Quantum state smoothing: Why the types of observed and unobserved measurements matter
QQuantum state smoothing: Why the types ofobserved and unobserved measurements matter
Areeya Chantasri, Ivonne Guevara, Howard M. Wiseman
Centre for Quantum Computation and Communication Technology(Australian Research Council),Centre for Quantum Dynamics,Griffith University, Nathan, Queensland 4111, AustraliaE-mail: [email protected], [email protected],[email protected]
Abstract.
We investigate the estimation technique called quantum state smoothingintroduced by Guevara and Wiseman [Phys. Rev. Lett. , 180407 (2015)], whichoffers a valid quantum state estimate for a partially monitored system, conditioned onthe observed record both prior and posterior to an estimation time. Partial monitoringby an observer implies that there may exist records unobserved by that observer. Itwas shown that, given only the observed record, the observer can better estimate theunderlying true quantum states, by inferring the unobserved record and using quantumstate smoothing, rather than the usual quantum filtering approach. However, theimprovement in estimation fidelity, originally examined for a resonantly driven qubitcoupled to two vacuum baths, was also shown to vary depending on the types ofdetection used for the qubit’s fluorescence. In this work, we analyse this variation ina systematic way for the first time. We first define smoothing power using an averagepurity recovery and a relative average purity recovery, of smoothing over filtering.Then, we explore the power for various combinations of fluorescence detection forboth observed and unobserved channels. We next propose a method to explain thevariation of the smoothing power, based on multi-time correlation strength betweenfluorescence detection records. The method gives a prediction of smoothing power fordifferent combinations, which is remarkably successful in comparison with numericallysimulated qubit trajectories.
1. Introduction
Theories for open quantum systems and quantum measurements [1, 2, 3, 4, 5] developedduring the past decades are now becoming standard tools in analysing and designingexperiments for quantum technologies. Decoherence from uncontrollable or undetectablenoise from the environment or measurement backaction is ubiquitous in currentexperiments in different platforms, such as in superconducting circuits [6, 7, 8], ion-trapexperiments [9, 10], and NV-centers [11]. However, given records from measurementsthat can be made on such systems, there can be different estimation strategies to best a r X i v : . [ qu a n t - ph ] A p r uantum state smoothing: Why the types of measurements matter prior to the estimationtime, has an analogue in the well-known quantum trajectory theory [1, 2, 3], describingstochastically evolving quantum states.From the classical approach to statistics, an estimated state, i.e., a PDF, of adynamical system can be conditioned on measurement results in three different ways:conditioning on the results prior to an estimation time (filtering), posterior to theestimation time (retrofiltering), and both (smoothing). In the regime where real-timestate estimation is not required, smoothing is optimal in the sense that the whole(all-time) measurement records are used in the estimation. Carrying these ideas tothe quantum regime, the analogues of the classical filtered state and the retrofilteredcounterpart are the quantum trajectory and retrodictive effect [15], respectively. Forthe smoothing, there have been various approaches to applying classical smoothingto quantum systems in the literature. Smoothing was first used for hybrid classical-quantum systems in Tsang’s work [16], where an unknown classical random process wasestimated from monitoring a quantum system affected by the classical process. In thelater work also by Tsang [17], smoothing for solely quantum systems was suggested inthe form of an expectation value of an observable of interest with a product of a quantumfiltered state and a retrodictive effect. This approach provided a smoothed estimate ofan unknown weak-measurement result of such observable, and the estimate was shownto be directly related to the weak value [18, 19]. This idea was further investigated foran estimation of a hidden but arbitrary-strength measurement result, via a formalismcalled past quantum states [20, 21, 22, 23, 24].However, as pointed out in [17, 20], the above methods could not provide a smoothedestimate of a quantum state in the usual sense, i.e., an estimated state that is Hermitian,positive semi-definite, and thus satisfies the Heisenberg uncertainty principle. Thisdeficiency was removed recently in Ref. [25], which proposed quantum state smoothing ,whereby a smoothed quantum state is introduced as a convex average of all possible true but unknown states weighted with a classical smoothed PDF. This allows one to assigna quantum state in the usual sense to a system of interest, conditioned on measurementrecords both prior and posterior to the estimation time. Inspired by the semiclassical“posterior decoding” of Ref. [26], Guevara and Wiseman introduced quantum statesmoothing in Ref. [25] by considering a situation in which a quantum system of interestis coupled to several baths (see also the subsequent related work of Ref. [27]). Anobserver, denoted by Alice, can measure only some of the baths, yielding an observed record O . It is then postulated that there exists another party, named Bob, who notonly has access to Alice’s record but also can monitor the rest of the baths, yielding ahidden or unobserved (by Alice) record U . Therefore, given both O and U , Bob knowsthe true state of the system, whereas Alice can only do her best to estimate Bob’s uantum state smoothing: Why the types of measurements matter Photon detection U nobserved channel Record: O bserved channel Record: ! O
Figure 1.
Schematic diagrams showing the main idea of this paper. A resonantlydriven two-level system (qubit) is coupled to two baths, one observed by Alice andanother unobserved by her. The fluorescence detection for the two channels can beone of the three types: (a) photon detection, (b) x -homodyne detection, and (c) y -homodyne detection. Quantum state smoothing (plots on the right) is appliedto the observed record, by computing a set of “possibly true” states { ρ ←— O , ←— U } andthe retrodictive operator ˆ E —→ O , and combining them according to Eq. (7) to obtain asmoothed state ρ S for Alice. state using her partially observed record. Using this Alice-Bob protocol, we can askquestions regarding the difference between Alice’s estimated states (using smoothingand filtering) and the underlying true quantum state (Bob’s state) of the system, interms of the fidelity or the purity of the estimated states.In the original work [25], a theoretical model for a resonantly driven two-levelsystem (a qubit with a Rabi oscillation) coupled to bosonic baths, was considered asan example. Part of the fluorescence of the driven qubit was monitored by Alice witha homodyne detection, and the remaining fluorescence was detected by Bob, using aphoton detector. It was shown that Alice’s smoothed states (using the quantum statesmoothing) have higher purity on average than her filtered states (i.e., the conventionalquantum trajectory approach [28, 29]), and that this equates to higher fidelity on averagewith the corresponding Bob’s true states. However, it was shown that there werenoticeable differences in the purity improvement, from using different phases in Alice’shomodyne detection setup. This suggested that the choices of Alice’s detection (andpossibly also Bob’s detection) could affect the purity improvement in using smoothingover filtering. In this paper, we take up this issue and further questions regarding the smoothing power . That is, we study qualitative measures for the improvement smoothingoffers over filtering, for various Alice-Bob detection choices. The specific questionswe ask are: How does the smoothing power depend on the types of measurementperformed on the qubit system? Can we qualitatively predict the smoothing powerfor all combinations of qubit measurements, without doing stochastic simulations?The specific strategy of this paper is to investigate the smoothing power for aresonantly driven qubit, considering all possible combinations of observed-unobserved uantum state smoothing: Why the types of measurements matter correlations betweenobserved and unobserved records, specifically, two-time and three-time correlators,which we calculate analytically. We suggest a systematic approach to decide howmuch each correlator is relevant to the smoothing power. On this basis, we made ourpredictions for the nine observed-unobserved combinations and then test them usingnumerical simulations of filtered and smoothed states.We organize the paper as the followings. We first briefly review, in Section 2,the quantum trajectory theory (quantum state filtering) and quantum state smoothingtheory, and introduce two measures for the smoothing power, in addition to the averagepurity recovery used in [25]. In Section 3, we introduce the theoretical model of aresonantly driven qubit coupled to bosonic baths, and present three possible detectionsfor bath monitoring (unravelling) along with the stochastic master equations, whichgenerate quantum trajectories for the filtered states. We then apply the quantumstate smoothing method to this model in Section 4, and display numerically simulatedfiltered and smoothed states, for a few different combinations of observed-unobservedmeasurements. The derivations and analyses of correlators between observed andunobserved records are presented in Section 5, where we also make predictions on thesmoothing power for all nine observed-unobserved combinations. Section 6 contains thenumerical results for the purity recoveries, including errors, for all nine combinations,for comparison with the predictions made in Section 5. The discussion and conclusioncomprise Section 7. We also include detailed analysis for the numerical errors inAppendix A.
2. Quantum trajectory theory and quantum state smoothing
Consider a dynamical quantum system interacting with multiple baths under thestrongest Markov assumption [30]. This allows the system’s state to be monitoredcontinuously in time via observation of the bath states. Whether the bath states arenot measured, or whether the results of the measurement are ignored, the dynamics of uantum state smoothing: Why the types of measurements matter ρ ( t ) = − i d t [ ˆ H, ρ ( t )] + d t L ∑ k = D[ ˆ c k ] ρ ( t ) , (1)where ρ is the system’s state matrix and ˆ H is a Hamiltonian describing any unitarydynamics of the system. The summation in the last term is over all the Lindbladsuperoperators D[ ˆ a ] ρ = ˆ aρ ˆ a † − ( ˆ a † ˆ aρ + ρ ˆ a † ˆ a ) given the Lindblad operators { ˆ c , ˆ c , ..., ˆ c L } which describe decoherence from L channels of the system-bath interaction.In the situation that the baths are measured continuously in time, and the resultsare not ignored, the system’s state ρ should reflect the information in the availablemeasurement records. So far, the most commonly used approach is the quantumtrajectory theory, where an estimated state of the system at any time t is obtainedby conditioning on the measurement records prior to that time. This is also knownas quantum state filtering [1, 2, 3, 4]. For now, let us consider, for simplicity, asingle decoherence channel ( L = ) and no unitary dynamics in Eq. (1), and denotea measurement result observed from time t to t + d t by r t . A quantum filtered stateis computed using a measurement operation via ˜ ρ ( t + d t ) = M r t ˜ ρ ( t ) , where ˜ ρ is theunnormalised state. Following the notation used in [25], we define the measurementoperation M r t , which acts on everything to its right, with an ostensible PDF ℘ ost such that the completeness relationship is given by ∫ d r t ℘ ost ( r t ) Tr [M r t ρ ] =
1, for anormalised state ρ . Given the measurement past record , ←— R τ ≡ { r t ∶ t ∈ [ t , τ )} , for a timeof interest τ and an initial state ρ ( t ) = ρ , we get an unnormalised quantum filteredtrajectory and its corresponding probability of the past record from,˜ ρ ←— R τ ( τ ) = M r τ − d t ⋯M r t ρ , (2) ℘(←— R τ ∣ ρ ) = Tr [ ˜ ρ ←— R τ ( τ )]℘ ost (←— R τ ) , (3)where we have used ℘ ost (←— R τ ) = ∏ τ − d tt = t ℘ ost ( r t ) as the ostensible PDF for the pastrecord ←— R τ . By dividing a solution of Eq. (2) with its trace, we obtain a normalisedtrajectory of the quantum filtered state ρ ←— R τ ( τ ) . The purity of this filtered statewill be unity, if the measurement results ←— R τ are from a perfect measurement setup(no loss and no other decoherence effects) and the initial state ρ is pure. On theother hand, if the measurement result r t is hidden or ignored, the system’s staterepresenting the lack of knowledge then decoheres according to the master equationEq. (1), i.e., ρ ( t + d t ) = ρ ( t ) + d t D[ ˆ c ] ρ ( t ) = ∫ d r t ℘ ost ( r t )M r t ρ ( t ) = ∫ d r t ℘( r t ∣ ρ ) ρ ( t ) .The last expression is an average over normalised quantum states weighted by the trueprobabilities of the measurement results. This is why the quantum trajectory is said togenerate unravellings of the Lindblad master equation. We will discuss more on differentunravellings for a driven qubit example in the next section. We also note that, for therest of the paper, we will use the word “trajectory” in a more general sense than thequantum filtered state trajectory, applying it to any path of a quantum state in time. uantum state smoothing: Why the types of measurements matter future record , —→ R τ ≡ { r t ∶ t ∈ [ τ, T )} , the information can be included in the estimation as an effectoperator denoted by ˆ E . The effect operator evolves in time according to an adjointmeasurement operation ˆ E ( t ) = M † r t ˆ E ( t + d t ) . The adjoint operation is applied in atime-backward direction to an uninformative final time condition ˆ E ( T ) = ˆ I , giving aretrofiltered matrix and corresponding probability of the future record,ˆ E —→ R τ ( τ ) = M † r τ ⋯M † r T − d t ˆ I, (4) ℘(—→ R τ ∣ ρ τ ) = Tr [ ˆ E —→ R τ ( τ ) ρ ( τ )]℘ ost (—→ R τ ) , (5)where the ostensible PDF ℘ ost (—→ R τ ) is for the future record —→ R τ and ρ ( τ ) is a system’sstate at time τ . It can be shown [17] that the effect operator can be considered as aquantum analogue of the classical retrodictive likelihood function, i.e., the PDF of thefuture measurement record given a state at an estimation time τ .Following Guevara-Wiseman quantum state smoothing [25], there are twomeasurement records; one is observed ( O ) and another is unobserved ( U ) by Alice.Note that we now consider a general case, where the bold letter ( O or U ) denotesmeasurement records arising from an arbitrary number L of channels in Eq. (1). Alicedoes not know Bob’s record U true , but she can guess a possible U . Let us assume thatan initial state of the system of interest is known. We can then construct a possible true(Bob’s) state ρ ←— O , ←— U ( t ) from the observed record O and a possible unobserved record U using Eq. (2), with ←— R replaced by ←— O and ←— U . Since Alice has access only to the observedpart, her filtered state, denoted by ρ F , will then be equivalent to ρ F ( t ) = ρ ←— O ( t ) = E ←— U ∣←— O [ ρ ←— O , ←— U ( t )] = ∑ ←— U ℘(←— U ∣←— O ) ρ ←— O , ←— U ( t ) , (6)where ℘(←— U ∣←— O ) ∝ ℘ ost (←— U ) Tr [ ˜ ρ ←— O , ←— U ( t )] . We have used E A ∣ B [ X ] to represent an expectedvalue of X averaged over A for a given B , with probability weight given by ℘( A ∣ B ) .We can interpret Eq. (6) as an average over all possible true states guessed by Alice,conditioning on her past record ←— O . This also coincides with a filtered state ρ ←— O ( t ) givenby Eq. (2), with M o t not being purity-preserving because of the extra decoherencecoming from the unobserved record U . In a similar spirit, the quantum smoothed stateis defined as ρ S ( t ) = E ←— U ∣←→ O [ ρ ←— O , ←— U ( t )] = ∑ ←— U ℘(←— U ∣←→ O ) ρ ←— O , ←— U ( t ) , (7)where the only difference from the filtered state Eq. (6) is the weighting probability ℘(←— U ∣←→ O ) ≡ ℘(←— U ∣←— O , —→ O ) , which is now conditioned on both the past and future observedrecords. To simulate the smoothed state, we require an expression for this conditionalPDF, which can be computed. This can be obtained from elementary manipulation of uantum state smoothing: Why the types of measurements matter ℘(←— U ∣←→ O ) ∝ ℘(—→ O ∣←— U , ←— O )℘(←— U , ←— O ) , ∝ ℘ ost (←— U ) Tr [ ˆ E —→ O ( t ) ˜ ρ ←— O , ←— U ( t )] , (8)where, in both lines, we have omitted any proportionality factors that are independentof the unobserved record. It is obvious that the smoothed state in Eq. (7) is Hermitianand positive semi-definite, as desired.From comparing Eq. (6) and (7), the smoothing equation uses information of thewhole observed record ←→ O . Therefore, one would expect Alice’s smoothed state ρ S toestimate Bob’s (true) state ρ T ≡ ρ ←— O , ←— U true better than Alice’s filtered state ρ F does. Toquantify the quality of conditioned estimates ρ C (which are either ρ F or ρ S ), we lookat the fidelity between them and the true state, and average over all possible observedrecords to get a figure of merit. In a specific case when the true state is pure, it hasbeen shown [25] that the average fidelity is equivalent to the average purity of Alice’sconditioned states, that isE O [ F [ ρ T ( t ) , ρ C ( t )]] = E O [ P [ ρ C ( t )]] . (9)This is because the fidelity, as first defined by Jozsa [32], reduced to F [ ρ T , ρ C ] ≡ Tr [ ρ T ρ C ] for pure state ρ T , and the purity is P [ ρ C ] ≡ Tr [ ρ ] . The expected value E O [⋅] iscalculated by averaging over all realisations of observed records O . From Eq. (9), wecan then use the average purity as a measure of how well a conditioned state ρ C estimatesthe unknown state ρ T , without the need to know what the actual ρ T is.For an individual observed record, the smoothed state purity at any given time canbe sometimes larger, sometimes smaller, than the filtered one, resulting from fluctuationin individual noises. However, on average, we expect the smoothed purity to perform atleast equal to, and generally better than, the filtered purity. In this paper, we use twomeasures to quantify the smoothing power, which are
1) Average Purity Recovery R A ,t = E O [ P [ ρ S ( t )]] − E O [ P [ ρ F ( t )]] , (10)
2) Relative Average Purity Recovery R R ,t = E O [ P [ ρ S ( t )]] − E O [ P [ ρ F ( t )]] E O [ P [ ρ T ( t )]] − E O [ P [ ρ F ( t )]] . (11)The first quantity describes how much purity is improved on average, from usingsmoothing over filtering, whereas the second quantity describes a ratio of the purityimprovement over the maximum improvement possible. We will see later in the exampleof the driven two-level system that the bound of the average purity set by E O [ P [ ρ T ( t )]] in (11) can be replaced by a different value, less than one, if the dynamics of the filteredand smoothed states are confined to some subspace of quantum states. We note thatthe discussion in this section is general and can be applied to any type of Markovianquantum systems. But in the next sections, we turn to the applications of the smoothingtechnique to the afore-mentioned example. uantum state smoothing: Why the types of measurements matter
3. Different unravellings for a resonantly driven qubit in vacuum bosonicbaths
We follow the application of quantum state smoothing in Ref. [25], considering aresonantly driven two-level system spontaneously emitting photons to coupled vacuumbosonic baths. The baths can be measured, for example, via photon detection or dynedetection [33]. The Lindblad operator for the spontaneous decay of the two-level systemis a lowering operator ˆ σ − , therefore, from Eq. (1) with L =
1, we obtain a Lindbladequation of the form, d ρ = − i d t [ ˆ H, ρ ( t )] + d t γ D[ ˆ σ − ] ρ ( t ) , (12)where we have used γ for the total decay rate and ˆ H = ( Ω / ) ˆ σ x describes resonantdriving in a rotating frame, where Ω is the Rabi frequency. This Lindblad evolutioncan be unravelled, giving a filtered state trajectory that is dependent on the type ofmeasurement applied on the bath and the particular measurement record obtained. Inthis work, we are interested in three types of bath detection (as shown in Figure 1): (a)direct photon detection (jump record), (b) x -quadrature homodyne detection, and (c) y -quadrature homodyne detection (where (b) and (c) give diffusive records). The lattertwo are orthogonal quadrature measurements defined by the homodyne local oscillatorphases, Φ = = π /
2, corresponding to measuring ˆ σ x and ˆ σ y observables of thequbit, respectively. These three detection schemes capture the most interesting typesof backaction, which have all been seen in experiments [34, 35, 36]. In this section,for a pedagogical purpose, we consider a perfectly observed decoherence channel, wherethe observed measurement rate is given by γ o = γ . The discussion below follows thetreatment presented in Ref. [4].For photon detection, since at most one photon can be detected during aninfinitesimal time d t , the measurement backaction on the system’s state is describedby a measurement operation, M d J N , where d J N = c = √ γ ˆ σ − as a jump Lindblad operator. If a photon is detected (d J N = M ρ = ˆ c ρ ˆ c † . For the zero-photon event, the measurement backaction is described by M ρ = ˆ M ρ ˆ M † , where ˆ M = ˆ1 − i ˆ H d t − ˆ c † ˆ c d t , which includes the unitary evolution forthe infinitesimal time d t . In the time-continuum limit with a continuous record d J N ( t ) ,one can write down a stochastic master equation for the photon detection,d ρ ( t ) = − i d t [ ˆ H, ρ ( t )] + d J N ( t )G[ ˆ c ] ρ ( t ) − d t H[ ˆ c † ˆ c ] ρ ( t ) , (13)where, following [4], we are using G[ ˆ a ] = ˆ a ρ ˆ a † Tr [ ˆ a ρ ˆ a † ] − ρ, (14) H[ ˆ a ] = ˆ aρ + ρ ˆ a † − Tr [ ˆ aρ + ρ ˆ a † ] ρ. (15) uantum state smoothing: Why the types of measurements matter J Q during an infinitesimal time d t can take any real value. Thesubscript ‘Q’ here stands for the quadrature measurement, which can be replaced by ‘X’or ‘Y’ for the measurement with the phase Φ = = π /
2, repectively. Let us defineˆ c Φ = √ γ o ˆ σ − e − i Φ as a Lindblad operator for the diffusive measurement. The measurementbackaction for this diffusive monitoring is described by an operation M d J Q ρ = ˆ M Q ρ ˆ M † Q where ˆ M Q = ˆ1 − i ˆ H d t − ˆ c † Φ ˆ c Φ d t + d J Q ˆ c Φ . The stochastic master equation for a recordd J Q ( t ) is given by,d ρ ( t ) = − i d t [ ˆ H, ρ ( t )] + d t D[ ˆ c Φ ] ρ ( t ) + d W ( t )H[ ˆ c Φ ] ρ ( t ) (16)where d W is an infinitesimal Wiener increment related to the measurement record asd J Q ( t ) = Tr [( ˆ c Φ + ˆ c † Φ ) ρ ( t )] d t + d W ( t ) .We note that by averaging the stochastic master equation Eq. (13) (or Eq. (16)))over all possible realisations of the measurement record d J N (or d J Q ), one obtains theconsistent Lindblad master equation Eq. (12).
4. Quantum state smoothing for a resonantly driven qubit in vacuumbosonic baths
Given the three possible types of bath detection in Section 3, we now consider theAlice-Bob smoothing protocol for the resonantly driven qubit example. Let us assumethat the qubit is coupled to two baths, where Alice measures only one bath (observedchannel) and Bob monitors the other bath, which is hidden from Alice (unobservedchannel). Since we here are interested only in how the type of bath observation affectsthe smoothing power, we choose the amount of information available on both channels(observed and unobserved) to be equal. That is, we consider a qubit-bath systemdescribed by the Lindblad equationd ρ ( t ) = L ρ ( t ) ≡ − i d t [ Ω2 ˆ σ x , ρ ( t )] + d t γ o D[ ˆ σ − ] ρ ( t ) + d t γ u D[ ˆ σ − ] ρ ( t ) , (17)where the single decoherence channel in Eq. (12) is now divided into two channels, onewith an observed measurement rate γ o and another with an unobserved measurementrate γ u . For simplicity, we choose γ o = γ u = γ /
2. Thus, now ˆ c = √ γ / σ − andˆ c Φ = √ γ / σ − e − i Φ .Now, let us define “dO” and “dU” as shorthand for observed and unobservedrecords, which can be one of the three options:dN ⇒ jump records, d J N , dX ⇒ x -homodyne (Φ =
0) records, d J X , dY ⇒ y -homodyne (Φ = π /
2) records, d J Y . (18)Therefore, in total, there are nine combinations, denoted by “dOdU” (observed-unobserved), which are dNdN, dNdX, dNdY, dXdN, dXdX, dXdY, dYdN, dYdX, dYdY. uantum state smoothing: Why the types of measurements matter X-coordinate Y-coordinate Z-coordinate Purity
SmoothedFiltered dOdU = dNdYdYdXdXdX (a)(e)(i) (b)(f)(j) (c)(g)(k) (d)(h)(l)
Time ( units of ) T
Three sample trajectories (filtered and smoothed) shown by their Blochsphere coordinates ( x , y , z ) and their purities. Panels (a)-(d) show filtered (blackdashed) and smoothed (solid red) trajectories, where the jump record is observedand the y -homodyne record is unobserved, i.e., dOdU = dNdY. Panels (e)-(h) are fordOdU = dYdX, and Panels (i)-(l) are for dOdU = dXdX. The trajectories are plottedas functions of time in the unit of the decay time defined as T γ = / γ . The Rabioscillation frequency around the x -axis is Ω = γ . The measurement rate for both dOand dU channels is γ /
2, the total time is T = T γ , and the initial state for all cases is thequbit’s excited state. To obtain each smoothed quantum state, we use 10 realisationsof unobserved records generated randomly with ostensible statistics [25, 37]. The errorsfrom using a finite-size ensemble of unobserved records are shown only for the purityplots (see Appendix A.1), in panels (d), (h), and (l). We first examine the filtered and smoothed trajectories of the qubit subject to a fewdifferent pairs of observed and unobserved measurements via numerical simulation. Weused the simulation technique of Ref. [25], detailed in Ref. [37]. We show examples ofindividual qubit trajectories in Figure 2 by their three Bloch sphere coordinates and theircorresponding purities. We chose three sample pairs of dOdU (observed-unobserved):dNdY, dYdX and dXdX.In the first example, dNdY [the first row, panels (a)-(d)], Alice observes onlyjump records; therefore, her filtered state evolution (dashed black) exhibits dampedoscillations with discontinuous jumps at certain times (in this case, jumps happen at t ≈ . T γ and t ≈ . T γ , where T γ = / γ ). The curves are described by non-unitarydynamics, as they contain effects from no-jump backaction and dephasing from theunobserved part. The smoothed trajectory (solid red) diverges noticeably from thefiltered one as time passes, though it resets to the same (ground) state after each jump.The purities in the panel (d) show the improvement from smoothing over filtering. Theimprovement tends to grow with time from the initial state or the jumps, because thefiltered and smoothed states are equal at those points. The states are also equal at the uantum state smoothing: Why the types of measurements matter possible true state trajectories, has almost no trace of the unobserved diffusiverecord.In the second and third rows, panels (e)-(l) of Figure 2, we present different scenariosinvolving the diffusive records. The panels (e)-(h) are for the case of dYdX, where thefiltered and smoothed trajectories show fluctuations from an observed diffusive recorddY. Theoretically, both the filtered and smoothed states should be confined to the y - z plane of the Bloch sphere, because the initial state, the Rabi oscillation, the backactionfrom the observed y -homodyne record, and the decay from the unobserved evolutiondo not break the symmetry around the x -axis. However, since the unobserved recorddX can bring the state outside of the y - z plane, one can see tiny non-zero values inthe x -coordinate of the smoothed state in the panel (e), resulting from finite size ofensemble of unobserved record used in averaging the possible true states in Eq. (7). Inthe last row, panels (i)-(l), an explicit non-zero component in x -coordinate is presentedas a result of observing a dX record.From the three row of examples, one can see that the improvement in the purityof the smoothed states from the filtered states are noticeably different in the differentcases. Note that in the latter two cases (dYdX and dXdX), the smoothed states arenot always more pure than their corresponding filtered states. This phenomena waspreviously found in Ref. [25] for other cases (which are dYdN and dXdN) where diffusionis observed.We then consider average dynamics from 3000 sets of observed-unobserved recordsand compute average purities of smoothed and filtered states, i.e., E O [ P [ ρ S ]] andE O [ P [ ρ F ]] . The average purities, for the three example cases (dNdY, dYdX anddXdX) are presented in Figure 3(a), where we also compare them to the purity ofthe Lindblad dynamics solution. For two cases (dNdY and dXdX), we can now see aclear improvement in the average purity of the smoothed states over the filtered statesat all time (except the end points), in contrast to the result for individual trajectoriesin Figure 2. For the third case, dYdX, a very slight improvement is visible, but it iswithin the error bars of the simulations. One can also say with confidence that thedNdY combination shows the largest purity recovery of the three.In Figure 3(b), we show the average purity of filtered states of all nine dOdUcombinations, for two reasons. First, it is to verify that the type of unobservedmeasurements does not affect the purity of the filtered states. This can be seen fromthe plot in that the averages can be categorized into three groups (label group 1,2,3)according to the type of the observed record. These coincide, within error bars (of sizecomparable to those shown in Figure 3(a), with each group being distinct. Second, it isto serve as a reference guide when one considers the relative purity recovery Eq. (11),as one of the two measures of the smoothing power. The average purity for the filteredstate has its highest value when observing dX, second highest when observing dY, andlowest when observing dN. uantum state smoothing: Why the types of measurements matter dNdYdYdXdXdX ( units of T γ ) ( a ) ( units of T γ ) ( b ) Group 1: dO = dXGroup 2: dO = dYGroup 3: dO = dN Figure 3.
Average purity of filtered and smoothed trajectories from 3000 realisationsof observed records. (a) Average purity of smoothed trajectories (solid coloured lines)compared with their filtered counterpart (adjacent dashed black lines) for three dOdUcombinations: the top pair (dXdX), the middle pair (dYdX), and the bottom pair(dNdY), where associated error bars are shown in coloured bands behind these curves.(b) Average purity of filtered trajectories for all nine dOdU combinations, which canbe categorized into three groups based on dO alone, as expected. The colour legendis read according to the labels and the arrow, for example, dOdU = dXdX (green),dOdU = dYdN (dashed magenta), and dOdU = dNdX (solid orange). For referencing,the purity of the Lindblad dynamics solution (17) is shown in both panels (dotted graycurves). The system parameters used for these plots are the same as in Figure 2. Timeis shown in units of the decay time T γ = / γ .
5. Correlations between measurement records as predictors for smoothingpower
From the previous section, we have seen that the improvement in the average purityof the smoothed state in comparison to that of the filtered state is different for thethree examples dNdY, dYdX, and dXdX. However, before we delve into all the ninecombinations of dOdU, we should first try to understand what leads to the purityimprovement and see if we can predict the degree of recovery in a systematic way.As we learned from Section 2, the smoothed and filtered states are distinct preciselyin how the probability weights for unobserved records are conditioned on observedrecords. Therefore, one would expect the purity recovery obtained from using smoothingover filtering to be dependent on how the two records are correlated. As an extremescenario to build intuition, if both records were uncorrelated, i.e., all orders of correlationfunctions between records O and U , were zero, one would not expect the smoothingto give any advantage over filtering at all. This is because the weighting factors ℘(←— U ∣←— O ) = ℘(←— U ∣←→ O ) = ℘(←— U ) in both cases are the same, independent of whether itis conditioned on the whole record or only the past record. On the other hand, if O and U records were highly correlated, by conditioning on the whole observed record, thesmoothed state could estimate the true state state significantly better than the filteredone. This suggests that considering the correlation between the two records will be a uantum state smoothing: Why the types of measurements matter To assess the correlation between observed and unobserved measurement records, thesimplest quantity to analyze is the two-time correlation function. The functionswhich are higher order in time can be expected to be less relevant, but as we willsee, the three-time correlation functions are useful as tie-breakers . Correlation formeasurement records, by definition, can be obtained from averaging over all possiblerecord realisations, and both two- and three-time correlation functions have beenmeasured experimentally [38, 39, 40, 41]. Here we acquire analytical solutions, usingthe average dynamics described by the Lindblad master equation, Eq. (17). We firstconsider the usual correlation functions in terms of expected values of measurementrecords at two and three different times, and then construct specific correlators that areuseful in predicting the quality of the smoothing.Following the method in Ref. [4] for the derivation of autocorrelation functions,e.g., E [ d J N ( t + τ ) d J N ( t )] and E [ d J Q ( t + τ ) d J Q ( t )] , where E [⋅] represents an averageover all possible record realisations, we here derive the cross-correlation functions fortwo and three time arguments. First consider the two-time correlation between a jumprecord d J N and a diffusive record d J Q , which has been measured in quantum opticsexperiments [38, 42]. Recalling that the jump signal at a particular time can be either0 or 1, and that E [ d J Q ( t )] = Tr [( ˆ c Φ + ˆ c † Φ ) ρ ( t )] d t , we obtain (using the notation definedin Section 3),E [ d J Q ( t + τ ) d J N ( t )] = E [ d J Q ( t + τ )∣ d J N ( t ) = ] × Tr [ ˆ c † ˆ cρ ( t )] d t, = Tr [( ˆ c Φ + ˆ c † Φ ) e L τ ˆ cρ ( t ) ˆ c † ] d t , (19)knowing that the average has no contribution from when d J N ( t ) =
0. We note thatLindblad-evolution superoperator e L τ for a duration τ acts on the product of alloperators to its right. One can read the right hand side of the first line as a multiplicationof an average of the diffusive signal given jumps d J N ( t ) = t and the likelihoodof getting such jumps. The latter is equal to Tr [ ˆ c † ˆ cρ ( t )] d t . For an opposite ordering,where diffusive records are at time t and jump records are at time t + τ , the two-timecorrelator is different but can be obtained in a similar manner [4],E [ d J N ( t + τ ) d J Q ( t )] = E [ d J N ( t + τ )] Tr [( ˆ c Φ + ˆ c † Φ ) ρ ( t )] d t + E [ d J N ( t + τ ) d W ( t )] , = Tr [ ˆ c † ˆ c e L τ ( ˆ c Φ ρ ( t ) + ρ ( t ) ˆ c † Φ )] d t . (20)Combining the above solutions with the autocorrelators in [4], we can generalize theform of two- and three-time correlators for any types of measurement records denotedby d J K , d J M , and d J H . These dummy record variables can be any of the three typesdefined in Eq. (18), and we use dK, dM, dH as shorthand for the record types, which uantum state smoothing: Why the types of measurements matter [ d J M ( t ) d J K ( t )] ss = Tr [K M e L( t − t ) K K ρ ss ] d t , (21)E [ d J H ( t ) d J M ( t ) d J K ( t )] ss = Tr [K H e L( t − t ) K M e L( t − t ) K K d t , (22)where we introduced the superoperators: K N ρ = ˆ c ρ ˆ c † (for jump records dN), and K Q ρ = ˆ c Φ ρ + ρ ˆ c † Φ (for diffusive records, e.g., dX or dY). We have also replaced ρ ( t ) with ρ ss , i.e., a solution of L ρ ss =
0, since we are most interested in the steady-state(with the subscript ‘ss’) behavior of the system.The correlators or expected values presented above, however, are not yet suitableas a measure for correlation between different pairs of observed-unobserved records inthe smoothing problem. This is because the units of the jump and diffusive records arenot the same, and the correlators still contain contribution from averages not relevant tosmoothing. To define more meaningful formulas for correlation, we need to renormaliseEqs. (21)-(22) and subtract any contributions that should not be relevant in predictingthe smoothing power. For the two-time correlation, it is straightforward to identifythat an irrelevant quantity is the product of individual average records. Therefore, anormalised correlator that determines purely correlation between any two records (d J K and d J M , say) at two different times in the steady-state regime is, C [ d J K , d J M ]( τ ) = E [ d J K ( t + τ ) d J M ( t )] ss − E [ d J K ( t + τ )] ss E [ d J M ( t )] ss N K N M , (23)which is a function of time difference τ and is symmetric under the interchange of therecords, i.e., C [ d J K , d J M ] = C [ d J M , d J K ] . The normalisation factor N is defined as the root mean square signal multiplied by a factor d t / , giving N N = √ E [( d J N / d t ) ] d t = d t √ Tr [ ˆ c † ˆ cρ ss ] , (24) N Q = √ Tr [( d J Q / d t ) ] d t = d t √ Tr [( ˆ c Φ + ˆ c † Φ ) ρ ss ] d t + E [ d W ]/ d t ≈ d t, (25)for the jump and diffusive records respectively. Using these normalised factors, thecorrelators are now independent of the units of the measured photocurrents, as well asof the infinitesimal time d t .For the three-time correlator, since we are interested only in the correlation betweentwo records, we choose two out of the three signal variables in the expected valueEq. (22) to be from the same record with fixed time arguments between them. Then, theexpected value of interest is in this form: E [ d J K ( t + T ) d J M ( t + τ ) d J K ( t )] . Subtractingthe irrelevant correlation from the same record, we obtain the normalised three-timecorrelator that can capture the correlation between two records (d J K and d J M ) in thesteady-state limit, C [ d J K , d J M ]( τ, T ) = E [ d J K ( t + T ) d J M ( t + τ ) d J K ( t )] ss − E [ d J M ] ss E [ d J K ( t + T ) d J K ( t )] ss N N M , (26) uantum state smoothing: Why the types of measurements matter - - τ ( units of T Ω ) ( a ) - - - - τ ( units of T Ω ) τ = T Ω τ = ( b ) Figure 4.
Two-time and three-time correlators in the steady-state regime that canbe used to determine the smoothing power. (a) Two-time correlators C [ d J K , d J M ] =C [ d J M , d J K ] in Eq. (23) are shown as functions of τ . The non-vanishing correlatorsare for the following: C [ d J X , d J X ] (green), C [ d J Y , d J Y ] (blue), C [ d J N , d J N ] (red),and C [ d J N , d J Y ] (magenta). (b) Three-time correlators C [ d J K , d J M ] in Eq. (26) areshown as functions of τ , where T is chosen to be one Rabi period. The colour legendis read in the same way as in Figure 3, but with dK and dM, representing any twotypes of records. The values of correlators here are used for the analysis in Table 1.Time τ is presented in units of the Rabi period T Ω = π / Ω. using the steady-state correlators defined in Eq. (21)-(22). We note that the firstargument, d J K , of the C [⋅ , ⋅] definition is the record that appear twice in the correlator.Thus we now have correlators in Eqs. (23) and (26) defined with unit-lessmeasurement results and can capture solely the correlation between any two records. Weshow in Figure 4(a) and 4(b) the two-time and three-time correlators for all combinationsof records d J K and d J M , as functions of τ . There are only two out of six of the two-timecorrelators that are zero: C [ d J X , d J N ] , and C [ d J X , d J Y ] . For the three-time correlators,there are three out of nine, i.e., C [ d J X , d J X ] , C [ d J Y , d J X ] , C [ d J N , d J X ] , with vanishingcorrelation, shown in panel (b). The values of correlators, as shown in Figure 4, vary significantly as τ and T changeand it is not obvious how the variation could contribute to the purity improvement ofsmoothed states over filtered states in any logical ways. Therefore, we instead focus on aparameter-independent feature, which is the vanishing or non-vanishing property of thecorrelators. Some correlators, such as C [ d J X , d J N ] or C [ d J Y , d J X ] , are zero regardlessof the values of T and τ . Those that do not vanish identically are non-zero for almostall values of T and τ .In order to predict the power of quantum state smoothing offered by differentmeasurement unravelling combinations, we propose the following principles. Firstly,the stronger the correlation, the better the smoothing power. We quantify correlationstrength for a particular combination dOdU by postulating that the largest contribution uantum state smoothing: Why the types of measurements matter Table 1.
Prediction for the power of smoothing using correlation strength. For all ninecombinations of dO and dU, we show vanishing (struck-through) and non-vanishingcorrelators according to the definition in the text. We treat dO-dU correlator as givingthe highest contribution to the smoothing power, then dO-dU-dO, then dU-dO-dU.Four levels of smoothing power are predicted in the last column depending on howmany non-vanishing correlators are available and their predicted contribution. coming from its two-time correlator, then its three-time correlators. However, thereare two ways of writing the three-time correlators between dO and dU (see Eq. (26)), C ( d J O , d J U ) and C ( d J U , d J O ) for the observed (d J O ) and unobserved (d J U ) records.We postulate that the former one is more important for smoothing, because it allowsus to quantify the correlation of the dU record (at any time t ) with the dO recordboth before and after t . That is, the correlator C ( d J O , d J U ) can quantify the differencebetween using past-future dO record conditioning as opposed to only using the pastrecord. We then list the correlators in Table 1 according to their contribution tothe power of smoothing, indicating the vanishing property by striking through thesymbols. Using shorthand notations dO-dU, dO-dU-dO, and dU-dO-dU to representthe correlators C ( d J O , d J U ) , C ( d J O , d J U ) , and C ( d J U , d J O ) , respectively, we have ● Two-time correlators: – Non-vanishing: dX-dX, dY-dY, dN-dN, dY-dN – Vanishing: dX-dY, dX-dN ● Three-time correlators: – Non-vanishing: dY-dY-dY, dN-dN-dN, dY-dN-dY, dN-dY-dN, dX-dY-dX,dX-dN-dX – Vanishing: dX-dX-dX, dY-dX-dY, dN-dX-dNIn the last column of Table 1, we categorize all nine dOdU combinations into four levels according to their relevant strength of the correlators shown. The 4th level has itscorrelators all non-vanishing, as we expect from this the best purity improvement fromstate smoothing. At the other extreme, for the 1st level, only the three-time correlatordU-dO-dU is non-vanishing. Note that if we instead treat the three-time correlators uantum state smoothing: Why the types of measurements matter
6. Numerical investigation
To test the validity of the predictions made in the previous section, we analyze qubittrajectories, numerically generated, for all nine combinations of observed and unobservedrecord types, and compute their average purity recovery, Eq. (10), and relative averagepurity recovery, Eq. (11). For each of the nine combinations, the trajectory data includesin total 3000 sets, where each set contains one true state trajectory ρ T , one filteredstate trajectory ρ F , and one smoothed state trajectory ρ S . For the calculation of eachsmoothed trajectory, we follow Eq. (7), using 10 realisations of dU records randomlygenerated with ostensible statistics [25, 37]. More detail of the calculation and numericaltechniques used in the simulation is presented in [37].Numerical results are presented in Figure 5 showing the average purity recoveryin panels (a)-(b), and the relative average purity recovery in panels (c)-(d). In thepanels (a) and (c), the recoveries are plotted as functions of time. Thus the valuesare zero at both ends, as the filtered and smoothed states are identical at t = t = T (when ←→ O = ←— O in Eq. (7)). The recoveriesalso show transient behaviors during the time between t = t ≈ T γ , whereoscillations at approximately the Rabi frequency are still visible. During the time period T ss = [ . T γ , T γ ] , marked by vertical dashed gray lines in the plots, the results arerelatively flat indicating the steady-state behavior for this period of time, before therecoveries start to converge towards zero at the final time at T = T γ .Looking at the steady-state interval defined as T ss , it is remarkable that the recoveryresults in Figure 5 can be plausibly classified into 4 groups: the top one with highestvalues of R A ,t (dNdN, dYdN, dYdY, dNdY), the second group (dXdX), the third group(dXdN, dXdY), and the last group with lowest values of the recovery (dNdX, dYdX).These groups are perfectly correlated with the prediction of levels given in Table 1 (4thcolumn) using the record correlators and their vanishing/non-vanishing properties.The average purity recovery shows only the difference in the purity of the filteredand smoothed state on average, not taking into account the value of the purity of thefiltered state to begin with. In some cases, the purity recovery simply cannot be largebecause the filtered state is already highly pure, leaving little room for improvement bysmoothing. We thus also consider the relative average purity recovery Eq. (11), whichis the ratio of the recovery and the difference between the filtered state’s purity and thepurity of the underlying true state (unit purity). However, there can be some subtleties,for cases where the true state is not a good reference for purity recovery. Examples arethe two combinations dNdX and dYdX. There Alice’s smoothed and filtered states areconfined to the y - z plane, but the true states can be outside of the plane, on the Blochsphere, due to the dX record (which is unobserved by Alice). In these cases, Alice’ssmoothed and filtered states can never get close to the true state and therefore it makes uantum state smoothing: Why the types of measurements matter ( a ) dNdN dYdN dYdY dNdY dXdX dXdN dXdY dNdX dYdX0.000.010.020.030.04 ( b ) dOdU ( units of T γ ) ( c ) dNdN dYdN dYdY dNdY dXdX dXdN dXdY dNdX dYdX0.000.020.040.060.080.100.12 ( d ) dOdU Figure 5.
Average purity recovery and relative average purity recovery for all ninedOdU combinations of qubit measurements. (a) Average purity recovery, Eq. (10),as functions of time. (c) Relative average purity recovery, Eq. (11), as functions oftime, where, for dNdX (solid orange) and dYdX (solid turquoise), we have substituted P ( ρ T ) → P ( ρ YZT ) , i.e., the purity of the true state projection on the y - z plane, as it isan appropriate reference for the maximum purity recovery possible for these cases. (b)and (d) are the steady-state time averages for the average purity recovery Eqs. (27)as in (a), and for the relative average purity recovery (28) as in (c), where their errorbars (one standard derivation) are discussed in Appendix A. more sense to use the true state projection on the y - z plane as a reference in computingthe relative average purity recovery.We show in Figure 5(c), the relative recovery, using P ( ρ T ) = P ( ρ T ) → P ( ρ YZT ) withthe purity of the true state projection on the y - z plane. Comparing to the results of theaverage purity recovery, the last two groups in panel (a) are no longer distinguishablein panel (c). This suggests that the reason for the dNdX and dYdX combinations havelow purity recovery in (a) is because Bob was measuring dX record making his (true)state almost impossible to guess correctly by Alice, who could measure only dN or dY.Therefore, computing the purity recovery relative to the projection of Bob’s state on the y - z plane make the comparison to the dXdN and dXdY combinations more reasonable. uantum state smoothing: Why the types of measurements matter T ss , and itserror bar. The steady-state averages for the average purity recovery and relative averagepurity recovery are defined asE ss [R A ,t ] = ∣ T ss ∣/ d t ∑ t ∈ T ss E O [ P [ ρ S ( t )] − P [ ρ F ( t )]] , (27)E ss [R R ,t ] = ∣ T ss ∣/ d t ∑ t ∈ T ss E O [ P [ ρ S ( t )]] − E O [ P [ ρ F ( t )]] E O [ P [ ρ T ( t )]] − E O [ P [ ρ F ( t )]] , (28)where ∣ T ss ∣ is the length of the steady-state period. Since the purity of the filtered andsmoothed states at each time step have their own error bars from finite-size ensemble ofobserved realisations (and also finite-size hypothetical unobserved realisations, for thesmoothed state), we need to take all these errors into account and perform a full erroranalysis for the time-averaging. We present the details of the error analysis in AppendixA. The steady-state averaged recoveries with error bars in Figure 5(b) and (d) have madethe separate grouping much more apparent than the time-dependent plots in (a) and(c). We also found that the dYdN combination has an unexpectedly high smoothingpower; it could possibly be categorised as a separate group if one had a more refinedtheoretical prediction. However, the prediction so far has already been quite impressive,given that it only used the information of correlation strengths between observed andunobserved measurement records.
7. Discussion and conclusion
We have investigated quantum state smoothing [25], particularly for a resonantly drivenqubit coupled to bosonic baths, concentrating on calculating how much the smoothinghelps improve the quality of state estimation over the filtering. In particular, we analyse,for the first time, how the improvement depends on the types of bath measurement, byAlice and Bob, respectively. Here we are referring to the formulation of quantum statesmoothing as an Alice-Bob protocol, where the observer Alice has access to only theobserved record O , whereas the hypothetical observer Bob has access to both Alice’srecord and the record U unobserved by Alice. Alice is trying to estimate Bob’s state,which is pure, and which can thus be considered the true state of the system.We considered three detection schemes for the qubit’s fluorescence, and thereforenine combinations of the observed-unobserved (dOdU) records: dNdN, dNdX, dNdY,dXdN, dXdX, dXdY, dYdN, dYdX, and dYdY, where dN, dX, and dY refer tophoton detection records, x -homodyne, and y -homodyne detection records, respectively.Alice’s smoothed quantum states have higher purity on average than her correspondingfiltered quantum states, meaning that her smoothed states have higher fidelity to their uantum state smoothing: Why the types of measurements matter y -homodyne measurement and the photon detection. The worst combinations are onesthat consist of single x -homodyne measurement.We note that even though this work is only concerned with the particular exampleof a qubit coupled to bosonic baths, we expect that the insights gained here will beapplicable to other, quite different, physical systems, such as Linear Gaussian systems[21, 43]. Moreover, the ability to predict smoothing power beforehand could also behelpful in justifying the use of quantum state smoothing in estimating quantum state forexperimental systems in lossy environment. There are many other interesting questionsworth further investigating, such as: Would the smoothing power be maintained at thesame levels if Bob’s detection scheme was guessed wrongly by Alice? How does thefidelity of the smoothed state estimate compare with other state estimation methodsusing past-future information [44, 45]. Acknowledgments
We acknowledge the traditional owners of the land on which this work was undertakenat Griffith University, the Yuggera people. This research is funded by the AustralianResearch Council Centre of Excellence Program CE170100012. AC acknowledges thesupport of the Griffith University Postdoctoral Fellowship scheme.
Appendix A. Error analysis
We here discuss the error analysis used in computing the numerical error bars presentedin Figure 3 and 5. We discuss errors that could appear at any stages of the numericalaveraging with finite-size ensembles. We start with errors coming from the calculation ofan individual smoothed quantum state (involving an average over all possible unobservedrecords), then errors in the average purities (averaging over all possible observedrecords), and, finally, errors from averaging over the steady-state interval. uantum state smoothing: Why the types of measurements matter Appendix A.1. Covariance of smoothed states
The smoothed state is a result of a weighted average over all possible unobserved records,as in Eq. (7); therefore, using a finite-size ensemble of unobserved records (in this work,we used N U = ) should result in a finite error of the average. Considering a weightedaverage (mean) of a single random variable of the form: x = ∑ Ni = w i x i / ∑ i w i , the varianceof the mean (VOTM) is given by ( δx ) = ∑ i w i (∑ i w i ) σ , (A.1)where the variance of the population is σ = ∑ i w i x i ∑ i w i − x . (A.2)Note that if all the weights were the same, then we would arrive at a simple result ( δx ) = σ / N .Given the definition of the smoothed quantum state in Eq. (7) and the definitionof VOTM for a single variable above, we therefore can write a covariance matrix for thesmoothed state as,CoV ( ρ S ( t )) ≡ ∑ ←— U t ℘(←— U ∣←→ O ) ( ∑ ←— U t ℘(←— U ∣←→ O )) ⎛⎝∑ ←— U t ℘(←— U ∣←→ O ) ( ρ ←— O t , ←— U t ( t ) ⊗ ρ ←— O t , ←— U t ( t )) − ρ S ( t ) ⊗ ρ S ( t )⎞⎠ . (A.3)We can use this matrix to calculate uncertainties for any observables, for example, theVOTM for the y -component of the smoothed qubit’s state is given by ( δy S ) = Tr [( ˆ σ Y ⊗ ˆ σ Y ) CoV ( ρ S ( t ))] . (A.4)We can then calculate the variance of the smoothed state’s purity defined as P [ ρ S ] = Tr [ ρ ] = ( + x + y + z ) . From the error propagation rules, we get ( δP [ ρ S ( t )]) = x ( δx S ) + y ( δy S ) + z ( δz S ) , (A.5)where we have presented this error for the smoothed state’s purity in Figure 2(d), (h),and (l). Appendix A.2. Variance of averaged purities
The error bars presented in Figure 3 are for the average purities of the filtered andsmoothed states. To simplify the notations, let us use ¯ P F ( t ) = E O [ P [ ρ F ( t )]] and¯ P S ( t ) = E O [ P [ ρ S ( t )]] for the averages of the filtered and smoothed quantum states,respectively. Using (A.1), the variances of the averages are calculated from, ( δ ¯ P F ( t )) = N O E O [( P [ ρ F ( t )] − ¯ P F ( t )]) ] , (A.6) ( δ ¯ P S ( t )) = N O E O [( P [ ρ S ( t )] − ¯ P S ( t )]) ] + N O E O [( δP [ ρ S ( t )]) ] , (A.7) uantum state smoothing: Why the types of measurements matter t , with the N O = O in the ensemble. Appendix A.3. Variance of steady-state averages
In Figure 5(b) and (d), we plotted the steady-state averages of R A ,t and R R ,t andtheir associated error bars. The average purity recovery and the relative average purityrecovery are defined as in Eqs (27) and (28), R A ,t = E O [ P [ ρ S ( t )]] − E O [ P [ ρ F ( t )]] = ¯ P S ( t ) − ¯ P F ( t ) , (A.8) R R ,t = E O [ P [ ρ S ( t )]] − E O [ P [ ρ F ( t )]] E O [ P [ ρ T ( t )]] − E O [ P [ ρ F ( t )]] = ¯ P S ( t ) − ¯ P F ( t ) ¯ P T ( t ) − ¯ P F ( t ) . (A.9)Let us denote the steady-state averages by R A , ss = E ss [R A ,t ] and R R , ss = E ss [R R ,t ] . Theerror bars presented in Figure 5(b) and (d) are the square roots of the variances of thesteady-state averages obtained from ( δ R ss ) = N ss ∣ T ss ∣/ d t ∑ t ∈ T ss (R t − E ss [R t ]) + N ss ∣ T ss ∣/ d t ∑ t ∈ T ss ( δ R t ) , (A.10)where δ R ss ∈ { δ R A , ss , δ R R , ss } , R t ∈ {R A ,t , R R ,t } , and δ R t ∈ { δ R A ,t , δ R R ,t } . The first termof (A.10) describes the uncertainty contributed from the fluctuation of R t within thesteady-state duration, whereas the second term describes a contribution from variancesof individual R t ’s for t ∈ T ss . We also include an effective number of independentsamples N ss in calculating the variance of the steady-state average, because values inconsecutive timesteps are not completely independent of each other. Assuming thatqubit state trajectories are uncorrelated after one correlation time t corr , the effectivenumber of independent samples can then be approximated as N ss ≈ ∣ T ss ∣/ t corr +
1, where ∣ T ss ∣ is the steady-state duration. We approximate t corr by T γ = γ − , the inverse of thesystem’s decay rate.The variance ( δ R t ) in (A.10) for both R A ,t and R R ,t should be calculated carefully.The former one, the average purity recovery R A ,t , is a linear function of the purities;therefore, from the error propagation, we have ( δ R A ,t ) = N O E O [( P [ ρ S ( t )] − P [ ρ F ( t )] − R A ,t ) ] + N O E O [( δP [ ρ S ( t )]) ] . (A.11)We note that, interestingly, this is not equal to a simple sum of the variance, i.e., ( δ ¯ P S ( t )) + ( δ ¯ P F ( t )) , because the filtered and smoothed states (for each of the observedrecords) are strongly dependent on each other. For the latter quantity, the relativeaverage purity recovery, R R ,t , is a non-linear function of the three averages ¯ P F , ¯ P S ,and ¯ P T , which are also not independent of each other. We thus need to calculate thevariance from the full-form error propagation. Let us denote a quantity of interest by f ≡ R R ,t = ( a − b )/( c − b ) , which is a function of a , b , and c , where each has its own uantum state smoothing: Why the types of measurements matter ( δa ) , ( δb ) and ( δc ) , respectively. The variance of R R ,t can becomputed from ( δ R R ,t ) = ( δf ) = ( ∂f∂a ) ( δa ) + ( ∂f∂b ) ( δb ) + ( ∂f∂c ) ( δc ) + ( ∂f∂a ) ( ∂f∂b ) δ ( a, b )+ ( ∂f∂a ) ( ∂f∂c ) δ ( a, c ) + ( ∂f∂b ) ( ∂f∂c ) δ ( b, c ) , (A.12) = ( c − b ) ( δa ) + ( a − c ) ( c − b ) ( δb ) + ( b − a ) ( c − b ) ( δc ) + ( a − c )( c − b ) δ ( a, b )+ ( b − a )( c − b ) δ ( a, c ) + ( a − c )( b − a )( c − b ) δ ( b, c ) , (A.13)where a = ¯ P S ( t ) = E O [ P [ ρ S ( t )]] , (A.14) b = ¯ P F ( t ) = E O [ P [ ρ F ( t )]] , (A.15) c = ¯ P T ( t ) = E O [ P [ ρ T ( t )]] , (A.16) ( δa ) = ( δ ¯ P S ( t )) , (A.17) ( δb ) = ( δ ¯ P F ( t )) , (A.18) ( δc ) = ( δ ¯ P T ( t )) , (A.19) δ ( a, b ) ≡ E O [( P [ ρ S ( t )] − a )( P [ ρ F ( t )] − b )] , (A.20)and in the similar way for δ ( a, c ) and δ ( b, c ) . We note that ( δ ¯ P T ) is not zero for thecases of dOdU = dNdX and dYdX, where we have used P ( ρ T ) → P ( ρ YZT ) as discussedin the main text. Once we have the variances ( δ R A ,t ) and ( δ R R ,t ) , we can computethe steady-state variances ( δ R A , ss ) and ( δ R R , ss ) using (A.10). References [1] Davies E B 1969
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