Relativistic Quantum Metrology in Open System Dynamics

0.2 MB·English

by  Zehua Tian

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:[e-mail]
1
0 ⋆Correspondingauthor,e-mail:[e-mail]
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 detec-
tion 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 can-
not,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 differ-
ent 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 lo-
calized 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 populationmea-
surement 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 tempera-
ture: 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 measure-
ment,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