Dissipative Adiabatic Measurements: Beating the Quantum Cramér-Rao Bound

PHYSICAL REVIEW RESEARCH 2, 023418 (2020)

Dissipative adiabatic measurements: Beating the quantum Cramér-Rao bound

Da-Jian Zhang and Jiangbin Gong * Department of Physics, National University of Singapore, Singapore 117542

(Received 4 February 2020; accepted 11 June 2020; published 30 June 2020)

It is challenged only recently that the precision attainable in any measurement of a physical parameter is fundamentally limited by the quantum Cramér-Rao Bound (QCRB). Here, targeting at measuring parameters in strongly dissipative systems, we propose an innovative measurement scheme called dissipative adiabatic measurement and theoretically show that it can beat the QCRB. Unlike projective measurements, our measurement scheme, though consuming more time, does not collapse the measured state and, more importantly, yields the expectation value of an observable as its measurement outcome, which is directly connected to the parameter of interest. Such a direct connection allows us to extract the value of the parameter from the measurement outcomes in a straightforward manner, with no fundamental limitation on precision, in principle. Our ﬁndings not only provide a marked insight into quantum metrology but also are highly useful in dissipative quantum information processing.

DOI: 10.1103/PhysRevResearch.2.023418

I. INTRODUCTION measurements (PMs), a DAM therefore does not collapse the measured state, namely, the steady state. Moreover, as shown Improving precision of quantum measurements underlies below, its outcome is the expectation value of the measured both technological and scientific progress. Yet, it has never observable in the steady state, up to some fluctuations due to been doubted until recently [1] that the precision attainable in position uncertainties in the initial state of the apparatus. any quantum measurement of a physical parameter is bounded Aided by a simple yet well-known model, we show that by the inverse of the quantum Fisher information (QFI) mul- the QCRB can be beaten by DAMs although it does hold for tiplied by the number of the measurements in use [2]. Such a (ideal) PMs. So, despite the long-time requirement in DAMs, bound, known as the celebrated quantum Cramér-Rao Bound their ability of beating the QCRB comes as a surprise and (QCRB), has been deemed as the ultimate precision allowed constitutes a significant advancement in our understanding by quantum mechanics that cannot be surpassed under any of quantum metrology. Indeed, the outcomes of DAMs are circumstances [3,4]. As such, ever since its inception in 1967 directly connected to some parameter of interest whereas [5], the QCRB has been the cornerstone of quantum estima- this is never the case for PMs. Such a direct connection tion theory underpinning virtually all aspects of research in allows us to extract the value of the parameter from the quantum metrology [6–9]. outcomes in a straightforward manner, in principle, without In this paper, inspired by Aharonov and Vaidman’s adi- any fundamental limitation on precision. To solve the conflict abatic measurements [10], we propose an innovative mea- between our results and the QCRB, we resort to Ref. [1] surement scheme tailored for strongly dissipative systems. which shows that the QCRB relies on a previously overlooked Measurements based on our scheme are coined dissipative assumption. This assumption can be severely violated by adiabatic measurements (DAMs). The system to be measured DAMs. Our work therefore provides an intriguing measure- here is a strongly dissipative system initially prepared in its ment scheme that can surpass the so-called ultimate precision steady state and then coupled to a measuring apparatus via limit. an extremely weak but long-time interaction [see Fig. 1(a)]. Apart from being of fundamental interest, our results are The dynamics of the coupling procedure is dominated by the important from an application perspective as well. On one dissipative process, which continuously projects the system hand, DAMs allow for detecting steady states without per- into the steady state. Such a dissipation-induced “quantum turbing them. This somewhat exotic feature could be highly Zeno effect” [11] effectively decouples the system from the useful in dissipative quantum information processing [12–32], apparatus in the long-time limit, suppressing the so-called where steady states are typically entangled states or some quantum back action of measurements [6]. Unlike projective other desirable states. On the other hand, as shown below, decoherence and dissipation play an integral part in DAMs. As a consequence, if the effects of decoherence and dissipation *[email protected] become increasingly strong, the efficiency of DAMs would increase instead of declining. This theoretical view, verified Published by the American Physical Society under the terms of the by numerical simulations below, suggests the fascinating pos- Creative Commons Attribution 4.0 International license. Further sibility that decoherence and dissipation may be exploited distribution of this work must maintain attribution to the author(s) for good rather than causing detrimental effects in quantum and the published article’s title, journal citation, and DOI. measurements.

2643-1564/2020/2(2)/023418(14) 023418-1 Published by the American Physical Society DA-JIAN ZHANG AND JIANGBIN GONG PHYSICAL REVIEW RESEARCH 2, 023418 (2020)

(a) at the QCRB var(θˆ ) [NH(θ )]−1.Astheθ-independence Environment assumption has never been doubted until recent work [1], it has been believed for a long time that the QCRB is universally valid for all kinds of measurements and, therefore, repre- System sents the ultimate precision allowed by quantum mechanics. Moreover, the optimal POVM (asymptotically) saturating this bound has been shown to be the PM associated with the observable Aopt = θ1 + Lθ /H(θ )[4]. The proof of the QCRB is revisited in Appendix A. Shortly, we will show that DAMs violate the θ-independence assumption and can be exploited (b) (c) to beat the QCRB. To better digest the above knowledge, we may take the generalized amplitude damping process [33]asanexam- d ρ = ple. It is described by the Lindblad equation, dt (t ) 1 Lθ ρ Lθ ρ = γ θ σ−ρσ+ − {σ+σ−,ρ} + − (t ), where : [ ( 2 ) (1 1 θ σ+ρσ− − {σ−σ+,ρ} )( 2 )] is a Liouvillian superoperator depending on the parameter θ ∈ (0, 1), with γ denoting the decay rate, σ− =|01|, and σ+ =|10|. For this dissipative process, there is a unique steady state ρθ = diag(θ,1 − θ ), which can be approached if the evolution time is much longer FIG. 1. Schematic diagrams. (a) The setup: A dissipative system than the relaxation time of the process. Our purpose here coupled to a measuring apparatus. The coupling is extremely strong is to infer the value of θ from N repeated measurements but instantaneous for PMs whereas it is very weak but of long on ρθ . Considering that θ is a monotone function of the duration for DAMs. (b) An ideal PM corresponds to the strong temperature of environment [33], estimation of θ offers one coupling and short-time limit, with eigenvalues ai of the measured means of temperature estimation that has received much observable A as its outcomes. In contrast, an ideal DAM corresponds attention recently in quantum thermometry [34–36]. Using to the weak coupling and long-time limit, with the expectation ρ = θ, − θ = 1 , − 1 θ diag( 1 ), we have Lθ diag( θ −θ ) and fur- value Aθ as its outcome. Outcomes from both DAMs and PMs 1 θ = 1 θˆ in their ideal cases are still subject to fluctuations (represented by ther obtain H( ) θ(1−θ ) . So, the QCRB reads var( ) the shaded spread in the plotted probability distribution) originating θ(1 − θ )/N. On the other hand, direct calculations show that from position uncertainties in the initial state of the apparatus. (c) In Aopt =|00|. Clearly, the PM associated with Aopt has two practice, the coupling strength and coupling time are finite, causing potential outcomes 1 and 0, with the probabilities θ and some deviations from ideal cases for both PMs and DAMs. 1 − θ, respectively. Intuitively, one may think of 1 and 0 as the head and tail of a coin, with θ corresponding to the coin’s propensity to land heads. Performing N such PMs, we II. PRELIMINARIES PM,..., PM PM ∈{ , } can obtain a series of data, x1 xN , with xi 1 0 . θ θˆ PM,..., PM = We start with some preliminaries. Given a family of Then the value of can be estimated as, (x1 xN ): ρ N PM/ quantum states θ characterized by an unknown parameter i=1 xi N, amounting to the frequency of 1 appearing in θ, a basic task in quantum metrology is to estimate θ as the data. Using well-known results regarding N-trial coin flip precisely as possible by using N repeated measurements. Any experiments, we have measurement can be described by a positive operator-valued var(θˆ ) = θ(1 − θ )/N, (1) measure (POVM) x satisfying dx x = 1, with x labeling its outcome. To obtain an estimate of θ, one can input the indicating that the PM associated with A already reaches the ,..., opt outcomes x1 xN of N measurements into an estimator QCRB. In passing, given the same amount of resources, e.g., θˆ ,..., (x1 xN ), which is a map from the set of outcomes to the the number of measurements, one may wonder whether the parameter space. Upon optimizing over all unbiased estima- estimation error can be further reduced, e.g., by measuring a tors for a given measurement, one reaches the classical CCRB (possibly entangled) state other than ρθ . In Appendix B,we − var(θˆ ) [NF(θ )] 1. Here var(θˆ ) is the variance of θˆ and prove that the answer is negative if the QCRB is valid. To 2 F (θ ) = dxpθ (x)[∂θ ln pθ (x)] is the classical Fisher infor- simplify our paper as much as we can, throughout this paper, = ρ mation (CFI), with pθ (x) tr( x θ ) denoting the conditional we use the above simple example to demonstrate our findings, probability density of getting outcome x when the actual value although our findings are generally applicable to dissipative of the parameter is θ. To find the best precision, one needs to systems. further optimize the CFI F (θ ) over all possible measurements . In doing this, it has been implicitly assumed that is x x III. DISSIPATIVE ADIABATIC MEASUREMENT independent of θ [3–5]. Under this assumption, referred to as the θ-independence assumption hereafter, one can prove Keeping this example in mind, we proceed to develop 2 that F (θ ) H(θ ), with H(θ ) = tr(ρθ Lθ ) denoting the QFI DAMs. Suppose that we are given a dissipative system S expressed in terms of the symmetric logarithmic derivative with a Liouvillian superoperator Lθ .Herewetakeθ to be 1 θ ∂θ ρθ = θ ρθ + Lθ L , i.e., the Hermitian operator satisfying 2 (L a single parameter for simplicity. is assumed to be such ρθ Lθ ). Substituting F (θ ) H(θ ) into the CCRB, one arrives that (a) it admits a unique steady state ρθ and (b) the nonzero

023418-2 DISSIPATIVE ADIABATIC MEASUREMENTS: BEATING … PHYSICAL REVIEW RESEARCH 2, 023418 (2020) eigenvalues λh (h > 0) of Lθ have negative real parts, that g(t )dt = 1. Note that we have let g(t ) = 1/T for simplicity. is, there is a dissipative gap := minh>0 |Re(λh )| in the In PMs, this term is impulsive, that is, g(t ) = 1/T takes an Liouvillian spectrum. S is initially prepared in its steady state extremely large value but only for a very short time interval. ρ θ θ . Note that this is achievable even though is unknown, as So, the dominating term in Eq. (2)isHI . In the strong coupling S automatically approaches ρθ because of the dissipative gap and short-time limit, i.e., T → 0, the associated dynamical [12–32]. To simplify our discussion, we adopt the minimal map reads model of measurement that has been used time and again −iA⊗pˆ iA⊗pˆ lim ET (ρθ ⊗|φφ|) = e ρθ ⊗|φφ|e . (6) in the literature (see Appendix C for more details). That is, T →0 to measure an observable A, we add an interaction term, − S A H = T 1A ⊗ pˆ, coupling S to a measuring apparatus A, with Clearly, this map creates correlations between and ,giving I S coordinate and momentum denoted byx ˆ andp ˆ, respectively. rise to the quantum back action that the configuration of A Here, T is a positive real number, which, in the spirit of the after the measurement is determined by the outcome of . adiabatic theorem, will be eventually sent to infinity [10]. As Contrary to PMs, DAMs exploit the opposite limit of an →∞ usual, we are not interested in the dynamics of the apparatus extremely weak but long time interaction, i.e., T .For L itself and assume its free Hamiltonian to be zero [10]. In real this, the dissipative term θ dominates the interaction term situations, this could be achieved by switching to a rotating HI in Eq. (2). The former effectively eliminates correlations S frame. The dynamics of the coupling procedure is described created by the latter through continuously projecting into ρ by the equation (¯h = 1): its steady state θ . Resulted from this nontrivial interplay is the decoupling of S and A in the long-time limit, which d inhibits the state of S from any change or collapse. Such a ρ(t ) = Lθ ρ(t ) − i[HI ,ρ(t )] =: Lρ(t ). (2) dt mechanism is suggestive of the quantum Zeno effect, making If the coupling time is T (so the product of the coupling DAMs distinct from PMs in nature. strength, i.e., 1/T , and the coupling time is unity), S + A Now let us turn to the practical situation with the coupling LT undergoes the dynamical map, ET := e , transforming the strength and the coupling time being finite. In a PM, under the L initial state ρθ ⊗|φφ| to the state ET (ρθ ⊗|φφ|) at time influence of θ , i.e., decoherence and dissipation, the evolved T . Here, |φ denotes the initial state of A, which is set to be state ET (ρθ ⊗|φφ|) deviates from the ideal correlated state a Gaussian centered at x = 0. After the coupling procedure, given by Eq. (6). To ensure that such deviations are small the coordinatex ˆ is observed to determine the reading of the enough, it has to be required that HI is sufficiently strong pointer. The above minimal model can be implemented in a so decoherence and dissipation are comparatively weak. Ev- number of experimental setups, such as in cavity quantum idently, such a strong interaction requirement would be in- electrodynamics [37,38] and circuit quantum electrodynamics creasingly difficult to meet if decoherence and dissipation be- [39,40]. come increasingly strong. This is consistent with our intuition To figure out the effect of ET , we make use of the fact that decoherence and dissipation are detrimental in quantum L(ρ ⊗|pp |) = (Lp,p ρ) ⊗|pp |, with Lp,p ρ := Lθ ρ − measurements. Contrary to this understanding, DAMs appre- iT −1(pAρ − pρA). Here, |p denotes the eigenstate ofp ˆ, i.e., ciate decoherence and dissipation as useful resources because L , T pˆ|p=p|p. This leads to ET (ρθ ⊗|pp |) = (e p p ρθ ) ⊗ they are responsible for eliminating the system-apparatus |pp|. Note that a technical result in Ref. [41]is correlations. Certainly, in a nonideal DAM due to a finite T , the evolved state also deviates from the ideal decoupled L T L T e p,p Pθ − e p,p Pθ =O(1/T ), (3) state in Eq. (5)[seeFig.1(c)]. Nevertheless, the stronger L T L T decoherence and dissipation are, the more efficient the cor- indicating that e p,p Pθ gets closer and closer to e p,p Pθ as relation elimination is and, therefore, the shorter the coupling T approaches infinity [42]. Here, L , := Pθ L , Pθ , and Pθ p p p p time required to execute a DAM. That is, unlike the strong denotes the projection, Pθ (X ):= (trS X )ρθ , mapping an arbi- interaction requirement in PMs, the long-time requirement in trary operator X into ρθ .UsingEq.(3) and noting that the ex- −1 DAMs would be increasingly easy to meet if decoherence and plicit expression of L , reads L , =−iT (p − p )Aθ Pθ , p p p p dissipation are increasingly strong. This point can be placed where Aθ := tr(Aρθ ), we obtain on more solid ground by virtue of perturbation theory [43]. −i(p−p )Aθ lim E (ρθ ⊗|pp |) = ρθ ⊗ e |pp |. (4) To illustrate the above point, we come back to the forego- →∞ T T ing example. Here, γ represents the strength of decoherence Now, expressing |φ in ET (ρθ ⊗|φφ|)as|φ= and dissipation; more precisely, the dissipative gap = γ/2. φ(p)|pdp and using the linearity of the map ET as Note that the deviations in PM and DAM from their ideal well as Eq. (4), we arrive at the main formula of this paper: results can be quantified by the measures

−iAθ pˆ iAθ pˆ lim E (ρθ ⊗|φφ|) = ρθ ⊗ e |φφ|e . (5) E ρ ⊗|φφ| − E ρ ⊗|φφ| →∞ T T ( θ ) lim T ( θ ) (7) T T →0 Formula Eq. (5) shows that in the weak coupling and long time and limit, the steady state ρθ does not collapse and the pointer is shifted by the expectation value rather than eigenvalues of A E (ρθ ⊗|φφ|) − lim E (ρθ ⊗|φφ|), (8) T →∞ T [see Fig. 1(b)]. T To gain physical insight into the above result, we compare respectively. Figure 2 shows these two measures for different DAMs with PMs. Both DAMs and PMs utilize interaction γ , where we set, without loss of generality, A = Aopt =|00|, terms of the form HI = g(t )A ⊗ pˆ, with g(t ) normalized to θ = 1/2, and σ = 1/5, with σ denoting the standard deviation

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PM (a) (b) the probability density of getting xi in the (ideal) PM is (see Appendix F)

2 θ (xPM−1)2 − θ xPM PM − i 1 − i = √ 2σ 2 + √ 2σ 2 . pθ xi e e (11) 2πσ2 2πσ2 Equation (11) is, as a matter of fact, a continuous probability distribution but can be treated as the discrete probability distribution giving rise to Eq. (1) in the limit of σ → 0 (see Appendix F). In contrast to Eq. (1), var(θˆ )inEq.(10) approaches zero as σ → 0, indicating that there is no fundamental limitation on precision in the DAM. Here we point out that var(θˆ )inEq.(10) is, as a matter of fact, limited by the the standard deviation of the normal distribution Eq. (9), however, this is not a fundamental limitation but a limitation stemming from nonideal experimental conditions. Furthermore, it can be FIG. 2. Numerical results of the measures in Eqs. (7)and(8)for shown that the POVM operator associated with the DAM is − (xDAM −θ )2 (a) PM and (b) DAM with different γ . √ 1 2 DAM = e 2σ 1 (see Appendix F). Evidently, it vi- x 2πσ2 olates the θ-independence assumption, which is the technical of the Gaussian |φ. The characteristic scales of the physical reason for the DAM to be able to beat the QCRB. quantities used in Fig. 2 are chosen with a realistic model in Lastly, it may be instructive to give a simple physical mind [44]. As can be easily seen from Fig. 2(a), the slope picture for comprehending the result that the QCRB can be of Eq. (7) as a function of T increases as γ increases. This beaten by the DAM. To do this, suppose that we are given indicates that 1/T has to be increasingly large in PM for PM DAM PM only one datum, xi or xi . Roughly speaking, xi is either achieving a certain desired tolerance of the deviations. In 1 or 0, which is unrelated to θ. Thus, there is no way for contrast, as shown in Fig. 2(b), the slope of Eq. (8) as a func- θ PM σ us to accurately infer from xi . However, for a small , tion of 1/T decreases as γ increases, implying that T can be DAM = θ xi takes a value equal to or close to Aopt θ , which increasingly small in DAM for achieving the same tolerance. is directly related to θ. Based on this direct relationship, one For instance, if the desired tolerance is set to be 0.01, 1/T DAM can obtain a fairly good estimate from xi . In this sense, in PM are 162, 483, 802, 1119 MHz whereas T in DAM are DAM PM xi is more informative than xi . To further conﬁrm this 78.7, 26.6, 16.0, 11.5 μs, for γ = 5, 15, 25, 35 MHz, respec- point, one can work out their CFI. The CFI associated with tively. More numerical results revealing effects arising from Eq. (9) reads F DAM (θ ) = 1/σ 2, whereas that associated with an intermediate coupling time are presented in Appendix D. Eq. (11) satisﬁes F PM(θ ) H(θ ), no matter how small σ is Besides, we have examined one of the experimental se- (see Appendix F). That is, F PM(θ ) is bounded by the QFI but tups [44] used to implement the minimal model of mea- F DAM (θ ) is unbounded in principle. surement adopted here and obtained analogous results (see Appendix E). V. CONCLUDING REMARKS

IV. BEATING THE QUANTUM CRAMÉR-RAO BOUND Before concluding, we make a few remarks. Equation (10) is nothing but the CCRB for the normal distribution Eq. (9). Having proposed DAMs, we now show that they can beat Our finding here is that the CCRB Eq. (10) can be infinitely the QCRB. Imagine that the PM used in the foregoing ex- small in principle and, in particular, can be smaller than =| | ample is replaced by the DAM associated with Aopt 0 0 . the QCRB Eq. (1). This clearly demonstrates that there is Analogous to N PMs, N such DAMs produce a series of data, no intrinsic limitation on precision, which is contrary to the DAM = ,..., DAM xi , i 1 N, where xi denotes the outcome of the quantum Cramér-Rao theorem [3,4]. On the other hand, the ith DAM. As before, these data can be further input into the steady state in our example is a classical state and, therefore, θˆ DAM,..., DAM = N DAM/ estimator, (x1 xN ) i=1 xi N, to obtain an does not display any coherent nature. We emphasize that θ DAM estimate of . Note that the probability density of getting xi the reason of choosing this example is due to its simplic- in the (ideal) DAM is ity. Our measurement is certainly capable of revealing the (xDAM −θ )2 coherent nature of a quantum system as long as the steady DAM 1 − i = √ 2σ 2 , pθ (xi ) e (9) state in question is genuinely quantum. Any system studied 2πσ2 in Refs. [14,16–18,22–25,28–31], as far as we can see, may which can be verified by using Eq. (5) and not- serve as one illustrative example. = θ ing that Aopt θ (see Appendix F). It follows that In conclusion, we have proposed an innovative measure- θˆ DAM,..., DAM (x1 xN ) is an unbiased estimator with ment tailored for strongly dissipative systems. In our measurement, the dissipative dynamics continuously eliminates var(θˆ ) = σ 2/N. (10) correlations created by the weak system-apparatus interaction, For the sake of comparing Eq. (10) with Eq. (1), we resulting in the decoupling of the system from the apparatus stress that Eq. (1) is, strictly speaking, obtained under the in the long-time limit. Unlike PMs, our measurement there- limiting condition that the PM is ideal and σ → 0. Indeed, fore does not collapse the measured state and, moreover, its

023418-4 DISSIPATIVE ADIABATIC MEASUREMENTS: BEATING … PHYSICAL REVIEW RESEARCH 2, 023418 (2020) outcome is the expectation value of an observable in the indicating that (δθ)2 is simply the variance adopted in the steady state, which is directly connected to the parameter of main text, interest. By virtue of this, our measurement is able to beat the δθ 2= θˆ , QCRB, without suffering from any fundamental limitation on ( ) var( ) (A6) precision. These findings, solidified by a simple but delightful for an unbiased estimator. example, provide a revolutionary insight into quantum metrol- To estimate θ as precisely as possible, one needs to op- ogy. We highlight that our measurement works in a state- timize the estimation error (δθ)2 over all estimators and protective fashion and embraces decoherence and dissipation measurements. This is exactly the way that the QCRB was as useful resources. Such exotic features could be highly proven, which can be stated via the following two steps [3,4]. useful in dissipative quantum information processing. The first step is to optimize (δθ)2 over all estimators for a given measurement. This leads to the CCRB, ACKNOWLEDGMENTS (δθ)2 [NF(θ )]−1, (A7) D.-J.Z. thanks Dianmin Tong, Ya Xiao, Xizheng Zhang, and Sen Mu for helpful discussions. J.G. is supported by where F (θ ) is the CFI defined as the Singapore NRF Grant No. NRF-NRFI2017-04 (WBS No. ∂ 2 R-144-000-378-281). D.-J.Z. acknowledges support from the F (θ ) = dx pθ (x) ln pθ (x) National Natural Science Foundation of China through Grant ∂θ No. 11705105 before he joined NUS. 1 ∂ 2 = dx pθ (x) . (A8) pθ (x) ∂θ APPENDIX A: REVISITING THE PROOF OF THE QCRB The CCRB has been widely used in classical estimation Here, focusing on the single-parameter case, we revisit the theory. Its proof can be found in many articles (e.g., Ref. [3]) proof of the QCRB [3–5]. Suppose that one is given many and is definitely correct. We omit it here. The second step is to ρ copies of a quantum state θ characterized by an unknown optimize F (θ ) over all measurements to get the QCRB. This θ θ parameter and wishes to estimate as precisely as possible step is based on the following equality [3,4]: by using N repeated measurements. In general, a quantum ∂ ∂ measurement can be described by a POVM{x}. Here, x is pθ (x) = tr x ρθ , (A9) a positive-semidefinite operator satisfying dxx = 1, with ∂θ ∂θ 1 denoting the identity operator. x labels the “results” of the θ measurement. Although written here as a single continuous which is valid if x satisfies the -independence assumption, real variable, x can be discrete or multivariate. Accordingly, as can be seen from Eq. (A3). the outcomes of N repeated measurements can be expressed To get the QCRB from Eq. (A9) (see also Ref. [4]), one needs to introduce the symmetric logarithmic dDerivative, as x1, x2,...,xN . To extract the value of θ from these data, which is defined as the Hermitian operator Lθ satisfying one can resort to an estimator, θˆ (x1,...,xN ), which is a map from the data x1,...,xN to the parameter space. Lθ ρθ + ρθ Lθ ∂ρθ ,..., = . (A10) Given a set of outcomes x1 xN , one can obtain an 2 ∂θ estimate θˆ (x1,...,xN )ofθ. The deviation of the estimate from the true value of θ can be quantified by [3] Substituting Eq. (A10) into Eq. (A9) yields θˆ ,..., ∂ (x1 xN ) pθ (x) = [tr(ρθ xLθ )]. (A11) δθ(x1,...,xN ):= − θ. (A1) ∂θ |dθˆ/dθ| Inserting Eq. (A11) into Eq. (A8) and using Eq. (A3), one has Here, θˆ denotes the statistical average of θˆ (x1,...,xN ) over 2 potential outcomes x ,...,x , [tr(ρθ xLθ )] 1 N F (θ ) = dx , (A12) tr(xρθ ) θˆ= dx1 ···dxN pθ (x1) ···pθ (xN )θˆ (x1,...,xN ), (A2) which further leads to 2 where ρθ Lθ θ tr(√ x ) = ρ . F ( ) dx pθ (x) tr( x θ ) (A3) tr(xρθ ) √ The derivative dθˆ/dθ appearing in Eq. (A1)isusedto √ 2 ρθ x √ remove the local difference in the “units” of the estimate and = dxtr √ Lθ ρθ . (A13) ρ x the parameter [3]. The estimation error can be then defined as tr( x θ ) [3] Using the Cauchy-Schwarz inequality |tr(X †Y )|2 (δθ)2, (A4) tr(X †X )tr(Y †Y ) for two matrices X and Y , one can deduce δθ2 ,..., from Eq. (A13) that i.e., the statistical average of (x1 xN ) over potential outcomes x ,...,x . In particular, if θˆ (x ,...,x ) is unbi- 1 N 1 N θ ρ = ρ = ρ 2 . ased, i.e., θˆ=θ, there is F ( ) dxtr( xLθ θ Lθ ) tr(Lθ θ Lθ ) tr θ Lθ

δθ(x1,...,xN ) = θˆ (x1,...,xN ) −θˆ, (A5) (A14)

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So, the CFI is bounded from above by the so-called QFI: x = (x1,...,xN ). Therefore, a general scheme of quantum 2 metrology consists of three ingredients: an input state of N F (θ ) H(θ ):= tr ρθ L . (A15) θ probes, a measurement at the output, and an inference rule. Substituting Eq. (A15) into Eq. (A7), one arrives at the In the following, we do not impose any restriction on these QCRB: ing3redients. That is, the input state can be an arbitrary, possi- δθ 2 θ −1. bly highly entangled, state; the measurement is a general, not ( ) [NH( )] (A16) necessarily local, POVM (which is, of course, assumed to be θ It has never been doubted until recently [1] that the QCRB independent so the QCRB can be applied); and the inference is the ultimate precision allowed by quantum mechanics that rule can be biased or unbiased. Additionally, t appearing in cannot be surpassed under any circumstances [3,4]. As can be Eq. (B1) can take any non-negative value; it is unnecessary to seen from the above proof, Eqs. (A15) and (A16)aswellasthis fulfill the condition assumed in the main text, which requires belief are based on the θ-independence assumption. At first that the evolution time t is much longer than the relaxation glance, this assumption seems reasonable as it echoes with our time of the channel. quick impressions obtained from textbook materials regard- Using the QCRB [3], we have that the estimation error is ing PMs. Indeed, the textbook physics says that the POVM lower bounded by operators associated with a PM are the eigen-projections of 1 the measured observable. An incautious follow-up inference (δθ)2 . (B2) ⊗N ρ might be that the POVM operators associated with any types H θ (t )[ (0)] of measurements are only determined by the measured ob- Here, H[ ⊗N (t )[ρ(0)]] represents the QFI and ⊗N (t )[ρ(0)] θ θ θ servable and, therefore, independent. However, as critically is the output state of the N probes, where ρ(0) denotes the examined below, the POVM operators of a measurement may ⊗N input state. To evaluate the QFI H[ θ (t )[ρ(0)]], we intro- be determined by many factors in practice, even possibly duce two amplitude damping channels [33], (t ), i = 0, 1, θ θ i including the parameter itself. That is, the -independence defined as two completely positive trace-preserving (CPTP) assumption may not be true in practice. Therefore, although maps transforming the Bloch vector as the CCRB is always true, the QCRB as a universally valid bound is questionable. 0(t ): (rx, ry, rz ) γ γ − t − t −γ t −γ t → (e 2 rx, e 2 ry, 1 − e + e rz ), (B3) APPENDIX B: PROOF OF OPTIMALITY OF EQ. (1) and Here,weprovethatEq.(1) is optimal if the QCRB is valid. Let 1(t ): (rx, ry, rz )

Lθ t γ t γ t θ = − − −γ t −γ t (t ): e (B1) → (e 2 rx, e 2 ry, e − 1 + e rz ), (B4) be the quantum channel associated with the generalized am- respectively. Noting that the effect of θ (t )is plitude damping process. To estimate the parameter θ char- acterizing this channel, a general strategy [7]istosendN θ (t ): (rx, ry, rz ) γ γ probes through N parallel channels θ (t ), measure them at the − t − t −γ t −γ t → e 2 r , e 2 r , (2θ − 1)(1 − e ) + e r , (B5) output, and use an inference rule θˆ (x) to extract the value of x y z θ from the measurement results x (see Fig. 3). Here, we have we have collectively denoted the measurement outcomes as x, i.e., θ (t ) = θ 0(t ) + (1 − θ ) 1(t ). (B6)

t Equation (B6) enables us to rewrite θ (t )asaθ-independent quantum channel acting on a larger input space:

θ (t )[ρ] = (t )[ρ ⊗ ρθ ]. (B7)

t Here, ρθ = diag(θ,1 − θ ) is the steady state, and (t )is defined as (t )[ρ ⊗ σ ]:= i(t ) ⊗ Ei[ρ ⊗ σ ] i=0,1 = i(t )[ρ] ⊗ Ei[σ ], (B8) measurement i=0,1 where E [σ ]:=i|σ |i, i = 0, 1. It is not difficult to see that t i (t ) thus defined is a CPTP map. Using Eq. (B7), we have ⊗N ⊗N ⊗N H θ (t )[ρ(0)] = H (t ) ρ(0) ⊗ ρθ FIG. 3. General scheme for quantum metrology. N probes, prepared in an initial state, are sent through N parallel channels θ (t ). ⊗N H ρ(0) ⊗ ρθ A measurement is performed on the final state, from which the ⊗N parameter θ is estimated via an inference rule θˆ. = H ρθ , (B9)

023418-6 DISSIPATIVE ADIABATIC MEASUREMENTS: BEATING … PHYSICAL REVIEW RESEARCH 2, 023418 (2020) where we have used the monotonicity of the QFI un- Qubit der parameter-independent CPTP maps [3]. Noting that a 0 a ⊗N 1 1 2 x ρ = ρθ ρθ = H[ θ ] NH[ ] and H[ ] θ(1−θ ) , we further have ⊗ N H N (t )[ρ(0)] . (B10) Free Particle Screen θ θ(1 − θ ) Substituting Eq. (B10) into Eq. (B2) yields θ(1 − θ ) (δθ)2 , (B11) FIG. 4. A toy setup used to demonstrate the implementation of N PM (see p. 73 of Ref. [45]). A free particle passes by and interacts indicating that Eq. (1) is optimal if the QCRB is valid. with a qubit. The state of the free particle is described by a Gaussian wave packet moving along z direction. Upon interacting with the qubit, the particle is pulled or pushed along x direction, depending APPENDIX C: MORE DETAILS ON THE MINIMAL on the state of the qubit. In a PM, the position of the particle hitting MODEL OF QUANTUM MEASUREMENT the screen tells us about the eigenvalues of the measured observable Suppose that S is a qubit undergoing the generalized A. However, in a DAM, the shift is the expectation value Aθ . amplitude damping process, d In particular, the above minimal model was also adopted in ρS (t ) = Lθ ρS (t ), (C1) dt Aharonov and Vaidman’s proposal of adiabatic measurements L [10], which has been experimentally realized in optical setups with θ defined in the main text. Equation (C1) can be used [46]. to describe the dynamics of a qubit interacting with a Bosonic It may be helpful to recall a toy setup used to demonstrate thermal environment at finite temperature [33]. In this sce- θ the implementation of PM [45]. As schematically shown in nario, is a monotone function of the temperature of the Fig. 4, a free particle passing by the qubit plays the role of the environment, which characterizes losses of energy from the measuring apparatus A. Initially, it is prepared in a Gaussian qubit, i.e., effects of energy dissipation [33]. So, determining wave packet with a nonzero momentum along z direction, θ amounts to determining the temperature of the environment. 2+ 2+ 2 Considering that temperature estimation has received much 1 − x y z ip z φ(r, 0) = e 4σ 2 e z , (C4) attention recently in quantum thermometry [34–36], we aim (2πσ2 )3/4 to estimate θ in the main text. We assume that the dynamics where pz is a fixed positive number describing the z compo- described by Eq. (C1) is always on. Such an assumption is, nent of the momentum of the particle. Then the wave packet of course, reasonable in many physical scenarios/applications, moves along z direction with the group velocity e.g., in quantum computation, where effects of decoherence pz and dissipation are always on whenever one implements uni- vg = , (C5) tary gates or performs quantum measurements on qubits. m To simplify our discussion here as well as in the main text, where m denotes the mass of the particle. That is, we adopt the minimal model of quantum measurement, which x2+y2+(z−v t )2 1 − g i(p z−ωt ) has been used time and again in the literature [45]. That is, to φ(r, t ) = e 4σ 2 e z , (C6) πσ2 3/4 measure an observable A, we add an interaction term, (2 ) − with ω = p2/(2m)(¯h = 1). Here, we have neglected the H = T 1A ⊗ pˆ, (C2) z I spread of the wave packet by assuming that m is very large. coupling S to a measuring apparatus A. Here and in the main Note that vg can take an arbitrary value because of the freedom text, we have suppressed the subscript x inp ˆ for ease of in choosing pz. Upon interacting with the qubit, the particle notation. The initial state |φ of A is set to be a Gaussian gets pulled or pushed along x direction, depending on the centered at x = 0, state of the qubit. That is, the interaction is of the form HI = g(t )A ⊗ pˆx.[InEq.(C2), g(t ) is assumed to be time 1 − x2 |φ= dx e 4σ 2 |x, (C3) independent for simplicity.] Taking A = σ as an example, we πσ2 1/4 z (2 ) see that whether the particle gets pulled or pushed depends where σ denotes the standard deviation. Here and henceforth, on whether the state of the qubit is |0 or |1. The coupling we omit the two limits of a integral whenever they are −∞ strength is determined by the distance of the particle from the and +∞ for ease of notation. As usual, we are not interested qubit, whereas the coupling time is determined by the value of in the free dynamics of A itself. So, we require the free vg as well as the distance between the particle and the screen. Hamiltonian of A to be zero. Such a requirement is often In a PM, the coupling strength is large and the coupling time is imposed in proposals of quantum measurements and may be small, so HI is dominant whereas Lθ can be neglected. Under satisfied if one goes to a frame that rotates with the free Hamil- the influence of HI , the particle hits the screen with some shift tonian of A in the rest frame (see p. 15 of Ref. [45]).Itisworth along x direction. The shift tells us about an eigenvalue ai of noting that there are a number of experimental setups (such A. On the other hand, in a DAM, we consider the scenario as in cavity quantum electrodynamics and circuit quantum that Lθ is dominant but HI is comparatively weak. The former electrodynamics) that are physically equivalent and therefore effectively reshapes the latter to be g(t )Aθ pˆx. So, the shift is can be used to implement the above minimal model [45]. now the expectation value Aθ rather than an eigenvalue ai.

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According to the minimal model of measurement and noting that the dynamics Eq. (C1) is always on, we have that the dynamical equation describing the coupling of S and A reads d ρSA(t ) = Lθ ρSA(t ) − i[H ,ρSA(t )] =: LρSA(t ). (C7) dt I Evidently, the dynamics map associated with the coupling procedure is

LT ET = e , (C8) transforming the initial state ρθ ⊗|φφ| to the state ET (ρθ ⊗ |φφ|) at time T . In the main text, we mainly focus on the two limits T → 0 and T →∞, which correspond to PMs and DAMs, respectively. Besides, although T is set to be ∞ in DAMs for the sake of mathematical convenience, it is not very large for strongly dissipative systems. Moreover, the larger the dissipative gap is, the shorter T can be. This is analogous to the well-known result regarding the adiabatic theorem, where T is determined by the energy gap of the Hamiltonian of a closed system.

APPENDIX D: DISCUSSION ON INTERMEDIATE T In the main text, we discussed the two limiting cases of T → 0 and T →∞, which correspond to PM and DAM, respectively. The former is the best known type of quantum measurement, which has been extensively studied in the literature. The physical significance of the latter, however, has not been widely appreciated or even fully recognized. The aim of the present paper is to study the limiting case of T →∞and show how this type of quantum measurement is applied to dissipative systems. Here, we would like to go one step further and briefly discuss the case of intermediate T , although such a discussion is beyond the focus of this paper. To do this, we numerically compute the profile of the probability distribution pθ (x) for various values of T . Some typical examples of the profile are shown in Fig. 5. Here, we choose θ = 1/2 without loss of generality. The profile associated with the PM is shown in Fig. 5(a), in which there are two peaks corresponding to the two eigenvalues 0 and 1 of the measured observable Aopt.AsT increases, the probability density at x =Aoptθ = 1/2 increases whereas those at x = 0 and x = 1 decrease, as shown in Figs. 5(a)–5(d). This leads to the fact that the two peaks in Fig. 5(a) finally merge into one peak, as shown in Fig. 5(d). After that, the probability density at x =Aoptθ = 1/2 continues to increase as T increases, so the peak becomes sharper and sharper, as can be seen from Figs. 5(d)–5(h). Finally, when T 30 μs, the profile rarely FIG. 5. Plots of the probability distribution p1/2(x)for(a)T = changes even though we continue to enlarge T , as can be seen μ = . μ = . μ = . μ = = μ 0 s, (b) T 0 2 s, (c) T 0 4 s, (d) T 0 6 s, (e) T from Figs. 5(h)–5(j). This indicates that T 30 s can be 0.8 μs, (f) T = 1 μs, (g) T = 10 μs, (h) T = 30 μs, (i) T = 60 μs, already thought of as being sufficiently large so the profile and (j) T = 90 μs. Parameters used are σ = 0.2andγ = 5MHz. now corresponds to the DAM. rotating at the frequency of the driving pulse, the Hamil- ω APPENDIX E: DISCUSSION ON A REALISTIC MODEL =− R0 ασ + ασ − tonian of the qubit reads Hq 2 (cos x sin y ) ω σ σ = , , ω Here we examine the realistic model of a driven supercon- 2 z, where i, i x y z, denote Pauli matrices, and R0, ducting qubit dispersively coupled to a microwave resonator ω, and α are the Rabi frequency, the detuning, and the phase [39,40]. It is equivalent to the adopted minimal model of mea- of the driving pulse, respectively [47]. The qubit is exposed to surement and has been used to realize PMs [44]. In a frame decoherence and dissipation described by Ld = γ1D[σ−] +

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are 13.6, 10.4, 4.4, 2.0 μs, for γ1 = γ2 = 5, 15, 25, 35 MHz, respectively. This is consistent with the results found in the main text.

APPENDIX F: DETAILS ON EXAMINATION OF THE θ-INDEPENDENCE ASSUMPTION To examine the θ-independence assumption, we need to ﬁgure out an expression for the POVM operator x. Noting that the state of S + A immediately after the coupling procedure is ET (ρθ ⊗|φφ|), we have that the probability density of getting the pointer reading x is given by

pθ (x) = trSA[|xx|ET (ρθ ⊗|φφ|)]. (F1) Besides, as shown in the main text, there is

p2 γ 1 − 2 D[σ ] with D[o]ρ = oρo† − 1 {o†o,ρ}, where γ , i = 1, 2, φ(p) = e 4σ 2 (F4) 2 z 2 i (2πσ2 )1/4 are the relaxation and dephasing rates, respectively. The total Liouvillian describing the dynamics of the qubit therefore being the momentum representation of |φ, where σ = is L :=−i[H , •] + L , which plays the role of Lθ in 1/(2σ ), we have total q d Eq. (C7). The interaction between the qubit and the resonator 1 ∗ i(p−p )x L T reads H = χa†aσ , where χ represents the dispersive cou- pθ (x) = dpdp φ(p)φ (p )e trS [e p,p ρθ ]. I z 2π pling, and a† and a are the creation and annihilation operators (F5) for the resonator. The Hamiltonian of the resonator is Hr = ω † ω ra a, where r is the resonator frequency. In PMs, under Note that any linear map defined over the operator space of ∗ the influence of HI , the frequency of the resonator is shifted S has a dual map , which is the map such that as ω ± χ, depending on whether the state of the qubit is |0 r † ∗ † trS [X (Y )] = trS [ (X ) Y ], (F6) or |1. (In DAMs, it should be ωr + χA.) The shift can be read out by coupling the resonator to transmission lines. The where X and Y are two linear operators acting on the average number n of photons in the resonator is determined by Hilbert space of S. To digest the above definition, one can the power of the readout pulse as well as the decoherence and rewrite Eq. (F6)asX, (Y )= ∗(X ),Y , where X,Y := † dissipation experienced by the resonator. Here, we consider trS [X Y ] is known as the Hilbert-Schmidt inner product [33]. the scenario that the decoherence and dissipation of the qubit So, the dual map ∗ of a map is defined in a way that is is much stronger than those of the resonator so we can neglect completely analogous to how the Hermitian conjugate X † of the latter within a certain time window. On the other hand, a linear operator X is defined. It is not difficult to see that the L∗ we can suppress the term H by switching to a rotating L , T , T r dual map of e p p is e p p with frame, that is, we are in a doubly rotating frame. So, we may L∗ = γ θ σ σ − 1 {σ σ , } only consider the two terms L and H as in Eq. (C7)to p,p X +X − + − X total I 2 figure out the frequency shift resulting from the nontrivial 1 + (1 − θ ) σ−Xσ+ − {σ−σ+, X} interplay between them. A full analysis taking into account the 2 decoherence and dissipation of the resonator (possibly with + iT −1(pAX − pXA). (F7) experimental demonstration) is left to a future work. Figure 6 With the above knowledge, we can rewrite the term shows the numerical results of the measures in Eqs. (7) and L∗ L , T T † trS [e p p ρθ ] appearing in Eq. (F5)astrS [(e p,p I) ρθ ]. (8) for different γ1 and γ2. Here, we set γ1 = γ2 without loss of generality. Also, we identify χ with T −1 for the sake of From this fact and Eq. (F5), it follows that γ = γ notational consistency. As can be seen from Fig. 6,as 1 2 ∗ 1 ∗ i(p−p )x L T † pθ (x) = trS dpdp φ(p)φ (p )e (e p,p I) ρθ . increases, the slope of measure Eq. (7) as a function of T 2π increases, but that of measure Eq. (8)asafunctionof1/T decreases. This indicates that the strong interaction requirement (F8) in PMs is increasingly difficult to meet if decoherence and Comparing Eq. (F8) with Eq. (A3), we arrive at the expression dissipation become increasingly strong, whereas this is not the of x, case for the long-time requirement in DAMs. For instance, − 1 L∗ 5 ∗ i(p−p )x , T † if the desired tolerance of the deviations is set to be 10 , x = dpdp φ(p)φ (p )e (e p p I) . (F9) 1/T in PM are 114, 230, 267, 279 MHz whereas T in DAM 2π

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L∗ θ θ As can be seen from Eq. (F7), the term p,p appearing in Eq. (F9)is dependent. Hence, it is expected that x is dependent too. To conﬁrm that x is indeed θ dependent, let us take a closer look at Eq. (F9) in the following. For concreteness, we set the L∗ , T † measured observable to be Aopt =|00|.UsingMATHEMATICA, we can ﬁnd out the explicit expression of (e p p I) appearing in Eq. (F9), sinh(S/2) ∗ / + γ − ν L T † −(T γ +iν)/2 cosh(S 2) (T i ) 0 e p,p I = e S , (F10) / + γ + ν sinh(S/2) 0 cosh(S 2) (T i ) S

with the integration g (ν)dν, i = 1, 2, in general. To bypass this i T → S = T 2γ 2 + 2iT γν(1 − 2θ ) − ν2. (F11) difficulty, we consider the two limiting cases of 0 and T →∞. Here, for ease of notation, we have introduced the new vari- Consider first the limiting case of T → 0. Using Taylor- ables: series expansions for g1(ν) and g2(ν) about T = 0, we have, up to first order, μ = p + p ,ν= p − p . (F12)

ν2 i(x−1)ν − Substituting Eq. (F10) into Eq. (F9) and noting that g1(ν) = e e 8σ 2 − T γ (1 − θ ) φ φ∗ = 1 − μ2+ν2 μ ν = (p) (p ) πσ2 1/2 exp[ σ 2 ] and d d 2dpdp,we ∞ − (2 ) 8 ν2 ν n 1 i(x−1)ν − (i ) have × e e 8σ 2 1 − , (F17) n! g (ν)dν 0 n=1 = f μ dμ 1 . x ( ) ν ν (F13) 0 g2( )d and

Here, ν2 ixν − g2(ν) = e e 8σ 2 − T γθ 1 1 μ2 μ = − , ∞ − f ( ): / exp (F14) ν2 − ν n 1 π πσ 2 1 2 σ 2 ixν − ( i ) 4 (2 ) 8 × e e 8σ 2 1 − . (F18) n! ν2 T γ + iν n=1 g (ν):= exp − + iνx − 1 8σ 2 2 Substituting Eqs. (F17) and (F18) into Eq. (F13), we obtain, sinh(S/2) after some algebra, × cosh(S/2) + (T γ − iν) , (F15) S = (0) + (1), x x T x (F19) ν2 T γ + iν g (ν):= exp − + iνx − 2 8σ 2 2 with ⎛ ⎞ − (x−1)2 sinh(S/2) 1 2 × / + γ + ν . √ e 2σ 0 cosh(S 2) (T i ) (F16) (0) = ⎝ 2πσ2 ⎠ x x2 (F20) S 1 − 0 √ e 2σ 2 2πσ2 While it is easy to perform the integration f (μ)dμ, i.e., μ μ = 1 f ( )d 2π , it is quite difﬁcult to analytically work out and

⎛ ⎞ x x−1 (x−1)2 erf √ −erf √ 1 − 2σ 2σ ⎜−γ (1 − θ ) √ e 2σ 2 − 0 ⎟ (1) = ⎝ 2πσ2 2 ⎠. x x x−1 (F21) x2 erf √ −erf √ 1 − 2σ 2σ 0 −γθ √ e 2σ 2 − 2πσ2 2

Here, erf(x) is known as the Gauss error function, defined as has the effect of making x θ-dependent. In the limit of = √1 x −t 2 (0) θ → erf(x) π −x e dt. Evidently, x is independent but T 0, the coupling strength is infinitely strong and the (1) is θ dependent. coupling time is infinitely short, indicating that HI dominates x L In the limit of T → 0, which corresponds to the PM, we Eq. (C7) whereas θ can be omitted from Eq. (C7). As a = (0) θ have that result, x x is, of course, independent. However, in reality, any interaction is of finite strength and lasts for a finite = (0), x x (F22) time interval, implying that Lθ cannot be completely ignored and plays some role in the measurement. So, in addition to thereby reaching the textbook physics that the POVM operator (0), the (small) θ-dependent term T (1) resulting from the is θ independent. This can be understood on an intuitive level. x x effect of Lθ appears in Eq. (F19). This leads to the fact that Note that the interaction term HI in Eq. (C7) is unrelated x is θ dependent in practice. Therefore, strictly speaking, to θ, whereas the dissipative term Lθ is related to θ and

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θ the -independence assumption may not hold even for PMs in p1/2(x) practice. 2.0 =0.4 What happens if we enlarge T ?AsHI becomes weaker as T increases, Lθ plays an increasingly important role in the =0.3 measurement. So, it can be expected that x depends on θ 1.5 =0.2 increasingly heavily as T increases. Roughly speaking, the =0.1 degree of the θ dependence of x achieves its maximum in →∞ →∞ the limit of T . Motivated by this, we let T in our 1.0 proposal of measurements and expect that x depends on θ so heavily that the QCRB can be beaten by our measurement. This is the basic idea underlying our proposal. By the way, as a 0.5 matter of fact, the QCRB can be (slightly) beaten even by PMs with small but nonzero T , as shown below. Let us now ﬁgure out the expression of x in the limiting case of T →∞.Itis x − − not difﬁcult to see that 2 1 1 2 3

−θ ν − ν2 FIG. 7. Probability density p1/2(x) of getting the pointer reading ν = ν = i(x ) σ 2 , g1( ) g2( ) e e 8 (F23) x in the impulsive limit with different standard deviations σ . in this limit. Substituting Eq. (F23) into Eq. (F13), we have ⎛ ⎞ It is easy to see that the QFI associated with ρθ is given by −θ 2 1 − (x ) √ e 2σ 2 0 = ⎝ 2πσ2 ⎠, x −θ 2 (F24) 1 1 − (x ) H(θ ) = . (F26) 0 √ e 2σ 2 θ − θ 2πσ2 (1 ) in the limit of T →∞. Evidently, x in Eq. (F24)isθ de- Equation (F26) can be obtained by first solving Eq. (A10)to 1 1 θ = , − θ pendent. As an immediate consequence, the θ-independence get L diag( θ 1−θ ) and then inserting L into Eq. (A15). assumption does not hold for our proposal of measurements. Denote by Fσ (θ ) the CFI associated with pθ (x)inEq.(F25). Here, the subscript σ is used to indicate that the CFI is dependent of θ because of the θ dependence of pθ (x). Noting → 1. Projective measurement: T 0 that the θ-independence assumption and, therefore, the QCRB We now inspect more carefully the limiting case of T → 0. holds in the impulsive limit, we have It corresponds to the case that HI is extremely strong but lasts θ θ . for a very short time interval, i.e., HI is impulsive. Hereafter, Fσ ( ) H( ) (F27) to make our discussion conceptually clear, we refer to the limit of T → 0 as the impulsive limit. In contrast, the limit On the other hand, it is not difficult to see that of T →∞ is referred to as the adiabatic limit hereafter, lim Fσ (θ ) = H(θ ). (F28) which corresponds to DAMs to be discussed later on. In the σ →0 impulsive limit, the probability density of getting the pointer σ → reading x is given by Indeed, in the limit of 0, pθ (x)inEq.(F25) can be treated as the discrete probability distribution with the proba- θ − (x−1)2 1 − θ − x2 bility of getting 1 being θ and that of getting 0 being 1 − θ. pθ (x) = √ e 2σ 2 + √ e 2σ 2 , (F25) 2πσ2 2πσ2 Using Eq. (A8), one can confirm that the CFI associated with this discrete probability distribution is exactly the QFI which can be obtained by substituting Eq. (F20) into Eq. (A3). given by Eq. (F26). Note that the observable corresponding For the reader’s information, we plot in Fig. 7 the probability to pθ (x)inEq.(F25)isAopt =|00|. The above point in- distribution p1/2(x) with different standard deviations σ .As dicates that Aopt =|00| is optimal, since the QCRB can can be seen from Fig. 7 as well as Eq. (F25), for a small σ , be achieved if one performs the PM associated with Aopt. say, σ = 0.1, pθ (x) is only significantly different from zero if Combing Eqs. (F27) and (F28), we have x ≈ 0orx ≈ 1. Noting that 0 and 1 are the two eigenvalues of A =|00|, we deduce that, roughly speaking, the pointer Fσ (θ ) lim Fσ (θ ) = H(θ ). (F29) opt σ →0 reading x, as a random variable, is most likely to be one of these two eigenvalues in the PM with a small σ. This is just Figure 8 shows the numerical results of Fσ (θ ) for four dif- the well-known textbook physics that the potential outcome ferent σ , namely, σ = 0.4, 0.3, 0.2, 0.1. As can be seen of a PM is one of the eigenvalues of the measured observable. from Fig. 8,1/Fσ (θ ) is strictly larger than 1/H(θ )forσ>0 Strictly speaking, this textbook physics is valid only in the but approximately equals to 1/H(θ ) for a small enough σ , mathematical limit of σ → 0, for which the continuous proba- say, σ = 0.2. Therefore, to achieve the QCRB, in addition bility distribution pθ (x) can be treated as a discrete probability to choosing an optimal observable, one needs to prepare the distribution. To put it differently, in practice, where σ is small measuring apparatus A in a state with a small σ . but nonzero, the pointer reading x may not be exactly one of Consider now the PM with a small but nonzero T ,for the eigenvalues; there could be small fluctuations due to the which the θ-independence assumption does not hold. Substi- uncertaintyof initial position of the pointer. tuting Eq. (F19) into Eq. (A3), we have that the probability

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FIG. 8. Numerical results of the classical Fisher information Fσ (θ ) associated with pθ (x)inEq.(F25) for four different σ .The abscissa and the ordinate are θ and 1/Fσ (θ ), respectively. The black solid curve represents 1/H(θ ). For a given θ,1/Fσ (θ ) gradually decreases as σ decreases and ﬁnally approaches the limiting position speciﬁed by 1/H(θ ) in the limit of σ → 0. density of getting the pointer reading x is given by

θ − T γθ(1 − θ ) − (x−1)2 1 − θ − T γθ(1 − θ ) pθ (x) = √ e 2σ 2 + √ 2πσ2 2πσ2 − x2 x x − 1 × e 2σ 2 + T γθ(1 − θ ) erf √ − erf √ . 2σ 2σ (F30)

To examine whether the QCRB can be beaten in this case, we numerically compute 1/Fσ (θ ) associated with the pθ (x)in Eq. (F30) for three different θ and four different σ , namely, θ = 0.5, 0.4, 0.3, and σ = 1/5, 1/6, 1/7, 1/8. The numerical results are shown in Fig. 9, where we set γ = 5MHz.The three black solid lines depicted in Figs. 9(a)–9(c) represent the QCRBs 1/H(θ ) associated with these three values of θ, respectively. As can be seen from Fig. 9,1/Fσ (θ ) can be strictly less than 1/H(θ ) for some small but nonzero T , indicating that the QCRB can be beaten for the PM with a small but nonzero T , as claimed in the previous paragraph.

2. Dissipative adiabatic measurement: T →∞ Consider now the adiabatic limit. In this limit, the probability density of getting outcome x reads

1 − (x−θ )2 pθ (x) = √ e 2σ 2 , (F31) 2πσ2 FIG. 9. Numerical results of 1/Fσ (θ ) with small values of T for three different θ and four different σ .(a)θ = 0.5. (b) θ = 0.4. which can be obtained by substituting Eq. (F24) into Eq. (A3). (c) θ = 0.3. The three black solid lines in Figs. 9(a)–9(c) represent For the reader’s information, we plot in Fig. 10 the probability the quantum Cramér-Rao bounds 1/H(θ ) associated with these three distribution p1/2(x) with different standard deviations σ .As θ, respectively. Here, γ is set to be 5 MHz. can be seen from Fig. 10 as well as Eq. (F31), for a small σ , say, σ = 0.1, pθ (x) is only signiﬁcantly different from zero the expectation value of the measured observable, which is in if x ≈ θ. Note that θ is the expectation value of Aopt in the sharp contrast to that of the PM, i.e., an individual eigenvalue state ρθ . So, roughly speaking, the outcome of the DAM is of the measured observable.

023418-12 DISSIPATIVE ADIABATIC MEASUREMENTS: BEATING … PHYSICAL REVIEW RESEARCH 2, 023418 (2020)

p1/2(x) 4 =0.4 =0.3

3 =0.2 =0.1

x −2 −1 1 2 3

FIG. 10. Probability density p1/2(x) of getting the pointer reading x in the adiabatic limit with different standard deviations σ . FIG. 11. Classical Fisher information Fσ (θ ) associated with pθ (x)inEq.(F31) for four different σ . The abscissa and the ordinate are θ and 1/Fσ (θ ), respectively. For a given θ,1/Fσ (θ ) decreases as Substituting Eq. (F31) into Eq. (A8), we have that the σ decreases and ﬁnally approaches zero in the limit of σ → 0. associated CFI reads

1 measurement is a PM, this data, denoted as xPM, is either 1 or Fσ (θ ) = . (F32) σ 2 0, provided that σ is very small. Evidently, the data xPM itself is unrelated to the parameter θ. Therefore, given only one PM PM Figure 11 shows the proﬁles of 1/Fσ (θ ) for four different datum x , there is no way to deﬁnitely determine θ from x , σ . As can be easily seen from Eq. (F32)aswellasFig.11, even in principle. However, if the measurement is a DAM, this DAM DAM Fσ (θ ) can be arbitrarily large so long as σ is small enough. data, denoted as x , is approximately equal to θ. So, x Therefore, there is no intrinsic bound on precision in the is more informative than xPM,asxDAM is directly related to θ. DAM, in sharp contrast to the ideal PM case (see Fig. 8). Such a direct relationship allows one to extract the value of To better digest this point, one may proceed as follows. θ from xDAM straightforwardly, leading to the fact that θ can Suppose that we are only given one datum, which is obtained be determined to any degree of accuracy so long as σ is small from the measurement associated with Aopt =|00|.Ifthe enough.

[1] L. Seveso, M. A. C. Rossi, and M. G. A. Paris, Phys. Rev. A 95, [13] S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. Büchler, and 012111 (2017). P. Zoller, Nat. Phys. 4, 878 (2008). [2] C. W. Helstrom, Quantum Detection and Estimation Theory [14] F. Verstraete, M. M. Wolf, and J. I. Cirac, Nat. Phys. 5, 633 (Academic, New York, 1976). (2009). [3]S.L.BraunsteinandC.M.Caves,Phys. Rev. Lett. 72, 3439 [15]H.Weimer,M.Müller,I.Lesanovsky,P.Zoller,andH.P. (1994). Büchler, Nat. Phys. 6, 382 (2010). [4] M. G. A. Paris, Int. J. Quantum Inform. 07, 125 (2009). [16] M. J. Kastoryano, F. Reiter, and A. S. Sørensen, Phys. Rev. Lett. [5] C. W. Helstrom, Phys. Lett. 25A, 101 (1967). 106, 090502 (2011). [6] V. Giovannetti, S. Lloyd, and L. Maccone, Science 306, 1330 [17] J. Cho, S. Bose, and M. S. Kim, Phys.Rev.Lett.106, 020504 (2004). (2011). [7] V. Giovannetti, S. Lloyd, and L. Maccone, Phys.Rev.Lett.96, [18] H. Krauter, C. A. Muschik, K. Jensen, W. Wasilewski, J. M. 010401 (2006). Petersen, J. I. Cirac, and E. S. Polzik, Phys.Rev.Lett.107, [8] V. Giovannetti, S. Lloyd, and L. Maccone, Nat. Photonics 5, 080503 (2011). 222 (2011). [19] J. T. Barreiro, M. Müller, P. Schindler, D. Nigg, T. Monz, M. [9] D. Braun, G. Adesso, F. Benatti, R. Floreanini, U. Marzolino, Chwalla, M. Hennrich, C. F. Roos, P. Zoller, and R. Blatt, M. W. Mitchell, and S. Pirandola, Rev. Mod. Phys. 90, 035006 Nature (London) 470, 486 (2011). (2018). [20] Karl Gerd H. Vollbrecht, C. A. Muschik, and J. I. Cirac, Phys. [10] Y. Aharonov and L. Vaidman, Phys. Lett. A 178, 38 (1993). Rev. Lett. 107, 120502 (2011). [11] A. Beige, D. Braun, B. Tregenna, and P. L. Knight, Phys. Rev. [21] M. J. Kastoryano, M. M. Wolf, and J. Eisert, Phys. Rev. Lett. Lett. 85, 1762 (2000). 110, 110501 (2013). [12] M. B. Plenio, S. F. Huelga, A. Beige, and P. L. Knight, Phys. [22] A. W. Carr and M. Saffman, Phys.Rev.Lett.111, 033607 Rev. A 59, 2468 (1999). (2013).

023418-13 DA-JIAN ZHANG AND JIANGBIN GONG PHYSICAL REVIEW RESEARCH 2, 023418 (2020)

[23] E. G. Dalla Torre, J. Otterbach, E. Demler, V. Vuletic, and M. D. [36] S. Seah, S. Nimmrichter, D. Grimmer, J. P. Santos, V. Lukin, Phys.Rev.Lett.110, 120402 (2013). Scarani, and G. T. Landi, Phys. Rev. Lett. 123, 180602 [24] D. D. Bhaktavatsala Rao and K. Mølmer, Phys.Rev.Lett.111, (2019). 033606 (2013). [37] C. J. Hood, T. W. Lynn, A. C. Doherty, A. S. Parkins, and H. J. [25] C. D. B. Bentley, A. R. R. Carvalho, D. Kielpinski, and J. J. Kimble, Science 287, 1447 (2000). Hope, Phys. Rev. Lett. 113, 040501 (2014). [38] H. Mabuchi and A. C. Doherty, Science 298, 1372 (2002). [26] D.-J. Zhang, H.-L. Huang, and D. M. Tong, Phys. Rev. A 93, [39] A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. 012117 (2016). Schoelkopf, Phys. Rev. A 69, 062320 (2004). [27] D.-J. Zhang, X.-D. Yu, H.-L. Huang, and D. M. Tong, Phys. [40] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, Rev. A 94, 052132 (2016). J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Nature [28] M. Abdi, P. Degenfeld-Schonburg, M. Sameti, C. Navarrete- 431, 162 (2004). Benlloch, and M. J. Hartmann, Phys. Rev. Lett. 116, 233604 [41] P. Zanardi and L. Campos Venuti, Phys.Rev.Lett.113, 240406 (2016). (2014). [29] M. E. Kimchi-Schwartz, L. Martin, E. Flurin, C. Aron, M. [42] Unless otherwise stated, the Hilbert-Schmidt norm is adopted. Kulkarni, H. E. Tureci, and I. Siddiqi, Phys. Rev. Lett. 116, That is, for an operator X, the norm reads X := tr(X †X ), 240503 (2016). while for a superoperator E, it is the induced norm deﬁned as E = E [30] F. Reiter, D. Reeb, and A. S. Sørensen, Phys. Rev. Lett. 117, : supX1 (X ) . 040501 (2016). [43] D.-J. Zhang and J. Gong, arXiv:1907.10340. [31] M. Žnidaric,ˇ Phys.Rev.Lett.116, 030403 (2016). [44] M. Hatridge, S. Shankar, M. Mirrahimi, F. Schackert, K. [32] F. Reiter, A. S. Sørensen, P. Zoller, and C. A. Muschik, Nat. Geerlings, T. Brecht, K. M. Sliwa, B. Abdo, L. Frunzio, S. M. Commun. 8, 1822 (2017). Girvin, R. J. Schoelkopf, and M. H. Devoret, Science 339, 178 [33] M. A. Nielsen and I. L. Chuang, Quantum Computa- (2013). tion and Quantum Information (Cambridge University Press, [45] M. Naghiloo, arXiv:1904.09291. Cambridge, England, 2010). [46] F. Piacentini, A. Avella, E. Rebufello, R. Lussana, F. Villa, [34] L. Mancino, M. Sbroscia, I. Gianani, E. Roccia, and M. A. Tosi, M. Gramegna, G. Brida, E. Cohen, L. Vaidman, I. P. Barbieri, Phys. Rev. Lett. 118, 130502 (2017). Degiovanni, and M. Genovese, Nat. Phys. 13, 1191 (2017). [35] M. Mehboudi, A. Lampo, C. Charalambous, L. A. Correa, [47] G. Ithier, E. Collin, P. Joyez, P. J. Meeson, D. Vion, D. Esteve, F. M. A. García-March, and M. Lewenstein, Phys. Rev. Lett. 122, Chiarello, A. Shnirman, Y. Makhlin, J. Schrieﬂ, and G. Schön, 030403 (2019). Phys. Rev. B 72, 134519 (2005).

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