Abstract
Developing efficient framework for quantum measurements is of essential importance to quantum science and technology. In this work, for the important superconducting circuitQED 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 circuitQED measurements. The new result improves both the diagonal and offdiagonal 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.
Introduction
Continuous quantum weak measurement, which stretches the “process” of the Copenhagen’s instantaneous projective measurement, offers special opportunities to control and steer quantum state^{1,2}. This type of measurement is essentially a process of realtime monitoring of environment with stochastic measurement records and causing backaction onto the measured state. However, remarkably, the stochastic change of the measured state can be faithfully tracked. Because of this unique feature, quantum continuous weak measurement can be used, for instance, for quantum feedback of error correction and state stabilization, generating pre and postselected (PPS) quantum ensembles to improve quantum state preparation, smoothing and highfidelity readout and developing novel schemes of quantum metrology^{1,2}.
On the other aspect, the superconducting circuit quantum electrodynamics (cQED) system^{3,4,5} is currently an important platform for quantum measurement and control studies^{6,7,8,9,10,11,12,13,14,15}. In particular, the continuous weak measurements in this system, as schematically shown in Fig. 1(a), have been demonstrated in experiments^{12,13,14}, together with feedback control^{15}.
For continuous weak measurements, the most celebrated formulation is the quantum trajectory equation (QTE) theory, which has been broadly applied in quantum optics and quantum control studies^{1,2}. However, in some cases of real experiment, numerical integration of the QTE is not efficient and alternatively, the onestep quantum Bayesian approach has been employed to update the quantum state based on the integrated output currents^{12,13,14,15}, as illustrated in Fig. 1(b). Meanwhile, one may notice that the Bayes’ formula is the fundamental tool for classical noisy measurements^{2}. In this work we carry out the analytic and exact solution of the effective QTE of the cQED system. Desirably, we find that the solution is a generalization of the elegant quantum Bayesian approach developed in ref. 16 by Korotokov and currently applied to circuitQED experiments^{13,14,15}. The new result is not bounded by the “bad”cavity and weakresponse limits as in ref. 16 and promises higher efficiency than numerically integrating the QTE. The Bayesian rule also allows more efficient and analytical studies for some interesting problems such as quantum weak values^{17,18}, past quantum state^{19,20,21} and quantum state smoothing^{22,23}, etc.
Results
Exact Bayesian rule for cQED measurement
We consider the cQED architecture consisting of a superconducting transmon qubit dispersively coupled to a waveguide cavity, see Fig. 1(a). The qubitcavity interaction is given by the Hamiltonian^{3,4,5}, H_{int} = χa^{†}aσ_{z}, where χ is the dispersive coupling rate, a^{†} and a are respectively the creation and annihilation operators of the cavity mode and σ_{z} is the qubit Pauli operator. This interaction Hamiltonian characterizes a qubitstatedependent frequency shift of the cavity which is used to perform quantum state measurement. After a qubitstatedependent 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 quantumjumps, is a Gaussian white noise and satisfies the ensembleaverage 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 noinformation backaction 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 with α_{1}(t) and α_{2}(t) the cavity fields associated with the respective qubit states and .
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, characterizes the backaction of the measurement device not associated with information gain of the qubit state. And the overall decoherence rate, Γ_{d}, describes the ensemble average effect of the measurement on the qubit state, over large number of quantum trajectories.
Within the framework of “polaron” transformation, the effective quantum trajectory equation (QTE) for qubit state alone can be expressed as^{24}
for the diagonal and offdiagonal elements of the density matrix, respectively. Here we have also absorbed a generalized dynamic acStark shift, into the effective frequency of the qubit, .
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 Stratonovichtype form. We thus have
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. (7) and (8). For the first equation of ρ_{11}(t), perform on both sides. Noting that , we obtain , where . In deriving, we have used the property ρ_{11} + ρ_{22} = 1. Using this property again, we may split the solution into two equations
where . Further, integrating the second equation for ρ_{12}(t), we obtain
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 offdiagonal element ρ_{12}(t_{m}) in a compact form as
where we have introduced
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 offdiagonal 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 and , where N(t_{m}) = ρ_{11}(0)P_{1}(t_{m}) + ρ_{22}(0)P_{2}(t_{m}) and the current distribution probabilities read
where and 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 , 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 offdiagonal element,
Second, for the diagonal elements,
where j = 1, 2. 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 measured by pointcontact detector^{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 informationgain 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 informationgain rates (constants). And, this equation is conditioned on the measurement current , 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 , owing to the constant γ. Also, now the dephasing factor simply reads where . 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 in Eqs. (9)(10) and (13), the distribution probabilities of the standard Gaussian form
where and . With these identifications, one recovers the quantum Bayesian rule proposed by Korotkov in ref. 28.
Similarly, for the cQED setup, from , simple experience likely tells us that, corresponding to the qubit states and , the stochastic current I_{m} (average of the stochastic I(t) over t_{m}) should be respectively centered at , in terms of the Gaussian distribution as given by Eq. (21). However, out of our expectation, this is not true. Below we display numerical results to show the exactness of the Bayesian rule Eqs. (18) and (19) when associated with (16), rather than with the Gaussian formula (20).
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 acStark shift B(t). From the expressions of these quantities, we know that the key knowledge required is the coherentstate 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 and , satisfying , where . The simple analytic solutions read
In this solution, are the steadystate 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 weakresponse 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 timedependent 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 badcavity and weakresponse 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 badcavity and weakresponse 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 badcavity and weakresponse 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 circuitQED, 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 circuitQED experiments^{13,14,15}. The new result is not bounded by the “bad”cavity and weakresponse 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 stepbystep state estimations (over many infinitesimal time intervals of dt) using QTE, the Bayesian rule offers the great advantage of onestep 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 socalled (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
CircuitQED setup and measurements
Under reasonable approximations, the circuitQED system resembles the conventional atomic cavityQED system which has been extensively studied in quantum optics. Both can be well described by the JaynesCummings Hamiltonian. In dispersive regime^{3,4,5}, 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^{3,4,5}
where Δ_{r} = ω_{r} − ω_{m} and , with χ = g^{2}/Δ the dispersive shift of the qubit energy. In Eq. (23), a^{†} (a) and σ_{z} are respectively the creation (annihilation) operator of cavity photon and the quasispin operator (Pauli matrix) for the qubit. ε_{m} is the microwave drive amplitude for continuous measurements. For singlequadrature (with reference local oscillator phase φ) homodyne detection of the cavity photons, the measurement output can be expressed as . After eliminating the cavity degrees of freedom via the “polaron”transformation, this output current turns to Eq. (1). Moreover, the conditional qubitcavity joint state ρ(t), which originally satisfies the standard optical quantum trajectory equation^{1,2}, becomes now after eliminating the cavity states^{24}:
In addition to the Lindblad term (the second one on the r.h.s), the other superoperator is introduced through , where 〈σ^{z}〉 = Tr[σ^{z}ρ]. In this result, B(t) is the dynamic acStark shift and Γ_{d}, Γ_{ci} and Γ_{ba} are, respectively, the overall measurement decoherence, information gain and noinformation backaction rates (with explicit expressions given by Eqs. (2)–(4). In the qubitstate basis, Eq. (24) gives Eqs. (5) and (6).
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 Itótype stochastic equations
with j = 1, 2, ···, K. The corresponding Stratonovichtype equations read
Comparing Eq. (25) with Eqs.(5) and (6), we identify Y_{1} = ρ_{11} and Y_{2} = ρ_{12}. Then we have and . Applying the conversion rule Eq. (26), one can convert Eqs. (5) and (6) to Eqs. (7) and (8) by completing the following algebraic manipulations:
and
In the above manipulations, we have used 〈σ_{z}〉 = ρ_{11} − ρ_{22} = 2ρ_{11} − 1, 1 − 〈σ_{z}〉^{2} = 4ρ_{11}ρ_{22} and Γ_{m} = Γ_{ci} + Γ_{ba}.
Additional Information
How to cite this article: Feng, W. et al. Exact quantum Bayesian rule for qubit measurements in circuit QED. Sci. Rep. 6, 20492; doi: 10.1038/srep20492 (2016).
References
Wiseman, H. M. & Milburn, G. J. Quantum Measurement and Control (Cambridge Univ. Press, Cambridge, 2009).
Jacobs, K. Quantum Measurement Theory and Its Applications (Cambridge Univ. Press, Cambridge, 2014).
Blais, A. et al. Cavity quantum electrodynamics for superconducting electrical circuits: An architecture for quantum computation. Phys. Rev. A 69, 062320 (2004).
Wallraff, A. et al. Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature (London) 431, 162 (2004).
Chiorescu, I. et al. Coherent dynamics of a flux qubit coupled to a harmonic oscillator. Nature (London) 431, 159 (2004).
Schoelkopf, R. J. & Girvin, S. M. Wiring up quantum systems. Nature 451, 664 (2008).
PalaciosLaloy, A. et al. Experimental violation of a Bell’s inequality in time with weak measurement. Nat. Phys. 6, 442 (2010).
Groen, J. P. et al. Partialmeasurement backaction and nonclassical weak values in a superconducting circuit. Phys. Rev. Lett. 111, 090506 (2013).
Mariantoni, M. et al. Photon shell game in threeresonator circuit quantum electrodynamics. Nat. Phys. 7, 287 (2011).
CampagneIbarcq, P. et al. Persistent control of a superconducting qubit by stroboscopic measurement feedback. Phys. Rev. X 3, 021008 (2013).
Risté, D. et al. Initialization by Measurement of a Superconducting Quantum Bit Circuit. Phys. Rev. Lett. 109, 050507 (2012).
Hatridge, M. et al. Quantum BackAction of an Individual VariableStrength Measurement. Science 339, 178 (2013).
Murch, K. W., Weber, S. J., Macklin, C. & Siddiqi, I. Observing single quantum trajectories of a superconducting quantum bit. Nature 502, 211 (2013).
Tan, D., Weber, S. J., Siddiqi, I., Molmer, K. & Murch, K. W. Prediction and Retrodiction for a Continuously Monitored Superconducting Qubit. Phys. Rev. Lett. 114, 090403 (2015).
Vijay, R. et al. Stabilizing Rabi oscillations in a superconducting qubit using quantum feedback. Nature 490, 77 (2012).
Korotkov, A. N. Quantum Bayesian approach to circuit QED measurement. arXiv:1111.4016 (2011).
Williams, N. S. & Jordan, A. N. Weak values and the LeggettGarg inequality in solidstate qubits. Phys. Rev. Lett. 100, 026804 (2008).
Qin, L., Liang, P. & Li, X. Q. Weak values in continuous weak measurement of qubits. Phys. Rev. A 92, 012119 (2015).
Gammelmark, S., Julsgaard, B. & Molmer, K. Past Quantum States of a Monitored System. Phys. Rev. Lett. 111, 160401 (2013).
Tan, D., Weber, S. J., Siddiqi, I., Molmer, K. & Murch, K. W. Prediction and Retrodiction for a Continuously Monitored Superconducting Qubit. Phys. Rev. Lett. 114, 090403 (2015).
Rybarczyk, T. et al. Past quantum state analysis of the photon number evolution in a cavity. arXiv:1409.0958 (2014).
Tsang, M. Optimal waveform estimation for classical and quantum systems via timesymmetric smoothing. Phys. Rev. A 80, 033840 (2009).
Guevara, I. & Wiseman, H. Quantum State Smoothing. arXiv:1503.02799v2 (2015).
Gambetta, J. et al. Quantum trajectory approach to circuit QED: Quantum jumps and the Zeno effect. Phys. Rev. A 77, 012112 (2008).
Kloeden, P. E. & Platen, E. Numerical Solution of Stochastic Differential Equations (SpringerVerlag, Berlin, 1992).
Goan, H. S., Milburn, G. J., Wiseman, H. M. & Sun, H. B. Continuous quantum measurement of two coupled quantum dots using a point contact: A quantum trajectory approach. Phys. Rev. B 63, 125326 (2001).
Wang, P., Qin, L. & Li, X. Q. Quantum Bayesian rule for weak measurements of qubits in superconducting circuit QED. New J. Phys. 16, 123047 (2014); Corrigendum: Quantum Bayesian rule for weak measurements of qubits in superconducting circuit QED. ibid.17, 059501 (2015).
Korotkov, A. N. Continuous quantum measurement of a double dot. Phys. Rev. B 60, 5737 (1999).
Acknowledgements
This work was supported by the NNSF of China under grants No. 101202101 & 10874176 and the State “973” Project under grants No. 2011CB808502 & 2012CB932704.
Author information
Affiliations
Contributions
X.Q.L. supervised the work; W.F., P.F.L. and L.P.Q. carried out the calculations; X.Q.L. wrote the paper and all authors reviewed it. W.F. and P.F.L. equally contributed to this work.
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Cite this article
Feng, W., Liang, P., Qin, L. et al. Exact quantum Bayesian rule for qubit measurements in circuit QED. Sci Rep 6, 20492 (2016). https://doi.org/10.1038/srep20492
Received:
Accepted:
Published:
Further reading

Weakvalueamplification analysis beyond the AharonovAlbertVaidman limit
Physical Review A (2020)

Estimation of parameters in circuit QED by continuous quantum measurement
Physical Review A (2019)

Quantum parameter estimation via dispersive measurement in circuit QED
Quantum Information Processing (2018)

Realtime quantum state estimation in circuit QED via the Bayesian approach
Physical Review A (2018)

Gradual partialcollapse theory for ideal nondemolition longitudinal readout of qubits in circuit QED
Physical Review A (2018)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.