Exact quantum Bayesian rule for qubit measurements in circuit QED

Developing efficient framework for quantum measurements is of essential importance to quantum science and technology. In this work, for the important superconducting circuit-QED setup, we present a rigorous and analytic solution for the effective quantum trajectory equation (QTE) after polaron transformation and converted to the form of Stratonovich calculus. We find that the solution is a generalization of the elegant quantum Bayesian approach developed in arXiv:1111.4016 by Korotokov and currently applied to circuit-QED measurements. The new result improves both the diagonal and off-diagonal elements of the qubit density matrix, via amending the distribution probabilities of the output currents and several important phase factors. Compared to numerical integration of the QTE, the resultant quantum Bayesian rule promises higher efficiency to update the measured state, and allows more efficient and analytical studies for some interesting problems such as quantum weak values, past quantum state, and quantum state smoothing. The method of this work opens also a new way to obtain quantum Bayesian formulas for other systems and in more complicated cases.

quantum state measurement. After a qubit-state-dependent displacement transformation (called also "polaron" transformation) to eliminate the cavity degrees of freedom 24 , the single quadrature homodyne current can be reexpressed as In this result, ξ(t), originated from the fundamental quantum-jumps, is a Gaussian white noise and satisfies the ensemble-average properties of E[ξ(t)] = 0 and E[ξ(t)ξ(t′ )] = δ(t − t′ ). Γ ci (t) is the coherent information gain rate which, together with the other two, say, the no-information back-action rate Γ ba (t) and the overall measurement decoherence rate Γ d (t), is given by 24  Here we have denoted the local oscillator's (LO) phase in the homodyne measurement by ϕ, the cavity photon leaky rate by κ, and β α α β ( ) = ( ) − ( ) ≡ ( ) θ β t t t t e i 2 1 with α 1 (t) and α 2 (t) the cavity fields associated with the respective qubit states 1 and 2 .
In this work, for brevity, we directly quote the transformed QTE results from ref. 24 where the derivation and explanation of the above rates can be found with details. Briefly speaking, the coherent information gain rate, Γ ci , describes the backaction of information gain during the measurement, which corresponds to the "spooky" backaction rate termed in ref. 16 by Korotkov. Γ ba , corresponding to the "realistic" backaction rate in ref. 16 for the diagonal and off-diagonal elements of the density matrix, respectively. Here we have also absorbed a generalized dynamic ac-Stark shift, into the effective frequency of the qubit, Ω ∼ q . Eqs. (5) and (6) are defined on the basis of Itó calculus and is the form for numerical simulations 25,26 . However, as to be clear in the following, in order to get the correct result of the quantum Bayesian rule, one needs to convert it into the Stratonovich-type form. We thus have ci z t ba 12 12 12 12 12 We see that, compared to the Itó type Eqs. (5) and (6), the Gaussian noise ξ(t) in all the noisy terms has been replaced now by the homodyne current I(t) which is given by Eq. (1). Now we are ready to integrate Eqs.
In this solution, we introduced the measurement rate Γ m = Γ ba + Γ ci ; and all the factors except the last one have been in explicit forms of integration with known integrands. For the last factor, since , substituting the explicit solution of ρ 11 (t) to complete the integration yields Then we reexpress the solution of the off-diagonal element ρ 12 (t m ) in a compact form as Desirably, these factors recover those proposed in ref. 27 from different insight and analysis. As demonstrated in ref. 27, these factors have important effects to correct the "bare" Bayesian rule for the off-diagonal elements. As to be seen soon (after simple algebra), the above treatment provides also a reliable method allowing us to obtain new and precise expressions for the prior distribution knowledge of the output currents which are of essential importance to the Bayesian inference for the cQED measurement.
To make the results derived above in the standard form of Bayesian rule, let us rewrite , where N(t m ) = ρ 11 (0)P 1 (t m ) + ρ 22 (0)P 2 (t m ) and the current distribution probabilities read m m , and V = 1/t m characterizes the distribution variance. For Bayesian inference, the "prior" knowledge of the distribution probabilities P 1(2) (t m ) associated with qubit state Scientific RepoRts | 6:20492 | DOI: 10.1038/srep20492 1 ( ) 2 , should be known in advance. Then, one utilizes them to update the measured state based on the collected output currents, using the following exact quantum Bayesian rule. First, for the off-diagonal element, Second, for the diagonal elements, The results of Eq. (14)- (19) constitute the main contribution of this work.
Result of simpler case. The original work of quantum Bayesian approach considered a charge qubit 28 . This problem can be described by the following QTE which has been applied broadly in quantum optics 1,2 In this equation the Lindblad term takes D[σ z ]ρ = σ z ρσ z − ρ, and the information-gain backaction term reads H[σ z ]ρ = σ z ρ + ρσ z − 2Tr[σ z ρ]ρ. γ′ and γ are respectively the measurement decoherence and information-gain rates (constants). And, this equation is conditioned on the measurement current γ σ ρ ξ Tr [ ] z , since both share the same stochastic noise ξ(t).
Following precisely the same procedures of solving the above cQED system, one can arrive to similar results as Eqs. (9)(10) and (13) with, however, simpler X(t m ) which is now given by . For this simpler setup, the phase factor Φ 1 takes the trivial result Φ 1 (t m ) = ω q t m and the factor Φ 2 vanishes. The simpler result of X(t m ) allows us to introduce, from the factors ± ( ) e X t m in Eqs. (9)(10) and (13), the distribution probabilities of the standard Gaussian form Numerical results. To implement the above exact Bayesian rule for state inference in practice, we should carry out in advance the rates Γ ci (t), Γ ba (t), Γ d (t) and as well the ac-Stark shift B(t). From the expressions of these quantities, we know that the key knowledge required is the coherent-state parameters α 1 (t) and α 2 (t) which are, respectively, the consequence of the interplay of external driving and cavity damping for qubit states 1 and 2 , . The simple analytic solutions read are the steady-state cavity fields; and α 0 is the initial cavity field before measurement which is zero if we start with a vacuum.
In Fig. 2, taking a specific trajectory for each case as an example, we compare the results from different approaches. In this plot we adopt a reduced system of units by scaling energy and frequency with the microwave drive strength (ε m ). We have assumed two types of parameters in the simulation: for the upper row (weaker response) we assumed χ = 0.1 while for the lower one (stronger response) we assumed χ = 0.5, both violating the joint condition of "bad"-cavity and weak-response since we commonly used κ = 2. (Other parameters are referred to the figure caption). The basic requirement of the "bad"-cavity and weak response limits is κ  χ. This has a couple of consequences: (i) The time-dependent factor in Eq. (22), e ±iχt e −κt/2 , would become less important so that we can neglect the transient dynamics of the cavity field and all the rates (Γ d , Γ ba and Γ ci ) can be treated as (steady state) constants 16 . (ii) It makes the measurement rate (Γ m ) much smaller than κ, and the purity factor D(t m ) almost unity. This means that, with respect to the cavity photon's leakage, the (gradual collapse) measurement process is slow, and the qubit state remains almost pure in the ideal case (in the absence of photon loss and amplification noise). One can check that the parameters used in Fig. 2 (especially the case χ = 0.5) violate these criteria.
We find that for both cases the results from our exact Bayesian rule, Eqs. (18) and (19) together with (16), precisely coincides with those from simulating the QTE. As a comparison, in Fig. 2 we plot also the results from other two approximate Bayesian approaches. One is the Bayesian approach constructed by Korotkov in ref. 16 under the bad-cavity and weak-response limits, which is labeled in Fig. 2 by "K". Another is the Bayesian rule obtained in ref. 27, which holds the same factors as shown in Eqs. (14)-(16) but involves the usual Gaussian distribution of the type of Eq. (21). The results are labeled by "G" in Fig. 2. We find from this plot that, for the modest violation of the bad-cavity and weak-response limits, the "G" results are better than the "K" ones. However, for strong violation, the "G" results will become unreliable as well. In contrast, as we observe here and have checked for many more trajectories in the case of violating the bad-cavity and weak-response limits, the exact Bayesian rule can always work precisely while both the "K" and "G" approaches failed.

Discussion
To summarize, for continuous weak measurements in circuit-QED, we carry out the analytic and exact solution of the effective QTE which generalizes the quantum Bayesian approach developed in ref. 16 and applied in circuit-QED experiments [13][14][15] . The new result is not bounded by the "bad"-cavity and weak-response limits as in ref. 16, and improves the quantum Bayesian rule via amending the distribution probabilities of the output currents and several important phase factors 16,27 .
The efficiency of quantum Bayesian approach can be understood with the illustrative Fig. 1. Instead of the successive step-by-step state estimations (over many infinitesimal time intervals of dt) using QTE, the Bayesian rule offers the great advantage of one-step estimation, with a major job by inserting the continuous measurement outcomes (currents) into the simple expressions of the prior knowledge P 1,2 (t m ) given by Eq. (17) and the phase correction factor Φ 2 (t m ). Other α 1,2 (t)-dependent factors are independent of the measurement outcomes and can be carried out in advance. The time integration in P 1,2 (t m ) and Φ 2 (t m ) can be completed as soon as the continuous measurement over (0, t m ) is finished. Therefore, the state estimation can be accomplished using Eqs. (18) and (19), with similar efficiency as using the usual simple Gaussian distribution Eq. (21).
In this work we have focused on the single quadrature measurement. For the so-called (I, Q) two quadrature measurement, following the same procedures of this work or even simply based on the characteristic structure of the final results, Eqs. (14)- (19), one can obtain the associated quantum Bayesian rule as well 16,27 .
We finally remark that, just as Bayesian formalism is the central theory for noisy measurements in classical control and information processing problems, quantum Bayesian approach can be taken as an efficient theoretical framework for continuous quantum weak measurements. Compared to the "construction" method 16 , the present work provides a direct and more reliable route to formulate the quantum Bayesian rule. Applying similar method to more complicated cases such as joint measurement of multiple qubits or to other systems is of interest for further studies.

Methods
Circuit-QED setup and measurements. Under reasonable approximations, the circuit-QED system resembles the conventional atomic cavity-QED system which has been extensively studied in quantum optics. Both can be well described by the Jaynes-Cummings Hamiltonian. In dispersive regime 3-5 , i.e., the detuning between the cavity frequency (ω r ) and qubit energy (ω q ), Δ = ω r − ω q , being much larger than the coupling strength g, the system Hamiltonian (in the rotating frame with the microwave driving frequency ω m ) reads 3 ,e,f) The curves "E" (red), "G" (green), and "K" (blue) denote, respectively, our exact Bayesian rule, Eqs. (18) and (19) together with (17), the approximate one involving instead the usual Gaussian distribution of Eq. (21), and that proposed by Korotkov 16 under the bad-cavity and weak-response limits. In each figure, the lower panel plots also the difference from the quantum trajectory equation result, indicating that the BR proposed in this work is indeed exact. In all these numerical simulations, we chose the LO's phase ϕ = π/4 and adopted a system of reduced units with parameters Δ r = 0, ε m = 1.0, and κ = 2.0.

Conversion rule.
In order to convert Eqs. (5) and (6) to Eqs. (7) and (8), in this Appendix we specify the conversion rule from Itó to Stratonovich stochastic equations 25,26 . Suppose for instance we have a set of Itó-type stochastic equations