Simultaneous continuous measurement of non-commuting observables: quantum state correlations
Areeya Chantasri, Juan Atalaya, Shay Hacohen-Gourgy, Leigh S. Martin, Irfan Siddiqi, Andrew N. Jordan
SSimultaneous continuous measurement of non-commuting observables:quantum state correlations
Areeya Chantasri ∗ ,
1, 2
Juan Atalaya ∗ , Shay Hacohen-Gourgy,
4, 5
Leigh S. Martin,
4, 5
Irfan Siddiqi,
4, 5 and Andrew N. Jordan
1, 2, 6 Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA Center for Coherence and Quantum Optics, University of Rochester, Rochester, New York 14627, USA Department of Electrical Engineering, University of California, Riverside, California 92521, USA Quantum Nanoelectronics Laboratory, Department of Physics,University of California, Berkeley CA 94720, USA Center for Quantum Coherent Science, University of California, Berkeley CA 94720, USA Institute for Quantum Studies, Chapman University,1 University Drive, Orange, California 92866, USA (Dated: September 19, 2018)We consider the temporal correlations of the quantum state of a qubit subject to simultaneouscontinuous measurement of two non-commuting qubit observables. Such qubit state correlators aredefined for an ensemble of qubit trajectories, which has the same fixed initial state and can also beoptionally constrained by a fixed final state. We develop a stochastic path integral description for thecontinuous quantum measurement and use it to calculate the considered correlators. Exact analyticresults are possible in the case of ideal measurements of equal strength and are also shown to agreewith solutions obtained using the Fokker-Planck equation. For a more general case with decoherenceeffects and inefficiency, we use a diagrammatic approach to find the correlators perturbatively inthe quantum efficiency. We also calculate the state correlators for the quantum trajectories whichare extracted from readout signals measured in a transmon qubit experiment, by means of thequantum Bayesian state update. We find an excellent agreement between the correlators based onthe experimental data and those obtained from our analytical and numerical results.
I. INTRODUCTION
Continuous weak quantum measurement (CWQM) hasattracted much attention in the quantum information sci-ence community. This topic has been discussed theoret-ically in the past few decades [1–10], and its experimen-tal study has been motivated by recent developments insuperconducting qubit technology [11–17]. Such exper-imental and theoretical efforts have paved the way forinteresting applications of CWQMs such as rapid statepurification [18], quantum feedback [9, 13, 16, 19, 20],and preparation of entangled states [21–24]. WithCWQMs, it is also possible to simultaneously measurenon-commuting observables [25–27], and the first exper-imental demonstration of such measurement on a super-conducting qubit was realized only last year [28].Precise simultaneous measurement of non-commutingobservables of a quantum system is forbidden by text-book quantum mechanics. This is so because if one of themeasurements were to collapse the system wavefunctionto an eigenstate of a measurement operator, a precise orstrong measurement of another non-commuting observ-able would produce an uncertain result. Such measure-ment incompatibility with strong measurements, how-ever, can be bypassed by using CWQMs. The reasonis that the latter are rather imprecise (weak) measure-ments such that the readout signals have small signal-to-noise ratios (SNRs), and, therefore, the two compet-ing measurements collapse the system state only partiallyover time, resulting in a quantum state that continuouslyevolves in a diffusive manner. Simultaneous continuous measurement of two qubitobservables, σ z and σ ϕ ≡ σ z cos ϕ + σ x sin ϕ , was imple-mented by weakly coupling a superconducting transmonqubit to two intracavity modes [28]. The measurementwas effectively a stroboscopic measurement of a fast ro-tating qubit. In that experiment, the considered anglesincluded ϕ = ϕ ≠ ϕ = π / σ x and σ z ). The corresponding quan-tum trajectories were inferred from the measurementreadouts and verified using state tomography techniques.In the case of simultaneous measurement of commutingobservables ( ϕ = σ z (i.e., measurement induces twostate attractors at the Bloch points z = ± σ x and σ z , the qubitstate does not collapse to any eigenstates of such ob-servables. Instead, the quantum trajectories exhibit freediffusion in the Bloch sphere [27]. The detector read-out signals also continuously vary in time in a randomfashion and their temporal correlations were analyzed inRef. [29].The self-correlation function of the detector readoutsdoes not depend on the measurement quantum efficiency, η , if we consider temporal correlations at sufficiently longtimes (larger than the reciprocal bandwidth of the de-tector). The cross-correlation function is not affectedby η even at vanishing times [29]. This can be under-stood physically by considering the readout of a non-ideal detector (with η <
1) as the sum of two contribu-tions; namely, one from the readout of an ideal detector a r X i v : . [ qu a n t - ph ] J un ( η =
1) and another from a source of uncorrelated noise,whose strength is proportional to η − −
1. Both contri-butions then contribute additively to the readout corre-lators. In contrast, quantum state correlators depend onthe quantum efficiency, and their measurement requiresdetectors with not too small η , which is nowadays possi-ble with experimental setups based on superconductingqubits weakly coupled to microwave cavities [15, 28]. Inview of this experimental capability, the goal of this pa-per is to discuss the temporal correlations of the quantumstate itself during weak continuous measurements.In this work, we consider the statistical properties ofthe state of a qubit subject to simultaneous continuousmeasurement of two non-commuting observables. We areinterested in the temporal correlations of state-dependentquantities over certain sub-ensembles of quantum trajec-tories. We consider two scenarios (i) quantum state cor-relators with pre-selection (fixed initial state), and (ii)quantum state correlators with pre- and post-selection(fixed initial and final states). We adopt the calculationtechnique based on stochastic path integrals [30] to qubittrajectories in the Bloch sphere. The stochastic path in-tegral description provides a convenient way to explorethe statistical properties of sub-ensembles of quantumtrajectories by allowing us to impose boundary condi-tions at the beginning and at the end of each trajectory.We obtain analytical results for the case of ideal measure-ments of σ x and σ z of equal strength. In the non-idealcase, η <
1, we develop a diagrammatic perturbation the-ory, similar to the loop expansion in quantum field the-ory, to calculate the considered sub-ensemble averagesin the limit of small η . Moreover, we calculate the quan-tum state correlators from the quantum state trajectoriesmonitored in Ref. [28] and show that they agree well withour analytical and numerical results.We begin our analysis with a review of the quantumtrajectory approach to quantum measurement and thestochastic master equation in Sec. II A, and then intro-duce the stochastic path integral formalism in Sec. II B.The stochastic path integral formalism is then used tocompute statistical quantities such as conditional aver-ages and correlation functions, presented specifically forthe “XZ measurement” (the joint simultaneous measure-ment of σ x and σ z observables) in Sec. III. In Sec. III A,we investigate the simplest case, that of an ideal XZ mea-surement with equal measurement strength, where thequantum state dynamics resembles diffusion on a sphere.The non-ideal case is considered in III B, using the per-turbative expansion of the path integral in terms of di-agrams to approximate the statistical averages. A com-plementary approach to this physics using the Fokker-Planck is demonstrated in Sec. IV. We compare our re-sults with the correlators constructed from the experi-ment data using superconducting circuits [28] in Sec. V,and conclude in Sec. VI. II. BACKGROUNDA. The stochastic master equation
We first discuss the stochastic master equation de-scribing the evolution of a qubit subject to simultaneousCWQM of two non-commuting observables. Any qubitobservable can be decomposed in terms of the Pauli ma-trices σ x , σ y and σ z . Thus, without loss of generality, weassume that the measured qubit observables are σ z and σ ϕ ≡ σ z cos ϕ + σ x sin ϕ. (1)The cases of ϕ = ϕ = π correspond to simultaneousmeasurement of two commuting observables with corre-lated and anti-correlated measurement results, respec-tively. For other angles ϕ , the two measured observablesdo not commute.Simultaneous continuous measurement of the observ-ables σ z and σ ϕ in the weak coupling regime, given a sys-tem state ρ ( t ) , produces the detector readouts [29, 31, 32] r z ( t ) and r ϕ ( t ) respectively, r z ( t ) = ∆ r z ( Tr [ σ z ρ ( t )] + √ τ z ξ z ( t )) ,r ϕ ( t ) = ∆ r ϕ ( Tr [ σ ϕ ρ ( t )] + √ τ ϕ ξ ϕ ( t )) , (2)where ∆ r z and ∆ r ϕ are the responses of the z - and ϕ -detectors, which we can rescale for simplicity so that∆ r z,ϕ =
2. The parameters τ z and τ ϕ are the detector“characteristic measurement times”, which are definedas the integration time necessary to obtain a SNR of onefor the time-averaged measurement readouts [33]. Thequantum efficiency for each measurement channel is de-fined as η i = /( i τ i ) , where i = z, ϕ and Γ z,ϕ are thetotal (ensemble averaged) dephasing rates for the twomeasurements. Ideal measurements correspond to theunit quantum efficiency η i =
1, whereas, non-ideal mea-surements correspond to the case of η i <
1. The noises ξ z ( t ) and ξ ϕ ( t ) are assumed uncorrelated Gaussian whitenoises with the two-time correlation functions ⟨ ξ z ( t ) ξ z ( t ′ )⟩ = ⟨ ξ ϕ ( t ) ξ ϕ ( t ′ )⟩ = δ ( t − t ′ ) . (3)The qubit trajectory is described by the stochastic masterequation (Itˆo interpretation [34]) [29, 31, 32]˙ x = − ( Γ z + Γ ϕ cos ϕ ) x + Γ ϕ sin 2 ϕ z + ( − x ) ξ ϕ sin ϕ √ τ ϕ − xz ( ξ ϕ cos ϕ √ τ ϕ + ξ z √ τ z ) , (4a)˙ y = − ( Γ z + Γ ϕ ) y − xy ξ ϕ sin ϕ √ τ ϕ − yz ( ξ ϕ cos ϕ √ τ ϕ + ξ z √ τ z ) , (4b)˙ z = − Γ ϕ sin ϕ z + Γ ϕ sin 2 ϕ x + ( − z ) ( ξ ϕ cos ϕ √ τ ϕ + ξ z √ τ z )− xz ξ ϕ sin ϕ √ τ ϕ , (4c)where we have used the Bloch parametrization of thequbit density matrix, ρ ( t ) = [ ˆ + x ( t ) σ x + y ( t ) σ z + z ( t ) σ z ]/
2, where ˆ ρ ens ( t ) , can be obtained from the above Itˆo equations bysimply dropping the noise terms, which takes the Lind-blad form˙ ρ ens = Γ z ( σ z ρ ens σ z − ρ ens ) + Γ ϕ ( σ ϕ ρ ens σ ϕ − ρ ens ) . (5)We may also include additional terms to Eqs. (4) and (5),in the usual way, to account for coherent evolution ofthe qubit due to a Hamiltonian, H q , and environmen-tal decoherence due to a weak coupling of the qubit tounmonitored degrees of freedom [29]. B. Stochastic path integral for XZ measurements
An alternative description of the stochasticity of thequbit trajectories can be obtained by writing the jointprobability density function, P , of the noises in themeasurement readouts and the quantum state trajec-tory at all times as a (stochastic) path integral. Thegeneral treatment of the stochastic path integral formal-ism for continuous weak measurements is discussed indetail in Refs. [30, 35]. Here, we consider an exam-ple for the XZ measurement, where the measured qubitobservables are σ x and σ z . The more general case ofarbitrary angle ϕ is briefly discussed in Sec. III C. Toconstruct such stochastic path integral, we need two in-gredients. One is the probability densities of the mea-surement readout noises at each time step, dt , and theother element is the deterministic evolution of the quan-tum state given particular values of the noises. Consider-ing a qubit measurement of duration T , which is dividedinto N steps of duration dt , the joint probability densityof the noises and the quantum states (at all times) isgiven by P = ∏ N − k = P ( q k + ∣ q k , ξ x,k , ξ z,k ) P ( ξ x,k ) P ( ξ z,k ) .The terms ξ x,k and ξ z,k are the values of the whitenoises at the time t k = t + k dt , with the probabil-ity density P ( ξ x,k ) = √ dt / π exp (− ξ x,k dt / ) . The vec-tor q k = { x k , y k , z k } denotes the Bloch vector at thattime. The transition probability P ( q k + ∣ q k , ξ x,k , ξ z,k ) = δ ( q k + − E[ q k , ξ x,k , ξ z,k ]) describes the deterministic evo-lution of the qubit state for given ξ x,k and ξ z,k .For the XZ measurement, the qubit evolution is dic-tated by the stochastic master equation (4) with ϕ = π / x = − Γ z x + ( − x ) ξ x √ τ x − xz ξ z √ τ z , (6a)˙ z = − Γ x z + ( − z ) ξ z √ τ z − xz ξ x √ τ x , (6b)where Γ x denotes the dephasing rate due to measurementof σ x . Because the Bloch y -coordinate does not appearin Eq. (6), we can disregard its evolution as long as we are interested in the qubit state evolution on the Bloch xz plane. This is indeed the case if the initial value ofsuch variable is y ( t = t ) =
0. Then, from Eq. (4b), y ( t ) = q k = { x k , z k } . The form of E[ q k , ξ z,k , ξ x,k ] is obtained bywriting Eq. (6) in an explicit time-discretized form, e.g.,˙ x = ( x k + − x k )/ dt = f x ( x k , z k , ξ x,k , ξ z,k ) .Following the outline presented in Refs. [30, 35],we write the delta functions for the deterministicevolution as Fourier integrals to express the jointprobability density of the noises and qubit state as P ({ q k } N , { ξ x,k } N − , { ξ z,k } N − ∣ q ) ∝ ∫ D p e −S . Notethat the latter depends on the initial qubit state, q , atthe time t . We refer to the exponent, S , as the stochas-tic action and ∫ D p ≡ ∫ i ∞− i ∞ ⋯ ∫ i ∞− i ∞ ∏ N − k = ( π ) − dp x,k dp z,k .Here, p k = { p x,k , p z,k } , introduced by the Fourier rep-resentation of the delta functions, are considered asauxiliary integration variables which are regarded aspure imaginary (so that S is real). Hereon, our nota-tion will be in the time-continuous form, i.e., p x ( t ) = lim dt → { p x,k } , for simplicity and we will use the time-discrete form whenever necessary. The stochastic action S for the XZ measurement of a qubit is given by S = ∫ Tt dt { p x ( t ) ˙ x ( t ) + p z ( t ) ˙ z ( t ) − H} , (7)where H = − Γ z p x ( t ) x ( t ) − Γ x p z ( t ) z ( t ) + H x + H z giventhat H x = √ τ x [( − x ) ξ x p x − xzξ x p z ] − ξ x , (8a) H z = √ τ z [( − z ) ξ z p z − xzξ z p x ] − ξ z , (8b)where the time argument of the noise and qubit variablesare omitted for simplicity.The joint probability density, P , mentioned abovecan be used to compute statistical averages of state-dependent quantities, A[ q ( t )] , which may depend on q ( t ) at various times; e.g., A[ q ( t )] = z ( t ) z ( t ) or x ( t ) z ( t ) where t , t are some intermediate times be-tween t and T . In particular, the conditional averages with fixed boundary conditions, q ( t ) = q ≡ q in and q ( T ) = q N ≡ q f , of the quantum trajectories can be cal-culated as follows q f ⟨A[ q ( t )]⟩ q in =∫ q ( T )= q f q ( t )= q in D q ∫ D ξ A[ q ( t )]P[ q ( t ) , ξ ( t )∣ q in ] P ( q f , T ∣ q in , t ) , (9)where the denominator P ( q f , T ∣ q in , t ) = ∫ q ( T )= q f q ( t )= q in D q ∫ D ξ P[ q ( t ) , ξ ( t )∣ q in ] , (10)is the conditional probability density to obtain the fi-nal state q f at time T given the initial state, q in ,at the time t . We use the notation ∫ q ( T )= q f q ( )= q in D q ≡∫ ⋯ ∫ ∏ N − k = dx k dz k , which implies integration over theintermediate qubit states except over the initial and fi-nal states, q and q N , respectively. Similarly, ∫ D ξ ≡∫ ⋯ ∫ ∏ N − k = dξ x,k dξ z,k . III. STATE CORRELATIONS OF JOINTLYMEASURED, NON-COMMUTINGOBSERVABLES
In this section we show how to calculate conditional av-erages with pre- and/or post-selection using the stochas-tic path integral formalism. We consider conditional av-erages of the form, e.g., q f ⟨ z ( t ) x ( t )⟩ q in . The latterrepresents the two-time quantum state correlator of thequbit z -coordinate at time t and the x -coordinate attime t , and the average is over quantum trajectorieswith initial state q ( t ) = q in and final state q ( T ) = q f .Conditional averages with only pre-selection can be ob-tained by taking the limit T → ∞ . We first consider thecase which is simple to treat analytically; namely, idealXZ measurement with detectors of equal measurementstrengths (Γ x = Γ z ). Then, we discuss a diagrammaticperturbation theory for non-ideal measurements. A. Conditional averages for ideal XZ measurement
We will assume that the initial state of the qubit atthe time t = xz great circle: x + z =
1; i.e., y ( ) =
0. From Eq. (4b),we notice that the y -coordinate will remain zero duringthe XZ measurement. Moreover, ideal measurements arecharacterized by the fact that pure states remain pureduring the measurement, even in the case of simultaneousweak measurement of non-commuting observables [27].Thus, for the considered ideal XZ measurement, thequantum trajectories can be parametrized by only thepolar coordinate, θ ( t ) , x ( t ) = sin θ ( t ) , y ( t ) = z ( t ) = cos θ ( t ) . (11)From Eqs. (4a) and (4c), we obtain the following equationfor the polar coordinate, θ ( t ) , in the Itˆo interpretation˙ θ ( t ) = ( Γ x − Γ z ) sin θ ( t ) cos θ ( t )− sin θ ( t ) ξ z ( t )√ τ z + cos θ ( t ) ξ x ( t )√ τ x , (12)where τ x = / x and τ z = / z . In particular, forthe case of interest of detectors of equal measurementstrengths, the above equation reduces to (Γ x = Γ z = Γ m )˙ θ ( t ) = ξ θ ( t )√ τ m , (13)where τ m = τ x = τ z is the measurement time ofboth measurement channels, and ξ θ ( t ) ≡ cos θ ( t ) ξ x ( t ) − sin θ ( t ) ξ z ( t ) can be regarded as a single Gaussian noiseterm with two-time correlation function: ⟨ ξ θ ( t ) ξ θ ( t ′ )⟩ = δ ( t − t ′ ) . Equation (13) describes free diffusion of thequbit state on the xz great circle.The joint probability density of the qubit state,parametrized by the angle θ ( t ) , and the effective noise, ξ θ ( t ) , can be obtained by following the steps discussed insection II B. We find P[ θ ( t ) , ξ θ ( t )] ∝ ∫ D p θ exp ( − S[ p θ ( t ) , θ ( t ) , ξ θ ( t )]) , (14)where the exponent is equal to (assuming that the initialtime is t = S = ∫ Tt = dt { ip θ ( t )[ ˙ θ ( t ) − τ − / ξ θ ( t )] + ξ θ ( t ) } , (15)and the auxiliary integration variables, p θ ( t ) , are real.From Eq. (9), we notice that to calculate conditional av-erages of state-dependent quantities only, A[ q ( t )] , it isconvenient to integrate out the noise ξ θ ( t ) and get P[ θ ( t )] ∝ ∫ D ξ θ P[ θ ( t ) , ξ θ ( t )] , (16) ∝ exp (− τ m ∫ T dt ˙ θ ( t )) , (17)which is the probability density functional for each real-ization of θ ( t ) , omitting a trivial proportional constant.In Eq. (17), the angle coordinate should be treated as acoordinate on the real axis; i.e., θ ∈ (−∞ , ∞) . Then, theconditional average, Eq. (9), for XZ measurements withdetectors of equal measurement strength, can be writtenas q f ⟨A[ q ( θ )]⟩ q in = ∑ n ∈ Z ∫ θ ( T )= θ f + πnθ ( )= θ in D θ A[ q ( θ )]P[ θ ( t )] P ( θ f , T ∣ θ in , ) , (18)where the initial state is q in = ( sin θ in , , cos θ in ) andthe final state is q f = ( sin θ f , , cos θ f ) ; such states areparametrized by the angles θ in , θ f ∈ [ , π ) . The sum inEq. (18) is over all angles, θ ( T ) , corresponding to thesame physical state q f . The denominator in Eq. (18) isthe transition probability, P ( θ f , T ∣ θ in , ) = ∑ n ∈ Z ∫ θ ( T )= θ f + πnθ ( )= θ in D θ P[ θ ( t )] . (19)We are interested in calculating the quantum state cor-relators with pre- and post-selection C zz ( t , t ∣ q f , T ; q in , ) ≡ q f ⟨ z ( t ) z ( t )⟩ q in , (20) C zx ( t , t ∣ q f , T ; q in , ) ≡ q f ⟨ z ( t ) x ( t )⟩ q in . (21)Because of the rotational symmetry, the correlator C xx ( t , t ∣ q f , T ; q in , ) can be obtained from the resultfor C zz by changing θ in → θ in − π / θ f → θ f − π / A s s [ q ( θ )] ≡ exp [ i ∫ T dt J s s ( t ) θ ( t )] (22)where J s s ( t ) = s δ ( t − t ) + s δ ( t − t ) can be regardedas a source field, see below, and s , s = ±
1. Then, thestate correlators of interest can be expressed as C zz ( t , t ∣ q f , T ; q in , ) = ∑ s ,s =± q f ⟨A s s [ q ( θ )]⟩ q in , (23) C xz ( t , t ∣ q f , T ; q in , ) = i ∑ s ,s =± s q f ⟨A s s [ q ( θ )]⟩ q in . (24)In Eq. (24), the coefficient s in each term of thesum arises because we are interested in the corre-lation of the x -coordinate at the time t , x ( t ) =∑ s =± s exp [ is θ ( t )]/ i , and the z -coordinate at thetime t , z ( t ) = ∑ s =± exp [ is θ ( t )]/ ∫ D θ P[ θ ( t )]A s s [ q ( θ )] , see also Eq. (18). Such aGaussian path integral can be straightforwardly calcu-lated [36]. We find ∫ θ ( T )= θ T θ ( )= θ D θ P[ θ ( t )]A s s [ q ( θ )] = ( τ m πT ) / × exp {− ∫ T dt [ τ m θ ( t ) − iJ s s ( t ) ¯ θ ( t )]} , (25)where ¯ θ ( t ) satisfies the (saddle-point) equation¨¯ θ = − iτ m J s s ( t ) , (26)with the boundary conditions ¯ θ ( ) = θ and ¯ θ ( T ) = θ T .The solution of Eq. (26) can be written in terms ofthe corresponding Green’s function, G ( t, t ′ ) , which satis-fies the equation ∂ t G ( t, t ′ ) = δ ( t − t ′ ) with homogeneousboundary conditions: G ( , t ′ ) = G ( T, t ′ ) = θ ( t ) = − iτ m [ s G ( t, t ) + s G ( t, t )] + θ T − θ T t + θ . (27)The Green’s function reads explicitly as G ( t, t ′ ) = ( t − t ′ ) Θ ( t − t ′ ) − ( − t ′ / T ) t, (28)where Θ (⋅) is the Heaviside step function. Note thatthe Green’s function is symmetric, G ( t, t ′ ) = G ( t ′ , t ) , and G ( t, t ) = − t ( − t / T ) . By inserting Eq. (27) into Eq. (25),we find that the sought path integral is given by ∫ θ ( T )= θ T θ ( )= θ D θ P[ θ ( t )]A s s [ q ( θ )] = √ τ m πT F s s ( t , t )× exp {− τ m ( θ T − θ ) T + i θ T − θ T ( s t + s t ) + i ( s + s ) θ } , (29) where F s s ( t , t ) ≡ exp {∑ a,b = , s a s b G ( t a , t b )/ τ m } isa coefficient independent of θ and θ T . Using the re-sult (29) in Eq. (18), we obtain q f ⟨A s s [ q ( θ )]⟩ q in = F s s ( t , t ) e iθ in ( s + s ) × ∑ n ∈ Z e −( ∆ θ + πn ) τ m / T + i ( ∆ θ + πn )( s t + s t )/ T ∑ n ∈ Z e −( ∆ θ + πn ) τ m / T , (30)where ∆ θ = θ f − θ in . From Eqs. (30) and (23)–(24), thecorrelators of interest can be found.The method discussed above to calculate two-timequantum state correlators can be easily generalized tocalculate any n -point correlation function of x or z . Insuch cases, we would need to introduce A s ...s n as a gen-eralization of Eq. (22) with a source field J s ...s n ( t ) =∑ nj = s j δ ( t − t j ) and s j = ±
1. The correlators of interestcan be written as, for instance, q f ⟨ x ( t ) z ( t ) x ( t )⟩ q in =∑ s s s s s q f ⟨A s s s ⟩ q in / ( i ) . Each term in the lat-ter sum is evaluated following a procedure similar to thecalculation of q f ⟨A s s ⟩ q in .
1. Conditional quantum correlators without post-selection
Thus far we have discussed quantum state correlatorsover sub-ensemble of quantum trajectories with pre- andpost-selected states. Now, we consider quantum statecorrelators without post-selection. Such correlators canbe obtained by taking the limit T → ∞ on the previousresults. Specifically, ⟨A s s [ q ( θ )]⟩ q in = lim T →∞ q f ⟨A s s [ q ( θ )]⟩ q in (31) = e −( s t + s t + s s t min )/ τ m + iθ in ( s + s ) , (32)where t min = min { t , t } . We obtain the result (32)by first applying Poisson’s resummation formulato the numerator and denominator in Eq. (30);for instance, ∑ n ∈ Z exp [−( ∆ θ + πn ) τ m / T ] =√ T / πτ m ∑ n ∈ Z exp [− n T / τ m + ∆ θni ] . Then, inthe limit T → ∞ , only the n = ⟨ z ( t ) z ( t )⟩ q in = e − t + t τ m { cos θ in cosh ( t min / τ m )+ sin θ in sinh ( t min / τ m )} , (33a) ⟨ z ( t ) x ( t )⟩ q in = e − t + t τ m e − t min τ m sin ( θ in )/ . (33b)Notice that the sign of the cross-correlation is determinedentirely by the initial angle. It simply indicates whetherthe ( x, z ) coordinates start out as of the same or differentsign to give either positive correlation (+ , +) , (− , −) ornegative correlation (+ , −) , (− , +) . 𝒒 in 𝒒 f 𝑥𝑧 𝑇 = 10𝜏 m (a) 𝑡 /𝜏 𝑚 𝐶 𝑥𝑥 (𝑡 , 𝑡 )𝐶 𝑧𝑧 (𝑡 , 𝑡 )𝐶 𝑧𝑥 (𝑡 , 𝑡 ) 𝑡 = 1.2𝜏 m (b) 𝑇 = 3.5𝜏 𝑚 𝐶 𝑎 𝑏 𝑡 , 𝑡 FIG. 1. Conditional averages for ideal XZ measurement.Panel (a) shows the qubit state averaged over a sub-ensembleof quantum trajectories, which begin at the initial pure state, q in = { cos ( π / ) , , sin ( π / )} at time t = q f = { sin ( π / ) , , cos ( π / )} at time t = T . Weshow the sub-ensemble average trajectories for three differentperiods T = τ m , 3 . τ m and 10 τ m . Solid lines depict our ana-lytical results, Eq. (34), and the crosses depict Monte Carlosimulation results. The dashed line, connecting the initialstate and the fully mixed state, shows the conventional ex-ponential decay of the (total) ensemble average state due to XZ measurement. In panel (b), solid lines show our analyt-ical results for the quantum state correlators, Eqs. (23)–(24)and (30). The boundary conditions on the quantum trajecto-ries are the same as in panel (a). The crosses represent MonteCarlo simulation results.
2. Qubit state averaged over sub-ensemble with fixed initialand final states
We know the general fact that measurement inducesexponential decay of the ensemble average qubit state.In particular, for the considered ideal XZ measure-ment, the Bloch state decays as x ens ( t ) = x in exp (− Γ m t ) , y ens ( t ) = y in exp (− m t ) and z ens ( t ) = z in exp (− Γ m t ) , seeEq. (5). An interesting question to discuss is how thesub-ensemble average state evolves from a fixed initialstate, q in , at time t to a fixed final state q f at time T .To answer the above question, we need to calculate q sub − ens − avg ( t ) = q f ⟨ q ( t )⟩ q in . For the considered ideal XZmeasurement, the components q f ⟨ z ( t )⟩ q in and q f ⟨ x ( t )⟩ q in can be calculated from the real and imaginary parts of q f ⟨ exp [ iθ ( t )]⟩ q in , which in turn can be obtained from theresult (30) with s =
0. We obtain q f ⟨ z ( t )⟩ q in = e − t ( − t / T )/ τ m × ∑ n ∈ Z e −( ∆ θ + πn ) τ m / T cos ( θ in + ( ∆ θ + πn ) t / T )∑ n ∈ Z e −( ∆ θ + πn ) τ m / T , q f ⟨ x ( t )⟩ q in = e − t ( − t / T )/ τ m × ∑ n ∈ Z e −( ∆ θ + πn ) τ m / T sin ( θ in + ( ∆ θ + πn ) t / T )∑ n ∈ Z e −( ∆ θ + πn ) τ m / T , (34)and q f ⟨ y ( t )⟩ q in = xz plane.In Fig. 1 we show some of the results obtained in thissection. Figure 1(a) depicts the sub-ensemble averagestate for various values of T with boundary conditions onthe quantum trajectories specified by the angles θ in = π / θ f = π /
8. We notice that for T = τ m , the aver-age q sub − ens − avg ( t ) first becomes mixed (i.e., states lyinginside the Bloch sphere) and approaches the fully mixedstate ( x = y = z =
0) at around t = T /
2, and then itunwinds itself such that the subensemble average statereaches the target pure state q f at the final time T . Thisturning behavior is less obvious for shorter post-selectiontime T = . τ m and T = τ m as the trajectories in thesubensemble do not have enough time to wander aroundthe Bloch sphere to contribute to the mixedness of thestate. We point out that this evolution is rather differentfrom the conventional exponential decay of the (total)ensemble average evolution induced by the XZ measure-ment (shown as the dashed line in Figure 1(a)). Fig-ure 1(b) shows the quantum state correlators for the idealXZ measurement with the same boundary conditions asin Fig. 1(a) and T = . τ m . Solid lines depict our ana-lytical formulas, Eqs. (23)–(24) and (30), and the crossesdepict Monte Carlo simulation results. The percentage ofsimulated trajectories which satisfy the boundary condi-tions are 0 . .
30% and 0 .
34% for T = τ m , . τ m and10 τ m , respectively. B. Perturbative solutions of conditional averagesfor non-ideal XZ measurements
In the case of non-ideal XZ measurements, where thereexist measurement inefficiencies, qubit energy relaxationand dephasing due to unwanted coupling with the envi-ronment, our knowledge about the qubit state comes inthe form of a mixed state. In order to describe its evolu-tion, we need to consider two coordinates; for example,the x and z Bloch coordinates. In this case, an ana-lytic solution for the correlators is not forthcoming, so weapply the stochastic path integral perturbatively to com-pute the conditional averages for the quantum state vari-ables. Following the method presented in Refs. [30, 37],
Type Labels of vertices Value DiagramsInitial x in , z in x in , z in p x ξ x , p z ξ z √ τ x , √ τ z p z zxξ x , p x x ξ x , p x xzξ z , p z z ξ z − √ τ x , − √ τ z TABLE I. Different possible vertices and associated diagrams for correlation functions of x and z in joint continuous XZmeasurement. We note the different measurement times τ x,z go with the appropriate x or z diagram vertex or propagator. the stochastic action discussed in Section II B, Eqs. (7)–(8), is first rearranged into a free action and an interac-tion action, S = S F + S I , where, S F = ∫ T dt { ip x ( ˙ x + Γ z x ) + ip z ( ˙ z + Γ x z ) + ξ x + ξ z } , (35) S I = ∫ T dt [{ ip x xz − ip z ( − z )} ξ z √ τ z + {− ip x ( − x ) + ip z xz } ξ x √ τ x ] . (36)The free action includes only the bilinear terms in x, z, p x , p z , ξ x and ξ z variables and therefore can berewritten in terms of the free Green’s function, e.g., S F = i ∑ a = x,z ∫ dtdt ′ p a ( t ) G − a ( t, t ′ ) a ( t ′ ) , where a = x, z .The rest of the terms in the action define the interactionaction S I . The free Green’s functions for the variables x and z are given by G x ( t, t ′ ) = exp {− Γ z ( t − t ′ )} Θ ( t − t ′ ) ,G z ( t, t ′ ) = exp {− Γ x ( t − t ′ )} Θ ( t − t ′ ) . (37)The function Θ ( t ) is a left continuous Heaviside stepfunction (Θ ( ) = t → + Θ ( t ) =
1, see Ref. [30]).The Green’s functions for the noise terms in the free ac-tion are simply the delta functions G ξ ( t, t ′ ) = δ ( t − t ′ ) ,for both noises ξ x and ξ z . The perturbative expansioncomes from expanding the exponential, e S I , in powers of S I , where one can construct diagrammatic rules to keeptrack of nonvanishing terms. In the diagrammatic expan-sion, the terms in the interaction action S I determine ver-tices, whereas, the Green’s functions determine propaga-tors. The vertices and propagators are shown in Table I.We follow the diagrammatic rules explained in full detailin Ref. [30] and note that the type of expansion presentedin the reference is similar to the loop expansion in quan-tum theory. However, one can show that, in this case,the order of the expansion can be equivalently controlledby a small noise parameter which is related to the mea-surement efficiency η x,z . Even though the measurementefficiency is not shown explicitly in the equations we usedso far, they are actually contained in the definition of thecharacteristic measurement time τ x,z = /( x,z η x,z ) .We illustrate the diagrammatic approach by comput-ing the correlators of this type: C AB ( t , t ∣ q in , ) ≡ ⟨ A ( t ) B ( t )⟩ q in , where A, B are any two of the Blochsphere coordinates x, z and the conditioning is only onthe initial state q in = { x in , z in } at time t =
0. We notethat conditioning on both initial and final states is alsopossible, however the diagrammatic rules are far morecomplicated and we are not considering that case here.We begin with the cross-correlation function, C zx ( t , t ∣ q in , ) = ⟨ z ( t ) x ( t )⟩ q in = x t x in z t z in + x t zz t x in z in + x t xz t x in z in + x t zz t z in z in x in z in + x t xz t x in z in x in x in , (38)showing how the ending vertices z t and x t can be con-nected to other vertices in Table I only up to 0 loops(tree-level diagrams). Importantly, we note that thenoise propagators (shown as wavy lines) can only con-nect vertices with the same noise flavors, as a result ofthe independence of the two noises ξ x and ξ z . The math-ematical expressions corresponding to the diagrams aregiven by C zx ( t , t ∣ q in , ) = − ( τ z + τ x ) z in x in × ∫ T dt ′ G z ( t , t ′ ) G x ( t , t ′ ) G x ( t ′ , t ) G z ( t ′ , t )+ x in z τ z ∫ T dt ′ G z ( t , t ′ ) G x ( t , t ′ ) G z ( t ′ , t ) G x ( t ′ , t )+ x z in τ x ∫ T dt ′ G z ( t , t ′ ) G x ( t , t ′ ) G z ( t ′ , t ) G x ( t ′ , t ) . (39)We evaluate above integrals and obtain C zx = e − Γ x t − Γ z t { − x in z in ( Γ z η z + Γ x η x ) t min + x in z Γ z η z Γ x ( − e − x t min ) + z in x Γ x η x Γ z ( − e − z t min ) } . (40)Other examples are the self-correlators C zz and C xx . Thediagrams for the z - z correlator conditioning on an initialstate are given by C zz ( t , t ∣ q in , ) = ⟨ z ( t ) z ( t )⟩ q in = z t z in z t z in + z t zz t + z t zz t z in z in + z t zz t z in z in + z t zz t z in z in z in z in + z t xz t x in z in x in z in , (41)which can be evaluated to give C zz = e − Γ x ( t + t ) {− z η z z τ z t min + Γ z η z Γ x ( e x t min − )+ x z Γ x η x Γ z ( − e − z t min ) + z Γ z η z Γ x ( − e − x t min )} . (42)For the other self-correlator C xx , the calculation is in thesame way, with the replacements x in ↔ z in , and z ↔ x .Figure 2 shows the comparison between our analyt-ical results from the diagrammatic perturbation the-ory and Monte Carlo simulations. We show theresults for the covariance and variance functions ofthe Bloch coordinates, defined as Cov [ A ( t ) B ( t )] ≡⟨ A ( t ) B ( t )⟩ q in − ⟨ A ( t )⟩ q in ⟨ B ( t )⟩ q in and Var [ A ( t )] ≡⟨[ A ( t ) − ⟨ A ( t )⟩ q in ] ⟩ q in , respectively. We consider twocases: measurement efficiencies η x = η z = . η x,z are small, corresponding to the smallnoise limit where the noisy trajectories are not too faraway from its averaged trajectory [30]. C. Application to general cases withnon-commuting observables σ z and σ ϕ The analysis of the perturbative diagrams for XZ mea-surement can be generalized to the case of simultaneousand continuous measurement of σ z and σ ϕ with arbitrary ++++++++++++++++++++++++++++++++++++++++++×××××××××××××××××××××××××××××××××××××××××× ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ t = τ m - C o v a r i an c e ××××××××××××××××××××××××××××××××++++++++++++++++++++++++++++++++ V a r i an c e + Var [ z ( t )] × Var [ x ( t )] ( a ) + Cov [ z ( t ) z ( t )] × Cov [ x ( t ) x ( t )] ◦ Cov [ z ( t ) x ( t )] η = / τ m ++++++++++++++++++++++++++++++++++++++++++×××××××××××××××××××××××××××××××××××××××××× ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ t = τ m - C o v a r i an c e ××××××××××××××××××××××××××××××××++++++++++++++++++++++++++++++++ V a r i an c e + Var [ z ( t )] × Var [ x ( t )] ( b ) + Cov [ z ( t ) z ( t )] × Cov [ x ( t ) x ( t )] ◦ Cov [ z ( t ) x ( t )] η = / τ m t / τ m FIG. 2. Covariance and variance functions of the Bloch co-ordinates for non-ideal XZ measurements. The analytical re-sults obtained from the diagrammatic perturbative expansionare presented in solid curves and the Monte Carlo simulationresults are presented with markers. The results from the di-agrammatic expansion agree with the numerical data muchbetter for the case of small measurement quantum efficiency η = .
05, as shown in panel (b), than for the case with η = . η x = η z = η ). angle, ϕ , between the measurement axes. The stochas-tic equations for this case are given in Eq. (4) (consideronly equations for x and z ), and we can use them to con-struct a stochastic path integral with a stochastic action,which can be separated into free and interaction parts,˜ S = ˜ S F + ˜ S I . We notice that the terms in the free ac-tion for XZ measurement, Eq. (35), apart from the noiseterms, are of the form: p a ( ˙ a + Γ a ) where a = x, z , leadingto the Green’s functions of Eq. (37). In order to be ableto use the diagrammatic rules of XZ measurement casealso in the case of arbitrary ϕ , we need to keep the samestructure of the free action in both cases. We do this bytransforming the variables x, z in the stochastic masterequations (4) to a new set of variables u, v that diagonal-izes the linear parts of the evolution equations. That is,we find eigenvectors and eigenvalues of the decoherencematrix M , M = (−( Γ z + Γ ϕ cos ϕ ) Γ ϕ sin 2 ϕ ϕ sin 2 ϕ − Γ ϕ sin ϕ ) . (43)The eigenvalues of the decoherence matrix are λ + and λ − , λ ± = − Γ z − Γ ϕ ± √ Ξ2 , (44)where Ξ = Γ ϕ + Γ z + ϕ Γ z cos ( ϕ ) . The stochastic dif-ferential equations after the variable transformation areof the form ˙ u = λ − u + κ zu ξ z + κ xu ξ ϕ + α x u ξ ϕ + α x uvξ ϕ + α z u ξ z + α z uvξ z , (45a)˙ v = λ + v + κ zv ξ z + κ xv ξ ϕ + α x v ξ ϕ + α x uvξ ϕ + α z v ξ z + α z uvξ z , (45b)where the α - and κ -coefficients are functions of Γ z , Γ ϕ and ϕ . The final form of the free action, constructedfrom these stochastic master equations, is given by ˜ S F =∫ T dt { ip u ( ˙ u + λ − u ) + ip v ( ˙ v + λ + v ) + ξ ϕ / + ξ z / } which issimilar to Eq. (35). Therefore, we can apply the samediagrammatic rules used in the XZ measurement case andonly need to consider the new set of vertices built fromthe rest of the terms in the interaction action [those termsproportional to the α - and κ -coefficients in Eq. (45)]. IV. CONDITIONAL AVERAGES USING THEFOKKER-PLANCK EQUATION
In this section we will use the Fokker-Planck equa-tion for the transition probability, P ( q , t ∣ q ′ , t ′ ) , to cal-culate the conditional quantum state correlators definedin Eqs. (23)–(24). To find such correlators, we introducethe two-sided joint probability density W for the quan-tum state to reach the states q at time t and q at time t ≥ t given that the initial state is q in at time t andthe final state is q f at time T , W ( q f , T ∣ q , t ; q , t ∣ q in , t ) = P ( q f , T ∣ q , t ) P ( q , t ∣ q , t ) P ( q , t ∣ q in , t ) P ( q f , T ∣ q in , t ) . (46)The transition probability P ( q , t ∣ q ′ , t ′ ) , with t > t ′ , canbe obtained from the Fokker-Planck equation associatedto the Itˆo equations (4a)–(4c). In particular, such transi-tion probability can be easily calculated analytically forthe ideal XZ measurement case with detectors of equalmeasurement strengths. As in section III, we assumethat the initial state is such that y ( ) = y ( t ) = θ ( t ) , which wewill use it to denote the Bloch state q = { sin θ, , cos θ } .The Fokker-Planck equation associated to Eq. (13) readsas ∂ t P ( θ, t ∣ θ ′ , t ′ ) = D∂ θ P ( θ, t ∣ θ ′ , t ′ ) , (47)where the diffusion coefficient is D = ( τ m ) − and the ini-tial condition is P ( θ, t ′ ∣ θ ′ , t ′ ) = δ ( θ − θ ′ ) . The solution of 𝜃/𝜋 𝑊 ( 𝜃 f , 𝑇 𝜃 , 𝑡 𝜃 i n , ) 𝑡 = 0.5𝜏 m m m m m 𝜃 in = 𝜋/4 𝜃 f = 7𝜋/ 𝑇 = 10𝜏 m FIG. 3. Two-sided probability distribution of the polar coor-dinate, θ , at the time t . Such distribution takes into accountonly quantum trajectories with fixed initial and final states,parametrized by the angles θ in = π / θ f = π /
8, at thetimes t = T = τ m , respectively. The distribution isshown for various times: t = . τ m , 2 . τ m , 5 τ m , 7 . τ m and9 . τ m . The circles represent Monte Carlo simulation results. Eq. (47) is given by P ( θ, t ∣ θ ′ , t ′ ) = ( π ) − ∑ n ∈ Z exp [ in ( θ − θ ′ ) − Dn ( t − t ′ )] . The conditional average quantities, q f ⟨A s s [ θ ( t )]⟩ q in , in Eqs. (23)–(24) are now expressedin terms of the two-sided joint probability density as q f ⟨A s s [ q ( θ )]⟩ q in = ∫ π dθ ∫ π dθ e is θ + is θ × P ( θ f , T ∣ θ , t ) P ( θ , t ∣ θ , t ) P ( θ , t ∣ θ in , t ) P ( θ f , T ∣ θ in , t ) . (48)It can be shown that Eq. (48) leads to the same resultas Eq. (30). To show this, we evaluate the integrals inEq. (48) and obtain (assuming t = q f ⟨A s s [ q ( θ )]⟩ q in = e iθ in ( s + s )− D ( s t + s t + s s t ) ×∑ n ∈ Z e − Dn T +[ ∆ θ − D ( s t + s t ) i ] ni ∑ n ∈ Z e − Dn T + ∆ θni . (49)We then obtain the result (30) after applying Poisson’sresummation formula to the sums in the numerator anddenominator in Eq. (49).The Fokker-Planck approach also enables us to findanalytically the two-sided probability density W of θ atthe time t in terms of the transition probability, obtainedabove, W ( θ f , T ∣ θ, t ∣ θ in , t ) = P ( θ f , T ∣ θ, t ) P ( θ, t ∣ θ in , t ) P ( θ f , T ∣ θ in , t ) . (50)Figure 3 shows the above distribution for various timesbetween t = T = τ m , and θ in = π / θ f = π / θ in , then it tends to flatten out until the time t = T / t > T /
2, thedistribution recovers a Gaussian-like shape centered atthe angle θ f . From Fig. 3, it is easy to understand the0 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦
0. 1.0 2.0 3.0 4.000.20.4 C o v a r i an c e ( a ) t = τ m + Cov [ z ( t ) z ( t )] × Cov [ x ( t ) x ( t )] ◦ Cov [ z ( t ) x ( t )] t / τ m + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×
0. 1.0 2.0 3.0 4.000.20.40.6 V a r i an c e ( b ) × Var [ x ( t )] + Var [ z ( t )] t / τ m FIG. 4. Comparison of theory and experiment for quantumstate correlators: (a) covariances and (b) variances, for thequbit under the simultaneous measurement of σ x and σ z . Thecorrelators derived from experimental qubit trajectories areshown in solid curves, whereas the Monte-Carlo simulationresults are shown with markers, showing excellent agreementwith the experimental data. As expected from Fig. 2, thefirst order perturbative solutions, shown in dashed gray, whilegiving the correct qualitative behaviour, deviate from theexperimental results in the moderately high efficiency limit( η z = .
54 and η x = . behavior exhibited by the sub-ensemble average state, q sub − ens − avg ( t ) , shown in Fig. 1(a). Specifically, the de-cay of q sub − ens − avg toward the center of the Bloch sphereis due to the fact that the distribution, W , becomes in-creasingly flat until the time t = T / V. QUANTUM STATE CORRELATORS FORTHE TRANSMON QUBIT EXPERIMENT
We now discuss quantum state correlators for the ex-periment described in Ref. [28]. In this experiment, si-multaneous measurement of qubit observables σ z and σ x is implemented on a qubit rotating at the Rabi frequency,Ω R = π ⋅ xz Bloch plane at a frequency Ω rf close to Ω R [29].The measurement is a stroboscopic measurement wherethe measured qubit is an effective qubit living in the ro-tating frame defined by the frequency Ω rf . As discussedin Ref. [29], the effective qubit is subject to slow Rabioscillations on the xz plane because of the frequency mis- match: ˜Ω R = Ω R − Ω rf ≈ π ⋅ H q = ̵ h ˜Ω R σ y /
2. Moreover, the T -and T -relaxation processes of the physical qubit inducean effective depolarization channel on the effective qubiton the xz plane, with a decay rate γ = ( T − + T − )/ T = µ s and T = µ s [29]. The residual Rabioscillations and the xz -depolarization induces the qubitevolution as described by˙ x = − γx + ˜Ω R z (51a)˙ z = − γz − ˜Ω R x. (51b)assuming no measurement occurred. Therefore, to in-clude these effects to our theoretical model presented inSection II A, the terms on the right side of the aboveequations can be simply added to the right side ofEqs. (4).However, since the experimental measurement read-outs have finite integration time ( dt = t + dt is obtained from a state at thetime t by means of the composed mapping: ρ ( t + dt ) =L env ○M[ ρ ( t )] . Here, the operation M[ ρ ] = M x ○M z [ ρ ] represents the evolution due to measurement of σ z and σ x during the time period dt (neglecting errors of order dt due to the non-commutativity of the measured oper-ators). The evolution due to measurement of σ z alone isdefined as (M z [ ρ ]) ij = ( M z ρM † z ) ij Tr [ M z ρM † z ] exp (− γ ij dt ) (52)where M z = ( πτ z / dt ) − / exp [−( r z ( t ) − σ z ) dt / τ z ] andthe matrix ˆ γ has vanishing diagonal elements ( γ = γ =
0) and off-diagonal elements equal to γ = γ = Γ z − / τ z . Note that ˆ γ vanishes in the case of idealmeasurements. The state update Eq. (52) represents thequantum Bayesian update for a non-ideal measurementof σ z . The Bayesian update for σ x -measurement is sim-ilar to the former one. First, we rotate the state alongthe y -axis by − π / ρ → exp ( iπσ y / ) ρ exp (− iπσ y / ) ],we apply the Bayesian update defined in Eq. (52), andthen we rotate back the state along the y -axis by π / τ z = /( η z Γ z ) = . µ s and τ x = /( η x Γ x ) = . µ s(given that the measurement-induced dephasing rates areΓ z = Γ x = / . µ s). Another operation L env [ ρ ] rep-resents the evolution due to environmental decoherencewhich in this case is obtained by evolving Eq. (51a) overthe time step, dt . In our case, the environment deco-herence includes the effect of the quantum efficiencies η z = . , η x = .
41. In the experiment, the initial Blochvector is q in = { sin ( π / ) , , cos ( π / )} , and the total num-ber of readout traces is approximately 2 × for eachmeasurement channel. After calculating the quantumtrajectories, we calculate the quantum correlators with1only the pre-selection condition. The results are shownin Fig. 4.Figure 4(a) shows the comparison between state cor-relators obtained from recorded readouts, Monte-Carlosimulations, and the perturbation theory discussed insection III B. In the Monte-Carlo simulations, we use thesame set of parameters used in constructing the qubit tra-jectories from the recorded readouts. We find excellentagreement between the simulation results and the resultsbased on the recorded data. The perturbative results arestill able to capture the correct trend of the correlators,cf. Fig. 2(a) even as a first order result. We note that bykeeping higher order terms in the diagrammatic expan-sion will definitely improve the quantitative agreement.Fig. 4(b) shows the variances of the Bloch coordinates x and z . Here, we also find good agreement between thenumerical simulations and the results obtained from therecorded readouts. VI. CONCLUSIONS
We have investigated the conditional averages and tem-poral correlation functions of the qubit state evolutionunder simultaneous non-commuting observable measure-ment. We consider the joint measurement of the non-commuting pseudo-spin observables σ x and σ z . Underthese continuous measurements, the qubit state trajecto-ries in time are described by the stochastic master equa-tion, and the associated stochastic path integral. In theideal quantum limit measurement case of equal measure- ment rates, closed-form solutions of any multi-time cor-relation function can be found using the stochastic pathintegral formalism, even when conditioning on the ini-tial and final states. We also presented a complimentarymethod for finding the conditional correlators using theFokker-Planck equation. For the non-ideal measurementcase, where the measurement is inefficient and environ-ment dephasing is present, the conditional correlators canbe obtained perturbatively using the diagrammatic ap-proach. We have shown that the perturbative results,to first order in the efficiency, are in good agreementwith the numerically simulated trajectories in the smallquantum efficiency regime. Most importantly, we havecompared the results of this theoretical analysis with thequbit trajectory data inferred and tomographically ver-ified from a superconducting circuit coupled to multi-mode cavity, and have found excellent agreement betweenthe correlation functions of the experimental data andour theoretical treatment. ACKNOWLEDGMENTS
This work was supported by US Army Research Of-fice Grant No. W911NF-15-1-0496, by the National Sci-ence Foundation grant DMR-1506081, and the Temple-ton Foundation grant ID 58558. LM was supportedby the National Science Foundation Graduate Fellow-ship Grant No. 1106400 and the Berkeley Fellowship forGraduate Study.*A.C. and J.A. contributed equally to this work. [1] K. Kraus, A. B¨ohm, J. D. Dollard, and W. H. Wootters,