Connecting measurement invasiveness to optimal metrological scenarios
Saulo V. Moreira, Gerardo Adesso, Luis A. Correa, Thomas Coudreau, Arne Keller, Perola Milman
CConnecting measurement invasiveness to optimal metrological scenarios
Saulo V. Moreira , Gerardo Adesso , Luis A. Correa , Thomas Coudreau , Arne Keller and P´erola Milman Univ. Paris Diderot, Sorbonne Paris Cit´e, Laboratoire Mat´eriaux etPh´enom`enes Quantiques, UMR 7162, CNRS, F-75205, Paris, France Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems (CQNE),School of Mathematical Sciences, The University of Nottingham,University Park, Nottingham NG7 2RD, United Kingdom and Univ. Paris-Sud, Univ. Paris-Saclay 91405 Orsay, Franceand Laboratoire Mat´eriaux et Ph´enom`enes Quantiques, UMR 7162 CNRS, F-75205 Paris, France.
The connection between the Leggett-Garg inequality and optimal scenarios from the point of viewof quantum metrology is investigated for perfect and noisy general dichotomic measurements. Inthis context, we show that the Fisher information can be expressed in terms of quantum temporalcorrelations. This connection allows us to associate scenarios with relatively high Fisher informationto scenarios in which the Leggett-Garg inequality is violated. We thus demonstrate a qualitative and,to some extent, quantitative link between measurement invasiveness and metrological performance.Finally, we illustrate our results by using a specific model for spin systems.
I. INTRODUCTION
The term “macroscopic” has always been intuitivelyassociated with classical physics.
Macroscopic objects ,for instance, are the ones observed in our everyday lifescale, and are expected to behave according to the lawsof classical physics. It is known that classical physicsfails to provide a description of phenomena at the micro-scopic level, which demand the application of quantummechanical principles, such as the superposition princi-ple and entanglement. Therefore, one is naturally led tothe question of whether such quantum mechanical prin-ciples could also be observed at the macroscopic scale.This fundamental question concerning the validity of ex-trapolating quantum mechanics to the macroscopic world[1] was already pictured in 1935 in the Schr¨odinger’s cat
Gedanken experiment [2], where superpositions of states(“dead” and “alive”) of a macroscopic object (the cat)are at stake.Aiming to propose a test capable of experimentallyruling out the classical perspective of how macroscopicsystems are expected to behave, Leggett and Garg pro-posed the Leggett-Garg inequality (LGI) [3, 4]. Theauthors considered measurements of two-valued quanti-ties Q ( t i ) in macroscopic systems at four different times { t , · · · , t } . From measurement outcomes, one can com-pute the correlations C kl ≡ (cid:104) Q ( t k ) Q ( t l ) (cid:105) . The LGI canbe expressed as − ≤ K LG ≡ C + C + C − C ≤ , (1)and it holds under the following assumptions:(i) macroscopic realism: a macroscopic system withtwo or more macroscopically distinct states avail-able to it will at all times be in one of those states;and(ii) noninvasive measurability: it is possible, in prin-ciple, to determine the state of the system witharbitrarily small perturbation to its subsequent dy-namics. Therefore, according to Leggett and Garg, the violationof (1) witnesses the “ nonclassicality ” of the system con-sidered, in line with the definition of classicality providedby (i) and (ii) above.The LGI and the meaning of its violation have been thesubject of recent debates in the literature [5–9]. In Ref.[5] it is shown that only the assumption of noninvasivemeasurability is tested by the LGI in a model indepen-dent way. Given this, in Ref. [9], some of the authorsintroduced an operational model relating the LGI viola-tion with a parameter called the measurability of physicalsystems. The results were illustrated using perfect andnoisy parity measurements performed in spin- J systems.According to our model, the more the system is “measur-able,” i.e., the more one is able to faithfully distinguishbetween its different possible outcomes, the more the LGIis violated.Maximum measurability corresponds to projectivemeasurements. As measurability decreases and the mea-surements become weaker, LGI violation progressivelydiminishes, eventually vanishing at some point. There-fore, measurability is clearly associated with the inva-siveness of measurements, which in turn can depend one.g. measurement errors or on a dimension-dependentcoarse graining [10]. According to this model, the viola-tion of the LGI does not intrinsically depend on the sys-tem’s size, a notion that lacks itself of precise definitionwhenever quantum systems are concerned [11–13]. Re-cently, remarkable experimental achievements as well asexperimental proposals regarding LGI violation for sys-tems which can be reasonably considered macroscopicwere presented in Refs. [14–16].Seemingly unconnected, the field of quantum metrol-ogy has recently attracted considerable attention [17–24].The use of some quantum mechanical states as probes forthe sake of estimating a parameter θ has been shown tolead to a better scaling, with the dimension of the state,of the precision in the parameter’s estimation than us-ing classical resources only. For noisy systems, it wasshown that this scaling actually depends on the system’s a r X i v : . [ qu a n t - ph ] A ug size, the noise parameter, and the noise model [18]. Ul-timately, for a fixed dimension of the probe state, theprecision of the estimation of θ usually decreases as thenoise parameter increases, unless one resorts to appropri-ate control or error-correcting methods [24].In light of these elements, it thus seems natural toinvestigate connections between the LGI violation andquantum metrology. In this work, we do so by identify-ing each step of the LGI test with the steps of a metro-logical scenario. Assuming unbiased measurements (im-plying that the average of the estimated value over allexperimental results coincides with the true value of theparameter), we first introduce the definition of classical Fisher information (called, from now on, Fisher informa-tion), which bounds the standard deviation ∆ θ of theestimate of θ as ∆ θ ≥ / (cid:112) νF ( θ ) [25–27], where ν is thenumber of realizations of the experiment. For a givenmeasurement, F is given by F ( θ ) = (cid:88) l P l ( θ ) (cid:20) ∂ ln P l ( θ ) ∂θ (cid:21) , (2)where the P l ( θ ) are the probabilities of obtaining eachone of the different outcomes l and thus, (cid:80) l P l ( θ ) = 1.The generalization of the Fisher information to quantummechanics is done by writing P l ( θ ) = Tr[ ρ ( θ ) E l ], where E l is a positive operator valued measure (POVM). Bymaximizing F ( θ ) over all quantum measurements, oneobtains the quantum Fisher information (QFI) F Q [28–31], associated with the minimum lower bound for ∆ θ ,and saturated when ν → ∞ . Hence, the QFI correspondsto the Fisher information associated with the optimalmeasurement, i.e., the one which gives the best estima-tion for θ . II. LGI AND METROLOGICAL PROTOCOLS
In this section we compare the LGI test scenario to aparameter estimation protocol and establish some gen-eral results. In a LGI test, a maximally mixed initialstate ˆ ρ = I /d is prepared, where d is the dimension ofthe underlying Hilbert space. The dichotomic observ-able measured in the LGI is denoted by ˆ A , and the uni-tary time evolution, generated by the Hamiltonian ˆ H ,ˆ U ( t i ) = e − iHt i (in what follows ¯ h = 1). From now on,we suppose that, by rescaling the energies, the times t i are dimensionless. Using these definitions, we have thatthe two-time correlation appearing in the LGI (1) can bewritten as C ij = Tr[ ˆ A ˆ U ( t j − t i ) ˆ A ˆ U ( t i )ˆ ρ ˆ U † i ( t i ) ˆ U † j ( t i − t j )] = C ( θ ij ), where θ ij ≡ t i − t j .From now on we take all the time intervals t − t = t − t = t − t ≡ θ to be equal. While our resultscan be extended to the case of different time intervals,this choice is particularly convenient since it will allow usto express the Leggett-Garg parameter K LG as a simplefunction of the parameter θ , which will be the objectof interest in our metrological investigation (as detailed in the following). Under this specification, we can thenwrite the correlation function C ( θ ) as C ( θ ) = 1 d Tr[ ˆ A ˆ U ( θ ) ˆ A ˆ U † ( θ )] . (3)Generally, to calculate the LGI (1), one must performfour independent experiments in order to measure eachone of the correlations C , C , C and C . However,since we assumed all equal time intervals, the correlationfunctions C ij are stationary [4], that is, they depend onlyon the time difference θ . The LGI can in fact be rewrittenas | K LG ( θ ) | = | C ( θ ) − C (3 θ ) | ≤ , (4)meaning that it suffices to determine only two terms: C ( θ ) and C (3 θ ). Therefore, only two independent exper-iments, in which one performs two subsequent measure-ments, are required in this case.Since we start from the maximally mixed state, thesystem will remain unchanged before the first measure-ment. After it, however, the system’s state will be oneof the two possible outcomes resulting from the measure-ment of ˆ A , i.e. either ˆ ρ + or ˆ ρ − . We shall thus refer tothe first measurement as the preparation procedure .We now introduce a metrological scenario which canbe related to the LGI protocol described above. We con-sider the problem of estimating the unknown parameter θ through a measurement of the same dichotomic observ-able ˆ A . This second measurement can be either projec-tive or noisy, and may be generally described by a two-valued POVM. Also because we consider a maximallymixed initial state, the result of the first measurement issymmetric. That is, each one of its possible outcomes canbe obtained with equal probabilities 1 /
2. At the time thesecond measurement is performed, the evolved system’sstate is ˆ ρ ± ( θ ) = ˆ U ( θ )ˆ ρ ± ˆ U † ( θ ).The precision of the estimation can be characterizedby the Fisher information F ( θ ) as given by Eq. (2), inwhich P l ( θ ), with l = ±
1, is the probability for measuringˆ ρ ± ( θ ). As shown in the Appendix A, both ˆ ρ + and ˆ ρ − yield identical F ( θ ). The latter may be written in termsof the correlation function C ( θ ) as F ( θ ) = 11 − C ( θ ) (cid:20) ∂C ( θ ) ∂θ (cid:21) . (5)We note that Eq. (5) does not depend on the specific as-sumption of equal time intervals θ , and can be straight-forwardly generalized to construct the Fisher informationmatrix F ij as a function of the correlation functions C ij for the multi-parameter estimation of all the time sep-arations θ ij . However, as anticipated we will focus onthe particularly instructive case of a single parameter θ .Equation (5) turns out to have several remarkable prop-erties and serves as a guideline to establish a connectionwith the LGI.First, we note that the extrema of C ( θ ) are also theextrema of F ( θ ). A priori , the former are not all extremaof the latter, but let us focus on their common extrema,which we will label by θ e . It is straightforward to showthat θ e corresponds to a maximum of F if and only if C ( θ e ) = 1. The value C = ± C at θ = 0, and if C ( θ ) is aperiodic function with the period denoted by T (which isthe case if the Bohr frequencies of H are commensurate),then θ = nT ( n ∈ N ) will also correspond to extrema. Inthis last case, as C ( θ ) is an even function of θ , it can beshown that θ = nT / C ( θ ). Theglobal extremum corresponds to the value C ( nT / = 1only for ideal projective measurements.On the other hand, if an extremum of C is such that C ( θ e ) (cid:54) = 1, then F ( θ e ) = 0. Therefore, we find a verypeculiar situation, in which the estimation of θ e can beoptimal when the measurement is ideally projective butall information about θ e is lost when an infinitesimalamount of noise is added to the measurement (i.e., if C ( θ e ) = 1 − (cid:15) , then F ( θ e ) = 0 for arbitrarily small (cid:15) ).In other words, the maximum of F which is also an ex-tremum of C is not robust against noisy measurementsfor parameter estimation. III. PARITY MEASUREMENT ON A SPINSYSTEM
Equation (5) is quite general, and is based only on thefact that dichotomic measurements are performed in or-der to estimate the parameter θ . In the following, wewill consider a specific example which illustrates its con-sequences.We study the case of parity measurements performedin a spin- J system. Parity has been shown to be usefulin quantum optical metrology [32, 33], and has also beenused in [9] as part of a model where the LGI violation iscontrolled through a parameter determining the invasive-ness of a POVM. Let us briefly recall the main propertiesof this model. We consider a spin operator ˆ J , with spa-tial components ˆ J υ , υ = x, y, z . The ˆ J z eigenstates aredenoted as | m (cid:105) , − j ≤ m ≤ j , where j ( j + 1) ( j ∈ N )are the eigenvalues of ˆ J . The dynamics of the system isgoverned by the following Hamiltonian:ˆ H = Ωˆ J + ω ˆ J x , (6)where Ω and ω are constants with the dimension of fre-quency. In our setting, the initial state is ρ = 12 j + 1 j (cid:88) m = − j | m (cid:105) (cid:104) m | , (7)so that LGI violations can only arise from the measure-ments and system’s dynamics. We consider the two-valued POVM introduced in Ref. [9]ˆ E ± = ˆ M †± ˆ M ± = 12 ( I ± ˆ A ) , (8) where the dichotomic observable ˆ A takes the formˆ A ≡ (cid:88) µ (cid:88) m ∈ ∆ m µ ( − ( j − m ) f µ ( m, σ ) | m (cid:105) (cid:104) m | . (9)The functions f µ ( m, σ ) = e − ( m − µ )22 σ and ∆ m µ are dis-joint sets containing equally sized intervals of m . Theparameter σ can be interpreted as being associated withthe unfaithfulness of the measurement: for finite σ and m (cid:54) = µ , the particle is detected, but the value of m cannot be perfectly determined. Hence, σ → ∞ im-plies performing projective measurements, with perfectdetermination of the system’s parity as, for this case,ˆ A = ˆΠ z = (cid:80) m ( − j − m | m (cid:105) (cid:104) m | . Finally, the parameter∆ m µ determines the number N , among all the possiblevalues of m that the measurement apparatus can faith-fully detect, and therefore is called the resolution .We now study numerically the example of a spin 5 / b that is directly associated with themeasurability of the system or, alternatively, with theinvasiveness of a measurement and the width σ of thefunction f µ ( m, σ ) = e − ( m − µ )22 σ . By defining b ≡ e − / σ ,we have that σ → ∞ corresponds to b →
1, and σ → b →
0. In Eq. (9), we have considered only two pos-sible values of µ , µ ± = ± /
2, and the two correspondingintervals are ∆ m µ − = [ − / ,
0) and ∆ m µ + = (0 , / F ( θ ) and F Q . Asmentioned before, the first parity measurement of theLGI is identified as the state preparation in the quantummetrology protocol. The resulting state after this firstmeasurement is given by one of the two states:ˆ ρ ± ( t k ) = ( ˆ E ± ) ˆ ρ ( ˆ E ± ) p ± , (10)where p ± = Tr( ˆ E ± ˆ ρ ).Without loss of generality, we will work with ˆ ρ + (ˆ ρ − gives the same results). According to Eq. (6), the evolvedstate ˆ ρ + ( θ ), before the realization of the second paritymeasurement can be written asˆ ρ + ( θ ) = ˆ U ( θ )ˆ ρ + ˆ U † ( θ ) = e − iθ ˆ J x ˆ ρ + e iθ ˆ J x . (11)Combining Eqs. (2) and (8) we evaluated the Fisher in-formation F ( θ ) and the QFI F Q [34]. In the following,we split the analysis into two cases: projective and noisyparity measurements. A. Projective parity measurements ( b = 1 ) The results for b = 1, i.e. for noise-free parity mea-surements, are shown in Fig. 1(a). The Fisher informa-tion F ( θ ) and the quantum Fisher information F Q areboth plotted in this figure. We see that they coincidefor θ = nπ , showing that the measurement scenario isoptimal at this point. We also note that these maxima π / π / π / π θ | K L G | , | C | F , ℱ Q ( a ) π / π / π / π θ | K L G | , | C | F , ℱ Q ( b ) FIG. 1: (color online) Plots of the Fisher information F (solidmagenta line, scaled to the right vertical axis), the quantumFisher information F Q (dashed magenta line, scaled to theright vertical axis), the absolute value of the two-time corre-lation C (dotted black line, scaled to the left vertical axis)and absolute value of K LG (solid black line, scaled to theleft vertical axis), as a function of θ , for (a) b = 1 and (b) b = 0 .
99. The LGI violation region (relative to the left ver-tical axis) is shaded in light gray. All the plotted quantitiesare dimensionless. of F (1 , θ ) are also extrema of C ( θ ), and, as expected, thecorrelation function reaches its optimal value C = ± K LG de-fined in Eq. (1) as a function of θ . The point of maximalcorrelation cannot be a point of LGI violation, and this iswell illustrated in Fig. 1. Therefore, invasiveness cannotbe witnessed for θ = nπ . As we can see from Fig. 1(a),the region around θ = nπ corresponds to relatively highFisher information, and maximum LGI violation also oc-curs in this region. Therefore, the most favorable metro-logical scenario occurs in the same region where invasive-ness is witnessed through LGI violation. Nevertheless,the maximum of the Fisher information does not coin-cide with the maximum violation of the LGI. B. Noisy parity measurements ( b < ) We now examine the cases corresponding to limitedprecision, which corresponds to measurability b <
1. As b decreases and the measurements become noisier, bothLGI violation and the optimality of the metrological sce-nario are progressively degraded. In Fig. 1(b), we haveplotted F ( θ ), C ( θ ) and K LG ( θ ) for b = 0 .
99. We nowobserve that the Fisher information is zero at θ = nπ .Recall that this drastic transition follows from Eq. (5):as discussed above, the “collapse” of the Fisher informa-tion under the addition of noise occurs because θ = nπ correspond to common extrema of C and F . This ob-servation further suggests that the LGI violation at themaximum Fisher information is a hallmark of the robust-ness of the latter against noise.In this way, we have obtained, in the framework ofthis specific model, a connection between the pointswhere invasiveness is witnessed and those correspondingto favourable and noise-robust metrological scenarios. IV. DISCUSSION
Our model sheds light on the relationship between thequantum Fisher information and quantum invasiveness.Some physical insight about this connection has alreadybeen given in Ref. [21] where, by taking into account a“no-signaling in time condition” [35], the authors arguedthat quantum states with large F Q are necessary for LGIviolation with large measurement uncertainties.In order to further investigate this point, we plotted F , | K LG | and F Q as a function of b for fixed values of θ inFig. 2. Specifically, in Fig. 2(a), we have fixed θ/π = 0 . b > . F Q and F increase monotonically as b increases and F approaches its optimal value, F Q , inthe region where LGI is violated. On the other hand,in Fig. 2(b), we take θ/π = 0 .
34 so that no violation ofthe LGI can occur. Note that F Q remains the same as afunction of b , as F Q does not depend on θ . It is thus clearthat large QFI is not a sufficient condition for violationof the LGI.Note as well that, in Fig. 2(b), the Fisher informa-tion increases monotonically as b increases but it doesnot reach its optimal value, F Q . In order to explorein further detail the quantitative relationship betweenthe Fisher information and the LGI violation, we plot inFig. 3 the normalized Fisher information F/ F Q versusthe absolute value of the Leggett-Garg parameter | K LG | (see caption for details). We see that both the maxi-mum (normalized) Fisher information and the maximumof | K LG | monotonically increase with b . Furthermore, itis interesting to note that F/ F Q at the point of maximalviolation of the LGI (solid magenta line) is a monoton-ically increasing function of the violation itself. We canalso see that, even though the maximization of F and | K LG | are generally incompatible, violation of the LGI is b F , ℱ Q b | K L G | ( a ) b F , ℱ Q b | K L G | ( b ) FIG. 2: (color online) Plots of the Fisher information F (solidmagenta line), the quantum Fisher information F Q (dashedmagenta line), and the absolute value of the Leggett-Gargparameter | K LG | (inset, solid black line), as a function of b for (a) θ/π = 0 .
95 and (b) θ/π = 0 .
34. In the insets, the LGIviolation region is shaded in light gray; note that the LGI isviolated in the interval 0 . ≤ b ≤ necessary to access the nearly optimal regime of F/ F Q above ≈ .
82. Finally, we see that, when the LGI is vio-lated, there is a lower bound for the Fisher information,given by F/ F Q > ∼ .
27, thus LGI violation guarantees anon-trivial minimum metrological precision. Conversely,when the LGI is not violated, the Fisher information canbe arbitrarily small and vanish for specific parameter set-tings.
V. CONCLUSION
We have established a connection between temporalcorrelations, involved in Leggett-Garg inequality tests,and the Fisher information associated with a specificmetrological scenario. In particular, guided by the gen- eral expression of the Fisher information in terms of two-time correlation functions, we established that the preci-sion of the estimation is very fragile against noise unless | K LG | F / ℱ Q b = . b = . b = b = . b = . FIG. 3: (color online) Normalized Fisher information, F/ F Q ,versus the absolute value of the Leggett-Garg parameter | K LG | . The dashed gray lines are contours at fixed b for all θ ∈ [0 , π/ b = 0 . , . , . , . , . θ maximizing the LGI vi-olation, while the dashed magenta line connects the points atthe θ maximizing instead the Fisher information. The LGIviolation region is shaded in light gray. All the plotted quan-tities are dimensionless. accompanied by LGI violation. In addition, and lookingat a specific example, we showed that a large quantumFisher information is not sufficient for violating the LGI.We also illustrated how a violation of the LGI may seta non-trivial lower bound to the precision of parameterestimation while, on the other hand, large LGI viola-tions may enable nearly optimal parameter estimation.The ultimate precision limit in which Fisher informa-tion and quantum Fisher information coincide may onlybe achieved in the presence of a violation of the LGI.Generalizations of such intriguing connections betweenmeasurement invasiveness and sensitivity beyond specificmodels certainly deserve further investigation. Acknowlegments
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In the following, we will show that Eq. (5) can be de-rived by considering either ρ + ( θ ) or ρ − ( θ ) as the prepa-ration state. Recall that we can express the two-valuedPOVMs as ˆ E ± = ˆ M †± ˆ M ± = 12 (1 ± ˆ A ) . (A1)In this way, if ρ + ( θ ) is considered, the probabilitiesof obtaining the outcomes ± at the time at which thesecond measurement is performed can be written as P ± ( θ ) = Tr( ˆ E ± ρ + ( θ )) = 12 ± C ( θ ) , (A2)and, if one considers the preparation ρ − ( θ ), we have P ± ( θ ) = Tr( ˆ E ± ρ − ( θ )) = 12 ∓ C ( θ ) ..