Incorporating Heisenberg's Uncertainty Principle into Quantum Multiparameter Estimation
IIncorporating Heisenberg’s Uncertainty Principle into Quantum MultiparameterEstimation
Xiao-Ming Lu ∗ and Xiaoguang Wang † Department of Physics, Hangzhou Dianzi University, Hangzhou 310018, China Zhejiang Institute of Modern Physics and Department of Physics, Zhejiang University, Hangzhou 310027, China
The quantum multiparameter estimation is very different with the classical multiparameter esti-mation due to Heisenberg’s uncertainty principle in quantum mechanics. When the optimal mea-surements for different parameters are incompatible, they cannot be jointly performed. We find acorrespondence relationship between the inefficiency of a measurement for estimating the unknownparameter with the measurement error in the context of measurement uncertainty relations. Takingthis correspondence relationship as a bridge, we incorporate Heisenberg’s uncertainty principle intoquantum multiparameter estimation by giving a tradeoff relation between the measurement ineffi-ciencies for estimating different parameters. For pure quantum states, this tradeoff relation is tight,so it can reveal the true quantum limits on individual estimation errors in such cases. We applyour approach to derive the tradeoff between attainable errors of estimating the real and imaginaryparts of a complex signal encoded in coherent states and obtain the joint measurements attaining thetradeoff relation. We also show that our approach can be readily used to derive the tradeoff betweenthe errors of jointly estimating the phase shift and phase diffusion without explicitly parameterizingquantum measurements.
The random nature of quantum measurement imposesultimate limits on the precision of estimating unknownparameters with quantum systems. Quantum parame-ter estimation theory has been developing for more thanhalf a century to reveal and pursue the quantum-limitedmeasurement [1–9]. In classical parameter estimationtheory, the Cram´er-Rao bound (CRB) together with theasymptotic normality of the maximum likelihood esti-mator give a satisfactory approach to derive the asymp-totically attainable accuracy of estimation, where theFisher information matrix (FIM) plays a pivotal role [10–17]. The CRB and the FIM have been extended toquantum regime [1–6], where not only estimators—dataprocessing—but also quantum measurements are takeninto consideration in optimization.For single parameter estimation, Helstrom’s versionof quantum CRB can be attained at large samplesdue to the asymptotic efficiency of adaptive measure-ments [8, 18–20]. However, unlike the classical param-eter estimation, the quantum CRB does not possess theasymptotic attainability in general for multiparameterestimation. This can be understood as a consequenceof the fact that the optimal measurements for differ-ent parameters may be incompatible in quantum me-chanics so that they cannot be jointly performed accord-ing to Heisenberg’s uncertainty principle (HUP) [21, 22].Many application scenarios, e.g., superresolution imag-ing [23, 24], quantum enhanced estimation of a magneticfield [25, 26], and joint estimation of phase shift andphase diffusion [27], essentially belong to quantum mul-tiparameter estimation problems. Therefore, the char-acterization of the quantum-limited bound on the esti-mation errors is of great importance to many practicalapplications of quantum estimation. Nevertheless, it isstill challenging to derive, characterize, and understand the quantum limit on accuracies for the multiparameterestimation [9, 28–45].To approach the attainable bounds on the estimationerrors, various versions of quantum CRBs have been pro-posed [1–3, 6, 9, 34, 46–48]. These quantum CRBs areformulated by taking a scalar figure of merit, e.g., theweighted mean errors, and formulating the inequalitiesfor the scalar mean errors by utilizing the mathemati-cal structures of the operators on the Hilbert space (e.g.,the Cauchy-Schwarz inequality). It is still unclear howthe HUP affects the multiparameter estimation when theoptimal measurement for individual parameters are in-compatible . In others words, the HUP has not beenincorporated at the first place in quantum multiparame-ter estimation.In this work, we tackle the problem of the fundamen-tal limit on the errors of estimating multiple parametersby incorporating the HUP into quantum multiparameterestimation. We define the regret of Fisher informationfor a quantum measurement that is used to estimate anunknown parameter and shall derive the following corre-spondence relation:
Information Regret ↔ Measurement Error . Taking this relationship as a bridge, we obtain trade-off relations between the information regrets for differentparameters through Branciard’s and Ozawa’s versions ofmeasurement uncertainty relations in terms of the state-dependent measurement error defined by Ozawa [49–54].This tradeoff relation is tight for pure quantum states,so it can faithfully reveal the quantum limits on multipa-rameter estimation errors with pure quantum states. Weshall apply the regret tradeoff relation to the coherentstate estimation and the joint estimation of phase shiftand phase diffusion. a r X i v : . [ qu a n t - ph ] A ug Let us start with a brief introduction on quantum mul-tiparameter estimation. Let θ = ( θ , θ , . . . , θ n ) ∈ R n bean unknown vector parameter, which can be estimatedvia observing a quantum system. The state of the quan-tum system depends on the true value of θ and is de-scribed by a parametric density operator ρ θ . The quan-tum measurement can be characterized by a positive-operator-valued measure (POVM) M = { M x | M x ≥ , (cid:80) x M x = } , where x denotes the outcome and θ byˆ θ = (ˆ θ , ˆ θ , . . . , ˆ θ n ), which is a map from the observationdata to the estimates. The estimation error can be char-acterized by the error-covariance matrix defined by itsentries E jk = E θ [(ˆ θ j − θ j )(ˆ θ k − θ k )] , where the expectation E θ [ • ] is taken with respect to the observation data withthe joint probability mass function p θ ( x , x , . . . , x ν ) = (cid:81) νj =1 tr( M x ν ρ θ ) with ν being the number of experimentalruns. The error-covariance matrix of any unbiased esti-mator ˆ θ obeys the CRB E ≥ ν − F − in the sense thatthe matrix E − ν − F − is positive semi-definite [14–17],where F is the (classical) FIM for a single experimentalrun and defined by F jk = E θ (cid:20) ∂ ln p θ ( x ) ∂θ j ∂ ln p θ ( x ) ∂θ k (cid:21) (1)with p θ ( x ) = tr( M x ρ θ ). The CRB is asymptotically at-tainable by the maximum likelihood estimator [10, 11],whose distribution at a large ν is approximate to a multi-variate normal distribution with the mean being the truevalue of θ and the covariance matrix being ν − F − , ac-cording to the central limit theorem [15, Theorem 9.27].The FIM depends on the quantum measurement via p θ ( x ) = tr( M x ρ θ ), so does the CRB. We use F ( M ) toexplicitly indicate the dependence of F on a POVM M .Quantum parameter estimation takes into considerationthe optimization over quantum measurements. For anyquantum measurement, the FIM is bounded by the fol-lowing matrix inequality: [18, 55] F ( M ) ≤ F , (2)where F is the so-called quantum FIM, also known as theHelstrom information matrix [1, 2]. The quantum FIMis the real part of a Hermitian matrix Q (i.e., F = Re Q )defined by Q jk = tr( L j L k ρ θ ) , (3)where L j , the symmetric logarithmic derivative (SLD)operator for θ j , is a Hermitian operator satisfying ( L j ρ θ + ρ θ L j ) / ∂ρ θ / ∂θ j . Combining Eq. (2) with the CRByields the quantum CRB E ≥ ν − F − for any quantummeasurement and any unbiased estimator. This quan-tum CRB was first obtained by Helstrom with a differentmethod [1, 2].To characterize the efficiency of a quantum measure-ment for multiparameter estimation, we here define the regret of Fisher information by R ( M ) = F − F ( M ) . (4)This matrix R ( M ) is positive semi-definite due to Eq. (2)and real symmetric as both the quantum and classicalFIMs are real symmetric according to their definitions.For single-parameter estimation, Braunstein and Cavesproved that the classical Fisher information can equalthe quantum Fisher information with an optimal quan-tum measurement [18] and thus the regret R ( M ) thereofvanishes. In the multiparameter setting, for any col-umn vector v ∈ R n , there exist a quantum measurement M such that v (cid:62) R ( M ) v = 0, where (cid:62) denotes matrixtranspose. This is because v (cid:62) F ( M ) v and v (cid:62) F v canbe interpreted as the classical and quantum Fisher in-formation, respectively, about a parameter ϕ satisfying ∂ / ∂ϕ = (cid:80) j v j ∂ / ∂θ j . The POVM M making vR ( M ) v (cid:62) vanish can be considered as an optimal measurement forestimating ϕ and in general depends on v . For differ-ent parameters, the optimal measurement may be differ-ent and even incompatible. Consequently, the entries of R ( M ) in general cannot simultaneously vanish, which isa manifestation of HUP. In what follows, we shall give aquantitative characterization of the mechanism in whichthe HUP affects the regret matrix of Fisher information.Define by ∆ j = (cid:112) R jj / F jj the normalized-square-rootregret of Fisher information with respect to θ j . Note that∆ j takes value in the interval [0 , j + ∆ k + 2 (cid:113) − c jk ∆ j ∆ k ≥ c jk , (5)where c jk is a real number given by c jk = | Im Q jk | (cid:112) Re Q jj Re Q kk = | Im Q jk | (cid:112) F jj F kk (6)with Q jk being given by Eq. (3). For nonzero c jk , Eq. (5)describes the tradeoff between the regrets of Fisher infor-mation with respect to different parameters. For a family ρ θ of pure states, the inequality Eq. (5) is tight, in thesense that there exists a quantum measurement M suchthat the equality in Eq. (5) holds; In such a case, our re-sult fully reflects the tradeoff between different regrets ofFisher information. For mixed states ρ θ , the inequalityEq. (5) can be tightened by replacing c jk thereof by itsvariant (cid:101) c jk = tr (cid:12)(cid:12) √ ρ θ [ L j , L k ] √ ρ θ (cid:12)(cid:12) (cid:112) F jj F kk , (7)where | X | = √ X † X for an operator X . Note that the co-efficient (cid:101) c jk is not less than c jk for all quantum states andequal to c jk for all pure states. We also give the secondform of the tradeoff relation in terms of the estimationerrors: γ j + γ k − (cid:113) − (cid:101) c jk (cid:113) (1 − γ j )(1 − γ k ) ≤ − (cid:101) c jk , (8) ρ θ | i| i U Φ( ρ θ ) e L j e L k FIG. 1. Unitary implementation (the dashed box) of themeasurement channel. The thick red lines stand for the inputand output ports. The commuting observables (cid:101) L j and (cid:101) L k canbe jointly measured in the output state Φ( ρ θ ), which in theHeisenberg picture is equivalent to the joint measurement ofa pair of commuting observables L j = U † (1 s ⊗ (cid:101) L j ⊗ r ) U and L k = U † (1 s ⊗ (cid:101) L k ⊗ r ) U in the initial state ρ total = ρ θ ⊗ | (cid:105)(cid:104) | ⊗ | (cid:105)(cid:104) | of the entirety. where we have defined γ j = 1 / ( ν E jj F jj ) for simplicity.The above inequality is a result of combining Eq. (5) withthe classical CRB E jj ≥ ν − ( F − ) jj ≥ / ( νF jj ).We here outline the proof of Eq. (5) and leave the de-tails in the Supplemental Material [56]. Denote by H s the Hilbert space associated with the underlying quan-tum system. For a given POVM M on H s , we define ameasurement channel Φ( ρ ) = (cid:80) x tr( M x ρ ) | x (cid:105)(cid:104) x | , where {| x (cid:105)} is an orthonormal basis associated with the mea-surement outcomes x ’s and span another Hilbert space H r . Note that the density operators Φ( ρ θ ) are alwaysdiagonal with the basis {| x (cid:105)} . As a result, the SLD oper-ators of Φ( ρ θ ) are also diagonal with the basis {| x (cid:105)} andcan be represented as (cid:101) L j = (cid:88) x ∂ ln tr( M x ρ θ ) ∂θ j | x (cid:105)(cid:104) x | . (9)The measurement channel Φ can be implemented by aunitary operation U acting on H s ⊗ H r ⊗ H r such thatΦ( ρ ) = tr , (cid:2) U ( ρ ⊗ | (cid:105)(cid:104) | ⊗ | (cid:105)(cid:104) | ) U † (cid:3) (10)for all density operators ρ on H s , where tr , denotes thepartial trace over the first and third tensor factors of theHilbert space and | (cid:105) can be an arbitrary initial state [57,Chapter 2] (see Fig. 1 for a schematic illustration). Usingthe techniques developed in Ref. [58], we show that [56] R jj = tr (cid:2) ( L j − L j ⊗ r ⊗ r ) ρ total (cid:3) , (11)where L j = U † ( s ⊗ (cid:101) L j ⊗ r ) U with s and r beingthe identity operators on H s and H r , respectively, and ρ total = ρ θ ⊗ | (cid:105)(cid:104) | ⊗ | (cid:105)(cid:104) | .We observe that R jj expressed in Eq. (11) is of thesame form as the square of Ozawa’s definition of mea-surement error [49, 50, 59], when taking L j as the idealobservable we intend to measure and L j as the observableactually measured. We list in Table I the correspondencerelation between the parameter estimation scenario and TABLE I. Correspondence relation.Estimation-regret scenario Measurement-error scenarioregret of Fisher information measurement errorSLD L j of ρ θ ideal observable L j on ρ θ SLD (cid:101) L j of Φ( ρ θ ) approximate observable on Φ( ρ θ ) L j = U † (1 s ⊗ (cid:101) L j ⊗ r ) U approximate observable on ρ θ the measurement error scenario. Notice that the Her-mitian operators L j and L k always commutes, as both (cid:101) L j and (cid:101) L k are diagonal with the basis {| x (cid:105)} . There-fore, the observables L j and L k can be jointly mea-sured in quantum mechanics. When two ideal observ-ables L j and L k do not commute, it may be impossibleto make their measurement errors, which equals the re-grets R jj and R kk in our context, simultaneously vanish.By invoking the measurement uncertainty relations [49–51, 53, 54, 60] in terms of Ozawa’s definition of mea-surement error, we can derive the tradeoff relation be-tween the regrets of Fisher information with respect todifferent parameters. Concretely, the inequality Eq. (5)follows from Branciard’s version of measurement uncer-tainty relation, which is tight for pure states [53]. UsingOzawa’s work on strengthening Branciard’s inequality formixed states [60], the inequality Eq. (5) can be tightenedthrough replacing c jk by (cid:101) c jk .It is worthy to point out that we do not designate theSLD operator as the ideal observable in reality to opti-mally estimate an individual parameter. Although theeigenstates of the SLD operator, which possibly dependon the true value of the parameter, in principle consti-tute a measurement basis extracting the maximum Fisherinformation at a parameter point [18], it is possible forsome models to find a global optimal measurement thatis independent of the parameter [6, 19]; A global opti-mal measurement is often more ideal than a local one forestimating the unknown parameter.We can give an operational significance to the coeffi-cients (cid:101) c jk through the tradeoff relation Eq. (5) as follows.If the QFI about a parameter θ j is exhaustively extractedby a quantum measurement M , i.e., ∆ j = 0, then it fol-lows from Eq. (5) that the regret for any other parameter θ k obeys ∆ k ≥ (cid:101) c jk . That is, (cid:101) c jk is the lower bound onthe residual regret for θ k when there is no regret for θ j .For pure states, this lower bounds c jk can be attained asEq. (5) is tight in such cases.Let us now consider as an example the estimation of acomplex number α encoded in a coherent state [61] | α (cid:105) = e −| α | / (cid:80) ∞ n =0 ( α n / √ n !) | n (cid:105) , where | n (cid:105) ’s are the numberstates. The parameters of interest here are the real andimaginary parts of α , i.e., θ = Re α and θ = Im α . Aftersome algebras, we get Q = 4 (cid:0) i − i (cid:1) and thus c = 1.The regret tradeoff Eq. (5) then becomes ∆ + ∆ ≥ F + F ≤ . . . . . . ν E ν E HUP incorporatedRLD, geometric meanRLD, arithmetic meanSLD, harmonic mean
FIG. 2. Permissible errors of estimating a complex number α encoded in a coherent state. The regions below the curvesare forbidden by the corresponding inequalities. The blacksolid curve stands for the tradeoff relation obtained in thiswork, which can be asymptotically attained by an optimalmeasurement. The other three curves stand for the tradeoffrelations that have been systematically analyzed in Ref. [34];They are obtained by generalized-mean Cram´er-Rao boundsbased on the SLD and the right logarithmic derivative (RLD). information or 1 ν E + 1 ν E ≤ a the annihilation operators for themode for which the coherent state is defined. The mea-surements of the quadrature components Q = ( a + a † ) / P = ( a − a † ) / (2 i ) are natural for estimating the co-herent signal, as (cid:104) α | Q | α (cid:105) = Re α and (cid:104) α | P | α (cid:105) = Im α .Indeed, the maximum Fisher information about θ and θ can be obtained by measuring Q and P , respectively,corresponding to either F = 4 or F = 4. However, Q and P are not commuting so that they cannot be jointlymeasured. It is known that we can jointly measure thecommuting operators Q − Q (cid:48) and P + P (cid:48) , where Q (cid:48) and P (cid:48) are the quadrature components of an ancillary mode(whose annihilation operator is denoted by a (cid:48) ) in the vac-uum state, to estimate the real and imaginary parts of α , see Refs. [3, 5, 6]. This measurement strategy at-tains the minimum unweighted arithmetic mean error ofestimation with F = F = 2, see the blue circle in Fig. 2. We show in the Supplemental Material [56] thatother error combinations on the bound Eq. (12) can beasymptotically attained if we prepare the ancillary modein a squeezed vacuum state exp (cid:2) ( ra (cid:48) − ra (cid:48)† ) (cid:3) | (cid:105) with r ∈ R . In such case, the extracted FIM can be tuned bychanging r as F = 4 / ( e r + 1), F = 4 e r / ( e r + 1),and F = F = 0. Moreover, the joint probability den-sity function of outcomes of Q − Q (cid:48) and P + P (cid:48) are bothGaussian, so taking their sample means as the estimatesfor θ and θ asymptotically attains the classical CRB.In our second example, we consider the joint estimationof phase shift and phase diffusion [27]. For a two-modeprobe state, the parametric density operator can be effec-tively simplified as ρ = ( | (cid:105)(cid:104) | + | (cid:105)(cid:104) | + e − iθ − θ | (cid:105)(cid:104) | + e iθ − θ | (cid:105)(cid:104) | ), where θ stands for the phase shift and θ the phase diffusion. In Ref. [27], Vidrighin et al. ob-tained the tradeoff relation F / F + F / F ≤ etal. ’s tradeoff relations follows from our regret tradeoff re-lation Eq. (5) in a very easy way. We only need to show (cid:101) c = 1 by a straightforward calculation according toits definition (see the Supplemental Material [56] for thedetails). As a result, we get ∆ + ∆ ≥
1, which is equiv-alent to Vidrighin et al. ’s tradeoff relation by recognizing∆ j = 1 − F jj / F jj .In conclusion, we have incorporated the HUP intoquantum multiparameter estimation by deriving a trade-off relation between the regrets of Fisher informationabout different parameters. Unlike the quantum CRBson scalar mean errors, the regrets tradeoff quantitativelycharacterizes how the HUP affects the combinations ofestimation errors for multiple parameters. The corre-spondence relationship we found between information re-gret and measurement error also, as a bonus, supplies anoperational meaning to Ozawa’s definition of the state-dependent measurement error, on which there exists acontroversy for a long time [59, 62, 63].Our approach also opens a new perspective on quan-tum geometry. The matrix Q defined by Eq. (3) is knownas the quantum geometric tensor on the manifold of phys-ical quantum state, up to an insignificant constant fac-tor [64, 65]. The real part of Q —the quantum FIM—gives a Riemannian metric on the manifolds of quantumstates. The imaginary part of Q gives a curvature form ofBerry’s connection [65], which has relations to the quan-tum FIM [28, 66] and the density of quantum states [67].It is known that a zero curvature is necessary for thesimultaneous vanishing of the regrets of Fisher informa-tion about different parameter [8, 33, 36, 48]. Note thatin our tradeoff relation, c jk is the curvature divided by ascalar related to the entries of the quantum FIM. So ourtradeoff relation quantitatively characterize the intricatemechanism in which the simultaneous reduction of theregrets of Fisher information about different parametersis restricted by a nonzero quantum curvature, which isindicated as Information Regret ← Quantum Curvature . Carollo et al. has proposed an incompatibility index,which is similar to c jk , based on the ratio between thecurvature and the quantum FIM as a figure of merit forthe quantumness of a quantum multiparameter estima-tion model [28]. In addition, since (cid:101) c jk is better than c jk to manifest the regrets tradeoff for mixed states, it maybe possible to take the quantity tr |√ ρ θ [ L j , L k ] √ ρ θ | asan alternative form of quantum curvature.This work is supported by the National NaturalScience Foundation of China (Grants No. 61871162,No. 11805048, and No. 11935012) and Zhejiang Provin-cial Natural Science Foundation of China (GrantNo. LY18A050003). ∗ [email protected]; http://xmlu.me † [email protected][1] C.W. Helstrom, “Minimum mean-squared error of esti-mates in quantum statistics,” Phys. Lett. A , 101–102(1967).[2] C.W. Helstrom, “The minimum variance of estimates inquantum signal detection,” IEEE Trans. Inform. Theory , 234–242 (1968).[3] H. Yuen and M. Lax, “Multiple-parameter quantum esti-mation and measurement of nonselfadjoint observables,”IEEE Trans. Inform. Theory , 740–750 (1973).[4] V. P. Belavkin, “Generalized uncertainty relations and ef-ficient measurements in quantum systems,” Theor. Math.Phys. , 213–222 (1976).[5] Carl. W. Helstrom, Quantum Detection and EstimationTheory (Academic Press, New York, 1976).[6] A. S. Holevo,
Probabilistic and Statistical Aspects ofQuantum Theory (North-Holland, Amsterdam, 1982).[7] S.D. Personick, “Application of quantum estimation the-ory to analog communication over quantum channels,”IEEE Trans. Inform. Theory , 240–246 (1971).[8] Masahito Hayashi, ed., Asymptotic theory of quantumstatistical inference: Selected Papers (World ScientificPublishing Company, 2005).[9] Mankei Tsang, Francesco Albarelli, and Animesh Datta,“Quantum semiparametric estimation,” Phys. Rev. X ,031023 (2020).[10] R. A. Fisher, “On the mathematical foundations of the-oretical statistics,” Philosophical Transactions of theRoyal Society of London. Series A, Containing Papersof a Mathematical or Physical Character , 309–368(1922).[11] R. A. Fisher, “Theory of statistical estimation,” Mathe-matical Proceedings of the Cambridge Philosophical So-ciety , 700–725 (1925).[12] H. Cram´er, Mathematical Methods of Statistics (Prince-ton University Press, Princeton, 1946).[13] C. Radhakrishna Rao, “Information and the accuracyattainable in the estimation of statistical parameters,”
Breakthroughs in Statistics: Foundations and Basic The-ory , Bull. Calcutta Math. Soc. , 81–89 (1945).[14] Steven M. Kay, Fundamentals of Statistical Signal Pro-cessing, Volume I: Estimation Theory (Prentice Hall,1993).[15] Larry Wasserman,
All of Statistics: A Concise Coursein Statistical Inference (Springer Publishing Company,Incorporated, 2010).[16] George Casella and Roger L. Berger,
Statistical Inference ,2nd ed. (Duxbury Press, Pacific Grove, 2002).[17] E.L. Lehmann and George Casella,
Theory of Point Es-timation (Springer-Verlag, New York, 1998).[18] Samuel L. Braunstein and Carlton M. Caves, “Statisti-cal distance and the geometry of quantum states,” Phys.Rev. Lett. , 3439–3443 (1994).[19] Samuel L. Braunstein, Carlton M. Caves, and G.J. Mil-burn, “Generalized uncertainty relations: Theory, exam-ples, and lorentz invariance,” Ann. Phys. , 135 – 173(1996).[20] Akio Fujiwara, “Strong consistency and asymptotic ef-ficiency for adaptive quantum estimation problems,” J.Phys. A: Math. Gen. , 12489 (2006).[21] W. Heisenberg, “The physical content of quantum kine-matics and mechanics,” Z. Phys. , 172 (1927), Englishtranslation in Quantum Theory and Measurement , editedby J. A. Wheeler and W. H. Zurek (Princeton Univ.Press, Princeton, NJ, 1984), p. 62.[22] Paul Busch, Teiko Heinonen, and Pekka Lahti, “Heisen-berg’s uncertainty principle,” Phys. Rep. , 155 – 176(2007).[23] Mankei Tsang, Ranjith Nair, and Xiao-Ming Lu, “Quan-tum theory of superresolution for two incoherent opticalpoint sources,” Phys. Rev. X , 031033 (2016).[24] Mankei Tsang, “Resolving starlight: a quantum perspec-tive,” Contemp. Phys. , 279–298 (2019).[25] Tillmann Baumgratz and Animesh Datta, “Quantum en-hanced estimation of a multidimensional field,” Phys.Rev. Lett. , 030801 (2016).[26] Zhibo Hou, Zhao Zhang, Guo-Yong Xiang, Chuan-FengLi, Guang-Can Guo, Hongzhen Chen, Liqiang Liu, andHaidong Yuan, “Minimal tradeoff and ultimate preci-sion limit of multiparameter quantum magnetometry un-der the parallel scheme,” Phys. Rev. Lett. , 020501(2020).[27] Mihai D. Vidrighin, Gaia Donati, Marco G. Genoni,Xian-Min Jin, W. Steven Kolthammer, M.S. Kim, Ani-mesh Datta, Marco Barbieri, and Ian A. Walmsley,“Joint estimation of phase and phase diffusion for quan-tum metrology,” Nat. Commun. , – (2014).[28] Angelo Carollo, Bernardo Spagnolo, Alexander ADubkov, and Davide Valenti, “On quantumness in multi-parameter quantum estimation,” J. Stat. Mech: TheoryExp. , 094010 (2019).[29] Jes´us Rubio, Paul Knott, and Jacob Dunningham, “Non-asymptotic analysis of quantum metrology protocols be-yond the Cram´er–Rao bound,” Journal of Physics Com-munications , 015027 (2018).[30] Francesco Albarelli, Jamie F. Friel, and Animesh Datta,“Evaluating the Holevo Cram´er-Rao bound for multi-parameter quantum metrology,” Phys. Rev. Lett. ,200503 (2019).[31] Mankei Tsang, “The Holevo Cram´er-Rao bound is atmost thrice the Helstrom version,” arXiv:1911.08359.[32] Francesco Albarelli, Mankei Tsang, and Animesh Datta, “Upper bounds on the Holevo Cram´er-Rao bound formultiparameter quantum parametric and semiparamet-ric estimation,” arXiv:1911.11036v1.[33] Jasminder S. Sidhu and Pieter Kok, “Geometric perspec-tive on quantum parameter estimation,” AVS QuantumSci. , 014701 (2020).[34] Xiao-Ming Lu, Zhihao Ma, and Chengjie Zhang,“Generalized-mean Cram´er-Rao bounds for multiparam-eter quantum metrology,” Phys. Rev. A , 022303(2020).[35] Jasminder S. Sidhu, Yingkai Ouyang, Earl T. Campbell,and Pieter Kok, “Tight bounds on the simultaneous esti-mation of incompatible parameters,” arXiv:1912.09218.[36] Sammy Ragy, Marcin Jarzyna, and Rafa(cid:32)l Demkowicz-Dobrza´nski, “Compatibility in multiparameter quantummetrology,” Phys. Rev. A , 052108 (2016).[37] Nan Li, Christopher Ferrie, Jonathan A. Gross, AmirKalev, and Carlton M. Caves, “Fisher-symmetric in-formationally complete measurements for pure states,”Phys. Rev. Lett. , 180402 (2016).[38] Huangjun Zhu and Masahito Hayashi, “UniversallyFisher-symmetric informationally complete measure-ments,” Phys. Rev. Lett. , 030404 (2018).[39] Jun Suzuki, “Explicit formula for the Holevo bound fortwo-parameter qubit-state estimation problem,” J. Math.Phys. , 042201 (2016).[40] Jun Suzuki, “Information geometrical characterization ofquantum statistical models in quantum estimation the-ory,” Entropy , 703 (2019).[41] Jun Suzuki, Yuxiang Yang, and Masahito Hayashi,“Quantum state estimation with nuisance parameters,”arXiv:1911.02790v3.[42] Ilya Kull, Philippe Allard Gu´erin, and Frank Verstraete,“Uncertainty and trade-offs in quantum multiparameterestimation,” J. Phys. A: Math. Theor. , 244001 (2020).[43] Angelo Carollo, Davide Valenti, and Bernardo Spagnolo,“Geometry of quantum phase transitions,” Phys. Rep. , 1–72 (2020).[44] Nan Li and Shunlong Luo, “Fisher concord: Efficiencyof quantum measurement,” Quantum Measurements andQuantum Metrology , 44–52 (2017).[45] Jing Liu, Haidong Yuan, Xiao-Ming Lu, and XiaoguangWang, “Quantum Fisher information matrix and mul-tiparameter estimation,” J. Phys. A: Math. Theor. ,023001 (2020).[46] Richard D. Gill and Serge Massar, “State estimation forlarge ensembles,” Phys. Rev. A , 042312 (2000).[47] Hiroshi Nagaoka, “A new approach to Cram´er-Raobounds for quantum state estimation,” in AsymptoticTheory of Quantum Statistical Inference , edited byMasahito Hayashi (World Scientific Publishing Co. Pte.Ltd., Singapore, 2005) pp. 100–112.[48] K Matsumoto, “A new approach to the Cram´er-Rao-typebound of the pure-state model,” J. Phys. A: Math. Gen. , 3111 (2002).[49] Masanao Ozawa, “Universally valid reformulation of theHeisenberg uncertainty principle on noise and distur-bance in measurement,” Phys. Rev. A , 042105 (2003).[50] Masanao Ozawa, “Uncertainty relations for joint mea-surements of noncommuting observables,” Phys. Lett. A , 367 – 374 (2004).[51] Michael J. W. Hall, “Prior information: How to circum- vent the standard joint-measurement uncertainty rela-tion,” Phys. Rev. A , 052113 (2004).[52] Morgan M. Weston, Michael J. W. Hall, Matthew S. Pals-son, Howard M. Wiseman, and Geoff J. Pryde, “Ex-perimental test of universal complementarity relations,”Phys. Rev. Lett. , 220402 (2013).[53] Cyril Branciard, “Error-tradeoff and error-disturbancerelations for incompatible quantum measurements,”Proc. Natl. Acad. Sci. U.S.A. , 6742–6747 (2013).[54] Xiao-Ming Lu, Sixia Yu, Kazuo Fujikawa, and C. H. Oh,“Improved error-tradeoff and error-disturbance relationsin terms of measurement error components,” Phys. Rev.A , 042113 (2014).[55] Fumio Hiai and Dnes Petz, Introduction to Matrix Anal-ysis and Applications , 1st ed., Universitext (Springer In-ternational Publishing, Cham, 2014).[56] See Supplemental Material for detailed derivations.[57] M. M. Wolf, “Quantum channels & operations–guided tour,” Online available at .[58] Xiao-Ming Lu, Sixia Yu, and C. H. Oh, “Robust quan-tum metrological schemes based on protection of quan-tum Fisher information,” Nat. Commun. , 7282 (2015).[59] Masanao Ozawa, “Soundness and completeness of quan-tum root-mean-square errors,” npj Quantum Information , 1 (2019).[60] Masanao Ozawa, “Error-disturbance relations in mixedstates,” arXiv:1404.3388 [quant-th].[61] Roy J. Glauber, “Coherent and incoherent states of theradiation field,” Phys. Rev. , 2766–2788 (1963).[62] Paul Busch, Pekka Lahti, and Reinhard F. Werner, “Col-loquium: Quantum root-mean-square error and measure-ment uncertainty relations,” Rev. Mod. Phys. , 1261–1281 (2014).[63] David Marcus Appleby, “Quantum errors and distur-bances: Response to Busch, Lahti and Werner,” Entropy , 174 (2016).[64] J.P. Provost and G. Vallee, “Riemannian structure onmanifolds of quantum states,” Commun. Math. Phys. ,289–301 (1980).[65] M.V. Berry, “The quantum phase, five years after,” in Geometric Phases in Physics , edited by A. Shapere andF. Wilczek (World Scientific, Singapore, 1989) Chap. 1.1,pp. 7–28.[66] Wei Guo, Wei Zhong, Xiao-Xing Jing, Li-Bin Fu, andXiaoguang Wang, “Berry curvature as a lower bound formultiparameter estimation,” Phys. Rev. A , 042115(2016).[67] Haijun Xing and Libin Fu, “Measure of the density ofquantum states in information geometry and its ap-plication in the quantum multi-parameter estimation,”arXiv:2006.00203.[68] E. Arthurs and J. L. Kelly, “On the simultaneous mea-surement of a pair of conjugate observables,” Bell Syst.Tech. J. , 725 (1965).[69] Carl W. Helstrom, “Cram´er-Rao inequalities foroperator-valued measures in quantum mechanics,” Int.J. Theor. Phys. , 361–376 (1973).[70] J. C. Garrison and R. Y. Chiao, Quantum optics (OxfordUniversity Press, New York, 2008).
Supplemental material
DETAILED DERIVATION OF THE REGRET TRADEOFF RELATIONRegret of Fisher information
Firstly, let H s be the Hilbert space associated with the underlying quantum system, and S ( H ) the set of alldensity operators on a Hilbert space H . For a given POVM M acting on H s , define the measurement channelΦ : S ( H s ) → S ( H r ) by Φ : ρ (cid:55)→ (cid:88) x tr( M x ρ ) | x (cid:105)(cid:104) x | , (S1)where H r is the Hilbert space for a register that associates the outcomes x with an orthonormal basis {| x (cid:105)} . Theoutput density operators Φ( ρ θ ) are diagonal with the basis {| x (cid:105)} and its SLD operator with respect to θ j can beexpressed as (cid:101) L j = ∂ ln tr( M x ρ θ ) ∂θ j | x (cid:105)(cid:104) x | . (S2)Denote by (cid:101) F the quantum Fisher information matrix of Φ( ρ θ ). Note that (cid:101) F equals the classical Fisher informationmatrix under the measurement M , i.e., (cid:101) F = F ( M ).Secondly, it can be shown that (cid:101) F jk = Re tr (cid:104)(cid:101) L j (cid:101) L k Φ( ρ θ ) (cid:105) = tr (cid:34)(cid:101) L j (cid:101) L k Φ( ρ θ ) + Φ( ρ θ ) (cid:101) L k (cid:35) = tr (cid:20)(cid:101) L j ∂ Φ( ρ θ ) ∂θ k (cid:21) = tr (cid:20)(cid:101) L j Φ (cid:18) ∂ρ θ ∂θ k (cid:19)(cid:21) , (S3)where we have used the SLD equation ∂ Φ( ρ θ ) ∂θ j = 12 (cid:104)(cid:101) L j Φ( ρ θ ) + Φ( ρ θ ) (cid:101) L j (cid:105) (S4)in the third equality and ∂ Φ( ρ θ ) /∂θ j = Φ( ∂ρ θ /∂θ j ) in the fourth equality. Now, let us introduce the dual map Φ † that satisfies tr (cid:2) Φ † ( X ) ρ (cid:3) = tr[ X Φ( ρ )] (S5)for any density operator ρ on H s and any bounded operator X on H r . It then follows that (cid:101) F jk = tr (cid:20) Φ † ( (cid:101) L j ) ∂ρ θ ∂θ k (cid:21) = Re tr (cid:16) Φ † ( (cid:101) L j ) L k ρ θ (cid:17) , (S6)where we have used the SLD equation ∂ρ θ /∂θ k = ( L k ρ θ + ρ θ L k ) /
2. Since the quantum Fisher information matrix issymmetric, by interchanging the subscripts j and k in the right hand side of Eq. (S6), we can also get (cid:101) F jk = Re tr (cid:104) Φ † ( (cid:101) L k ) L j ρ θ (cid:105) . (S7)For the simplicity of notation, let us define (cid:104)•(cid:105) := tr( • ρ θ ). By noting that (cid:101) F jk = Re (cid:104) L j Φ † ( (cid:101) L k ) (cid:105) = Re (cid:104) Φ † ( (cid:101) L j ) L k (cid:105) = Re (cid:104) Φ † ( (cid:101) L j (cid:101) L k ) (cid:105) , we show that the entries of the regret matrix can be expressed as R jk := F jk − (cid:101) F jk = Re (cid:104) L j L k − L j Φ † ( (cid:101) L k ) − Φ † ( (cid:101) L j ) L k + Φ † ( (cid:101) L j (cid:101) L k ) (cid:105) . (S8)Thirdly, we shall show that the expression Eq. (S8) of the regret matrix entries can be rewritten in a more elegantway through the open-system representation of quantum channels. There always exist a unitary operator U on H s ⊗ H r ⊗ H r such that Φ( ρ ) = tr , [ U ( ρ ⊗ | (cid:105)(cid:104) | ⊗ | (cid:105)(cid:104) | ) U † ] (S9)for all density operators ρ on H s , where | (cid:105) can be any state in H r and tr , is the partial trace over the first andthird tensor factors of H s ⊗ H r ⊗ H r (see Ref. [57, Chapter 2]). With this open-system representation of Φ, it can beshown that tr[ X Φ( ρ )] = tr (cid:2) ( s ⊗ X ⊗ r ) U ( ρ ⊗ | (cid:105)(cid:104) | ⊗ | (cid:105)(cid:104) | ) U † (cid:3) , (S10)implying that Φ † ( X ) = tr , (cid:2) U † ( s ⊗ X ⊗ r ) U ( s ⊗ | (cid:105)(cid:104) | ⊗ | (cid:105)(cid:104) | ) (cid:3) . (S11)With this open-system representation of the dual map of Φ, Eq. (S8) can be expressed as R jk = Re tr[ N j N k ( ρ θ ⊗ | (cid:105)(cid:104) | ⊗ | (cid:105)(cid:104) | )] , (S12)where N j := L j − L j ⊗ r ⊗ r with L j := U † ( s ⊗ (cid:101) L j ⊗ r ) U . Measurement uncertainty relation
We here briefly introduce the measurement uncertainty relations, which will be invoked to derive the regret tradeoffrelation. Let A and B be two Hermitian operators standing for the ideal observables we intend to measure. Inquantum mechanics, when [ A, B ] (cid:54) = 0, these two observables cannot be jointly measured. To approximate the jointmeasurement of A and B when [ A, B ] (cid:54) = 0, we can measure another pair of commuting observables A and B acting onthe system possibly dilated by adding an ancilla whose state is denoted by a density operator η [68]. Ozawa proposedto quantify the (state-dependent) measurement errors for the ideal observables A and B in the quantum state ρ by (cid:15) A = (cid:112) tr[( A − A ⊗ ( ρ ⊗ η )] and (cid:15) B = (cid:112) tr[( B − B ⊗ ( ρ ⊗ η )] , (S13)respectively, and derived the following measurement uncertainty relation [49, 50]: (cid:15) A (cid:15) B + (cid:15) A σ B + (cid:15) B σ A ≥ C AB := 12 | tr([ A, B ] ρ ) | , (S14)where σ A := (cid:113) tr( A ρ ) − tr( Aρ ) and σ B := (cid:113) tr( B ρ ) − tr( Bρ ) . Branciard obtained a stronger inequality [53]: (cid:15) A σ B + (cid:15) B σ A + 2 (cid:113) σ A σ B − C AB (cid:15) A (cid:15) B ≥ C AB , (S15)which implies Ozawa’s inequality and is tight when ρ is a pure state. For mixed states, Ozawa showed that Branciard’sinequality can be strengthen by replacing C AB by D AB := 12 tr |√ ρ [ A, B ] √ ρ | (S16)with | X | := √ X † X for an operator X . Derivation of the regret tradeoff relation
Now, we derive our regret tradeoff relation. It follows from Eq. (S12) that R jj = tr (cid:2) ( L j − L j ⊗ r ⊗ r ) ( ρ ⊗ | (cid:105)(cid:104) | ⊗ | (cid:105)(cid:104) | ) (cid:3) , (S17)which is in the form of Ozawa’s definition of measurement error by taking L j → A , L j → A , and | (cid:105)(cid:104) | ⊗ | (cid:105)(cid:104) | → η .Correspondingly, we have σ A = (cid:113) tr (cid:0) L j ρ (cid:1) − tr( L j ρ ) = (cid:112) F jj . (S18)Let us consider another parameter θ k and take L k → B and L k → B . It is easy to see that [ L j , L k ] = 0 for [ (cid:101) L j , (cid:101) L k ] = 0.Consequently, the square roots of the regret of Fisher information for different parameters θ j and θ k , i.e., (cid:112) R jj and √ R kk , can be considered as the measurement errors of measuring a pair of commuting observable L j and L k in adilated system to approximate the measurement of L j and L k , which are unable to be jointly measured for the caseof [ L j , L k ] (cid:54) = 0. It then follows from Branciard’s inequality that R jj F kk + R kk F jj + 2 (cid:113) F jj F kk − C jk (cid:112) R jj R kk ≥ C jk , (S19)where C jk := | tr([ L j , L k ] ρ ) | . Dividing both sides of the above inequality by F jj F kk , we get our regret tradeoffrelation in the main text. ESTIMATING COHERENT STATE
Here, we give the detailed calculations for the first example in the main text, i.e., the joint estimation of parameters θ = Re α and θ = Im α in coherent states | α (cid:105) . Due to ∂ | α (cid:105) ∂θ = ( − θ + a † ) | ψ (cid:105) , ∂ | α (cid:105) ∂θ = ( − θ + ia † ) | ψ (cid:105) (S20)and Q jk = 4 (cid:18) ∂ (cid:104) α | ∂θ j (cid:19) ( − | α (cid:105)(cid:104) α | ) (cid:18) ∂ | α (cid:105) ∂θ k (cid:19) , (S21)it can be shown that Q = 4 (cid:18) i − i (cid:19) . (S22)We then get the regret tradeoff relation R + R ≥
4, which is equivalent to F + F ≤ θ and θ by using an ancillary mode whose annihilation operator is denoted by a (cid:48) and satisfies [ a (cid:48) , a † ] = 0. Define thedimensionless coordinate and momentum operators for these two modes by Q = a + a † , P = a − a † i ,Q (cid:48) = a (cid:48) + a (cid:48)† , P = a (cid:48) − a (cid:48)† i . (S23)Note that [ Q, P ] = i/ a, a † ] = 1. The two observables A = Q − Q (cid:48) and B = P + P (cid:48) can be jointlymeasured, as they are commuting. It can be shown that (cid:104)A(cid:105) = θ , (cid:104)B(cid:105) = θ , ∆ A = ∆ Q + ∆ Q (cid:48) = 1 /
2, and∆ B = ∆ P + ∆ P (cid:48) = 1 / Q − Q (cid:48) and P + P (cid:48) , we need obtain thejoint probability density function with respect to the corresponding outcomes, which are denoted by ξ and η . Denoteby | ξ, η (cid:105) the simultaneous eigenstates of the commuting observables Q − Q (cid:48) and P + P (cid:48) . It is known that [69] | ξ, η (cid:105) = π − / (cid:90) e iηq | q (cid:105) Q ⊗ | q − ξ (cid:105) Q (cid:48) d q , (S24)where | q (cid:105) Q and | q − x (cid:105) Q (cid:48) are the eigenstates of Q and Q (cid:48) with the eigenvalues q and q − ξ , respectively. Note that | ξ, η (cid:105) are normalized so that (cid:104) ξ (cid:48) , η (cid:48) | ξ, η (cid:105) = δ ( ξ − ξ (cid:48) ) δ ( η − η (cid:48) ). According to Born’s rule in quantum mechanics, the jointprobability density function of the outcomes of measuring Q − Q (cid:48) and P + P (cid:48) is given by p ( ξ, η ) = | (cid:104) ξ, η | | α (cid:105) ⊗ | (cid:105) | .The coherent state in the coordinate representation is given by the wave function [70, Section 5.1.1] ψ α ( q ) := (cid:104) q | α (cid:105) = (cid:18) π (cid:19) / exp (cid:2) − ( q − θ ) (cid:3) exp(2 iqθ ) . (S25)Therefore, p ( ξ, η ) = 1 π (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) e − iηq ψ α ( q ) ψ ( q − ξ ) d q (cid:12)(cid:12)(cid:12)(cid:12) (S26)= 2 π (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) e − iηq e − ( q − θ ) e iqθ e − ( q − ξ ) d q (cid:12)(cid:12)(cid:12)(cid:12) (S27)= 1 π exp (cid:2) − ( η − θ ) − ( ξ − θ ) (cid:3) . (S28)0The classical Fisher information matrix of p ( ξ, η ) is F = (cid:18) (cid:19) , (S29)which saturates the tradeoff F + F ≤ Q − Q (cid:48) and P + P (cid:48) with the ancillary mode being in the squeezedstate S ( ζ ) | (cid:105) , where S ( ζ ) = exp (cid:2) ( ζ ∗ a (cid:48) − ζa (cid:48)† ) / (cid:3) with ζ = re iϕ being an arbitrary complex number is the squeezeoperator. Here, we set ϕ = 0. It then can be shown that S ( ζ ) † a (cid:48) S ( ζ ) = a (cid:48) cosh r − a (cid:48)† sinh r, (S30)implying that S ( ζ ) † Q (cid:48) S ( ζ ) = Q (cid:48) e − r and S ( ζ ) † P (cid:48) S ( ζ ) = P (cid:48) e r . (S31)Therefore, jointly measuring Q − Q (cid:48) and P + P (cid:48) with the ancillary mode being in the squeezed state S ( ζ ) | (cid:105) isequivalent to jointly measuring A r := Q − e − r Q (cid:48) and B r := P + e r P (cid:48) with the ancillary mode being in the vacuumstate.The normalized simultaneous eigenstates of A r and B r are given by | ξ, η (cid:105) r = e r/ √ π (cid:90) e iηq | q (cid:105) Q ⊗ | e r ( q − ξ ) (cid:105) Q (cid:48) d q . (S32)It is easy to see that A r | ξ, η (cid:105) r = ξ | ξ, η (cid:105) r and (cid:104) ξ (cid:48) , η (cid:48) | ξ, η (cid:105) r = δ ( ξ − ξ (cid:48) ) δ ( η − η (cid:48) ). To show that | ξ, η (cid:105) r is the eigenstateof B r with the eigenvalue η , we need to write | ξ, η (cid:105) r with the momentum representation. Using P (cid:104) p | q (cid:105) Q = √ π e − ipq with | p (cid:105) P denoting the eigenstate of P , we get | ξ, η (cid:105) r = e r/ π / (cid:90) (cid:90) (cid:90) exp[2 iηq − ip q − ip e r ( q − ξ )] | p (cid:105) P ⊗ | p (cid:105) P (cid:48) d q d p d p (S33)= e r/ √ π (cid:90) (cid:90) δ ( η − p − p e r ) exp(2 ip ξe r ) | p (cid:105) P ⊗ | p (cid:105) P (cid:48) d p d p (S34)= e − r/ √ π (cid:90) exp[2 iξ ( η − p )] | p (cid:105) P ⊗ (cid:12)(cid:12) e − r ( η − p ) (cid:11) P (cid:48) d p . (S35)With Eq. (S35), it is easy to see that B r | ξ, η (cid:105) r = η | ξ, η (cid:105) r .To calculate the classical Fisher information matrix, we need the joint probability density function: p ( ξ, η ) = e r π (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) e − iηq ψ α ( q ) ψ ( e r ( q − ξ )) d q (cid:12)(cid:12)(cid:12)(cid:12) (S36)With the wave function of coherent state, namely, Eq. (S25), we get p ( ξ, η ) = 2 e r π (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) exp (cid:2) − iηq − ( q − θ ) + 2 iqθ − e r ( q − ξ ) (cid:3) d q (cid:12)(cid:12)(cid:12)(cid:12) (S37)= 1 π e r e r + 1 exp (cid:20) − η − θ ) e r + 1 − e r ( ξ − θ ) e r + 1 (cid:21) (S38)This probability density function is Gaussian with the covariance matrix as follows:Σ = (cid:32) e r +14 e r +14 e r (cid:33) . (S39)The classical Fisher information matrix with respect to θ and θ is then given by F = Σ − = (cid:32) e r +1 e r e r +1 (cid:33) , (S40)which saturates the tradeoff relation F + F ≤ JOINT ESTIMATION OF PHASE AND PHASE DIFFUSION
Here, we give the details of the calculation for the second example in the main text, i.e., the joint estimation ofphase shift and phase diffusion. The density operators of a two-level quantum system can always be represented by ρ = ( I + r · σ ), where I is the 2 × r = ( r , r , r ) ∈ R , and σ = ( σ , σ , σ ) is the vector ofPauli matrices. For the joint estimation of phase and phase diffusion [27], the Bloch vector r is given by r = e − θ n ,where n = (sin χ cos θ , sin χ sin θ , cos χ ) is a unit vector. Here, θ and θ are the parameters of interest and χ is aparameter determined by the initialization of the quantum system.To obtain the SLD operators, we use the eigenvalue decomposition of ρ . The eigen-projections of ρ θ are Π ± = ( I ± n · σ ) with the eigenvalues λ ± = [1 ± exp (cid:0) − θ (cid:1) ] /
2. The SLD operators about θ j can be expressed as L j = (cid:88) u,v = ± λ u + λ v Π u ( ∂ j ρ θ )Π v . = (cid:88) u,v = ± λ u + λ v Π u ( ∂ j r · σ )Π v . (S41)Substituting the expressions of λ ± , Π ± , and r into the above formula, we get L = (cid:32) − ie − θ − iθ ie − θ + iθ (cid:33) , L = 1 e θ − (cid:32) θ − θ e θ − iθ − θ e θ + iθ θ (cid:33) . (S42)Therefore, we get the quantum geometric tensor: Q = (cid:32) e − θ θ e θ − (cid:33) , (S43)from which we can see c = 0. To calculate (cid:101) c , we note that[ L , L ] = − ie θ − (cid:18) − θ
00 4 θ (cid:19) . (S44)After some algebras, we get tr |√ ρ [ L , L ] √ ρ | = 4 e − θ θ (cid:112) e θ − (cid:101) c = tr |√ ρ [ L , L ] √ ρ | √ Re Q Re Q = 1 ..