Table Of Content

Relativistic Quantum Metrology in Open System Dynam- ics ZehuaTian1, Jieci Wang1,2,†, Heng Fan2, andJiliangJing1,⋆ Quantum metrology studies the ultimate limit of precision in estimating a physical quantity ifquantumstrategiesareexploited. Hereweinvestigatetheevolutionofatwo-levelatomasa detector which interacts with a massless scalarfield using the master equation approach for open quantum system. We employ local quantum estimation theory to estimate the Unruh temperature whenprobedbyauniformlyaccelerateddetectorintheMinkowskivacuum. In 5 1 particular,weevaluatetheFisherinformation(FI)forpopulationmeasurement,maximizeits 0 value over all possible detector preparations and evolution times, and compare its behavior 2 with that of the quantum Fisher information (QFI). We find that the optimal precision of n a estimation is achieved when the detector evolves for a long enough time. Furthermore, we J findthatinthiscasetheFIforpopulationmeasurementisindependentofinitialpreparations 7 of the detector and is exactly equal to the QFI, which means that population measurement 2 is optimal. This result demonstrates that the achievement of the ultimate bound ofprecision ] imposed by quantum mechanics is possible. Finally, we note that the same configuration is c q alsoavailableto the maximum ofthe QFIitself. - r g [ 1DepartmentofPhysics,andKeyLaboratoryofLowDimensional,QuantumStructuresandQuan- 1 tum Control of Ministry of Education, Hunan Normal University, Changsha, Hunan 410081, v China. 6 7 2Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese 6 6 AcademyofSciences,Beijing100190,China. 0 . †e-mail:[email protected] 1 0 ⋆Correspondingauthor,e-mail:[email protected] 5 1 : v i X It is well known that in the modern theory of quantum fields, the concept of particle is r observer-dependent 1. One of themost fundamental manifestationsofthis fact is theUnruh effect a 1,2, i.e., the inertial vacuum is perceived by a uniformly accelerated observer as populated by a thermalbathofradiation. Itisbelievedthatthiseffectisdeeplyconnectedwithimportantphysical phenomena such as Hawking radiation 3–6. Thus, its observation would be expected to provide experimental support for Hawking radiation and black hole evaporation. Furthermore, the detection of the Unruh effect would have an immediate impact in many fields such as astrophysics 7,8, cosmology9,blackholephysics10,particlephysics11,quantumgravity12 andrelativisticquantum information13,14. However,althoughalargenumberofdifferentschemesinvolvingBose-Einstein condensates 15–18 and superconducting circuits 19,20 have been proposed to detect the associated radiation effect, it remains an open research program to detect this effect in experiments, this is 1 because theassociated temperaturelies far belowany observablethreshold(smallerthan 1 Kelvin evenforaccelerations as highas 1021m/s2). SincetheUnruh effect isratherweak, high-precision quantum measurement is essential during its detection. On the other hand, due to the fact that natureisbothquantumandrelativistic,itcan beexpected bytheoretical argumentsthattheUnruh effect is incorporated intothequestionofhowto processinformationby usingquantumtechnolo- gieswhicharebeyondtheclassicalapproaches13,14. Thiscreativecombinationprovidesnotonlya morecompleteframetounderstandthetheoryofquantuminformationbutalsoperhapsanewway toaddresstheproblemof“informationloss”inblackholescenarios. Inparticular,withinthisarea attheoverlapofrelativityandquantummechanics,itseemsnaturaltoapplynovelapproachesand techniquesforquantummeasurements. Thismakestherelativisticaspectoftheeffects potentially moreaccessibleto detection. The Unruh temperature of interest to us is nonlinear function of the density matrix and cannot,eveninprinciple,correspondtoaproperquantumobservable. Therefore,itsdirectobservation is not accessible. In these situations one has to turn to indirect measurements, inferring the value of the quantity of interest by inspecting a set of data coming from the measurement of a different observable, or a set of observables. In this regard, let us note that any conceivable strategy aimed at evaluating the quantity of interest ultimately reduces to a parameter-estimation problem that may be properly addressed in the framework of quantum estimation theory (QET) 21–26. Rel- evant examples of this situation are given by discussions of quantum speed limits in open system dynamics 27–29, measurements of non-Markovianity of open quantum processes 30, estimation of quantum phase 24,25,31–33, qubit thermometry 34,35, and so on. For example, with the help of rigor- ousmethodsfromquantumstatisticsandestimation21,recentlyAspachsetal. 36 haveinvestigated the ultimate precision limits for the estimation of the Unruh-Hawking temperature. Shorter after that,anumberofanalogouspapershaveemergedtostudythetopicoftheestimationofrelativistic effects 36–42. Up to date, almost all work involving relativistic metrology is guided by an interesting link between field theory and quantum information: The change of coordinates between an inertial observer and a noninertial observer in the description of the state of a scalar field is equivalent to the transformation that affects a light beam undergoing parametric down-conversion in an optic parametric oscillator 1,43. The parameters encoded in quantum fields are assumed to be directly estimated without any scheme that investigates how to extract this information from the fields (relevant processes involveto howto introducea probeand prepare what kind ofprobe). Besides, thequantumstatestoproberelativisticeffectsaredirectlypreparedwiththefreefieldmode36,39–42, which, as we all known, is spatiallynot localized and thus cannot be experimentallyaccessed and 2 measured bylocalized apparatuses44. Motivated by these considerations, in this work we employ a uniformly accelerated and localized two-level atom as the probe to detect the Unruh temperature. We aim at estimating the inverse Unruh temperature β = 1/T and try to address the following questions: (1) Which is the best probe state? (2) Which is the optimal measurement that should be performed at the output probe state? (3) Which is the minimum fluctuation in the temperature estimation, as well as the ultimateboundto precisionimposedbyquantummechanics. Results Physicalmodelandprobestatepreparation. Weconsideratwo-levelatomasthedetectorwhich interactswithafluctuatingvacuumscalarfield. Thismodelassumesthat thedetectorbehaves like an open system, i.e., a system immersed in an external field. Therefore, in the following we will treatthedetectoras anopenquantumsystemandthevacuumwiththefluctuationsofthequantum field as theenvironment. Let us first introduce the total Hamiltonian of the total system, detector plus field. Without lossofgenerality,itistaken as H = H +H +H , (1) s Φ(x) I where H = 1ω σ and H are respectively the Hamiltonian of the detector and scalar field, s 2 0 z Φ(x) and H = µ(σ +σ )Φ(x(τ)) represents their interaction. Note that ω is the detector’s energy- I + − 0 level spacing, σ is the Pauli matrix, σ (σ ) is the atomic rasing (lowering) operator, and Φ(x) z + − corresponds to the scalar field operator. Here, the two-level atom can be fully described in terms of a two-dimensional Hilbert space. Its quantum state, with respect to a fixed and arbitrary basis in this space, will be represented by a 2 × 2 density matrix ρ, which is Hermitian ρ† = ρ, and normalized Tr(ρ) = 1 with det(ρ) ≥ 0. On the other hand, the equation of motion of the scalar field is ((cid:3) + m2)Φ = 0 with (cid:3)Φ = gµν∇ ∇ Φ = (−g)−1/2∂ [(−g)−1/2gµν∂ Φ], where m is µ ν µ ν the mass of the field, and g is the determinant of the metric g . For the Minkowski spacetime µν case, one set of solutions of this equation of motion is uk(t,x) = [2ω(2π)3]−1/2eik·x−iωt with ω = (k2 +m2)1/2. The field modes uk and their respective complex conjugates form a complete orthonormalbasis, so Φ may beexpanded as Φ(x) = k[akuk(t,x)+a†ku∗k(t,x)], which plays a crucial roleinthefollowingcalculationofthetwo pointfunctionofquantumfield. P Initially,thetotalquantumsystemisdescribedbythedensitymatrixρ = ρ(0)⊗|−ih−|,in tot 3 whichρ(0)isthereduceddensitymatrixofthedetector,and|−iisthevacuumofthefielddefined by ak|−i = 0 for all k. In the frame of the detector, theevolutionin the proper timeτ of the total densitymatrixρ satisfies tot ∂ρ (τ) tot = −iL [ρ (τ)], (2) H tot ∂τ where the symbol L represents the Liouville operator associated with H, L [S] = [H,S]. To H H obtainthedynamicsofthedetector,wemusttraceoverthefielddegreesoffreedom. Afterthat,in the limit of weak coupling the evolving density matrix ρ(τ) of the detector obeys an equation in theLindbladform 45–47 ∂ρ(τ) = −i[H ,ρ(τ)]+L[ρ(τ)] (3) eff ∂τ with 1 1 H = Ωσ = {ω +µ2Im(Γ +Γ )}σ , eff z 0 + − z 2 2 3 L[ρ(τ)] = [2L ρL† −L†L ρ−ρL†L ], (4) j j j j j j j=1 X whereΓ = ∞eiω0sG+(s±iǫ)ds, L = γ−σ , L = γ+σ , L = γzσ , γ = 2µ2ReΓ , ± 0 1 2 − 2 2 + 3 2 z ± ± γ = 0, G+(x−x′) = h0|Φ(x)Φ(x′)|0i is the field correlation function, and s = τ −τ′. Eq. (3) z R p p p characterizes the evolution of the detector. In particular, the second on its right hand side denotes the dissipation resulting from the external environment, i.e., the scalar field that the detector cou- plesto. ItiscalledtheLindbladtermanddescribestheresponseofthedetectortotheenvironment. Alltheinformationthatweareinterestedinandwanttoestimateinthefollowingisencodedinits relevantparameters. In order to solve the Eq. (3), let us express the reduced density matrix in terms of the Pauli matrices, 3 1 ρ(τ) = 1+ ρ (τ)σ . (5) i i 2 ! i=1 X If we choose the initial state of the detector as |ψ(0)i = sin θ|0i+e−iφcos θ|1i, substitutingEq. 2 2 (5)into(3), wecan obtainitsanalyticalevolvingmatrix, ρ (τ) ρ (τ) 1 ee eg ρ(τ) =   (6) 2 ρ (τ) ρ (τ)  ge gg      4 with B ρ (τ) = 1+e−Aτ cosθ+ (1−e−Aτ), ee A B ρ (τ) = 1−e−Aτ cosθ − (1−e−Aτ), gg A ρeg(τ) = ρ∗ge(τ) = e−21Aτ−i(Ωτ+φ)sinθ, (7) where A = γ + γ and B = γ − γ . Moreover, the state of the detector can be diagonalized + − + − and decomposedas ρ(τ) = λ |ψ (τ)ihψ (τ)i|+λ |ψ (τ)ihψ (τ)i| with + + + − − − 1 λ = (1±η), ± 2 [|ρ (τ)||0i+e−i(Ωτ+φ)(ρ (τ)−2λ )|1i] |ψ (τ)i = eg ee ∓ , (8) ± (ρ (τ)−2λ )2 +|ρ (τ)|2 ee ∓ eg p whereη = (ρ (τ)−1)2 +|ρ (τ)|2. ee eg p FromEqs. (3)and(4)weknowthattheWightmanfunctionforthescalarfieldthatthedetec- torcouplestoplaysanimportantroleintheevolutionofthedetector. Inthisregard,letusnotethat ifauniformlyaccelerateddetectorwithtrajectory,t(τ) = 1 sinh(aτ), x(τ) = 1 cosh(aτ), y(τ) = a a z(τ) = 0, is coupled to a massless scalar field in the Minkowski vacuum, then the corresponding Wightmanfunctionshouldbe1 a2 a(τ −τ′) G+(x,x′) = − sinh−2 −iε . (9) 16π2 2 (cid:20) (cid:21) In thiscase, itis easy toobtain µ2ω e2πω0/a +1 µ2ω A = 0 , B = − 0. (10) 2π e2πω0/a −1 2π (cid:18) (cid:19) Substituting Eq. (10) into (6), it is easy to check that when evolving long enough time, i.e., τ ≫ 1 with 1 being the time scale for atomic transition, the detector eventually γ++γ− γ++γ− approaches tothestate e−βHs ρ(∞) = . (11) Tr[e−βHs] Here let us remark that the state in Eq. (11) is a thermal state with a temperature T = 1/β. Thus, theaccelerated detectorfeels as ifitwere immersedin athermal bathwithtemperatureT = a/2π 1. Wewillestimatethisrelativisticparameterin thefollowing. 5 Fisherinformationbasedonpopulationmeasurement. AswestatedintheDiscussionsection, theQFIdeterminestheultimateboundontheprecisionoftheestimatoralthoughitisthendifficult tofindoutwhichmeasurementisoptimaltoachievesuchultimatebound. Thisoccursbecausethe QFIdoesnotdependonanymeasurements,foritisobtainedbymaximizingtheFIoverallpossible quantum measurements on the quantum system. Thus, to find out the optimal measurement to estimate the Unruh temperature, we first calculate the FI for the population measurement, and then compare the FI with the QFI to determine whether the population measurement is optimal according to the condition of optimal quantum measurement, i.e., POVM with a FI equal to the QFI. Forthepopulationmeasurement,theFI, according toEqs. (6)and (17), isgivenby [∂ p(e|β)]2 [∂ p(g|β)]2 β β F(β) = + p(e|β) p(g|β) 1 [∂ ρ (τ)]2 [∂ ρ (τ)]2 = β ee + β gg . (12) 2 ρ (τ) ρ (τ) (cid:20) ee gg (cid:21) Substituting Eqs. (7) and (10) into Eq. (12), we can obtain the detailed formula of the FI. It is interestingtonotethattheFIisindependentofquantumphaseφ. Itonlydependsontheparameters τ,θ and ω . Thus, the FI in fact should be written as F(β,τ,θ,ω ), while we adopt the notation 0 0 F(β)forconveniencehere. Inthefollowing,byevaluatingtheFIwewanttofindboththeoptimal initial detector preparation and the smallest temperature value that can be discriminated. We will work withdimensionlessquantitiesbyrescaling timeandtemperature τ 7−→ τ ≡ γ τ, β 7−→ β ≡ βω , (13) 0 0 where γ0 = µ22πω0 is the spontaneouseemission rate of tehe atom. For convenience, we continue to term β and τ, respectively,as β and τ. e Let useconsider that the detector is uniformly accelerated with proper acceleration a and Unruh temperature T proportional to a 2. We assume that the inverse temperature has the value β = 10. The probabilities p(j|β) = ρ (τ) evolve according to Eq. (7). The corresponding jj behavior of the FI is shown in the top panel of Fig. 1. We can see that for θ = π the FI is larger thantheFI ofothercases duringinitialperiod, butwhen τ ≫ 1 alltheFI aresaturated γ++γ− and equal to each other. This means that the FI displays a robust maximum at the optimal time τ ≫ 1 for all θ. We can also obtain the same results from the bottom panel of Fig. 1. It is γ++γ− shownthattheFIevolvesperiodicallyasafunctionoftheinitialstateparameterθandforanytime themaximalFIisalwaysobtainedbytakingθ = π,i.e.,bypreparingthedetectorintheground max state. Furthermore,forsmalltimetheFIsuddenlydropstozero,exceptforasharppeakcenteredat θ ,asθ varies,butforlongtimetheFIchangeslesswithrespect toθ. Thus,wecan arriveatthe max conclusion that the maximum sensitivity in the predictions for the inverse Unruh temperature can 6 beobtained by initiallypreparing thedetectorin itsground state. However,ifthedetector evolves for a long enough time, the maximum sensitivity in the predictions is independent on the initial stateinwhichthedetectorisprepared. Itisnosurprisebecausetheaccelerateddetectoreventually evolvesto athermalstateregardlessofitsinitialstate48. In Fig. 2 we plot the FI for different fixed temperatures as a function of time τ. We can see that the FI approaches its maximum value when the detector evolves for a long enough time. Also its value for different β varies over several orders of magnitude, changing from 10−5 to 0.1, which means the FI is very dependent on the temperature itself. As we demonstrated above, the reasonforsaturationisthattheaccelerateddetectoreventuallyevolvestoathermalstateregardless of its initial state 48. Furthermore, in this case, the thermal state only depends on the thermal temperature felt by the detector, i.e., the acceleration of the detector. On the other hand, the higher the temperature, the bigger the FI is, i.e., the easier it is to achieve a given precision in theestimationoftemperature. QuantumFisherinformation Inordertoassesstheperformanceofthepopulationmeasurement in theestimation of theUnruh temperature we have evaluated theQFI of the family of states ρ(τ) in Eq. (6). Substituting Eqs. (8) and (10) into (19), it is easy to obtain the QFI. Let us note that the QFI depends on β, τ and θ, but is independent of the phase φ of the detector. Thus, to find out the optimalworking regimes we have to maximizethe valueof the QFI overall three relevant parameters. SimilartotheanalysesoftheFIshownabove,wefirstfixtheUnruhtemperaturebyassuming β = 10,anddiscusshowtheeffectivetimeτ (initialstateparameterθ)affectstheQFIfordifferent initialstateparameters θ (effectivetimeτ). Obviously,fromFig. 3 weknowthat themaximumof theQFIisachievedbyinitiallypreparingthedetectorinthegroundstate. However,iftheeffective timeislong enough, i.e., thedetectorevolvesfor along enough time,τ ≫ 1 , no matterwhat γ++γ− the initial state is prepared in, the QFI always achieves the maximum, which means the optimal sensitivityinestimationofβ isindependentoftheinitialpreparationofthedetectoriftheeffective time is long enough. Besides, in Fig. 4 we plot the QFI for different fixed temperatures as a function of the effective time τ. We find that, if the detector evolves for a long enough time, the QFI we computed abovefor different Unruh temperatures saturates at different values which vary over several orders of magnitude. Furthermore, the higher the temperature, the bigger the QFI is, i.e., the easier it is to achieve a given precision in the estimation of temperature. Thus, we can arrive at the conclusion that the maximum sensitivity in the predictions for the inverse Unruh temperature can be obtained when thedetector evolves for a long enough time, and the maximum 7 sensitivity in the predictions is independent on the initial state in which the detector is prepared. Inthiscasewewanttoemphasizethatthisstrategyprovidesoptimalityinthesensethatinequality (18)issaturated and thevarianceVar(β)is assmallas possible. Wefindthatforτ ≫ 1 boththeFIandQFItakethemaximumlimit. Interestingly,upon γ++γ− inspectingthetemporalevolutionoftheexcitedstateprobability,p(e|β)hasaminimumunderthis condition (also the quantum state of the detector is thermal discussed in Eq. (11)). Thus, we can giveaphysicalexplanationtotheFIandQFIbehavior. Becausewewanttoestimateatinyquantity that carries information about thermal disorder, of course, only when the external environment is mostly occupied by the Unruh thermal particle, and the more the better, we then could expect to find the maximum sensitivity in the predictions. This condition corresponds to the probability p(e|β)achievingitsminimum. In the above analysis, we have shown the behaviors of the FI and QFI, and obtained the conditionsthat how toachievethemaximumFI and QFI. It is interestingto notethatthebehavior of H(β) is identical to that of F(β), as is apparent by comparing Fig. 1 and 3. Besides, under the same condition (θ,φ,τ) = (θ,φ,∞) both the FI and QFI obtain the maximum value when β is fixed. In order to find out whetherthe populationmeasurement is optimalduring theestimation processoftheUnruhtemperature,wewillcheckwhetherthemaximizedFIisequaltotheoptimal QFI. Thus, we prepare the detector in its ground state, i.e., θ = π, and assume that the detector evolvesforalongenoughtime. Thisallowsustoeasilyfindthatthedetectoreventuallyevolvesto a thermal state. In this case, the off-diagonal terms of state (6) vanish and it is diagonal with two eigenvalues 1 λ = , + e2πω0/a +1 e2πω0/a λ = , (14) − e2πω0/a +1 andcorrespondingeigenvectors|eiand|gi. Forthisquantumstatisticmodel,wefindthattheFIis equal totheQFIgivenby (∂ λ )2 (∂ λ )2 F(β) = H(β) = β + + β − . (15) λ λ + − It means that the estimation of β via the population measurement is optimal. Eq. (15) is the ultimatebound to precision of estimationof theUnruh temperature. Because the populationmeasurement is optimal, our results in this regard suggest that the achievement of the ultimate bound toprecisionofestimationoftheUnruhtemperatureallowedbyquantummechanicsisinthecapa- bilityofcurrent technology. 8 Discussion Weintroducedadetector,i.e.,atwo-levelatom,whichisuniformlyacceleratedandinteractswitha masslessscalarfieldintheMinkowskivacuum,andemployittodetecttheUnruhtemperature. By employinglocal quantumestimationtheory wehavestudiedtheestimationoftheUnruh tempera- tureviaquantum-limitedmeasurementsperformedonthedetector. Inparticular,wehaveanalyzed theprecisionofestimationasafunctionofboththedetectorinitialpreparationsandtheinteraction parameters, and evaluatedthelimitsofprecisionposedby quantummechanics. It is shown that the FI for the population measurement, which establishes a classical bound onprecision,takesthemaximumlimitwhenthedetectorevolvesforalongenoughtimecompared with the time scale for atomic transition, 1 , i.e., when τ ≫ 1 . In this case, the FI for γ++γ− γ++γ− population measurement is independent of any initial preparations of the detector. Furthermore, wefind that thesameconfigurationis also correspondingto themaximumofthe QFIbased on all possiblequantummeasurements,whichestablishestheultimateboundtotheprecisionallowedby quantum mechanics. Interestingly,themaximumFI is equal to the maximumQFI under thesame conditions, which means the optimal measurement for the estimation of the Unruh temperature correspondsto thepopulationmeasurement. Thus,duringthedetectionoftheUnruhtemperature, we can achieve the ultimate bound to the precision by performing a population measurement on the detector, and the ultimatebound is givenby Eq. (15). Because the populationmeasurement is allowedbythecurrenttechnology49–55,ourresults,inthisregard,indicatethattheultimatebound toprecisionofestimationofUnruhtemperatureimposedbyquantummechanicscaninprinciplebe achievedunderthecurrenttechnology. Ontheotherhand,ourresultsdemonstratethatthermalized quantumstatisticmodel,Eq. (11),playsanoptimalroleintheestimationoftheUnruhtemperature. This occurs because we want to estimate a tiny quantity that carries information about thermal disorder. Therefore, it is natural to expect to find the maximum sensitivity in the predictions when theexternal environment,thatis coupledwiththedetector,is mostlyoccupiedby theUnruh thermal particle, and the more the better. This condition corresponds to when τ ≫ 1 , i.e., γ++γ− when thedetectorstateisthermalized. Our model avoids two critical technical difficulties in the estimation of the Unruh temperature: aphysicallyunfeasibledetectionofglobalfreemodeinthefullspace36 andanon-analytical expression of QFI due to the boundary conditions of the moving cavity 37,38. Recently, the open quantum system approach has been used to understand the Hawking effect of black hole 56 and Gibbons-Hawking effect of de sitter universe 57. Thus, our above analysis can also be applied to discussing the estimation of Hawking temperature and Gibbons-Hawking temperature. Also we 9 couldturntotheestimationofotherparameters,suchastheatomicfrequencyandphase,analyzing what kind of role that the relativistic effects play in this metrology. In particular, the simulation ofrelativisticallyacceleratingatomsintrappedionsystemsand superconductingcircuitshasbeen studiedinRef. 58. ThesimulationsproposedinRef. 58 arepreciseanaloguesofthephysicalsetting required here. Ourtechniquescouldpossiblybeimplementedduringsuch simulations. Methods Usually, two main steps are contained in estimation process: at first we has to choose a measurement,and then,after collectingasampleofoutcomes,weshouldfind an estimator,i.e., afunction toprocess dataandto inferthevalueofthequantityofinterest. Foragivenmeasurementscheme, themean squareerror Var(β) = E [(βˆ−β)2]ofanyestimatoroftheparameter, β, isboundedby β theCrame´r-Rao inequality21 1 Var(β) ≥ , (16) MF(β) whereM isthenumberofidenticalmeasurementsrepeated and F(β)is theFI givenby |∂ p(j|β)|2 F(β) = p(j|β)(∂ lnp(j|β))2 = β . (17) β p(j|β) j j X X EfficientestimatorsarethosesaturatingtheCrame´r-Rao inequality. In ordertoobtaintheultimate boundtoprecision,i.e.,thesmallestvalueoftheparameterthatcanbediscriminated,theoptimiza- tion of FI is needed via a suitable choice of all its dependent parameters. From Eqs. (7) and (17), the FI obviously depends on the detector initial state parameters and evolving time, and so on. In thisregard,letusnotethattheinitialstatesofthedetectorandevolvingtimeplayanimportantrole inthismetrologyprocess,which essentiallydeterminetheultimateboundonprecision. On the other hand, we can also maximize the FI over all possible quantum measurements on the quantum system. By introducing the symmetric Logarithmic Derivative (SLD) satisfying Lβρβ+ρβLβ = ∂ρβ, the FI of any quantum measurement is upper bounded by the so-called QFI 2 ∂β givenby F(β) ≤ H(β) = Tr ρ L2 . (18) β β Here, it is interesting to note that the QFI does not d(cid:0)epend(cid:1)on any measurements carried on the detector, indeed being obtained by maximizingoverall possiblemeasurements26. Further studies showthatthedetailedformulafortheQFIis of26 H(β) = (∂βλk)2 +2 (λk −λk′)2 hψk|∂βψk′i 2, (19) k=± λk k6=k′=± λk +λk′ (cid:12) (cid:12) X X (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 10