Precise Phase Measurements using an Entangled Coherent State

0.33 MB·

by  P. A. Knott

Precise Phase Measurements using an Entangled Coherent State
P.A.Knott1 andJ.A.Dunningham1,2
1SchoolofPhysicsandAstronomy,UniversityofLeeds,LeedsLS29JT,UnitedKingdom
2DepartmentofPhysicsandAstronomy,UniversityofSussex,Falmer,BrightonBN19QH,UnitedKingdom
[e-mail],[e-mail]
Keywords: Coherentstate,metrology,phasemeasurement.
Abstract: Quantum entanglement offers the possibility of making measurements beyond the classical limit, however
4
some issues still need to be overcome before it can be applied in realistic lossy systems. Recent work has
1
usedquantumFisherinformation(QFI)toshowthatentangledcoherentstates(ECSs)maybeusefulforthis
0
purposeastheycombinesub-classicalphaseprecisioncapabilitieswithrobustness(Jooetal.,2011).However,
2
todatenoeffectiveschemeformeasuringaphaseinlossysystemsusinganECShasbeendevised. Herewe

n presentaschemethatdoesjustthis.Weshowhowonecouldmeasureaphasetoaprecisionsignificantlybetter
a thanthatattainablebybothunentangled‘classical’statesandhighly-entangledNOONstatesoverawiderange
J
ofdifferentlosses.Thisbringsquantummetrologyclosertobeingarealisticandpracticaltechnology.
6
1

h] 1 INTRODUCTION shows remarkable robustness to photon losses: en-
p tangled coherent states ECSs. We discuss how these
- states can be created, and present a scheme which
t Quantummetrologyistheartofmakingprecision
n allows us to measure a phase using an ECS to sub-
measurements by taking advantage of the properties
a classicalprecision,evenwhenawiderangeofdiffer-
u ofquantummechanics. Themainadvantageofquan-
entlossratesareaccountedfor.
q tum metrology over classical metrology is that it al-
[ lows usto achievethe same precisionwith fewer re-

sources. Making more precise measurements with
1
v limited numbers of particles or photons has many 2 PRECISE PHASE
9 important applications, including microscopy, grav- MEASUREMENTS
6 itational wave detection, measurements of material
9 properties, and medical and biological sensing (Na-
3 gataetal.,2007). Manyoftheseexamplescouldben- 2.1 EnhancementUsingEntanglement
.
1 efitfromadevicewhichoperateswithalowerphoton
0 flux, for example in biological sensing, where dis- Throughout this paper we will generally be con-
4 turbing the system too much can damage the sam- cerned with measuring phases in a device using the
1
ple. Another important reason for developing quan- sameprinciplesofaMach-Zehnderinterferometer,as
:
v tum metrology is that it provides a stepping stone shown in Fig. 1. The first step in this device is to
Xi to more complicated quantum technologies such as combine the two input states at a beam splitter. One
quantumcomputers: ifwecanbuildmetrologicalde- of the paths then picks up a phase, φ, and the two
r
a vices that can beat the classical limit by exploiting paths are recombined at a second beam splitter. The
entanglement, then this puts us in a good position to resultingoutputfromthesecondbeamsplittercanbe
begintotacklemoreadvancedmanipulationsofquan- measuredbydetectorsD1andD2toextractthephase
tum properties. Furthermore, measurements are cru- information. A‘classical-like’stateoflight,whichis
cial to the development of science, and any way to equivalenttosendingasequenceofindependentsin-
improvethemisawelcomedevelopment. gle particles (SP) through the interferometer allows
One of the main stumbling points in quantum onetomeasurethephasewithaprecisionthatscales
metrologyiscreatingastatethatisrobusttoparticle withthetotalnumberofparticlesnasthe“shotnoise
√
losses,whichwillalwaysbeaconcerninanyrealistic limit” 1/ n (Gkortsilas et al., 2012). However by
device. In this paper we present an optical state that makinguseofanentangledstate,theprecisioncanbe
improved to 1/n: the “Heisenberg limit” (Dunning- included,theyarestillfragile,andwithlargeamounts
hamandKim,2006). of loss they are outperformed by classical strategies.
Wewillnowturntoastatewhichshowshugepoten-
Input 1 tial, as it is intrinsically robust to the effects of loss:
D1 coherentstates.
φ
Beam Beam 2.2 EntangledCoherentStates
Splitter Splitter
Acoherentstateisdefinedas:
D2
Input 2
Fusigedureto1m:eAasuMreacahp-Zhaeshendφe.rPinhtoetrofnesroamreetseernwdhthicrhoucgahnthbee |α(cid:105)=e−|α2|2 ∑∞ √αn |n(cid:105), (2)
n!
beamsplittersandphaseshift,thenmeasuredatthedetec- n=0
torsD1andD2. whereαisacomplexamplitudeandnistheparticle
number. Inordertoachievequantumenhancedmea-
In order to create an entangled number state we
surement,westillneedtocreateanentangled coher-
replacethefirstbeamsplitterintheinterferometerin
entstate(ECS)(Sanders,2012;Hirotaetal.,2011):
Fig.1bya“quantumbeamsplitter”(QBS).Aquan-
tum beam splitter is like an ordinary interferometer,
(cid:16) (cid:17)
butwithanonlinearityinonearm(Dunninghamand N |α,0(cid:105)+eiθ|0,α(cid:105) (3)
Kim, 2006)insteadofthephase, andhasthefollow-
ingeffectonstate|n,0(cid:105): (cid:112)
where N =1/ 2+2e−|α|2cosθ. We could create
the ECS with θ=π/2 by sending input state |α,0(cid:105)
QBS 1 through the QBS. Alternatively, the ECS with θ=0
|n,0(cid:105)−−→ √ [|n,0(cid:105)+i|0,n(cid:105)] (1)
2 canbecreatedbyinteractinga“catstate”Nα(|+α(cid:105)+
|−α(cid:105))withacoherentstate|α(cid:105)atabeamsplitter(Joo
The state on the right hand side is known as a
et al., 2011). This latter scheme is likely to be more
NOON state (Dowling, 2008). After creating the
experimentallyfeasible. However,theissueofexper-
NOON state, we then apply the phase shift to one
imentalimplementationswillbelefttolaterwork.
ofthepathsintheinterferometer, aswedidwiththe
InordertoinvestigatethepotentialoftheECSin
SP state. We can then send this state through a sec-
quantum metrology, Joo et. al. (Joo et al., 2011)
ondQBS,andmeasurethenumberofparticlesatthe
calculateditsquantumFisherinformation(QFI).The
detectors. If we send a NOON state of n particles
QFIforageneralstateρ(BraunsteinandCaves,1994;
through through this scheme, then the precision of
Boixoetal.,2009)isgivenby:
phasemeasurementcanbeseentovaryasδφ=1/n,
theHeisenberglimit(DunninghamandKim,2006).
However, there is a problem with such an ap- F =Tr(cid:0)ρA2(cid:1) (4)
Q
proachbecauseNOONstatesarehighlyfragiletopar-
where A is found from solving the symmetric loga-
ticle loss. Losing just one particle from a NOON
rithmicderivative∂ρ/∂φ=1/2[Aρ+ρA]. Thepreci-
state (Eq. 1) will project the state onto either
einφ|n,0(cid:105) or i|0,n(cid:105). The global phase in each of sioninthephasemeasurement(morespecificallythe
lowerboundonthestandarddeviation)isgivenbythe
these cases is not physical and cannot be measured,
quantum Crame´r-Rao bound (Braunstein and Caves,
so all the phase information is lost when we lose
1994):
just one particle. Despite this, a number of clever
schemes have been devised with robustness to loss 1
δφ≥ , (5)
which stillcapture subshot noiselimit precision, al- (cid:112)
µF
Q
beitnotquiteattheHeisenberglimit. Anexampleof
one of these schemes is a NOON “chopping” strat- whereµisthenumberofcopiesofthestate(i.e.times
egy (Dorner et al., 2009), in which multiple smaller that the measurement is independently repeated).
NOONs are sent through an interferometer instead This gives the best possible precision with which a
of one big one. Other examples include unbalanced statecanmeasureaphase. ForNOONstatesandSP
NOON states (Demkowicz-Dobrzanski et al., 2009), states the quantum Crame´r-Rao bound gives us the
BAT states (Gerrits et al., 2010) and mixtures of SP Heisenbergandshotnoiselimitsrespectively.
and NOON states (Gkortsilas et al., 2012). While Joo et. al. used the QFI to show that with and
thesestatescanbeattheshotnoiselimitwhenlossis without loss the ECS can achieve better precisions
than SP, NOON, and some other candidate states. ofdifferentnumbersofphotonsbeingdetected,given
Zhang et. al. (Zhang et al., 2013) derived an ex- thatthephaseintheinterferometerisφ :
1
pression for the QFI with loss for arbitrary α, and
confirmed the potential of ECSs for robust quan- P(n ,n |φ=φ )=|(cid:104)n ,n |ψ (cid:105)|2 (8)
tum metrology. What is still missing, however, is a 1 2 1 1 2 4
concrete way for converting this promise into a real Usingthisconditionalprobabilitydistributionwe
schemeformakingthephasemeasurement. can apply Bayesian statistics to build up our knowl-
edge of the phase φ as we repeat the process with a
streamofECSs(Gkortsilasetal.,2012).Fig.2shows
that this scheme, with no loss, allows us to beat the
3 MEASURING A PHASE USING
bestpossibleprecisionobtainableusingNOONstates
AN ENTANGLED COHERENT
of comparable sizes. For small α we do not saturate
STATE the QFI, but we significantly improve upon the best
possiblemeasurementusingaNOONstate. Forlarge
αthisschemecomesveryclosetosaturatingtheQFI,
3.1 ASimpleSchemewithNoLoss
butitcanbeseenthatinthisregionECSsandNOON
statesoperateataverysimilarprecision.
Wewillnowlookattheeffectofsendinginputstate
|ψ (cid:105)=|α,0(cid:105)throughtheinterferometerinFig.1,but 0.04
1 δφCM
wofitthhethfiersbteQamBSspislitttoerpsroredpulcaecethdebEyCQSB|Sψs.(cid:105)T: heeffect 0.035 δδφφCNFF
2
0.03
QBS 1 0.025
|ψ (cid:105)=|α,0(cid:105)−−→ √ (|α,0(cid:105)+i|0,α(cid:105))=|ψ (cid:105). (6)
1 2
2 δφ 0.02
We then perform a phase shift which gives us the
0.015
state:
0.01
|ψ (cid:105)= e−√|α2|2 ∑∞ √αn (cid:2)einφ|n,0(cid:105)+i|0,n(cid:105)(cid:3) 0.005
3 0
2 n! 1 2 3 4 5
n=0 α
= √1 (cid:0)|αeiφ,0(cid:105)+i|0,α(cid:105)(cid:1), (7) Figure 2: The phase precision of the scheme described in
2 section3.1fornolossisshownhere.Weplotdifferentsized
ECSs against their precision. Here (and in later figures)
followedbythesecondQBS: δφCM istheECSusingourmeasurementscheme, δφCF is
the QFI for the ECS and δφNF is the QFI for the NOON
state(ofequivalentsizeaseachECS).SPstatesmeasureat
|ψ4(cid:105)=e−|α2|2 ∑∞ √αn iein2φ(cid:18)|n,0(cid:105)sinnφ+|0,n(cid:105)cosnφ(cid:19). precision0.0354.
n! 2 2
n=0 We will now briefly discuss the details of our
schemethatallowustocompareourresultstoNOON
Fromthiswecancalculatetheprobabilityampli-
and SP states. In many applications of quantum
tudeofdetectingdifferentnumbersofphotonsatthe
metrologysuchasbiologicalsensing,probingmateri-
outputs. Todothiswefirsttaketheinnerproductof
|ψ (cid:105)with|n (cid:105) |n (cid:105) =|n ,n (cid:105),i.e. thestatewithn alspropertiesandgravitationalwavedetection,weare
ph4otonsatd1eteDc1to2rDD21and1n p2hotonsatdetectorD21. concerned with the number of photons (or particles)
2
passingthroughthephaseshiftitself,andwouldlike
Thisgivesus:
to minimise this number when possible. We would
thereforeliketocomparethephasemeasurementpre-
(cid:104)n1,n2|ψ4(cid:105)= cision of different states when a set number of pho-
ie−|α2|2 (cid:20)√αnn1!ein21φsinn21φδn2,0+√αnn2!ein22φcosn22φδn1,0(cid:21).twoenssiRmpelnytesrenthdethpehasstaet.e tFhororutghhe tuhneenintatenrgfelerdomsetateter
1 2 2R times, as in each run an average of 1/2 a pho-
Thedeltafunctionsheretellusthatitisimpossible ton enters the phase shift. For the NOON state in
todetectphotonsatbothoutputs. Thisisclearlytrue equation 1, each run sends n/2 photons through the
asanyphotondetectioncollapsesthestateintoeither phase shift, and so we simply send the NOON state
|n,0(cid:105)or|0,n(cid:105). Wecannowcalculatetheprobabilities throughtheinterferometer2R/ntimes. ForECSsthe
situation is slightly different, as each run contains a
differentnumberofphotons. Wecancalculatetheav-
erage number of photons passing through the phase ρ =c (|ψ (cid:105)(cid:104)ψ |)+
2 1 2 2
faonrdthweegtehneererafolrEeCsSenidntehqeuaEtCioSnt3hrtooubgehn¯th=eNint2e|rαfe|2r-, 1c2(cid:0)|αeiφ√η,0(cid:105)(cid:104)αeiφ√η,0|+|0,α√η(cid:105)(cid:104)0,α√η|(cid:1)
2
ometer R/n¯ times. For most of our results we have
used R = 400 as this allows us to consistently cal-
culate the precision of the phase measurement with wherec1=e|α|2(η−1),c2=1−c1and:
differentstates.
3.2 IntroducingLoss |ψ (cid:105)= √1 (cid:2)|αeiφ√η,0(cid:105)+i|0,α√η(cid:105)(cid:3). (11)
2
2
Measurement by Theresultingstateisamixtureoflossandnoloss
Incoming the environment
state components. ForNOONstatesweknowduringeach
runiftherehasbeenlosssimplybycountingthenum-
bersofparticlesattheoutputs.However,forECSswe
BS Fictional beam can no longer do this, as we don’t know the number
splitter
ofparticlesinanECStobeginwith. This,combined
with the fact that the size of the coherent state has
decreased after the loss, means that the ECS in this
Figure3: Thisshowsa“fictional”beamsplitterandmea-
simple interferometer loses its phase precision more
surementbytheenvironmentasamodelforloss.
quickly than NOON states, as shown in Fig. 4. This
To simulate the effects of loss we introduce “fic- agreeswiththeworkby(Zhangetal.,2013)whocal-
tional”beamsplittersafterthephaseshift(Gkortsilas culated the Fisher information of ECSs with loss for
etal.,2012;Jooetal.,2011;Demkowicz-Dobrzanski any value of α. They showed that the quantum en-
etal.,2009)asshowninFig.3,whichhaveprobabil- hancement of the ECS decreases more quickly than
ity of transmission η≡cos2θ. After the phase shift NOONstateswithloss.Despitethistheyalsoshowed
andincludingthevacuumstatesusedtosimulateloss, thattheECSsstillcontainsomephaseinformationaf-
theECSweareconcernedwithisgivenby: ter loss (unlike NOON states) and this should allow
ustorecoverthephaseinformationandthereforeend
upbeatingNOONstatesinthelongrun. Oursimple
|ψ (cid:105)= √1 (cid:2)|αeiφ,0,0,0(cid:105)+i|0,0,α,0(cid:105)(cid:3), (9) schemeclearlydoesnotdothis,butinthenextsection
0
2 wewillpresentaschemethatdoes.
where the second and fourth terms in the kets repre-
sentthemodesintowhichparticlesarelostfromthe 0.07 δφNF n=4
firstandthirdtermsrespectively. δφSF
The effect of the “fictional” beam splitters then 0.06 δφCM α=2
leavesusinthestate|Ψ (cid:105)givenby:
1 0.05
√1 (cid:2)|αeiφsinθ,αeiφcosθ,0,0(cid:105)+i|0,0,αsinθ,αcosθ(cid:105)(cid:3). φ0.04
2 δ
0.03
We then take the density matrix ρ = |ψ (cid:105)(cid:104)ψ |
1 1 1
andtorepresentmeasurementbytheenvironmentwe 0.02
traceovertheenvironmentalmodesasfollows:
0.01
ρ =∑∑(cid:104)e |(cid:104)e |ρ |e (cid:105)|e (cid:105). (10) 0
2 1 2 1 2 1 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
e1 e2 η
Figure4: HerethelegendreferstostatesasinFig.2,with
Using ∑e(cid:104)e|X(cid:105)(cid:104)Y|e(cid:105) = (cid:104)Y|X(cid:105) and the δφSF theQFIforSPst√ates. WecanseethatECSsdegrade
nonorthogonality of coherent states (cid:104)α|β(cid:105) = quicklywithloss(α= 2here).ForlargerαtheECSloses
exp(−1|α|2+α∗β−1|β|2) it can be shown that precisionwithlossevenquicker.
2 2
ρ isreducedto:
2
4 IMPROVED SCHEME WITH to|Ψ (cid:105).Wethentraceovertheenvironmentaldegrees
2
LOSS of freedom to give ρ1 =∑e(cid:104)e2|Ψ2(cid:105)(cid:104)Ψ2|e(cid:105) where |e(cid:105)
represents all four environmental modes. This gives
us:
Despite the fact that an entangled coherent state
canstillretainsomephaseinformationafterloss,we
haveseenthatwithasimplemeasurementschemethe
ρ =|Φ (cid:105)(cid:104)Φ |+|Φ (cid:105)(cid:104)Φ | (14)
2 1η 1η 2η 2η
phase information cannot be recovered, and we end
updoingevenworsethanNOONstates. Wehavede- +e−|α0µ|2(|Φ (cid:105)(cid:104)Φ |+|Φ (cid:105)(cid:104)Φ |), (15)
1η 2η 2η 1η
visedascheme,showninFig.5,whichcanbeusedto
where η is the transmission rate through the inter-
recoverthisdesiredphaseinformation. Thekeyisto
useextra“reference”coherentstatesaboveandbelow ferometer, |Φ1η(cid:105)= √12|α1η,α0√ηeiφ,0,α1η(cid:105), |√Φ2η(cid:105)=
themaininterferometerwhichcanbeusedtoperform √i |α1η,0,α0η,α1η(cid:105),α0η=α0 η,α1η=α1 ηand
2 √
a homodyne measurement and recover the phase in-
α =α 1−η. Thisstatecanalsobewrittenas:
0µ 0
formation. When a photon is lost from the ECS in
equation 7 the state collapses into |αeiφ,0(cid:105) or |0,α(cid:105).
Ifweareleftwiththesecondstate,thenthephasein- ρ =c |Ψ (cid:105)(cid:104)Ψ |+c (|Φ (cid:105)(cid:104)Φ |+|Φ (cid:105)(cid:104)Φ |)
2 1 1η 1η 2 1η 1η 2η 2η
formation is irretrievable, but if we are left with the
state|αeiφ,0(cid:105)thenthephaseinformationisstillthere. where |Ψ1η(cid:105) √is the state in equation 12 with all the
αreducedto ηα. Inthisformitiseasytoseethat
However, in order to extract it we need a reference
themixedstateafterlossiscomprisedofapurestate
state |α(cid:105) to “compare” it to, hence including the up-
perandlowerarmsinourinterferometer. |Ψ1η(cid:105)withcoefficientc1=e−|α0µ|2,acollapsedstate
|Φ (cid:105)whichcontainsthephase,andacollapsedstate
1η
Loss after the phase shift |Φ (cid:105)thatdoesnotcontainthephase,bothwithcoef-
2η
ficientc =1−c . Wecanthensendρ throughthe
2 1 2
remainderoftheinterferometer,givingtheprobabili-
tiesattheoutputsas:
P(#)=(cid:104)#|Φ (cid:105)(cid:104)Φ |#(cid:105)+(cid:104)#|Φ (cid:105)(cid:104)Φ |#(cid:105)
1η 1η 2η 2η
φ BS D1 +e−|α0µ|2(cid:2)(cid:104)#|Φ1η(cid:105)(cid:104)Φ2η|#(cid:105)+(cid:104)#|Φ2η(cid:105)(cid:104)Φ1η|#(cid:105)(cid:3),
where|#(cid:105)=|k,l,m,n(cid:105),thestatewithkparticlesinthe
QBS QBS D2 first output, l in the second and so on. The barred
states|Φ (cid:105)and|Φ (cid:105)canbefoundbysending|Φ (cid:105)
1η 2η 1η
and |Φ (cid:105) through the remainder of the interferome-
2η
D3
ter.Weinitiallytooktheobviouschoiceforthe“refer-
BS
ence”statesasα =α . However,wefoundthatthis
1 0
schem√egaveuspoorresults,asshowninFig.6√where
D4 α= 2. For this very small choice of α = 2 we
0
canbeattheNOON andSPstatesuptoaround 15%
loss, which is indeed a very positive find. But after
thispointitismorebeneficialtouseeitherNOONor
Figure 5: Quantum interferometer with extra arms to re-
SP states. If we increase α then the results soon get
coverphaseinformationwithloss.
much worse and before long we cannot beat either
The state in this “long arm” interferometer after NOONorSPstatesifthereisanylossatall.
thephaseshiftis: Despitetheseshortcomings,theclearpotentialof
ECSs warranted a more rigorous search for changes
that can optimise our scheme. Indeed, if we in-
|Ψ (cid:105)= √1 (cid:0)|α ,α eiφ,0,α (cid:105)+i|α ,0,α ,α (cid:105)(cid:1) stead use the initial ECS N(|α,0(cid:105)+|0,α(cid:105)) where
1 1 0 1 1 0 1
2 (cid:113)
(12) N =1/ 2(1+e−|α|2), as used by Joo et. al. (Joo
etal.,2011),webegintogetmorepositiveresults. A
=|Φ (cid:105)+|Φ (cid:105). (13)
1 2
muchmoresignificantchangewecanmakeistovary
Afterbeingactedonbythefictionalbeamsplitters thesizeofα fordifferentlossvalues,insomecases
1
thatsimulateloss,thisstateistransformedfrom|Ψ (cid:105) uptoaroundα =2.4α –adetailedstudyofwhythis
1 1 0
0.07
δφCM showedthatforlargeαthereisasmallregionwhere
δφNF the NOON state performs better than the ECS be-
0.06 δφSF
cause “although the classical term of the ECS is ro-
bust against the photon losses, the Heisenberg term
0.05
decays about twice as quick as that of the NOON
state”(Zhangetal.,2013). Thisagreeswellwithour
δφ0.04
results. Ourschemedoesn’tsaturatetheFisherinfor-
mation,butcomereasonablyclose,andthisisenough
0.03
tobeattheNOONandSPstatesmuchofthetime.
Future work will include examining different
0.02
ECSs with different QBSs in order to try and come
closer to saturating the QFI. We would also like to
0.010. 4 0.5 0.6 0.7 0.8 0.9 1
η look how this measurement scheme could be carried
Figure 6: This scheme clearly doesn’t perform well com- out in an experiment. Despite the fact that we have
paredtoNOONandSPwhenthereismorethan15%loss. looked at how to measure the phase, there are still
parts of our scheme that are not easily achievable in
experiment,andwewouldliketoironthesepartsout
is the case is the subject of ongoing work. The pre-
sothatwehaveafullyrealisableschemetomeasurea
cisionwithwhichthisschemecanmeasurethephase
phasetoasignificantlyhigherprecisionthatthecom-
alsodependsonthe(approximate)phasebeingmea-
petingstates.
sured(thisistrueformostschemes).Nonethelessthis
shouldnotposemuchofaproblemaswecanjustput
0.07
avariablephaseshiftinthelower-middlepath,which δφCM
δφSF
allows us to vary the phase difference so that effec- 0.06 δφNF
tivelyφcanbewhateverwechoose. δφCF
0.07 0.05
δφ
CM
δφNF δφ0.04
0.06 δφ
SF
δφ
CF
0.03
0.05
0.02
φ
δ
0.04
0.01
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
η
0.03 Figure 8: Here α=2. Our scheme beats NOON and SP
statesformostlossvalues.
0.02
0.15
0. 3 0.4 0.5 0.6 0.7 0.8 0.9 1 δφCM
η δφSF
Figure7: Withalargeα1w√ecanbeatboththeNOONand 0.12 δδφφNCFF
SPstates. Here,forα0= 2webeatbothNOONandSP
allofthetime.
0.09
Withthesechanges,andaftercarefullyoptimising
φ
δ
over φ an√d α1, we then obtain the results in Fig. 7
0.06
for α = 2. It can be seen that our state now out-
0
performs the NOON and SP states for all values of
loss. Figures8and9showtheresultsforα0=2and 0.03
α =5respectively. Wecanseethatfortheselarger
0
valuesofα ourschemestillbeatsthecompetitorsfor
0 0
themajorityofηvalues. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
η
Our results fit well with the Fisher information
Figure9:Hereα=5.AgainweperformbetterthanNOON
given by Zhang et. al. which is shown as the red andSPmostofthetime.
solid line δφ on all three figures. The authors
CF
5 CONCLUSIONS REFERENCES
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measurement, up to this point it has not been clear
80(3):032103.
howthephaseinformationcanactuallybemeasured.
Braunstein, S. and Caves, C. (1994). Statistical distance
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