, V + i (cid:126) Σ / ≥
0. Here Σ kl = − i [ˆ x k , ˆ x l ]is the symplectic matrix and V is the covariance matrix V kl = (cid:104) ˆ x k ˆ x l + ˆ x l ˆ x k (cid:105) / − (cid:104) ˆ x k (cid:105)(cid:104) ˆ x l (cid:105) , for ˆ x k being an ele-ment of ˆ x and (cid:104)·(cid:105) being the usual quantum expectationvalue. These let us represent the quantum state of anLGQ system by its Gaussian Wigner function [21] de-fined as W (ˇ x ) = g (ˇ x ; (cid:104) ˆ x (cid:105) , V ), using dummy variable ˇ x . Quantum state smoothing for LGQ systems. — We nowapply the quantum state smoothing technique [22] toLGQ systems. Following the Alice-Bob protocol intro-duced in Eq. (1), a true state of the LGQ system, denotedby the mean (cid:104) ˆ x (cid:105) T and a variance V T , is obtained givenboth ←− O and ←− U records. That is, the filtering equations(6)-(7) apply, but conditioned both on Alice’s observedrecord (of the form similar to (4)) y o d t = C o (cid:104) ˆ x (cid:105) T d t + d w o , (12)and on Bob’s record, unobserved by Alice, y u d t = C u (cid:104) ˆ x (cid:105) T d t + d w u , with independent Wiener noises. Theequations for the true state ared (cid:104) ˆ x (cid:105) T = A (cid:104) ˆ x (cid:105) T d t + K +o [ V T ]d w o + K +u [ V T ]d w u , (13)d V T d t = AV T + V T A (cid:62) + D − K +o [ V T ] K +o [ V T ] (cid:62) − K +u [ V T ] K +u [ V T ] (cid:62) , (14)where K ± r [ V ] = V C (cid:62) r + Γ (cid:62) r , for r ∈ { o,u } .Since Alice has no access to Bob’s record, her con-ditioned state (filtered or smoothed) is obtained bysumming over all possible true states of the system,with probability weights conditional on Alice’s observedrecords ( ←− O or ←→ O , respectively) as in Eq. (1). For LGQsystems, the state depends on ←− U only via the mean,Eq. (13). Therefore, we can replace the (symbolic) sumin Eq. (1) by an integral: ρ C = (cid:90) ℘ C ( (cid:104) ˆ x (cid:105) T ) ρ T ( (cid:104) ˆ x (cid:105) T )d (cid:104) ˆ x (cid:105) T . (15)Now let us define a “haloed” variable ◦ x = (cid:104) ˆ x (cid:105) T for no-tational simplicity. We can replace the conditional state ρ C and true state ρ T with their Wigner functions. Thelatter is Gaussian: g (ˇ x ; ◦ x , V T ). The integral in Eq. (15)convolves this with the PDF ℘ C ( ◦ x ) conditioned on theobserved records. This PDF is a conditioned (filtered orsmoothed) LG distribution for ◦ x , based on the observeddata, ℘ C ( ◦ x ) = g ( ◦ x ; (cid:104) ◦ x (cid:105) C , ◦ V C ), where ◦ V C is the condi-tional variance for the variable ◦ x [50]. As both functionsinside the integral Eq. (15) are Gaussian, the Wignerfunction for ρ C is also Gaussian: g (ˇ x ; (cid:104) ˆ x (cid:105) C , V C ) = (cid:90) g ( ◦ x ; (cid:104) ◦ x (cid:105) C , ◦ V C ) g (ˇ x ; ◦ x , V T )d ◦ x . (16)By elementary properties of convolutions, we get the con-ditioned mean (cid:104) ˆ x (cid:105) C = (cid:104) ◦ x (cid:105) C and the conditioned variance V C = ◦ V C + V T . This will allow us to solve for the filteredand smoothed quantum states for LGQ systems.Now, all that remains is to apply classical LG estima-tion theory (filtering or smoothing) to determine (cid:104) ◦ x (cid:105) C and ◦ V C . We first obtain [50] filtering equations for ◦ x ,using the past observed record Eq. (12),d (cid:104) ◦ x (cid:105) F = A (cid:104) ◦ x (cid:105) F d t + K +o [ ◦ V F + V T ]d ◦ w F , (17)d ◦ V F d t = A ◦ V F + ◦ V F A (cid:62) + ◦ D − K +o [ ◦ V F + V T ] K +o [ ◦ V F + V T ] (cid:62) , (18)where we have defined ◦ D = (cid:80) r ∈{ o , u } K +r [ V T ] K +r [ V T ] (cid:62) ,and d ◦ w F = y o d t − C o (cid:104) ◦ x (cid:105) F d t . We also show in [50] thatthis haloed filtered variance is related to the variance ofthe usual quantum filtered state V F (computed withoutinvoking the unobserved record) via V F = ◦ V F + V T withthe same mean (cid:104) ˆ x (cid:105) F = (cid:104) ◦ x (cid:105) F , consistent with the convo-lution (16). For the retrofiltering equations for ◦ x , usingthe future record, we have − d (cid:104) ◦ x (cid:105) R = − A (cid:104) ◦ x (cid:105) R d t + K − o [ ◦ V R − V T ]d ◦ w R , (19) − d ◦ V R d t = − A ◦ V R − ◦ V R A (cid:62) + ◦ D − K − o [ ◦ V R − V T ] K − o [ ◦ V R − V T ] , (20)which lead to a similar variance relation V R = ◦ V R − V T [50]. However, the minus sign in the ◦ V R relation indicatesthat the convolution (16) does not apply for retrofiltering,which propagates in the backward direction in time.We then combine the haloed filtering and retrofilteringequations, as in Eq. (10) and (11), to obtain the haloedsmoothing equations, and using (16), we arrive at the FIG. 1. (Colour online) Various long-time states of the on-threshold OPO system in Eq. (24), represented by their 1-SD Wigner function contours in phase space, centred at theorigin. The homodyne angles used by Alice and Bob ( θ o , θ u ) are at the black dot in Fig. 2. The unconditional state(solid black) shows infinite and finite variances in q and p ,respectively, as a result of the damping and squeezing. Al-ice’s filtered and smoothed states, are blue (filled grey) anddashed-red ellipses, respectively. The dotted-black ellipseshows the (pure) true state, conditioned on both Alice’s andBob’s results, while the dot-dashed green ellipse shows theSWV “state.” LGQ state smoothing equations (cid:104) ˆ x (cid:105) S = ( V S − V T )[( V F − V T ) − (cid:104) ◦ x (cid:105) F + ( V R + V T ) − (cid:104) ◦ x (cid:105) R ] , (21) V S = (cid:2) ( V F − V T ) − + ( V R + V T ) − (cid:3) − + V T , (22)as the main result of this Letter. In the classical limit,where there is no uncertainty relation for V T and we canlet V T →
0, these reproduce classical LG smoothing,Eqs. (10)–(11), as expected.The advantages LGQ state smoothing offers over filter-ing are readily seen in Fig. 1, where we note that the pu-rity for a Gaussian state is defined as P = ( (cid:126) / (cid:112) | V | − [21] for a variance V . The smoothed state has a smallervariance (higher purity) than the filtered state, but has alarger variance than a pure state (purity less than unity).In contrast, the SWV state for the same system (i.e., us-ing Eqs. (10)–(11)) is unphysical (its ellipse is smallerthan that of a pure state).Now that we have the closed-form expression for thesmoothed LGQ state, we can investigate, in the steadystate, some interesting limits in Alice’s measurement ef-ficiency η o , the fraction of the system output which isobserved by Alice.If, as in the OPO system we will consider later, theunconditioned ( η o = 0) variance diverges, then Alice’sconditioned (filtered and retrofiltered) variances, if finite,must grow as η o →
0. From Eqs. (21)–(22), when V F and V R are large, compared to V T , the smoothed LGQ statereduces to the SWV state Eqs. (10)–(11). The SWV statehas the same form as classical smoothed states, which of-ten have the same scaling as filtered states, but with amultiplicative constant improvement [8, 14, 51]. Conse-quently, in the limit η o →
0, we expect P SWV = P S ∝ P F FIG. 2. (Colour online) (Top) Contour plots of the RPR,Eq. (23), for the OPO system for different values of observedand unobserved homodyne phases using η o = 0 .
5. The dashedline represents θ o = θ u and the solid line is the optimal θ u (that giving the highest RPR for each value of θ o ). The cir-cle and the star relate to Figs. 1 and 3, respectively. (Bot-tom) Purity for the OPO’s filtered (solid blue) and smoothed(dashed red) states, choosing the optimal θ optu for each θ o . as functions of η o .In the opposite limit, η o →
1, we analytically show [50]that the relative purity recovery (RPR), R = P S − P F − P F , (23)a measure of how much the purity is recovered fromsmoothing over filtering relative to the maximum recov-ery possible, usually scales with the unobserved efficiency.That is, R ∝ η u ≡ − η o . Example of the on-threshold OPO system. — We nowapply quantum state smoothing to the on-threshold OPO[21, 25], an LGQ system with N = 1 described by themaster equation (cid:126) ˙ ρ = − i [(ˆ q ˆ p + ˆ p ˆ q ) / , ρ ] + D [ˆ q + i ˆ p ] ρ . (24)The first term defines a Hamiltonian giving squeezingalong the p -quadrature, while the second term describesthe oscillator damping. Here, the drift and diffusion ma-trices are A = diag(0 , −
2) and D = (cid:126) I . Let us as-sume that Alice observes the damping channel via ho-modyne detection. Therefore, the matrix C o in (12)is C o = 2 (cid:112) η o / (cid:126) (cos θ o , sin θ o ), where θ o is the homo-dyne phase [21, 25]. For simplicity, we assume Bobalso performs a homodyne measurement, with a dif-ferent phase θ u , so that C u = 2 (cid:112) η u / (cid:126) (cos θ u , sin θ u ). R
1, the RPR is ∝ (1 − η o ) (dashed black onright). The measurement back-actions are described by matri-ces Γ r = − (cid:126) C r /
2, for r ∈ { o,u } .We now solve for filtered and smoothed states for theOPO in steady state. We are particularly interested inthe RPR (23) of smoothing over filtering, and in the com-binations of homodyne phases that result in the largestRPR. The RPR is always positive (see Fig. 2), meaningthat the smoothed quantum state always has higher pu-rity than the corresponding filtered one. If Alice’s phase θ o is fixed, one might guess that Bob’s phase giving thebest purity improvement should be the same, θ u = θ o .However, that is not at all true (see Fig. 2). The optimal θ optu is not a trivial function of θ o . Rather, θ optu ≈
0, i.e.,Bob should measure the q -quadrature, which is presum-ably related to the fact that, without measurement in,the variance in q diverges.We then examine, in Fig. 3, the low and high efficiencylimits for the OPO system at the starred point in Fig. 2.As predicted earlier, in the limit η o → P SWV begins to separate from P S when the purities are no longer small, as the former pro-ceeds to have purity greater than 1 when η o > .
06. Inthe limit η o → η u = 1 − η o , as expected. The approximationholds surprisingly well even when η u is not small.To conclude, we have developed the theory of quantumstate smoothing, which gives valid smoothed quantumstates, for LGQ systems, a class of systems with widephysical applicability. By utilizing the Gaussian proper-ties, we obtained closed-form smoothing solutions that donot require simulations of ensembles of unobserved mea-surement records and corresponding true states. Thisenabled us to perform detailed analysis of the smoothedquantum state for various measurement regimes. A ques-tion for future work is to understand the (numericallyfound) optimal strategy for greatest improvement in thepurity. There are also interesting questions regardinghow the smoothed LGQ variance (22) would react to in-serting an invalid true state (i.e., one that does not solve Eq. (14)). 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Appendix A: Haloed Filtering, Retrofiltering, and Smoothing
We begin by deriving the haloed filtering and retrofiltering equations, and then show that V F = ◦ V F + V T and V R = ◦ V R − V T . We start with the equations for the true state (Eq. (13)-(14) in the main text) given both theobserved and unobserved records,d (cid:104) ˆ x (cid:105) T = A (cid:104) ˆ x (cid:105) T d t + K +o [ V T ]d w o + K +u [ V T ]d w u , (A1)d V T d t = AV T + V T A (cid:62) + D − K +o [ V T ] K +o [ V T ] (cid:62) − K +u [ V T ] K +u [ V T ] (cid:62) . (A2)We then use ◦ x = (cid:104) ˆ x (cid:105) T and define ◦ E d ◦ v p = K +o [ V T ]d w o + K +u [ V T ]d w u , so that Eq. (A1) is recast in a simple form asd ◦ x = A ◦ x d t + ◦ E d ◦ v p . (A3)That is, Eq. (A3) has the same form as the classical linear Langevin equation (Eq. (2) in the main text) and ◦ x can beconsidered a classical parameter (a possible trajectory of the centroid of the true state). Moreover, Alice’s observedmeasurement current (Eq. (12) in the main text), y o d t = C o ◦ x d t + d w o , (A4)has the same form as Eq. (4) in the main text, with d w o identified as d ◦ v m . The correlation of the measurement noisewith the process noise is easily evaluated as ◦ Γ (cid:62) d t = ◦ E d ◦ v p (d ◦ v m ) (cid:62) = K +o [ V T ]d t. (A5)That is, ◦ Γ = Γ o + C o V T . Similarly, we can define ◦ D = ◦ E ◦ E (cid:62) = K +o [ V T ] K +o [ V T ] (cid:62) + K +u [ V T ] K +u [ V T ] (cid:62) . (A6)
1. Filtering
From the above, we obtain the haloed filtering equations in a similar way using Eqs. (6)–(7) in the main text,replacing x , D , C , and Γ with ◦ x , ◦ D , C o , and ◦ Γ, respectively. The haloed filtering equations are then given byd (cid:104) ◦ x (cid:105) F = A (cid:104) ◦ x (cid:105) F d t + K +o [ ◦ V F + V T ]d ◦ w F , (A7)d ◦ V F d t = A ◦ V F + ◦ V F A (cid:62) + ◦ D − K +o [ ◦ V F + V T ] K +o [ ◦ V F + V T ] (cid:62) , (A8)as shown in Eqs. (17)–(18) of the main text, where d ◦ w F ≡ y o d t − C o (cid:104) ◦ x (cid:105) F d t .To show the relation between the haloed filtering equation and the usual quantum filtering equations, we begin byrecognising a different form of ◦ D . From Eq. (A2) we see that ◦ D = AV T + V T A (cid:62) + D − d V T d t , (A9)and substituting this into Eq. (A8) to getdd t ( ◦ V F + V T ) = A ( ◦ V F + V T ) + ( ◦ V F + V T ) A (cid:62) + D − K +o [ ◦ V F + V T ] K +o [ ◦ V F + V T ] . (A10)If we use V F = ◦ V F + V T in the above equation, we obtain exactly Eq. (7) in the main text, i.e., an equation for thefiltered variance. Therefore, we can regard this variance V F as the variance of the usual filtering for an LGQ system,which one could derive without the Alice-Bob protocol. We can also check that the haloed filtered mean is identical tothe usual filtered mean (Eq. (6) of the main text, but for an LGQ system), as should be the case from the convolution.This can be seen by using V F = ◦ V F + V T in the haloed filtered estimate Eq. (A7), which givesd (cid:104) ◦ x (cid:105) F = A (cid:104) ◦ x (cid:105) F d t + K +o [ V F ]d ◦ w F . (A11)Considering the same initial conditions for (cid:104) ◦ x (cid:105) F and (cid:104) ˆ x (cid:105) F , the haloed filtered mean will remain identical to the filteredmean, (cid:104) ˆ x (cid:105) F = (cid:104) ◦ x (cid:105) F , so the innovations will also be identical, with d ◦ w F = d w F ≡ y o d t − C o (cid:104) ˆ x (cid:105) F . Note that neitherof these innovations is the same as the d w o = y o d t − C o (cid:104) ˆ x (cid:105) T in Eq. (A1), as the latter is the innovation in Alice’srecord defined using the true state, i.e., from Bob’s all-knowing point of view rather than Alice’s.
2. Retrofiltering and Smoothing
Similarly to the filtering case, we can get the haloed retrofiltering equations using Eqs. (8)–(9) in the main text − d (cid:104) ◦ x (cid:105) R = − A (cid:104) ◦ x (cid:105) R d t + K − o [ ◦ V R − V T ]d ◦ w R , (A12) − d ◦ V R d t = − A ◦ V R − ◦ V R A (cid:62) + ◦ D − K − o [ ◦ V R − V T ] K − o [ ◦ V R − V T ] , (A13)where d ◦ w R ≡ y o d t − C o (cid:104) ◦ x (cid:105) R . Now, adding Eq. (A9) to Eq. (A13), we arrive at − dd t ( ◦ V R − V T ) = − A ( ◦ V R − V T ) − ( ◦ V R − V T ) A (cid:62) + D − K − o [ ◦ V R − V T ] K − o [ ◦ V R − V T ] , (A14)which is the equation for the retrofiltered variance V R and as a result we have V R = ◦ V R − V T .The relation V R = ◦ V R − V T is interesting, as it shows the asymmetry between the filtered state and the retrofilteredeffect. Note that V R is not the variance for a state conditioned only on the future observed record. Rather it is thevariance of a POVM element for the future record. To obtain V R in general it will be easier to compute the inverse of ◦ V R , rather than ◦ V R itself, for calculating the smoothed state. This is because the final condition on the retrofilteredvariance (haloed or not) is often taken to be infinite, as mentioned in the main text. Defining the inverse ◦ Λ R = ◦ V − ,we use the relation dd t (cid:16) ◦ V R ◦ Λ R (cid:17) = 0 to get d ◦ Λ R d t = − ◦ Λ R d ◦ V R d t ◦ Λ R . (A15)From this we obtain − d ◦ Λ R d t = ¯ A ◦ Λ R + ◦ Λ R ¯ A (cid:62) − ◦ Λ R ¯ D ◦ Λ R + C (cid:62) o C o , (A16)with ¯ A = A − Γ o C o − V T C (cid:62) o C o and ¯ D = ◦ D − Γ (cid:62) o Γ o − Γ (cid:62) o C o V T − V T C (cid:62) o Γ o − V T C (cid:62) o C o V T . This way, the final conditionis ◦ Λ R ( T ) = 0 and the LGQ smoothed state in terms of ◦ Λ R and V F is given by (cid:104) ◦ x (cid:105) S = ( V S − V T )[( V F − V T ) − (cid:104) ◦ x (cid:105) F + ◦ Λ R (cid:104) ◦ x (cid:105) R ] , (A17) V S = (cid:104) ( V F − V T ) − + ◦ Λ R (cid:105) − + V T . (A18) Appendix B: Purities and RPR for different efficiency limits1. Low Efficiency Limit
For the low efficiency limit, specifically for the on-threshold OPO system, we will show here that the purity P C ∝ η / , where C ∈ { F , SWV } . We first consider the case η o = 0, i.e., no conditioning on measurement results,where the linear matrix equation for the steady state of the filtered variance V F is given by AV F + V F A (cid:62) + D = 0 , (B1)with A = diag(0 , −
2) and D = (cid:126) I . Since the matrix A for this case is not strictly stable (with one of its eigenvaluesbeing zero), there is no stationary matrix solution for (B1), but in a long-time limit, we have [21] V F → (cid:126) (cid:20) ∞
00 1 / (cid:21) . (B2)We then consider the low efficiency case, for a small but non-zero observed efficiency η o →
0. Now the filtered varianceis given by Eq. (7) in the main text. This equation leads to the variance in the q -quadrature (top-left element of thevariance matrix) becoming finite, as long as the homodyne current contains some information about this quadrature( θ o (cid:54) = ± π/ V F of the form V F = (cid:126) (cid:20) α F β F β F γ F (cid:21) , (B3)we can obtain three relations for α F , β F and γ F , using Eq. (7) (in the main text) and the matrices C o , Γ o for theOPO system defined there: 1 = η o [( α F −
1) cos θ o + β F sin θ o ] , (B4) − β F = η o (( α F −
1) cos θ o + β F sin θ o ) ( β F cos θ o + ( γ F −
1) sin θ o ) , (B5)1 − γ F = η o [ β F cos θ o + ( γ F −
1) sin θ o ] . (B6)To evaluate the purity of the filtered LGQ state, P F , in the low efficiency limit, we do not need to solve for the fullsolution of V F from the above equations. The major contribution the measurement has to the variance is to bringthe q -component, that is the top-left element, α F , from infinity to a large but finite value; whereas the γ F and β F elements, describing variance in p -quadrature and covariance between the two quadratures, should still have valuesclosed to those of the unconditional solution since they are finite even in the absence of any information, and thesmall amount of information in the low efficiency limit will make little difference. Consequently, we can treat α F asbeing much larger than γ F and β F , where the former scales as an inverse order of η o and the latter two are O ( η k o ) for k ≥
0. From this, we can only use (B4) and solve for α F to leading order in η o , giving α F ≈ | cos θ o | − η − / . (B7)We now calculate the filtered purity using an assumption that the variance in p -quadrature represented by γ F shouldstill stay close to its unconditional value 1 /
2, beginning with | V F | ≈ | θ o | − η − / − β ≈ | θ o | − η − / . (B8)Thus we get P F = (cid:126) (cid:112) | V F | − = (cid:112) | cos θ o | η / , (B9)where we can see the η / scaling.For the purity of the smoothed weak-valued state P SWV in the small efficiency limit, we can use the intuition thatthe information used in the classical smoothing is twice as much in the filtering (considering the steady-state case),which should result in reducing the large variance in q -quadrature by half, i.e., α SWV ≈ α F , still much larger than β SWV and γ SWV . As in the filtered case, we expect the latter two to remain little changed from their unconditionedvalues. Thus | V SWV | ≈ | θ o | − η − / and P SWV = (cid:126) (cid:113) | V − | ≈ (cid:112) | cos θ o | η / , (B10)where we see the constant factor of improvement √ P F and P SWV in this limit, η o → V F and P F to leading orders in η o , V F = (cid:126) (cid:34) | sec θ o | η − /
2o 12 | sin θ o | η / | sin θ o | η / / (cid:35) , P F = (cid:112) | cos θ o | η / , (B11)as expected. For the V SWV , we first need to calculate the retrofiltered variance V R . We solve for the full solution of V R from Eq. (9) in the main text (in the steady-state limit) to leading orders in η o , V R = (cid:126) (cid:34) | sec θ o | η − / | csc θ o | η − / | csc θ o | η − / | csc θ o | η − (cid:35) . (B12)Finally, we can calculate the SWV variance (using Eq. (11) in the main text) and its purity, arriving at V SWV = (cid:126) (cid:34) | sec θ o | η − /
2o 12 | sin θ o | η / | sin θ o | η / / (cid:35) , P SWV = 2 (cid:112) | cos θ o | η / , (B13)which are consistent with our intuitive approach.0
2. High Efficiency Limit
In this section we derive the η u scaling for the steady-state relative purity recovery (RPR) in the high efficiencylimit for Alice ( η o → η o = 1 (allavailable records are observed), the filtered and true variances are equal. So, if we express the filtered variance as V F = V T + Q , it will typically be the case that Q → η o →
1, and we assume this to be so in all that follows. Inthis limit, we can see that the variance of the smoothed state, Eq. (A18), is V S = (cid:104) Q − + ◦ Λ R (cid:105) − + V T (B14)= Q (cid:104) ◦ Λ R Q (cid:105) − + V T (B15) ≈ Q (cid:104) − ◦ Λ R Q (cid:105) + V T (B16)= V F − Q ◦ Λ R Q , (B17)where the approximation holds since Q is small. This nicely shows how information from the future, as expressed by ◦ Λ R (cid:54) = 0, makes the smoothed variance smaller than the filtered one.Now we show that the RPR (Eq. (23) of the main text) scales as O ( Q ). The purity of a LGQ state is given by P C = (cid:126) (cid:104)(cid:112) | V C | (cid:105) − , (B18)for C = F , S or T. The purity of the filtered state in the high efficiency limit is P F = (cid:126) (cid:104)(cid:112) | V T + Q | (cid:105) − , (B19)= (cid:126) (cid:112) | V T | (cid:20)(cid:113) | I + V − Q | (cid:21) − , (B20)= P T (cid:20)(cid:113) | I + V − Q | (cid:21) − . (B21)Now we need to evaluate | I + Y | , where Y = V − Q is small. Using the formula | e Y | = exp[Tr( Y )] [53] and expandingthe left and right exponential terms, we get, to leading order | I + Y | ≈ Y ) . (B22)The purity of the filtered state is thus given by P F ≈ P T (cid:20)(cid:113) V − Q ) (cid:21) − (B23) ≈ P T (cid:2) − Tr (cid:0) V − Q (cid:1) / (cid:3) . (B24)For the purity of the smoothed state, we express the smoothed variance as V S = V F − X , where X = Q ◦ Λ R Q .Following the similar derivation as for the purity of the filtered state, we obtain the purity of the smoothed state as P S ≈ P F (cid:104) (cid:16) V − Q ◦ Λ R Q (cid:17) / (cid:105) . The RPR is then given by R = P S − P F − P F (B25)= P F + P F Tr (cid:16) V − Q ◦ Λ R Q (cid:17) / − P F − P T + P T Tr (cid:0) V − Q ) (cid:1) / P F Tr (cid:16) V − Q ◦ Λ R Q (cid:17) / − P T + P T Tr (cid:0) V − Q ) (cid:1) / . (B27)1If we consider that Bob observes the part unobserved by Alice’s measurement, i.e., η u = 1 − η o , the true state will bea pure state ( P T = 1), as it is conditioned on all possible measurement records. The RPR then becomes R ≈ P F Tr (cid:16) ( V T + Q ) − Q ◦ Λ R Q (cid:17) Tr (cid:0) V − Q (cid:1) (B28) ≈ P F Tr (cid:104)(cid:0) V − − V − QV − (cid:1) Q ◦ Λ R Q (cid:105) Tr (cid:0) V − Q (cid:1) (B29) ≈ P F Tr (cid:16) V − Q ◦ Λ R Q (cid:17) Tr (cid:0) V − Q (cid:1) (B30)= O ( Q ) O ( Q ) = O ( Q ) . (B31)Finally, all that is left is to check how Q scales with the unobserved measurement efficiency η u . Substituting in V F = V T + Q into Eq. (7) in the main text, we obtain0 = A ( V T + Q ) + ( V T + Q ) A (cid:62) + D − K +o [ V T + Q ] K +o [ V T + Q ] (cid:62) , (B32)considering the system to be in the steady state. Rearranging the above terms and using Eq. (14) in the main text(also in the steady state), we arrive at − ¯ AQ − Q ¯ A (cid:62) + QC (cid:62) o C o Q = η u ¯ K +u [ V T ] ¯ K +u [ V T ] (cid:62) , (B33)where we are using ¯ A = A − Γ (cid:62) o C o − V T C (cid:62) o C o as in Sec. I B above, and we have defined ¯ K +r [ V ] = (cid:112) /η u K +u [ V ] sothat in the limit η u →
0, all matrices in Eq. (B33), excluding Q , are independent of Bob’s measurement efficiency η u . That is because the matrices that are proportional to some positive power of Alice’s efficiency η o = 1 − η u have alimit independent of η u in the limit η u →
0. Now it might be thought that we can immediately discard the bilinearterm in Eq. (B33), since Q is small. This results in the linear equation − ¯ AQ − Q ¯ A (cid:62) = η u ¯ K +u [ V T ] ¯ K +u [ V T ] (cid:62) . (B34)However this equation has a unique valid (positive semidefinite) solution for Q if and only if ¯ A is Hurwitz. (AHurwitz matrix is a real matrix where the real part of the eigenvalues are strictly negative.) Fortunately, we canexpect this to be the case, for the following reason. In the limit η u → V T = V F − Q → V F , and the matrix¯ A → M ≡ A − Γ (cid:62) o C o − V F C (cid:62) o C o . Now this matrix M is well studied in control theory [21]; when the stationaryfiltered variance V F makes M Hurwitz, it is said to be a stabilizing solution. There are well known conditions thatensure this to be the case [21] and these are satisfied for most systems of interest. Moreover, these conditions areweaker for the case of quantum systems [21]. Thus we will assume ¯ A to be Hurwitz. From Eq. (B34) we immediatelysee that Q , and consequently, from Eq. (B31), the relative purity recovery ( R ), scales as η u = 1 − η o in the highefficiency limit η o →
1. We can see this scaling explicitly in the 2-dimensional case, relevant to the OPO system,where the solution to the linear matrix equation Eq. (B34) is [21, 54] Q = η u | ¯ A | ¯ K +u [ V T ] ¯ K +u [ V T ] (cid:62) + ( ¯ A − I Tr[ ¯ A ]) ¯ K +u [ V T ] ¯ K +u [ V T ] (cid:62) ( ¯ A − I Tr[ ¯ A ]) (cid:62) A ] | ¯ A | ..