SSelf-Stabilizing Measurements for Noisy Metrology
Sai Vinjanampathy ∗ Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543.
We present a protocol to perform self-stabilizing measurements on noisy qubits. We employ rapid purificationin a rotating frame whose frequency is estimated and periodically updated via a Bayesian estimation scheme.The Bayesian estimation protocol employs the continuous measurement record to improve the estimate, whichin turn purifies the qubit more. This procedure stabilizes the qubit. Such an adaptive measurement schemeserves the purpose of purifying the state, while minimally interfering with the phase estimation.
INTRODUCTION
Parameter estimation in noise environments is the key chal-lenge for the practical realization of quantum metrology [1–6]. Depending on the choice of input states and Hamiltonians,quantum metrological schemes have demonstrated an advan-tage over classical metrological schemes [6]. The figure ofmerit to judge the goodness of a metrological scheme is thevariance of the estimated phase, denoted by ∆ ϕ est . One ex-pects that larger the number of particles N that are involved inacquiring the unknown phase, the smaller the phase varianceshould be. Furthermore, increasing the number of times ν thatthe measurement is repeated is also expected to decrease theestimated phase variance. This intuition is seen to be true,for instance from the quantum Cram´er-Rao bound [7–9] thatstates that the variance of unbiassed estimators of a parameter ϕ scales as ∆ ϕ est ≥ ν F Q . (1)Here, F Q stands for the quantum Fisher information (QFI) andis related to N . For instance, if coherent light ∣ α ⟩ is used in aninterferometric scheme with the Hamiltonian ϕ ˆ n being linearin the number operator ˆ n , F Q can be as big as ∣ α ∣ , the averagenumber of photons in the coherent state. On the other hand,quadratic scaling of QFI for linear Hamiltonian interactionshas been studied theoretically [4, 10, 11] and experimentally[12] (also see [11]). Two problems remain in the way of theimplementation of realistic quantum metrology in the pres-ence of noise. The first is a theoretical challenge concern-ing asymptotic theoretical bounds and the second challengeinvolves quantum metrology in the presence of decoheringenvironments. An asymptotic bound on the estimated phasevariance is placed by the quantum Cram´er-Rao bound notedabove. There has been some work to address realistic boundsin for finite number of measurements [13, 14]. If the parame-ter is acquired by the action of a Hamiltonian G, the quantumFisher information defined as F Q = ∑ j , k ( λ j − λ k ) λ j + λ k ∣⟨ Ψ j ∣ G ∣ Ψ k ⟩∣ . (2)This formula [8] involves the instantaneous eigenstates ofthe density operator {∣ Ψ j ⟩} and the corresponding eigenvaues { λ j } . Since we wish to decrease ∆ϕ est , the asymptotic anal-ysis indicates that increasing Fisher information will reduce ∆ϕ .To this end, we note three properties of F Q : firstly since F Q ∝ G, to increase quantum Fisher information, it suffices toincrease the Hamiltonian strength. Let this strength be repre-sented by the norm of the Hamiltonian ∥ G ∥ . But this involvesincreasing the energy needed to implement the Hamiltonian,which is undesirable. We hence set ∥ G ∥ =
1, corresponding toinvestigating schemes that have the same (fixed) Hamiltonianstrength. Secondly, we note that since F Q is convex, pure stateprobes are always better than mixed state probes. Thirdly, wenote that for a single qubit state ρ = ( I + r . σ )/
2, Eq.(2) sug-gests that the Hamiltonian that maximizes the quantum Fisherinformation is perpendicular to r , namely G = r ⊥ . σ . Since theHamiltonian is in general not controllable, we will, withoutloss of generalization, take the Hamiltonian to be σ x . Moti-vated by these three criteria, in this paper, we will propose amethod to perform self-stabilizing phase measurements on adecohering qubit.The scheme consists of actively purifying the qubit as itacquires an unknown phase. We employ continuous mea-surement feedback control [15, 16] to implement the scheme.Though we are motivated by the formula for quantum Fisherinformation in Eq.(2) to derive the three desired criterion fora “good” feedback controlled metrological scheme, we willnot employ quantum Fisher information as the figure of merit.This is due to the fact that quantum Fisher information isan asymptotic bound. The continuous measurements used toimplement the control scheme are represented in terms of astochastic master equation, namely, d ρ = − i ¯ h [ ϕ G , ρ ] dt + ∑ j = γ j D[ σ j ] ρ dt + D[ c ] ρ dt + √ η H[ c ] ρ d W . (3)While the first term in the equation above represents theHamiltonian evolution with the unknown phase, the next termrepresents decoherence of the qubit represented by the ac-tion of three Pauli operators with damping factors γ j . Here, D[ c ] ρ ∶= c ρ c † − ( c † c ρ + ρ c † c )/ H[ c ] ρ ∶= c ρ + ρ c † − ⟨ c + c † ⟩ ρ corresponds tothe information gain due to the measurement. η representsthe measurement’s detector efficiency, η = d W is a Wiener increment[17, 18] given by zero mean and ⟪ d W ⟫ = dt . The measure- a r X i v : . [ qu a n t - ph ] M a y ment record for this process can be written as dy ( t ) = ⟨ c + c † ⟩ dt + d W √ η . (4)Such a continuous measurement and control of a quantum sys-tem has been studied and demonstrated in a variety of physicalsystems including quantum dots [19], nano mechanics [20–22], circuit quantum electrodynamics (CQED) [23–26] andcavity quantum electrodynamics [27]. RAPID PURIFICATION AND SELF STABILIZATION
Our protocol will involve purifying a qubit that is decoher-ing as it gathers information about the unknown phase by im-proving its purity. If we were interested not in phase estima-tion, but in simply purification, efficient algorithms to purifyusing continuous measurement quantum control already exist.In particular, several authors have investigated the purificationspeed arising from rapid purification protocols [28–31]. Suchprotocols aim to purify a qubit using Hamiltonian feedbackand continuous measurements as quickly as possible. The Ja-cobs protocol involves an adaptive measurement scheme sothat the measurement is always perpendicular to the state. Theevolution of the linear entropy S L = − tr ( ρ ) for the evolutionof a qubit subject to an adaptive measurement is given by dS L = − ( r . dr dt + r . dr d W ) . (5)Here, we have written d ρ = dr . σ dt + dr . σ d W for brevity.From Eq.(3), it is clear that dr depends entirely on the choiceof the measurement operator. If this operator is chosen to beperpendicular to the instantaneous Bloch vector r , then theevolution of the linear entropy is deterministic and entirelydictated by dr . In the absence of decoherence ( i.e., γ i = c = √ κ / X ,the evolution of the linear entropyis given by [28] dS L = − κ tr [ X ρ X ρ ] dt . (6)This is solved to yield S L ( t ) = S L ( ) exp (− κ tr [ X ρ X ρ ] t ) , pu-rifying the qubit rapidly.If the phase ϕ were known, the rapid purification proto-col could be implemented in a rotating basis. But, since ϕ is the unknown phase we wish to measure, our scheme willinvolve implementing rapid purification in a rotating frame,whose frequency is the estimated phase ϕ est . To estimate thephase, we will employ a Bayesian parameter inference fromcontinuously monitored systems discussed in the next section.Every m cycles, the measurement record is used to perform aBayesian update and the updated estimator is used to calcu-late the average position of the density matrix for the next m cycles of measurements. This allows us to perform rapidpurification in the rotating frame of the estimated phase. Atthe beginning of the protocol, since the prior probability den-sity is assumed to be flat, corresponding to the absence of any knowledge about the unknown phase ϕ . We hence choosea fixed axis (the axis we prepared the state in) and performmeasurements perpendicular to that axis for the first m cycles.Since we do not wish to interfere with the Hamiltonian, wewill indeed pick an axis that is mutually perpendicular to thequbit state and the Hamiltonian at any given time. At the endof that block, we estimate the unknown phase by computingP [ ϕ ∣ y ( t )] , the conditional probability given the measurementrecord y ( t ) . If the corresponding variance ∆ϕ est is less than agiven tolerance ε , another block of simulations and measure-ments are performed. An alternative stopping criterion for thisprotocol involves a predetermined total number of steps. Thismight be suitable if γ j are especially strong causing the qubitto eventually decohere completely. Note that though the firstblock of measurements has the effect of not purifying the statein general, the data obtained via static measurements will aidin the implementation of a rapid purification scheme. Witheach block of evolution, the Bayesian estimate [32] of the un-known phase ϕ set will get closer to the true phase, causing thenext block of simulated adaptive measurements to be closerto the “ideal” adaptive measurement. This procedure hencehas the effect of purifying the qubit and making the variancein the estimated phase smaller with each passing block. Thesteps of the protocol can be summarized as follows:1. On the first block, simulate m cycles of the state evolu-tion with a static measurement operator, σ z .2. Estimate the phase at the end of the first block.3. Use the estimated phase to compute the average trajec-tory for the next block. Use this to identify the mea-surement vectors that are mutually perpendicular to Gand ρ .4. Repeat previous step until ∆ϕ est < ε .In the next section, we will take up the task of estimating theunknown phase. BAYESIAN ESTIMATION FROM A CONTINUOUSRECORD
The central task at the end of each block of evolution of theprotocol outlined in the previous section is the estimation ofan unknown phase ϕ , given a measurement record y ( t ) [33,34]. This issue was studied in [34] and is summarized in thissection. Baye’s law applied to the measurement record y ( t ) states that P [ ϕ ∣ y ( t )] = P [ y ( t )∣ ϕ ] P [ ϕ ] P [ y ( t )] . (7)Here P [ ϕ ∣ y ( t )] represents the conditional probability densityfor the parameter ϕ , given the data y ( t ) , P [ ϕ ] representsthe prior probability distribution of the unknown parameter ϕ and P [ y ( t )] = ∫ d ϕ P [ y ( t )∣ ϕ ] P [ ϕ ] . A log likelihood func-tion l ( ϕ ∣ y ( t )) = log ( L [ ϕ ∣ y ( t )]) can be defined in terms of thelikelihood function, given by L [ ϕ ∣ y ( t )] = P [ ϕ ∣ y ( t )] P [ ϕ ] . (8)Here P [ ϕ ] is a convenient choice of normalization. While theprobability distribution P [ y ( t )∣ ϕ ] informs us about the proba-bility of generating a measurement record for a given param-eter, likelihood functions inform us of the opposite: the like-lihood of a parameter given a measurement record. To applythis to continuous measurements, we first note that the effectof the measurement operator outcome x on the state can bewritten as ρ ∣ x = Ω ( x ) ρΩ † ( x ) p ( x ) , (9)where the probability of observing this outcome is given by p x = tr [ Ω ( x ) ρΩ † ( x )] . The probability operators Ω † ( x ) Ω ( x ) are normalized as ∫ dx Ω † ( x ) Ω ( x ) = I . (10)Furthermore, introducing an “ostensible probability” p ( x ) ,the authors in [34] define a new set of POVMS, namely Ω ( x ) → Ω ( x )/√ p ( x ) so that the normalization conditionabove is modified to ∫ dxp ( x ) Ω † ( x ) Ω ( x ) = I . (11)This allows us to define a new set of states ˜ ρ ∣ x = Ω ( x ) ρΩ † ( x ) ,whose trace now depends on p ( x ) . The role of p ( x ) dx isto provide a reference measure on the set of measurementoutcomes. Note that the trace of ˜ ρ ∣ x now explicitly dependson the measurement record and does not change its depen-dance on ϕ for various measurement outcomes. Hence, it waspointed out that it can serve as a good likelihood function.Hence, at a given time t , this analysis leads to the likelihood function being defined as L ( t ) = tr { ˜ ρ ( t )} . Here ˜ ρ ∣ x at the time t is written as ˜ ρ ( t ) for brevity. L ( t ) obeys the evolution equa-tion dL ( t ) = tr {H[ c ] ˜ ρ ( t )} dy ( t ) (12)Returning to the protocol described in the previous section, astatic (time-independant) measurement operator c = σ z is cho-sen for the first block of evolution. The continuous monitoringof the decohering qubit (assumed to decohere under thermalLindbladians at a temperature β − ) is simulated and the corre-sponding measurement record dy ( t ) is employed to update thelikelihood function. At the end of the first block of evolution,the unknown phase is estimated as ϕ est = ∫ d ϕϕ P [ ϕ ∣ y ( t )] , (13)where P [ ϕ ∣ y ( t )] , the probability density is given by P [ ϕ ∣ y ( t )] = L [ y ( t )∣ ϕ ] P [ ϕ ]∫ d ϕ L [ y ( t )∣ ϕ ] P [ ϕ ] . (14)Furthermore, the variance is estimated directly from the prob-ability density P [ ϕ ∣ y ( t )] as ∆ ϕ est = ∫ d ϕ { ϕ − ϕ est } P [ ϕ ∣ y ( t )] , (15)Now, given ϕ set , the next cycle of measurement directions issimulated to implement rapid purification over the next blockof evolution. Since we cannot know the precise trajectoryof the future blocks of evolution, we have to use the aver-age equation to simulate the evolution of the Bloch vector.Since this will only approximately be the “correct” feedbackscheme (both due to ∆ϕ set and due to the average equation),repeated cycles of estimation and evolution might be needed.We simulate such an evolution in the next section. RESULTS AND DISCUSSION
We consider a qubit undergoing evolution nuder a thermalLindbladian, namely d ρ = − i ¯ h [ ϕ G , ρ ] dt + γ ¯ n D[ σ + ] ρ dt + γ ( + ¯ n )D[ σ − ] ρ dt + D[ c ] ρ dt + √ η H[ c ] ρ d W . (16)Here ¯ n is the average number of thermal phonons in the qubitat thermal equilibrium. Fig.(1) shows the simulation of threecycles of evolution for realistic damping factors derived for agood CQED qubit. The damping corresponds to parametersgiven in [25]. The simulation reveals that for good control,where the measurement strength κ was much larger than thequbit damping rates, the Bayesian estimator is able to esti- mate the unknown phase within a couple of cycles. In con-trast, in the regime of bad control, when κ and γ are compara-ble, the Bayesian estimator needs more blocks to estimate theunknown phase. This is represented in Fig(2). We note thatthe feedback scheme could be implemented by employing fastcontrol via field programmable gate arrays for CQED devices.The essential part of this control scheme is the choice of ap- FIG. 1. Self-stabilizing phase measurements in the regime of goodcontrol: The qubit decoherence rates are two orders of magnitudesmaller than the measurement strength. In this regime, if the qubitmeasurement were to have a preferred direction, the qubit would un-dergo zeno-like dynamics. In this regime, it is seen that the unknownphase is estimated correctly by the Bayesian estimator discussed inthe text.FIG. 2. Self-stabilizing phase measurements in the regime of badcontrol: The qubit decoherence rates are each comparable to mea-surement strength. In this regime, the qubit suffers strong decoher-ence and several cycles are needed to estimate the unknown phase. proximately unbiassed measurements to rapidly purify a qubitin a rotating frame which is being estimated by a Bayesian es-timator. From the standpoint of implementation, we can alsointroduce an additional delay between the end of the blockand the implementation of the feedback loop. This wouldcorrespond to waiting for the feedback loop to be computed,a task that might not be slower than the ultrafast dynamicsof qubit implementations such as CQED. In this article, wehave demonstrated active feedback stabilization of a qubit de-cohering while acquiring an unknown phase. We employedcontinuous measurements to stabilize the qubit. By usingBayesian estimation in conjunction with rapid purification, wehave simulated a qubit that is stabilized in the regime of goodcontrol. Finally, whether there exist local measurement and feedback schemes capable of self-stable metrology not withsingle qubit state, but entangled states, is an open question.Centre for Quantum Technologies is a Research Centre ofExcellence funded by the Ministry of Education and the Na-tional Research Foundation of Singapore. ∗ [email protected][1] V. Giovannetti, S. Lloyd, and L. Maccone, Nature Photonics ,222 (2011).[2] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. ,010401 (2006).[3] J. K. Stockton, J. Geremia, A. C. Doherty, and H. Mabuchi,Physical Review A , 032109 (2004).[4] R. Demkowicz-Dobrzanski, U. Dorner, B. Smith, J. Lundeen,W. Wasilewski, K. Banaszek, and I. Walmsley, Physical Re-view A , 013825 (2009).[5] K. Modi, H. Cable, M. Williamson, and V. Vedral, PhysicalReview X , 021022 (2011).[6] J. P. Dowling, Contemporary physics , 125 (2008).[7] S. L. Braunstein and C. M. Caves, Physical Review Letters ,3439 (1994).[8] S. Luo, Letters in Mathematical Physics , 243 (2000).[9] M. G. Paris, International Journal of Quantum Information ,125 (2009).[10] P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick,S. D. Huver, H. Lee, and J. P. Dowling, Physical review letters , 103602 (2010).[11] R. Demkowicz-Dobrza´nski, J. Kołody´nski, and M. Gut¸˘a, Na-ture communications , 1063 (2012).[12] M. Napolitano, M. Koschorreck, B. Dubost, N. Behbood,R. Sewell, and M. W. Mitchell, Nature , 486 (2011).[13] M. Tsang, Physical review letters , 230401 (2012).[14] Y. Gao and H. Lee, Journal of Physics A: Mathematical andTheoretical , 415306 (2012).[15] K. Jacobs and D. A. Steck, Contemporary Physics , 279(2006).[16] H. M. Wiseman and G. J. Milburn, Quantum measurement andcontrol (Cambridge University Press, 2010).[17] K. Jacobs,
Stochastic processes for physicists: understandingnoisy systems (Cambridge University Press, 2010).[18] C. W. Gardiner et al. , Handbook of stochastic methods , Vol. 3(Springer Berlin, 1985).[19] A. N. Korotkov, Physical Review B , 5737 (1999).[20] P. A. Truitt, J. B. Hertzberg, C. Huang, K. L. Ekinci, and K. C.Schwab, Nano letters , 120 (2007).[21] J. Suh, M. D. LaHaye, P. M. Echternach, K. C. Schwab, andM. L. Roukes, Nano letters , 3990 (2010).[22] K. Jacobs, J. Finn, S. Vinjanampathy, et al. , Physical Review.A (2011).[23] A. Blais, R.-S. Huang, A. Wallraff, S. Girvin, and R. J.Schoelkopf, Physical Review A , 062320 (2004).[24] I. Siddiqi, R. Vijay, F. Pierre, C. Wilson, M. Metcalfe,C. Rigetti, L. Frunzio, and M. Devoret, Physical review letters , 207002 (2004).[25] R. Vijay, C. Macklin, D. Slichter, S. Weber, K. Murch, R. Naik,A. N. Korotkov, and I. Siddiqi, Nature , 77 (2012).[26] J. R. Friedman, V. Patel, W. Chen, S. Tolpygo, and J. E. Lukens,nature , 43 (2000).[27] M. Brune, S. Haroche, J. Raimond, L. Davidovich, and N. Za-gury, Physical Review A , 5193 (1992). [28] J. Combes and K. Jacobs, Physical review letters , 010504(2006).[29] J. Combes, H. M. Wiseman, and K. Jacobs, Physical ReviewLetters , 160503 (2008).[30] H. M. Wiseman and J. Ralph, New Journal of Physics , 90(2006).[31] J. Combes, H. M. Wiseman, and A. J. Scott, Physical ReviewA , 020301 (2010). [32] E. T. Jaynes, Probability theory: the logic of science (Cam-bridge university press, 2003).[33] J. Geremia, J. K. Stockton, A. C. Doherty, and H. Mabuchi,Physical review letters , 250801 (2003).[34] S. Gammelmark and K. Mølmer, Physical Review A87