Quantum sensors based on weak-value amplification cannot overcome decoherence

2.2 MB·English

by  George C. Knee

Quantum sensors based on weak-value amplification cannot overcome decoherence
George C. Knee∗ and G. Andrew D. Briggs
Department of Materials, University of Oxford, Oxford OX1 3PH, United Kingdom
Simon C. Benjamin
Department of Materials, University of Oxford, Oxford OX1 3PH, United Kingdom and
Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543
Erik M. Gauger†
Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 and
Department of Materials, University of Oxford, Oxford OX1 3PH, United Kingdom
(Dated: January 18, 2013)
Sensorsthatharnessexclusivelyquantumphenomena(suchasentanglement)canachievesuperior
3
performance compared to those employing only classical principles. Recently, a technique based
1
onpostselected,weakly-performedmeasurementshasemergedasamethodofovercomingtechnical
0
noiseinthedetectionandestimationofsmallinteractionparameters,particularlyinopticalsystems.
2
Thequestionofwhichothertypesofnoisemaybecombatedremainsopen. Wehereanalyzewhether

n the effect can overcome decoherence in a typical field sensing scenario. Benchmarking a weak,
a postselectedmeasurementstrategyagainstastrong,directstrategyweconcludethatnoadvantage
J is achievable, and that even a small amount of decoherence proves catastrophic to the weak-value

7 amplification technique.
1

] I. INTRODUCTION is chosen such that the deflection induced in the me-
h
ter is much less than its inherent uncertainty, and when
p
- The prospect of enhancing the sensitivity of nanoscale the pre- and postselected states are close to orthogonal,
nt detectorshasrecentlyemergedinlightofagreaterunder- (cid:104)ψf|ψi(cid:105) → 0, one may extract weak values Aw much
a standing of fundamental aspects of quantum theory [1]. largerthanmax((cid:104)A(cid:105))throughappropriatemeasurements
u In particular, the appropriate application of entangled of the meter. AAV’s expression for Aw has a limited
q states can offer sensing performance at the Heisenberg range of validity [6]: however, exact treatments reveal
[
limit: far superior to the standard quantum limit im- that the qualitative effect persists outside this range [7–

2 posed on separable states [1–4]. Another feature of 9]. Weak values have been obtained experimentally, see
v quantum physics not reproducible classically is an effect e.g. Refs. [10, 11].
1
known as weak-value amplification: in this work we ana- In this Article, we shall concentrate on technological
6
2 lyze its impact on the field of quantum metrology. applications inspired by AAV’s work: the use of the
0 In a seminal paper of 1988, Aharonov, Albert and larger-than-usual deflection of the meter for increasing
. Vaidman(AAV)presentedacuriousquantummechanical the sensitivity of a suitably designed detector. Although
1
thought experiment giving rise to a quantity the authors theideaofweak-valueamplification(WVA)ismentioned
1
2 called ‘the weak value of a quantum variable’ [5]. AAV in Ref. [5], its true utility has only recently begun to
1 defined the quantity transpire. AAVconsideredanamplificationofthedeflec-
: tion imparted to a beam of spin-1/2 particles passing a
v A :=(cid:104)ψ |A|ψ (cid:105)/(cid:104)ψ |ψ (cid:105),
i w f i f i Stern-Gerlachmagneticfieldgradient. Ayearlater,Duck
X
et al. described an analogous experiment involving the
generalizingtheusualexpectationvalue(cid:104)A(cid:105)=(cid:104)ψ |A|ψ (cid:105).
i i
r displacement of a coherent light beam by a birefringent
Obtaining A involves i) initializing a quantum system
a w
crystal [6]. In both cases the internal state of each ele-
of interest (henceforth ‘system’) into the state |ψ (cid:105); ii)
i
mentofthebeamensembleispreselected,coupledweakly
coupling the system weakly to an ancillary measuring
to its spatial wavefunction and finally postselected; leav-
device (henceforth ‘meter’) through the system operator
ing the meter (the element’s spatial wavefunction) in a
A; let the coupling strength be parameterized through a
characteristically broad distribution peaked around A .
constant G ∈ [0,1]; iii) post-selecting the system into a w
Ordinarily(withoutpostselection),theshiftinthemeter
definite final state |ψ (cid:105). The meter can then be interro-
f
wavefunction could well be too slight to detect, due to
gated at full strength to reveal something about the sys-
thefinitespatialresolutionormisalignmentofthedetec-
tem. A surprising interference effect arises when postse-
tion apparatus. Such imperfections can be thought of as
lection and weak measurement are combined; i.e. G<1
classical randomness occurring after the wave function
has collapsed, and are known as examples of technical
noise. This type of noise does not affect the quantum
∗ [email protected] state prior to measurement, but obscures the results af-
† [email protected] ter measurement. It can be mitigated by the increased
2
deflectionachievedbypostselectingthebeamintoanun- in Sections II and III we introduce a canonical phase es-
likely internal state. timationproblemandapplytheFisherinformationfunc-
In2008,atinylateraldisplacementofalightbeamwas tional to ascertain the performance of what we call the
promoted to a detectable shift in this way, revealing for ‘direct strategy’. In Section IV we explore the possibil-
the first time the spin Hall effect of light by improving ity of using weak measurements alone, combining this in
thesignal’sratiototechnicalnoisebyafactorof104 [12]. Section V with postselection, allowing us to evaluate the
In another experiment, a similar amplification was used performance of the ‘WVA strategy’. We summarize our
to boost the angular deflection of light in a Sagnac in- conclusions in Section VI.
terferometer [13]. These examples suggest that, if one is
limited by technical noise (such as finite detector reso-
lution or imperfect initialization of the meter [14]), the II. SPIN BASED SENSING OF MAGNETIC
FIELD
weak-valueinterferencephenomenoncanbeausefultool
to detect tiny shifts which are hard to resolve by other
means. The paradigmatic example of a phase parameter esti-
mation problem is a spin-1/2 particle in a strong mag-
Experimental setups are typically afflicted by varying
netic field B along the z-direction. The strong magnetic
typesandmagnitudesofnoise. Ifmanytypesofnoiseare
field leads to a Zeeman splitting of the spin’s energy lev-
present, one is motivated to overcome the noise that is
els, allowing it to be modelled as a quantum two level
actingasalimitingfactor. Whilstalargershiftintheme-
system commonly referred to as a ‘qubit’. If the Bloch
terwavefunctionisclearlybeneficialwhentechnicalnoise
vector describing the quantum state of the qubit is ini-
is the dominant source of error, other types of noise may
tializedintheequatorialplane(i.e. intoastateunbiased
notbeovercomesoeasilywiththeWVAtechnique. Shot
with respect to the ‘up’ |0(cid:105) and ‘down’ |1(cid:105) eigenstates)
noise,theintrinsicuncertaintyassociatedwithpurestate
it will precess around the z axis at a rate proportional
quantum mechanics, may not be defeated [15], although
to the strength of B. Moving into a frame that rotates
it has been shown that the weak-value technique can
with this evolution, small deviations δB in the magnetic
match the sensitivity of direct sensing strategies when
field strength are seen as slower evolutions of the Bloch
shot noise dominates, bringing the aforementioned ben-
vector around the z-axis. In this frame, and in the com-
efits (viz. suppression of technical noise), as it were, ‘at
putational {|0(cid:105),|1(cid:105)} basis, the evolution of the spin is
no extra cost’ [16]. What has remained unexplored is
described by
whether this remains true when a third type of noise
(otherthanshotnoiseandtechnicalnoise)isconsidered.
2ρ (t)=1
11
Decoherence is a type of noise associated with mixed
2ρ (t)=−ie−igδBtΞ(t); (1)
state quantum mechanics that arises when a system in- 12
teracts with an unknown environment [17]. It randomly ρ is the 2-dimensional density matrix of the spin, and g
affects the dynamics of the quantum state prior to mea- is its gyromagnetic ratio which we set to unity without
surementandimposeslimitsonthesensitivityofparam- lossofgenerality. Asusual,completenessdemandsρ =
22
eterestimationschemes[18,19]. CantheWVAeffectin- 1−ρ and ρ = ρ∗ . We take Ξ(t) ∈ [0,1] to be a
11 21 12
spire new detectors which resolve small parameter shifts real, non-negative function (assumed to be independent
more quickly than standard techniques, thereby defeat- ofδB)whichdescribestheattenuationoftheoff-diagonal
ing decoherence? terms. By choosing this function appropriately one can
The approach taken in this Article is different to the model, for example, pure dephasing (Ξ(t)=e−Γt) or 1/f
preceding examples in the following sense: instead of in- noisetypedecay(Ξ(t)=e−Γt2)withacharacteristicrate
vestigatingandamplifyingapreexistingweakinteraction Γ [21]. The dependence on t is important but sometimes
between two quantum degrees of freedom (the process suppressed in our notation. δB is an unknown quantity
of estimating an interaction parameter [20]), we are in- and is the subject of the parameter estimation problem.
terested in the amplification of an arbitrary phase shift Whilst for clarity we describe the tangible example of
caused by a classical field on a first quantum system by fieldsensingwithaspin,ourresultsapplytomanyother
couplingit(witharbitrarystrength)toasecond. Wecall phase estimation scenarios.
thistheprocessofestimatingaphaseparameter. Thede- Thedensitymatrix,throughexposuretothemagnetic
coherence of the first system then presents the dominant field,acquiresanincreasingamountofinformationabout
noiseintheproblem. Wedonotconsidertechnicalnoise, δB as time progresses. However, the build-up of useful
allowing ourselves ideal measurement and control of the information is in competition with injurious dephasing
quantum system but not its environment. In assessing mechanisms, which invariably wash out all information
theperformanceofWVA,weshallnotbesatisfiedmerely after a sufficiently long time. To estimate the value of
with anomalously large expectation values (as intimated the parameter δB, one exploits the causal influence of
by the weak value), but rather demand that the particu- δBonthestatisticsofoutcomeswhenmeasuringthespin
lar statistical model associated with it exhibits a higher after an appropriately chosen exposure time. Statistical
informational content than is otherwise possible. inferences about δB may then be made if one has access
The remainder of this Article is organized as follows: to many identical preparations of the density matrix.
3
III. QUANTUM FISHER INFORMATION To eliminate the dependence on the measurement
choiceonecandeploytheoptimalPOVM:sincetheesti-
mationprocedureinvolvesmanysamplesoftheprobabil-
Previous studies have commonly assessed the WVA
itydistribution,someofthemmaybeusedtoadaptively
approach (for the purpose of estimating an interaction
update the POVM after an initial guess, causing it to
parameter in the presence of technical noise) using the
rapidly converge on an optimum [33]. A quantity that
signal to noise ratio, or the ratio of meter deflection to
captures the maximum F in a variation over all POVMs
detector resolution [13, 14, 16, 22–26]. By contrast, the
is the quantum Fisher information (QFI), defined as
Fisher information [27–30] is used in parameter estima-
tion when one does not wish to make assumptions about
specific measurement limitations; it can be thought of H :=2(cid:88)|(cid:104)m|∂δBρ|n(cid:105)|2. (3)
p +p
as a measure of how much information about a given nm n m
parameter (in this case δB) is obtainable from a par-
Theabovesumonlyincludestermsforwhichp +p (cid:54)=0
ticular statistical model: the family of probability dis- n m
and where n,m index the basis states in the spectral
tributions generated by the density matrix ρ when it is (cid:80)
decomposition of ρ, ρ= p |i(cid:105)(cid:104)i| [34]. In our case
measured. Inthecontextofweak-valueamplificationthe i i
FisherinformationwaspreviouslyemployedbyRef.[20], H =Ξ2t2, (4)
d
andHofmannhasarguedforaformalconnectionbetween
the Fisher information and AAV’s weak value [31]. The whichis,notably,independentoftheparameterδB. The
Fisher information is defined as oscillationsintimehavedisappeared,andthisleavesonly
an envelope with turning points which can be found by
F :=(cid:88) 1 (∂ p(k|δB))2, solving Ξ˙t2 = −Ξ. For example when Ξ = e−Γt the
p(k|δB) δB maximum of H occurs at t∗ =1/Γ.
k
According to the Cramer-Rao bound [35], the mini-
where ∂ denotes the partial derivative with respect to mum variance of the parameter δB is given by 1/(NH)
δB
the parameter that is to be estimated (δB) and p(k|δB) whereN isthenumberoftrials. AlargervalueofH thus
is the conditional probability of getting outcome k given entails a smaller minimum variance, which is obtained
the value of δB. The parameter indexes a continuum efficiently through maximum likelihood estimation [30].
of differing probability distributions over measurement The canonical measure of the utility of a detector is the
outcomes: the outcomes in this context being e.g. ‘spin measurementsensitivityS,whichistheminimumuncer-
up’ or ‘spin down’ with the probabilities given by the tainty achievable in the parameter in a fixed amount of
generalized Born rule sensingtime,allowingfortheestimationproceduretobe
stopped (e.g. after t∗), reset and repeated an arbitrary
p(k|δB)=Tr[ρΠ ], number of times within the fixed duration. Novel ap-
k
proaches to quantum metrology must show an improve-
where Π is the POVM element associated with out- ment in S (which depends on H) to claim an advantage
k
come k. The maximum likelihood estimation procedure over established techniques. The aforementioned entan-
involves examining these probabilities and guessing a glement enhanced sensors have been shown to have a
value of δB that would generate the observed frequen- lower(readsuperior)sensitivity[1–3,36,37]byvirtueof
cies with the greatest probability, inverting the statistics achieving a square root improvement over the standard
in a Bayesian sense [32]. The Fisher information is then approach given the same resources.
inversely proportional to the variance in the estimate of
δB,andprovidesagoodindicationoftheperformanceof
any given statistical model. It is clear that the choice of IV. ARBITRARY STRENGTH ANCILLA
MEASUREMENT
measurement(POVM)willaffecttheFisherinformation.
FixingthePOVMtoasharpmeasurementintheσ :=
x
|0(cid:105)(cid:104)1|+|1(cid:105)(cid:104)0| basis, one finds for a spin in a field An ancillary spin can be used as part of the measure-
ment process, often bringing some advantage – for ex-
t2cos2(tδB)Ξ2 ample in the preamplification of a crystal defect based
Fd = 1−Ξ2sin2(tδB). (2) magnetic field sensor [38]. Consider an ancillary qubit
(the meter) initialized in the x-y plane, coupled to the
system qubit so that one has control over the joint sys-
Here, the subscript ‘d’ is used to denote quantities per-
tem. Weenvisageameasurementoperationwhicheffects
tainingtoadirectsensingofδB,i.e. throughthedensity
the following unitary transformation on the system and
matrixdescribedbyEq.(1). Thisexpressionexhibitsos-
meter
cillations over time as the angle between the quantum
state and the measurement basis varies between pessi- M(G)=Π ⊗I+Π ⊗exp(iGπσ /2). (5)
+ − z
mal (when the measurement basis is parallel to the final
state, tδB = [n+1/2]π) to optimal (when the measure- Π are projectors onto the ±1 eigenstates of a traceless
±
mentbasisisperpendiculartothefinalstate,tδB =nπ). ‘controlobservable’(ofthesystem)thatliesintheplane,
4
ment strength. Assuming the experimenter has control
straWteVgAy sdtirraetcetgy mpoesatssuerleection over the measurement operation, it is clear that she ar-
ment c rtoantgheescΘon=trotlδBob+seπrv/a2bfloerbbeeinstgruensublitasse(dwhwi.crh.tc.othrreessypsotnedms
o
ntro state), whence
l
METER measurement strength SYSTEM Hanc →Ξ2t2sin2(Gπ/2). (7)
This matches the performance of the direct strategy
when the measurement corresponds to a full controlled
π−rotation,G=1. TheWVAschemewouldseemtosac-
rificeperformancebyoperatingneartoG=0(wherethe
FIG. 1. (Color online) A schematic for two different ap- dependenceofH onGisroughlyquadratic),however,
anc
proachestoparameterestimationusingspins;theviewisinto we shall see that the postselection step of the protocol
theequatorialplaneoftheBlochsphere. Adirectstrategyre- leads to an amplification which mitigates this apparent
liesonlyoninterrogationsofthesystemspin,whichpicksup loss of information.
afielddependentphaseexp(itδB)overtime. AWVA(weak-
V. POSTSELECTED STRATEGY
value amplified) strategy makes use of an ancillary spin in
the hope of gaining an advantage. The latter technique in-
volves coupling the first spin with a second ‘meter’ spin with Nowweintroducepostselectionintotheprotocol. The
variablestrength,andtheninterrogatingthemeterspinonly system spin is allowed to pick up phase in the weak field
if the system spin is successfully postselected into a certain as usual. After a given time the measurement M is
final state. Such a postselection can be achieved by the use triggered: this entangles the system spin with the me-
of projective measurements in an appropriate basis given by ter, which is initialized into the −1 eigenstate of σ ,
y
θ, and is known to lead to a larger than expected deflection η(0) = |i−(cid:105)(cid:104)i−|. The system spin is then measured,
of the meter spin if the coupling G is weak. Note that whilst
and only if it is found in a certain postselection state
δB isanunknownquantityofinterest,θ,Θ,Garetunableby √
|ψ (cid:105)=(|0(cid:105)+eiθ|1(cid:105))/ 2,theancillaryspinisinterrogated
the experimenter in the WVA strategy (their meanings are f
in the usual manner (using an adaptive maximum likeli-
explained in the main text).
hoodestimationprocedure). Inthemostgeneralcasethe
angle between pre- and postselection can be varied inde-
pendentlyofthemeasurementcontrolangle(assketched
Π − Π = cosΘσ + sinΘσ . The choice of σ :=
+ − y x z
in Fig. 1). The density matrix of the meter after the
|1(cid:105)(cid:104)1|−|0(cid:105)(cid:104)0| as a meter observable in the second term
measurementinteractionandpost-selectionwillbegiven
is chosen because the ancilla is initialized in plane [39].
by [40]
Thissetupallowsthemeasurementstrengthtobevaried
independentlyoftheinitialstateofthemeter,andhence
captured by G alone: the measurement is strong when η(t)∝(cid:104)ψf|sM[ρ(t)⊗η(0)]M†|ψf(cid:105)s,
G=1 and gets weaker as G→0. After tracing out the
system spin, one finds that the QFI of the meter spin is withtheproportionalityconstantfixedbynormalization.
Undertheseoperations,theevolutionofthemeteriscon-
Ξ2t2sin2(Gπ/2)sin2(Θ−tδB) fined to the x-y plane, which is optimal for phase esti-
H = . (6)
anc 1−Ξ2cos2(Θ−tδB) mation [41]. The full evolution of η is given in Appendix
A, but here we specialize, without loss of generality, to
The subscript ‘anc’ denotes quantities pertaining to an θ =Θ+π/2 (corresponding to a postselection state that
indirect sensing protocol, involving the use of an ancilla is unbiased w.r.t. the control observable [42]). Thus we
spin coupled to the system spin with arbitrary measure- obtain, after normalizing, the simplified expression:
1
η (t)=
11 2
e−21iπG(cid:2)Ξsin(cid:0)Gπ(cid:1)sin(θ−tδB)+i(cid:0)cos(cid:0)Gπ(cid:1)+Ξcos(θ−tδB)(cid:1)(cid:3)
η (t)=η (t)= 2 2 . (8)
12 12 2Ξcos(cid:0)Gπ(cid:1)cos(θ−tδB)+2
2
Of course, η = 1−η and η = η∗ . Calculating the in the equatorial plane leads to
22 11 21 12
QFI can be difficult for arbitrary density matrices, but
we found that applying Eq. (3) to a general qubit state (y2−1)xˆ2−2xyxˆyˆ+(x2−1)yˆ2
H = , (9)
x2+y2−1
5
where x+iy =2η (t) and ‘ˆ’ denotes partial derivative G G
12
w.r.t.δB. Onestraightforwardlyobtains,takingrealand 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Π Π
imaginary parts of (8),
H =H sin2(Gπ/2)A. (10)
wva d
Thesubscript‘wva’denotestheuseofanindirectsensing ∆B Π Π ∆B
t 2 2 t
strategythroughthedensitymatrix(8);butnotethatwe
havetreatedthemeasurementstrengthentirelygenerally.
The expression admits the following interpretation: the
firsttermisthebareinformationavailablethroughdirect 0 HwvaHd HdHwva 0
techniques;thesecondtermrepresentsthecostofhaving
a finite strength measurement, and is present with and 0.0 0.2 0.4 0.6 0.8 1.0
without postselection. The final term Π Π
A=(1+Ξ(t)cos(Gπ/2)cos(θ−tδB))−2 (11)
is due to the weak-value amplification effect: it becomes ∆B Π Π ∆B
largewhenbothθ−tδBisclosetoanoddintegermultiple t 2 2 t
of π and cos(Gπ/2) ≈ 1, i.e. for a weak measurement
strength and almost orthogonal pre- and postselection.
The upper two panels of Fig. 2 demonstrate that in this q qHwvaHd
regimeHwva >Hd,andtheWVAstrategywouldseemto 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
outperform the direct one. However, the much reduced G G
postselection probability
FIG.2. (Coloronline)Contourlinesoffourimportantquan-
q =Tr[(|ψ (cid:105)(cid:104)ψ |⊗I)Mρ(t)M†] tities are plotted against measurement strength G and the
f f
angle (θ −tδB) between pre- and postselection. UL pane:
√
must be taken into account, evaluating to q =1/(2 A). Hwva : Hd uncorrected for the low postselection probability.
Note that q is nonzero when pre- and postselection are Whiteregionsshowaratiohigherthanoneandhencetheap-
parent superiority of the WVA scheme over a direct scheme;
orthogonal due to the back action of the weak measure-
URpane: theinverseratio,whiteregionsnowshowwherethe
ment[7]. Oncetheprobabilityisproperlyaccountedfor,
WVA scheme is inferior; LL pane: the postselection success
probabilityq;LRpane: showsthecorrectedratioqH :H ,
Ξ2t2sin2(Gπ/2) wva d
qH = . (12) note there are no longer white regions and the weak strategy
wva 2(1+Ξcos(Gπ/2)cos(θ−tδB)) is never better. The ∗ denotes a choice of (G,θ) that is gen-
eralized to include dephasing noise in Fig. 4.
A fair comparison between the efficiency of the weak-
value approach and the direct approach can be made by
considering the ratio of the two strategies. Before any
Toillustratethisbehaviorwithaconcreteexample,the
optimizationthis ratio, shownin thelowerright panelof
time dependence of H and H is shown in Fig. 4 for
d wva
Fig. 2, is given by phenomenological dephasing noise, Ξ(t) = e−Γt. In Ap-
pendix B we address the possibility of keeping all of the
qH sin2(Gπ/2)
wva = . (13) data after the postselection measurement. We also gen-
H 2(1+Ξcos(Gπ/2)cos(θ−tδB))
d eralize the decoherence model to incorporate, e.g. am-
plitude damping processes, and consider other scenarios
Note that by inspection this expression never exceeds
where WVA may be of benefit: namely those scenarios
unity for any function Ξ(t) ∈ [0,1]. Since the inequal-
wherethemeasurementitselfisnotimplementedcleanly,
ity qH < H ∀t implies S > S (where S is the
wva d wva d andwhentheinteractionisunavoidablyweak(Appendix
measurement sensitivity), this argument is sufficient to
D). None of these generalizations alter our conclusion.
establishthattheWVAtechniquecanneverreachabet-
ter (i.e. lower) sensitivity than a direct technique.
Note that when dephasing noise is completely absent
then one can reach a ratio of unity for the correct choice VI. CONCLUSION
of G and θ; this is in good agreement with the results of
Starling et al. [16] and Zhu et al. [15]. However, even a We have analyzed the utility of weak-value amplifi-
smallattenuation iscatastrophictotheweak-valuetech- cation for the purpose of estimating an unknown phase
nique because for small G there is a faster than expo- parameter appearing in the Hamiltonian of a two level
nential decay of the ratio qH :H as Ξ decreases (see quantum system, finding no advantage over strong and
wva d
Fig. 3), and a strong measurement quickly becomes fa- directtechniquesforthebroadclassofnoisemodelscap-
vorable. tured by Eq. (1). This includes any kind of dephasing
6
1 completely absent the WVA strategy can match the per-
1 formance of the direct approach, encouraging the motto:
stronger measurement
‘one postselected run acting as though many unpostse-
lected runs’ [26]. In contrast to entanglement or discord
0.99
enhanced sensing protocols, however, which are robust
0.9
against a degree of mixing of the quantum state [36],
0.5 0.7 anylevelofdephasingnoiseruinstheperformanceofthe
WVA approach.
0.1 While we have described the system-meter qubit pair
as spin-1/2 particles, they are isomorphic to many other
physical systems: for example one can use the polariza-
tionstatesofphotonstomeasureaphaseshiftintroduced
byacrystal,andcouplephotonstogethertoenableweak
G
0 0.25 0.5 0.75 1 measurement [11].
We reiterate that our results do not contradict stud-
FIG. 3. (Color online) Ratio of information available in the ies that have already put WVA to use experimen-
WVA strategy and in its direct counterpart, as a function of tally [12, 13, 16] or the many theoretical proposals for
themeasurementstrengthG. Thedifferentcurvescorrespond
improving signal to noise [14, 20, 25, 26, 43, 44] since
to various values of the attenuation function Ξ(t). The post-
in those cases the quantity of interest is an interaction
selection is fixed to θ = tδB+π. When G = 1, one should
parameter, and only technical noise is overcome. When
match the direct strategy, only the postselection probability
the limiting disturbance is to the quantum state how-
q = 1/2 implies by symmetry that half of the information is
ever, rather than to the classical information following
thrown away (see Appendix B).
themeasurement,thenthereisnoadvantagetobegained
by using a weak-value amplified approach.
Future work may elucidate whether imaginary weak
values can be more useful for metrology than real weak
0.04 values, as has been suggested by Refs. [22, 25]; it would
Quantum
be interesting to study how this might apply in finite
Fisher
dimensional meters [45]. The issue of technical noise in
0.02 Information
finite dimensional meters could also be studied.
0
0 3
0.05 2
0.1
t 0.15 1 (cid:31)
0 VII. ACKNOWLEDGEMENTS
FIG.4. (Coloronline)QuantumFisherinformation(inunits
This work was supported by the Engineering and
ofT−2 )forthetwocompetingstrategiesisplottedasafunc-
Physical Sciences Research Council (EPSRC), the John
tion of time t and the dephasing rate Γ (for Ξ=e−Γt). The
Templeton Foundation, and the National Research
upper green surface corresponds to the direct strategy H
d
Foundation, the Ministry of Education, Singapore.
and the lower purple surface corresponds to qH , the cor-
wva
rected weak-value amplified strategy. The postselection and
measurement strength are fixed to θ = tδB +π,G = 0.02,
respectively, corresponding to the ∗ in Fig. 2. Even a mod-
erate amount of dephasing has a catastrophic effect on the
weak-value scheme.
Appendix A: Full evolution of ancilla
noise,andweshowinAppendixCthatthequalityofour For a completely general choice of angles between pre-
conclusion is preserved when other imperfections (such andpostselectionandthemeasurementcontrolasdefined
as T processes) are considered. When decoherence is in Fig. 1, the evolution of the meter quit is described by
1
η = 1
11 2
4ie−21iGπ(cos(Gπ/2)+icos(θ−Θ)sin(Gπ/2)+(cos(Gπ/4)2cos(θ−tω)−cos(θ−2Θ+tω)sin(Gπ/4)2+icos(Θ−tω)sin(Gπ/2))Ξ)
η = .(A1)
12 8+8(cos(Gπ/4)2cos(θ−tω)+cos(θ−2Θ+tω)sin(Gπ/4)2)Ξ
InthemaintextafixedrelationshipbetweenΘandθwas cannot provide an advantage.
chosenforsimplicity,andbecausethisadditionalfreedom
7
Appendix B: Using all the data (a) (b)
Since the WVA technique can get close to the perfor-
mance of a direct strategy and involves discarding the
majority of experimental runs, one might imagine that
some of the discarded data may be used to increase the
information,perhapsevenallowingthetechniquetoout-
perform the direct strategy. In addition to the quantum
Fisher information arising from successful postselection
Eq. (12), one now has
(c)
(1−q)H⊥ = t2Ξ2sin2(Gπ/2) (B1) 1
wva 2−2Ξcos(Gπ/2)cos(θ−tω)
resulting from runs that would ordinarily be discarded. 0.5
Onecanseethatthetwoquantitiesarecomplementary
Π
in the following sense. In the regime where the WVA 0
1
effect is strongest, the discarded runs carry less and less
information. Since the total information is additive, one
Π
can achieve Θ(cid:29)t∆B 2 0.5
G
H := qH +(1−q)H⊥ (B2)
total wva wva
= t2sin2(Gπ/2) . (B3)
1−Ξ2cos2(Gπ/2)cos2(θ−tω)
0 0
This quantity cannot exceed H , but has some interest-
d
ing features. It is greater than qH in regions where
wva FIG. 5. (Color online) Quantum Fisher information for a
theWVAeffectissmall,notablyreachingqH /H =1
wva d weak-valueamplifiedsensingstrategy(a)whenthepostselec-
(rather than 1/2) when the measurement is fully strong.
tionis‘successful’(b)whenthepostselectionis‘unsuccessful’
It converges on qHwva when the WVA effect is pro- (c) when both are considered together; the plots shown cor-
nounced, see Fig. 5. respond to the decoherence-free case.
Appendix C: A more general noise model
thoughtofasderivingfromtheorderinwhichtheampli-
fication and noise act. Because the noise acts before the
One can allow the populations of the density matrix ρ measurement takes place, it gets amplified as much as
to have a time dependence, the signal does. Secondly, one can think of the failure as
deriving from the drop in signal strength incurred from
2ρ (t)=R(t)
11
the weak interaction strength.
2ρ (t)=−ie−igδBtΞ(t). (C1)
12
Note that R(t) and Ξ(t) are not independent and are
√
related by Ξ ≤ 2 R−R2, and that decoherence mod-
1. Noisy meter system
els such as relaxation can now be modelled R(t) =
e−t/T1, Ξ(t) = e−t/(2T1). This has no effect on Hd or
H (despite the ancilla qubit now having an evolution Itisnaturaltoinquireaboutthecasewhentheamplifi-
wva
out of plane that depends on δB). Notwithstanding fur- cationtakesplacebeforeanynoisehasacted. Ourmodel
theroptimizationsofthemeasurementinteractioninthis canbeeasilyadaptedtostudythispossibility. Oneimag-
scenario, the results presented here are conserved when ines a situation where a well-isolated quantum system
thenoisemodelisgeneralizedtoallowforrelaxation,po- (perhaps a nuclear spin with an infinite coherence time)
larization damping or other noise channels. isusedtosenseaparameter(amagneticfielddeviation),
but is inaccessible to direct measurement. Readout can
be performed by coupling to an ancillary spin (perhaps
Appendix D: Imperfections in the measurement an electron spin) as above, but now the ancilla suffers
device an attenuation of its coherences described by Σ(t). An
estimation scheme using a strong measurement G = 1
without postselection can achieve (with the usual opti-
For an exhaustive assessment of the utility of WVA,
mization over the measurement control)
one can consider more contrived situations. The failure
ofthetechniquetoprovideenhancementoverdirectmea-
surements on a noisy system is twofold. Firstly it can be H˜ =Σ2t2. (D1)
anc
8
The ‘ ˜ ’ denotes the decoherence acting on the me- 2. Limited measurement strength
ter rather than the system. Allowing the measurement
strength to vary and enabling postselection yields
H˜ = Σ2t2sin2(Gπ/2) (D2) Let us consider the advantage postselection alone can
wva (1+cos(Gπ/2)cos(θ−tδB))2 haveonanarbitrarystrengthmeasurementaccomplished
→ Σ2t2cot2(Gπ/4) (D3) via a meter qubit. Perhaps one has no control over the
valueofG,butisinfactforcedtomakeaweakmeasure-
when tuning the postselection as above. Unsurprisingly
ment. In this case the appropriate ratio is
the signatures of AAV’s effect persist as long as the at-
tenuation of coherence in the meter has not completely
annihilatedtheoffdiagonaldensitymatrixterms[46–48].
Once more, the weak value approach suggests an em-
qH 1
phatic improvement (the expression diverges as G→0). Hwva = 2(cid:0)1+Ξcos(cid:0)Gπ(cid:1)cos(θ−tδB)(cid:1). (D5)
Note, however, that assuming the postselection can be anc 2
implemented is not entirely consistent with the idea of
beingforcedtomeasureindirectlythroughanoisyqubit.
Nevertheless, with the postselection probability taken
into account one finds Now, there is a clear advantage as long as the dephasing
noise is not too aggressive. But again one has had to
qH˜ =Σ2t2cos2(Gπ/4). (D4) assume the use of strong measurements for the purpose
wva
ofpostselectionbutdenytheiruseintheestimationpart
In close resemblance to the above case, the QFI never of the protocol, which is not entirely consistent. This
exceeds the direct strategy benchmark; in fact, it varies situationmayappearsimilartothecaseofestimatingan
betweenatbestbeingequaltoandatworsthalfaslarge. interaction parameter: but there the unknown quantity
So, even when given this improbably favorable scenario, of interest is the measurement strength, and the Fisher
the WVA approach fails to offer an advantage. information must be calculated w.r.t. G [49].
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