Quantum Bayesian rule for weak measurements of qubits in superconducting circuit QED

Compared with the quantum trajectory equation, the quantum Bayesian approach has the advantage of being more efficient to infer quantum state under monitoring, based on the integrated output of measurement. For weak measurement of qubits in circuit quantum electrodynamics(cQED), properly accounting for the measurement backaction effects within the Bayesian framework is an important problem of current interest.Elegant work towards this task was carried out by Korotkov in"bad-cavity"and weak-response limits (arXiv:1111.4016).In the present work, based on insights from the cavity-field states (dynamics) and the help of an effective QTE, we generalize the results of arXiv:1111.4016 to more general system parameters.The obtained Bayesian rule is in full agreement with Korotkov's result in limiting cases and as well holds satisfactory accuracy in non-limiting cases in comparison with the QTE simulations. We expect the proposed Bayesian rule to be useful for future cQED measurement and control experiments.

In this context, rather than strong projective measurement, more interesting is the type of weak measurement [21][22][23] whose experimental realization is an extremely attracting subject [10,11]. In particular, this type of monitoring on quantum state is an essential prerequisite for measurement-based feedback control of quantum systems [23]. For continuous weak measurements most popular is the quantum trajectory theory [23], as broadly applied in quantum optics and quantum control problems. The quantum trajectory theory can also address the solid-state charge qubit measurements by mesoscopic quantum-point-contact and single-electron-transistor detectors [24,25]. For this setup, an equivalent scheme known as quantum Bayesian approach was proposed [26] and exploited for applications. For cQED, which is analogous to the conventional optical cavity-QED, the quantum trajectory approach seems the most natural choice [12-14, 19, 20]. Despite this, in a recent study [27], Korotkov developed a promising quantum Bayesian approach in the "bad-cavity" and weak-response limits. Owing to its competitive efficiency and advantage of accounting for realis- * Electronic address: lixinqi@bnu.edu.cn tic imperfections, this approach has been employed in recent experiments on quantum measurement [11] and feedback control [16].
Construction of the quantum Bayesian approach is largely based on the classical Bayes formula. For the diagonal elements of the qubit, the Bayes formula works perfectly; however, it does not work for the off-diagonal elements. One proceeds then by a purity consideration [26], together with some additional physical insights [27]. In this work, rather than using such type of considerations, we would like to fulfill a similar task by employing the quantum trajectory equation (QTE) approach. In order to gain necessary insights, our analysis will pay particular attention to the nature of the cavity field under continuous quadrature monitoring. This treatment permits us to avoid the bad-cavity and weak-response assumptions [27], making thus the obtained quantum Bayesian rule applicable to more general setup parameters.
The paper is organized as follows. We begin in Sec. II with a brief description of cQED and the optical QTE, before identifying the nature of the cavity state conditioned on weak measurements in Sec. III. We then construct in Sec. IV the quantum Bayesian rule for single-quadrature measurements and in Sec. V for two-quadrature measurements. In Sec. VI, numerical results and comparisons are presented, for both the limiting and non-limiting cases. Finally, we summarize the work with remarks in Sec. VII. In addition, two Appendices are provided, for the analytic solution of the cavity field and an equivalence proof of two expressions of the purity degradation factor.

II. MODEL AND QUANTUM TRAJECTORY EQUATION
Let us consider the simplest cQED setup with only one superconducting qubit in the resonator cavity [1]. In this setup, the central section of superconducting coplanar waveguide plays the role of an optical cavity, and the superconducting qubit the role of an (artificial) atom. The superconducting qubit is coupled to the one-dimensional transmission line (1DTL) cavity which acts as a simple harmonic oscillator. Therefore the qubit, the 1DTL cavity, and their mutual coupling can be well described by the Jaynes-Cummings Hamiltonian. Moreover, we consider the setup in a dispersive regime [1][2][3], i.e., with the detuning between the cavity frequency (ω r ) and qubit energy (ω q ), ∆ = ω r − ω q , much larger than the coupling strength g. In this limit and in the rotating frame with the microwave driving frequency ω m , the system can be described by an effective Hamiltonian [1,2] where ∆ r = ω r − ω m (for resonant drive, ∆ r = 0), and ω q = ω q +χ with χ = g 2 /∆ being a dispersive shift to the qubit energy and cavity frequency. In Eq. (1), a † (a) and σ z are respectively the creation (annihilation) operator of cavity photon and the quasi-spin operator (Pauli matrix) for the qubit. ǫ m is the microwave drive amplitude to the cavity. For measurements, in this work we will first consider the single quadrature (I ϕ ) measurement in detail, then convert the obtained results to the (I, Q) two-quadrature measurement. For the single quadrature homodyne measurement, one actually measuresÎ ϕ = 1 2 (ae −iϕ + a † e iϕ ), where ϕ is the local oscillator (LO) phase [23]. The measurement output can be expressed as where κ is the damping rate of the cavity photon and ξ(t) a Gaussian white noise originating from the stochastic quantum-jump, which satisfies the ensemble-average The quantum average · · · ̺(t) is defined by · · · ̺(t) = Tr[(· · · )̺(t)], with ̺(t) the qubit-cavity conditional state given by the quantum trajectory equation (QTE) [23]:

III. CAVITY STATE CONDITIONED ON WEAK MEASUREMENTS
To get necessary insights for constructing a quantum Bayesian rule for the qubit state, it is crucial to identify the nature of the cavity state conditioned on the quadrature outcomes. First, we notice that, for a specific qubit state |g or |e , the interplay of measurement drive and cavity loss would lead to the formation of a coherent state |α g (t) or |α e (t) for the cavity field, with α g(e) (t) determined by the following equations: Notice also that this result is associated with an ensemble average over the stochastic leakage of photons. In the stationary limit, the coherent-state parameter reads α g(e) = −iǫ m /(i∆ r,g(e) + κ/2), where ∆ r,g(e) = (ω r − ω m ) ∓ χ. The transient solution is also available (but with a lengthy expression, see Appendix A).
As a heuristic discussion for measurement principle, let us first consider a simpler model of qubit measurement by another two-state meter (e.g., a spin), which are prepared in an entangled initial state c g |g ⊗ |↑ + c e |e ⊗ |↓ .
Here |↑ and |↓ are the meter basis states (in the σ z representation). Then, let us consider a projective measurement on the spin in a different (e.g., x) direction. A specific result, for instance "+1" in the x-direction, would project the joint state onto ( is not parallel to |↑ (|↓ ), the strong projective measurement on the meter does not collapse the qubit state onto |g or |e . To the qubit state, this falls into the category of weak measurements. Now consider the qubit measurement in cQED. The qubit-cavity state is initially prepared as |Ψ(0) = (c g |g +c e |e )⊗|α 0 (α 0 = 0 if the cavity field is the vacuum). If one is faithfully tracking the emitted photon by continuous homodyne measurement, the subsequent timedependent state can be expressed as Here, instead of |α g(e) (t), we denote the respective cavity state as α g(e) (t) , indicating a lack of ensemble average. Based on the quantum measurement theory [23], this state is a stochastic and quantum pure state. That is, both the superposition coefficients c g(e) (t) and the cavity states α g(e) (t) are stochastic, depending on the random outputs of measurement. Now, consider a further weak measurement on this state, with an integrated quadrature output I m over the time interval (t, t + τ ). One may imagine that the joint state of the qubit-plus-cavity could be expressed as where the state update factors are given by d g(e) = ψ m (t + τ )|α g(e) (t) . The essential feature of this result is that the cavity field has collapsed onto a unique eigenstate |ψ m of the quadrature operator, conditioned on the measurement record I m . However, we will show that this is not true.
As an explicit demonstration, we performed a direct simulation of Eq. (3) for continuous homodyne measurements, starting with a superposition qubit state and a vacuum cavity state. Based on the conditional joint qubitplus-cavity state ̺(t), the cavity states |α g (t) and |α e (t) in Eq. (5) can be extracted from ̺(t), respectively, in terms of density matrix ̺ gg (t) = g|̺(t)|g /Tr[ g|̺(t)|g ] and ̺ ee (t) = e|̺(t)|e /Tr[ e|̺(t)|e ]. Here Tr[· · · ] is over the cavity degrees of freedom. Using a Q-function representation, in Fig. 1 we plot the difference of ̺ gg (t) and ̺ ee (t) at a given time. We find that this type of coexistence , with Tr[· · · ] over the cavity degrees of freedom. Plotting the Q-function α|(̺ee−̺gg)|α /π reveals that the cavity field does not collapse onto a unique eigenstate of the quadrature operator (i.e., the observable) after experiencing a weak quadrature-measurement. Parameters: ∆r = 0, χ = 0.1, ǫm = 1.0, κ = 2.0, ϕ = 0.
of distinct coherent states will persist along the whole continuous weak measurement process. This result indicates that the weak measurement (with an outcome I m ) during (t, t+τ ) does not collapse the cavity fields α g(e) (t) onto any common eigenstate |ψ m . This differs substantially from the simple two-state-meter example discussed earlier.
We may contrast the state structure revealed here with the assumption in Ref. [10]. In the Supplementary Materials (in Sec. 2.1.2, before Eq. (6)), the following statement was made for the two-quadrature measurement: the measurement process, with quadrature outcomes (I m , Q m ), would project the cavity modes onto a unique eigenstate. While the final result obtained there seems correct, it might be of interest to further clarify their treatment.
More careful inspection reveals that the states ̺ gg (t) and ̺ ee (t) are very close to the coherent states |α g (t) and |α e (t) , respectively. As we will see below, this identification can help us to determine the purity degradation factor in the Bayesian rule. Here, one may understand this essential result as follows. Rather than a direct point-process detection for the outgoing photon, the homodyne-type quadrature measurement is relatively soft to the cavity field. It does not drastically alter the number of cavity photons. The continuous quadrature measurement is mainly updating our knowledge about the superposition components, say, the coefficients c g (t) and c e (t) in Eq. (5), but not collapsing the cavity states ̺ gg (t) and ̺ ee (t) onto a common eigenstate |ψ m . This is the socalled informational evolution [28,29]. However, around α g (t) and α e (t), the cavity field does exist stochastic fluctuations, which will result in a stochastic phase factor to the qubit off-diagonal elements -as we will see in the following sections.
Finally, we remark that this cQED is an interesting way to understand the puzzle of Schrödinger's cat. That is, the cavity state α g(e) (t) corresponds to the macroscopic state ("dead" or "alive") of the cat, while the continuum of states outside the cavity corresponds to the infinite number of microscopic states of the cat. It is right these infinite number of microscopic degrees of freedom that destroy the coherence of the superposed entangled state.

IV. BAYESIAN RULE FOR SINGLE QUADRATURE MEASUREMENT
The conditional evolution of the qubit-plus-cavity under continuous measurements is well captured by Eq. (3), However, a full simulation of this equation is timeconsuming and almost intractable in practice (e.g., in the quantum feedback control experiment [16]). More efficient method is using a quantum Bayesian rule to update the qubit state, merely based on the measurement record I m over certain finite time interval t m . For the sake of clarity, below we present our construction procedures in order by three steps (addressed by three subsections).

A. Bare Bayesian Rule
To perform a Bayesian inference based on I m , which is defined by I m = 1 tm tm 0 dtI ϕ (t) and collected in experiment from the quadrature measurement records, we need to know in advance the distribution P g(e) (I m ) associated with the qubit state |g(e) . As given in Appendix A, a simple analysis shows that the distribution is Gaussian: with D = 1/t m being the variance. The average quadrature outcome,Ī g(e) , is given bȳ where α g(e) (t) is the cavity field discussed in Sec. III, with explicit solution presented in Appendix A.
With the knowledge of P g (I m ) and P e (I m ), using the classical Bayes formula one can determine |c g (t m )| 2 and |c e (t m )| 2 , in Eq. (5). Obviously, they coincide with the diagonal elements of the reduced density matrix of the qubit state. Therefore, we have where N = ρ gg (0)P g (I m ) + ρ ee (0)P e (I m ). One can examine that Eq. (9) is in full agreement with the quantum trajectory equation simulation, as expected.
Regarding the off-diagonal element ρ ge (t m ), the situation is subtle. No classical rule applies here. Following Ref. [26], based on purity consideration, we preliminarily havẽ ρ ge (t m ) = ρ ge (0)e −i ωqtm P g (I m )P e (I m )/N , (10) to approximate ρ ge (t m ). However, as shown in Fig. 2(a), this result differs considerably from the exact one from a direct simulation of Eq. (3).

B. Purity Degradation Factor
For further corrections toρ ge (t m ), let us look back to Eq. (5). Based on the joint-state structure, we propose to amend Eq. (10) by a purity degradation factor as The physical meaning of this correction factor is an account of the purity degradation, after partly averaging an entangled state. More specifically, as shown in Fig. 1 and discussed earlier around it, we know that the cavity state evolves along |α g(e) (t) , with some tiny stochastic fluctuations. Here, as a first step, we introduce the purity degradation factor, D(t m ) ≡ | α e (t m )|α g (t m ) |, to characterize the decrease of the qubit coherence owing to averaging the cavity state, while studying later the fluctuation effects on the qubit. Using the property of coherent state, an explicit result can be obtained as Moreover, as proved in Appendix B, this result is precisely equivalent to the following one by a more sophisticated analysis based on the quantum trajectory equation: where Γ d (t) is a time-dependent overall decoherence rate of the qubit caused by measurement, and Γ m (t) is the measurement (unraveling) rate. This identification reveals a deep connection between the two very different treatments, and provides an additional evidence for the validity of the purity degradation factor.

C. Effects of Dynamic and Stochastic Fluctuations of the Cavity Field
Equation (5) and Fig. 1 jointly indicate the qualitative feature of the cavity state and guide our construction for the quantum Bayesian rule. However, in order to quantify the fluctuation effects of the cavity field, more sophisticated skill is useful. Let us return to the quantum trajectory equation. Based on Eq. (3), via a qubit-statedependent displacement transformation, it is possible to eliminate the cavity degrees of freedom from this equation [19]. Below we continue our correction to ρ ge (t m ) based on the achievement of this technique. The transformed QTE reads [19] In this result, the effective magnetic field , describes a generalized ac-Stark shift of the qubit energy, as a consequence of dynamic fluctuations of the cavity field owing to dispersive coupling to the qubit. The superoperator is defined by M[σ z ]ρ ≡ (σ z ρ + ρσ z )/2 − σ z ρ, with σ z = Tr[σ z ρ], an average over the reduced density matrix of qubit. Additionally, characterize, respectively, the ensemble-average dephasing, single measurement information-gain, and backaction rates. Moreover, the sum of Γ ci and Γ ba , gives the measurement rate [19]. In the above rates, we have used the definition β(t) = α e (t) − α g (t) ≡ |β(t)|e iθ β . Returning to Eq. (14), one may notice two unitary terms on the r.h.s.: the first term, involving B(t); and the last with an effective stochastic field, − Γ ba (t)ξ(t). These two terms properly characterize the cavity-fieldfluctuation effects on the qubit and can be used to amend Eq. (11) further. We thus complete our correction to the off-diagonal element as follows: with the two additional phase factors For Φ 2 (t m ), I ϕ (t) in the integrand is determined from the following considerations. First, within the "polaron" transformation scheme, the output current can be reexpressed as [19]: Second, ensemble average of ρ ge (t m ) over the stochastic "field" ξ(t), or equivalently over the stochastic output current I ϕ (t), should be consistent with the result by averaging Eq. (14). Therefore, in the formal integrated solu- Equations (17) and (18), together with (9) and (10), constitute the quantum Bayesian rule we propose for qubit state under single-quadrature weak measurements. To implement the proposed Bayesian rule in real experiments, one can directly use the analytic solutions of α g (t) and α e (t), given in Appendix A, and compute all the relevant quantities including | α e (t m )|α g (t m ) | and Φ 1 (t m ). Provided the necessary setup parameters are determined (as discussed in further detail below), all these calculations can be fulfilled in advance. One can then update the qubit state. For further possible simplification, we propose here a scheme to approximate Φ 2 . From Fig. 1 and the related discussion, we know that Γ ba (t) is a slowly-varying function of time. Thus, we propose whereĪ As we will demonstrate, this approximation can work well and could be useful in practice.

V. BAYESIAN RULE FOR TWO-QUADRATURE MEASUREMENT
In practice, rather than the I ϕ single-quadrature measurement, the so-called (I, Q) two-quadrature measurement is another choice [10]. Among the various realizations [23], it can be implemented as follows (in terms of a quantum optics language, for convenience). Using a beam-splitter, the outgoing microwave field leaked from the cavity, √ κa, is split into two branches with amplitudes a 1 = κ/2a and a 2 = i κ/2a. Then, perform single-quadrature homodyne measurement on each branch for a 1 and a 2 , choosing the LO phases as ϕ = 0 and π, respectively. One can prove that, for the cavity field, this realizes an I-quadrature measurement in the first branch and a Q-quadrature measurement in the second branch, with measurement outputs described by I m (t) = κ/2 a + a † ̺(t) + ξ 1 (t), and Q m (t) = κ/2 −ia + ia † ̺(t) + ξ 2 (t). Conditioned on this type of (I, Q) two-quadrature measurements, the evolution of the qubit-cavity joint state follows the quantum trajectory equation:̺ To the entire qubit-plus-cavity state, the last two terms fully unravel the measurement with no information loss.
Since only the output of the first branch (leading to the third term) reveals the qubit state information, the information-gain rate is just half of the optimal single quadrature measurement. However, provided we keep track of the Q-quadrature output in the second branch, the qubit state can be maintained in high purity. In a similar manner to the single-quadrature measurement, applying the qubit-state dependent displacement transformation to the cavity field within the two-quadrature measurement framework also yields an effective quantum trajec-tory equation for the qubit state alone: Here, B(t) and Γ d (t) are the same as in the single quadrature measurement, see Eqs. (14) and (15). Comparing Eq. (22) with Eqs. (14) and (15) indicates that the Iquadrature measurement of the first branch is associated with Γ    ba (t) = (κ/2)|β(t)| 2 = Γ m (t)/2. Following the same procedures as for the singlequadrature measurement, we can construct a quantum Bayesian rule for the two-quadrature measurement. First, the integrated output distribution of the two-quadrature measurement is: with D = 1/t m the same as in the single-quadrature measurement. Respectively, the average of each quadrature output is given as withĪ g(e) (t) = √ 2κ Re[α g(e) (t)]; and withQ g(e) (t) = √ 2κ Im[α g(e) (t)]. As pointed out previously, tuning ∆ r = 0 and with an initial vacuum cavity state, the imaginary parts of α g (t) and α e (t) are equal. This means that the Q-quadrature output Q m does not provide qubit state information. However, in an ideal case, tracking this output simultaneously with the I-quadrature outcome (I m ) can maintain the joint state of the qubit-plus-cavity in a pure state. So, with respect to the optimal single-quadrature measurement (ϕ = 0), the (I, Q) two-quadrature measurement reduces the signal |Ī e (t m ) −Ī g (t m )| by a factor of 1/ √ 2. With the knowledge of P g(e) (I m , Q m ), the diagonal elements of the qubit state, ρ gg and ρ ee , can be determined straightforwardly using the Bayesian rule Eq. (9) for their informational evolution conditioned on the outcome (I m , Q m ). For the off-diagonal elements, the quantum Bayesian rule is the same as Eq. (17), with the phase factor Φ 1 (t m ) unchanged but Φ 2 (t m ) now given by , with λ1,2,3 = 0 or 1, andρge(t), Φ1(t) and Φ2(t) as defined in the main text. The qubit is assumed with an initial state (|e + |g )/ Q m (t) can be determined as follows. For the second branch Q-quadrature measurement, we have Q m (t) = I ϕ (t)| ϕ=π/2 = √ κ|µ(t)| sin θ µ + ξ 2 (t), by noting that ϕ = π/2 and Γ

A. Effects of Corrections
We show the correction effects in Fig. 2 and demonstrate the proposed Bayesian rule by comparison with the exact results from simulation of Eq. (3), in (a) for singlequadrature and (b) for two-quadrature measurements. For the case of the single-quadrature measurement, we have used Eq. (19) to approximate Φ 2 (t m ) for the purpose of revealing the quality of the rule using only the integrated quadrature. For both types of measurements, we find that the proposed quantum Bayesian rule can give reliable estimate for the qubit state.
In Fig. 2(a) each correction is presented individually for the single-quadrature measurement. Since the individual effect of each correction term is similar, only the total result is shown for the two-quadrature case illustrated in Fig.  2(b). Whereas the consequences of the phase factors are dramatic, one may notice that in this plot the correction from the purity-degradation-factor is very weak (almost negligible). However, its physical meaning is clear. It characterizes the intrinsic purity of the qubit state imposed by the cQED measurement in the ideal case. Actually, the qubit-state purity associated with Fig. 2 is about 0.97, but not unity. Changing the parameters, one can make this correction effect more prominent, as to be shown in Sec. VI C.

B. Limiting Cases
We now consider the three correction factors in limiting cases, making in particular a connection with the work by Korotkov [27]. First, for the purity degradation factor | α e (t m )|α g (t m ) |, based on the solution in Appendix A we obtain, in steady state: Further, in the bad-cavity and weak-response limit, κ ≫ χ, the result simplifies to | ᾱ e |ᾱ g | = exp[−8n(χ/κ) 2 ], wheren = |α 0 | 2 with α 0 = −iǫ m /( κ 2 ). Accordingly, only in the limit κ ≫ χ and with smalln (cavity photon number), can the purity factor be approximated to unity, implying a pure state of qubit under the quadrature measurement. Otherwise, this factor should be taken into account. In particular, this implies that for the not very bad cavity and withn not very small, the D factor cannot be treated as unity. For instance, as to be seen below in Fig.  4(a), the D factor will reduce to a value lower than 0.8 for χ = 0.1κ andn ≃ 4. We note that this factor was not addressed in Ref. [27]. But in a recent report [30] the similar D factors were included in the concurrence calculation for a two-qubit cQED system (see Eq. (12) in the Supplementary Information of Ref. [30]).
Let us now consider Φ 1 (t m ). The key quantity associated is the effective magnetic field B(t) = 2χRe[α g (t)α * e (t)]. Physically speaking, it describes a generalized (time dependent) ac-Stark effect. From the analytic solution in Appendix A, we formally reexpress α g(e) (t) = ∓a(t) + ib(t), which leads to α g α * e = −(a 2 − b 2 ) + 2iab. We see that the imaginary part (2ab) determines the dephasing rate Γ d , and the real part (b 2 − a 2 ) affects the qubit energy. Further, in steady state, we have where d 2 ≡ ∆ 2 r − χ 2 + κ 2 /4. Again, in the bad-cavity and weak-response limit, the result reduces to B ≃ 2χn. This is the standard ac-Stark shift to the qubit energy.
Finally, let us consider Φ 2 (t m ). Based on the steadystate solution of the cavity fields, we obtain the backaction rate as Moreover, in the limit κ ≫ χ, the result is further simplified as Γ ba ≃ 16nκ(χ/κ) 2 sin 2 ϕ. Substituting this result into the expression of Φ 2 (t m ), we find that we recover the "realistic"-back-action induced phase factor in the bad-cavity and weak-response limit, obtained in Ref. [27] by using photon-caused qubit rotation considerations. We note also that, in the context of (I, Q) two-quadrature measurements, the state-update rule constructed in Ref. [10] (in the Supplementary Materials) contains as well this same factor in the same limiting case. However, while our approach can derive theirs, it seems to be an open problem how to use their approaches to derive some results here.

C. Non-Limiting Cases
In this subsection we present some numerical results beyond the "bad-cavity" and weak-response limits, and compare with the Bayesian rule constructed in Ref. [27]. Cases that violate the restrictive limits can be the following: (i) the quality factor of the cavity is relatively high, as required in quantum information processing in order to employ the cavity photon as a data-bus; (ii) the qubitcavity coupling (χ in the dispersive regime) is strong, which is required for quantum information processing and would violate the weak-response assumption; and (iii) the average cavity photon numbern is not tiny, which has the advantage of enhancing the measurement signal to overcome the noise from the amplifiers and circuits.
For the sake of brevity, we denote the Bayesian rule proposed in Ref. [27] by BR-I, the rule constructed in the present work by BR-II, and the one involving the approximate Φ 2 of Eq. (19) by BR-II ′ . In Fig. 3 we display results for both the single-and two-quadrature measurements outside the "bad-cavity" and weak-response limits. Clearly, we find that in this case the purity-degradation factor, D(t) = | α e (t)|α g (t) |, is reduced to values obviously lower than unity, and for both measurements BR-II fits the (exact) QTE results (off-diagonal element of the qubit state) better than BR-I. For single-quadrature measurement in this non-limiting case, we find the use of the approximate phase factor Φ 2 causes some deviation from the precise one, as revealed in Fig. 3(c). However, combining with the other two corrections (D and Φ 1 ), BR-II ′ can give reasonable results as shown in Fig. 3(e).

D. Experimental Issues
In order to implement the Bayesian rule proposed in Sec. IV and V in experiments, the key quantities to be fixed are the cavity fields α g (t) and α e (t). Viewing the analytic solutions in Appendix A, the cavity damping rate κ  ) and (e) correspond to single-quadrature measurement, and the results in (d) and (f) to two-quadrature measurement. BR-I and BR-II denote the Bayesian rules proposed in Ref. [27] and in the present work, whereas BR-II ′ involves the approximation of Eq. (19). We assume the qubit in initial state (|e + |g )/ √ 2 and the setup parameters ∆r = 0, χ = 0.1κ and ǫm = κ. These parameters indicate the cavity photon numbern ≃ 4. and the dispersive coupling χ should be determined in advance. For a given detuning ∆ r and driving amplitude ǫ m , these two parameters can be extracted from the steadystate mean values of the quadrature measurements,Ī g and I e , which are related toᾱ g andᾱ e . With these extracted parameters at hand, one can accordingly implement the proposed Bayesian rule, with the cavity in an initial vacuum or certain known steady state (e.g., |ᾱ g ).
In experiments, one needs also to properly account for the unavoidable measurement inefficiencies, in particular the circuit and amplifier noises. This important issue has been addressed by Wiseman et al. in a series of papers [31][32][33], where the so-called realistic quantum trajectory equation has been developed. However, the resultant equation would be difficult to use in practice. Within the framework of the Baysian approach, it seems simpler to address this issue [34], since accounting for the extra noise only corresponds to a more Bayesian inferring. As a result, the effect of the extra noise requires us to partially average the ideal-measurement-result conditioned state. We will incorporate our present quantum Bayesian approach with this type of treatment in a separate work.
Finally, we mention that so far there have been a few experiments involving the quantum Bayesian rule in the bad-cavity and weak-response regime [10,11,16]. In the feedback control experiment [16], the phase-sensitive detection scheme corresponds to a case where the phase factor Φ 2 (t m ) in Eq. (17) vanishes. Moreover, in the bad-cavity and weak-response limit, the factor Φ 1 reduces to 2χnt m and the purity degradation factor is about unity. In two recent experiments [10,11], however, the phase factor e −iΦ2(tm) is present and was demonstrated with satisfactory accuracies. In view of these remarkable advances in cQED measurements, further experiments to demonstrate the quantum Bayesian rule in more general cases would be of great interest.

VII. CONCLUDING REMARKS
To summarize, we have constructed a quantum Bayesian rule for weak measurements of qubits in cQED. Our construction was guided by a microscopic analysis of the cavity field and the cavity-photon-eliminated effective QTE. This type of treatment provided a different route from Korotkov's method in Ref. [27]. The present work, for both the single-and two-quadrature measurements, generalizes the results in Ref. [27] from "bad-cavity" and weak-response limits to more general conditions. Numerical comparisons with the direct QTE simulations show that the proposed rule can work with high accuracy even in non-limiting cases.
We would like to remark that, originally, the Bayesian approach was not constructed or integrated from QTE [26,27]. Their mathematical connection is also not very straightforward. For infinitesimal short-time evolution, their equivalence can be proved. However, for longer time state update, there exist slight numerical differences, despite the reasonable agreements observed in Figs. 2 and 3. Rather than an exact derivation from QTE, we would like to regard the Bayesian rule as a construction. In particular, the cavity-photon-eliminated effective QTE contains unusual stochastic unitary term. As an ansatz, we inserted (integrated) it into the Bayesian dynamics of the off-diagonal elements and revealed an interesting connection with the results of Ref. [27] in limiting cases.
It has become clear that, in addition to the quantum trajectory equation, the empirical Bayesian formulas are very useful in experiments [10,11,16]. In connection with the experiment of Ref. [10], a POVM-type formalism was constructed for qubit state update in cQED, which is actually equivalent to the result obtained in Ref. [27]. Also, both results work in the "bad-cavity" and weak response limits. Their minor difference is: the (POVM) measurement operator (i.e., M Im,Qm in Eq. (6) in the Supplementary Materials of Ref. [10]), avoids the "purity" consideration in constructing the Bayesian rule and contains the phase factors discussed in Ref. [27] and in our present work. Finally, at the revision stage of this work, we became aware of the new publication [30], in which similar purity degradation factor was included in the concurrence calculation for a two-qubit cQED system (see Eq. (12) in the Supplementary Information). However, it seems that in this work the Bayesian rule proposed in Ref. [27], particularly the "realistic" back-action phase factor (Φ 2 denoted in our present work) was not involved. There, the quantum trajectory equation approach was also used to compare with the experimental quantum trajectories, see, Eqs. (14) and (22) in the Supplementary Information. Viewing these remarkable efforts and progress, we expect our proposed Bayesian rule to be useful, for both theoretical interests and future cQED measurement and control experiments.
to ϕ = θ β . In contrast, if we tune ϕ = θ β + π/2, the quadrature measurement does not reveal any information for the qubit-state [35]. More specifically, starting the measurement with an empty cavity and choosing a resonant measurement driving (∆ r = ω r − ω m = 0), one can easily prove that θ β = 0. This means that, choosing ϕ = 0, one can achieve maximal information for the qubit state, while no qubit state information can be inferred if choosing ϕ = π/2.
Moreover, for a single realization of quadrature measurement during (0, t m ), we introduce the mean integrated quadrature output I m = 1 tm tm 0 dtI ϕ (t), where the continuous outcome is described by Corresponding to qubit state |g(e) , the distribution P g(e) (I m ) of the integrated quadrature I m is Gaussian, P g(e) (I m ) = 1 √ 2πD exp[−(I m − I g(e) ) 2 /(2D)]. Here,Ī g(e) is the average quadrature outcome, given byĪ g(e) = 1 tm tm 0 dtĪ g(e) (t). The variance D can be analytically determined as follows. Consider the expression of the homodyne current. Noting that the first term describes the average current, the deviation of the individual quadrature output (I m ) from the averaged I g(e) is thus caused by the second term ξ(t). Denoting I(t) ≡ ξ(t), from definition we have On the other hand, via a direct calculation, we thus obtain D = 1/t m .

Appendix B: Revisit the Purity Degradation Factor
Based on the transformed Eq. (14), in this Appendix we revisit the purity degradation factor | α e (t m )|α g (t m ) |. First, we notice that the last term of Eq. (14) does not only play the usual role of unitary evolution, but also has an effect of unraveling the qubit state. This can be clearly seen by setting ϕ = π/2. In this case, Γ ci = 0, then one may expect that the r.h.s. second Lindblad term will gradually completely dephase the qubit state. However, interestingly, the last term will prevent this, owing to its certain unraveling ability. This feature is in agreement with the Bayesian rule for the case of identical P g (I m ) and P e (I m ). Both approaches predict that the qubit will be maintained in a superposition state with high purity. Even better, we can combine the last two terms into a single unraveling term: Γ m (t)/2H[Λσ z ]ρ(t)ξ(t), where Γ m (t) = Γ ci (t) + Γ ba (t) = κ|β(t)| 2 (the measurement rate) is independent of the choice of ϕ, and Λ = cos(ϕ − θ β ) − i sin(ϕ − θ β ) depends on ϕ. We may interpret this result as follows: Γ m determines the unraveling extent; and Λσ z describes the unraveling (measuring) means. We propose then a dephasing factor of the form Below, we prove that the two expressions of the purity degradation factor for the qubit state are identical, i.e., This is equivalent to proving the following: In obtaining this result, the property of coherent states has been used. Under the conditions ∆ r = 0 and α e (0) = α g (0) = 0, more explicitly we reexpress the solution of Eq. (A1) as α e (t) = iǫ m iχ + κ/2 e −(iχ+κ/2)t − 1 , α g (t) = iǫ m −iχ + κ/2 e (iχ−κ/2)t − 1 .