ARTICLE OPEN
Quan um-enhanced Dopple lida
Maximilian Reiche
1,2
✉, Robe o Di Candia
3,4
, Moe Z. Win
5
and Mikel Sanz
1,2,6,7
✉
We p opose a quan um-enhanced lida sys em o es ima e a a ge ’s adial eloci y, which employs squeezed and equency-
en angled signal and idle beams. We compa e i s pe o mance agains a classical p o ocol using a cohe en s a e wi h he same
pulse du a ion and ene gy, showing ha quan um esou ces p o ide a p ecision enhancemen in he es ima ion o he eloci y o
he objec . We iden i y h ee dis inc pa ame e egimes cha ac e ized by he amoun o squeezing and equency en anglemen . In
wo o hem, a quan um ad an age exceeding he s anda d quan um limi is achie ed assuming no pho on losses. Addi ionally, we
show ha an op imal measu emen o a ain hese esul s in he lossless case is equency- esol ed pho on coun ing. Finally, we
conside he e ec o pho on losses o he high-squeezing egime, which leads o a cons an ac o quan um ad an age highe
han 3 dB in he a iance o he es ima o , gi en a ound ip lida - o- a ge - o-lida ansmissi i y la ge han 50%.
npj Quan um In o ma ion (2022) 8:147 ; h ps://doi.o g/10.1038/s41534-022-00662-9
INTRODUCTION
Quan um me ology exploi s quan um mechanical esou ces, such
as en anglemen and squeezing, o measu e a physical pa ame e
wi h highe esolu ion han any s a egy wi h classical esou ces.
Many quan um me ology p o ocols in he pho onic egime
1
ha e
been p oposed such as quan um illumina ion (QI)
2–6
, quan um-
enhanced posi ion and eloci y es ima ion
7–13
, quan um phase
es ima ion
14,15
, ansmission pa ame e es ima ion
16–20
, noise
es ima ion
21
and es ima ion o sepa a ion be ween objec s
22,23
,
among o he s. In hese p o ocols, in o ma ion abou an objec is
e ie ed by in e oga ing i wi h a signal beam. In he mos
gene al s a egy, his signal is co ela ed o en angled wi h an idle
beam, which is e ained in he lab o pe o m a join measu emen
a he end o he p o ocol. Indeed, he scheme can be seen as an
in e e ome e se up, in which a channel depending on he
pa ame e o in e es is only applied o he signal mode.
O pa icula in e es o emo e sensing applica ions is he QI
p o ocol, whe e he aim is o de ec he p esence o a weakly
eflec ing a ge wi h an e o p obabili y smalle han using he
bes classical s a egy. He e, a quan um ad an age in he e o
p obabili y exponen can be achie ed by using a global
measu emen (up o 6 dB)
24–28
, o by using local measu emen s
(up o 3 dB)
29–31
. This ad an age is only achie ed in a e y noisy
en i onmen , such as he case o oom- empe a u e mic owa e
band, by a la ge bandwid h wo-mode squeezed- acuum s a e
3
.
This equi es a signal wi h a e y low pho on numbe pe mode,
which in he mic owa e egime is challenging o ansmi open-
ai . Since ampli ying he signal has been shown o b eak he
quan um ad an age
32,33
, QI as o iginally hough emains an
elusi e achie emen so a , e en hough ecen p og ess has been
made on elaxing he equi emen s o quan um ad an age
34
.
Once he p esence o a a ge is es ablished, p ope ies like i s
loca ion and eloci y a e also o in e es . These can be es ima ed
ia signal a i al ime and equency measu emen making use o
he Dopple e ec . Gio anne i, Lloyd and Maccone showed in
e .
7
, ha he GLM s a es, named a e hem, defined in he
equency domain can a ain he Heisenbe g limi (HL), which is a
1/Nscaling o he es ima ion e o o he a i al ime, whe e Nis
he o al numbe o pho ons. Equi alen ly, GLM s a es defined in
he ime domain each he HL o he es ima ion e o o
equency. This cons i u es a quad a ic imp o emen compa ed
wi h he s anda d quan um limi (SQL) achie ed by he classical
p o ocol. In e .
12
, he simul aneous es ima ion o loca ion and
adial eloci y was conside ed using wo GLM s a es in he
equency and ime domain, espec i ely, ha a e ans o med
in o wo en angled signal and idle beams ia a beam spli e . I
was shown ha he eloci y and he loca ion can simul aneously
be es ima ed achie ing he Heisenbe g limi . This p o es ha
equency en anglemen li s he A hu s-Kelly ela ion
35
, which
s a es ha he loca ion and eloci y o an objec canno be
es ima ed wi h a bi a y p ecision using unen angled ligh . The
wo k
13
u he ex ended his by add essing he simul aneous
es ima ion o ela i e loca ion and eloci y o wo a ge s by
means o wo-pho on en angled s a es. The main d awbacks o
hese p e ious wo ks a e he use o wo-pho on s a es, which
does no allow o a pho on-numbe -dependen analysis, and he
use o he mul ipho on GLM s a es, which a e non-no malizable
and hus no physical. In e .
36
, a no malized e sion o he GLM
s a e was in oduced o ange es ima ion. He e, i was shown ha
he Heisenbe g scaling pe sis s o he no malized e sion.
Howe e , hese GLM- ype s a es a e agile in lossy channels.
The loss o a single pho on ende s he s a e useless o e ie ing
in o ma ion abou he pa ame e . Al hough he obus ness
agains losses o hese GLM- ype s a es may be imp o ed by
educing hei en anglemen , his comes a he cos o dec easing
he enhancemen in he scaling o he e o es ima ion.
Fu he mo e, i is challenging o p oduce GLM s a es in he
labo a o y o pho on numbe s N>2
9
.
In his a icle, we p opose a p o ocol o a quan um Dopple
lida , which es ima es he adial eloci y o a eflec ing objec
using quan um ligh . As a p obe s a e, equency-en angled win-
beams a e used. The signal beam is sen agains he mo ing
objec , which causes a equency shi due o he Dopple e ec .
Finally, a measu emen o he e u ned signal and he idle is
1
Depa men o Physical Chemis y, Uni e si y o he Basque Coun y UPV/EHU, Apa ado 644, 48080 Bilbao, Spain.
2
EHU Quan um Cen e , Uni e si y o he Basque Coun y UPV/
EHU, Bilbao, Spain.
3
Depa men o In o ma ion and Communica ions Enginee ing, Aal o Uni e si y, Espoo 02150, Finland.
4
Depa men o Mic o echnology and Nanoscience
(MC2), Chalme s Uni e si y o Technology, SE-41296 Gö ebo g, Sweden.
5
Labo a o y o In o ma ion and Decision Sys ems, Massachuse s Ins i u e o Technology, Camb idge,
MA 02139, USA.
6
Basque Cen e o Applied Ma hema ics (BCAM), Alameda de Maza edo 14, 48009 Bilbao, Spain.
7
IKERBASQUE, Basque Founda ion o Science, Plaza Euskadi 5,
48009 Bilbao, Spain. ✉email: maximilian. eiche @ehu.eus; [email p o ec ed]
www.na u e.com/npjqi
Published in pa ne ship wi h The Uni e si y o New Sou h Wales
1234567890():,;
pe o med. We p opose o he p o ocol a mul imode p obe s a e
ha can be gene a ed by a pa ame ic downcon e e . The s a e is
composed o pho on pai s ha sha e equency en anglemen .
This pho on-pai s uc u e is esilien agains losses, since he loss
o a single pho on only e ec s i s pa ne , bu no he o he
pho on pai s. This is a c ucial di e ence wi h GLM s a es, whe e
he loss o a single pho on means he loss o all he in o ma ion
abou he pa ame e o in e es due o he global en anglemen .
The quan um p o ocol is benchma ked agains a classical p o ocol
shining he objec wi h he same ene gy and o he same ime
du a ion o make he compa ison ai . We employ he Quan um
Fishe in o ma ion (QFI) as he figu e o me i in he compa ison,
since i gi es he maximal amoun o ex ac able in o ma ion
abou he pa ame e o in e es . Calcula ing he QFI o his
mul imode s a e is challenging, bu by using p ope ies o
Gaussian s a es and in oducing Schmid modes, which e ec i ely
disc e izes he equency-con inuous p oblem, we de i e an
analy ical exp ession o he QFI. Two quan um esou ces can be
iden ified in ou esou ce quan um s a e, namely, squeezing and
equency en anglemen . The pe o mance o he quan um
p o ocol is s udied as a unc ion o he pho on numbe in h ee
di e en pa ame e egimes, called high- equency en anglemen ,
high-squeezing, and mixed egime. The la e , o which a
ema kable Heisenbe g scaling can be a ained, is called in his
manne because nei he squeezing no equency en anglemen
a e dominan . We p opose a measu emen se up ha a ains he
QFI, consequen ly achie ing he highes es ima ion accu acy o
he eloci y. I is no ewo hy ha he measu emen se up can be
pe o med sepa a ely in he signal and he idle , acili a ing he
expe imen al equi emen s.
The pape is s uc u ed as ollows. In Sec ions “Quan um
es ima ion heo y”and “Gaussian s a es”, he undamen als o
quan um es ima ion heo y and Gaussian s a es a e in oduced. In
Sec ion “Model o he mo ing a ge ”, we model he mo ing
a ge as a pe ec ly eflec i e mi o boos ed a a ela i e cons an
eloci y. A e wa ds in sec ions “Classical p o ocol and Quan um
p o ocol”, we in oduce he p obe s a es employed in bo h he
quan um and classical p o ocols. As a figu e o me i o bench-
ma k hei pe o mance, we make use o he QFI. Then, in Sec ion
“A ai compa ison”, we discuss he di e en pa ame e egimes
ob ained and s udy when quan um ad an age exis s and how i
beha es as a unc ion o he signal pho on numbe . In sec ion A
loss analysis o he high-squeezing egime, he p o ocol is s udied
in he p esence o losses in he signal beam. Finally, in sec ion A
loss analysis o he high-squeezing egime an op imal measu e-
men a aining he ul ima e p ecision se by he quan um C amé -
Rao bound is p o ided.
RESULTS
Model o he mo ing a ge
We model he objec o which we wish o es ima e i s cons an
adial eloci y ela i e o emi e as a pe ec mi o in a (1 +1)-
dimensional space ime. Fo now, we assume an absence o noise
and loss. As can be seen in Fig. 1, he quan um Dopple lida emi s
a signal beam owa ds he mo ing objec , while also emi ing an
idle beam which is e ained in he labo a o y, such ha a
measu emen can be pe o med o he e u ned signal and he
idle . The elec omagne ic field o he signal beam obeys he wa e
equa ion ð∂2
c2∂2
xÞϕð ;xÞ¼0, whe e cis he speed o ligh and
we only conside one pola iza ion o he field o he sake o
simplici y. The p esence o he a ge which is modelled as a
pe ec mi o imposes he bounda y condi ion ϕ( ,x
m
)=0, whe e
x
m
= is he loca ion o he mi o . We assume he emi e o be
o he igh o he mi o , which co esponds o i s spa ial
coo dina e > x
m
. The gene al solu ion o he wa e equa ion
sa is ying he bounda y condi ion is gi en by
ϕðx; Þ¼Z1
0
dω
ffiffiffiffiffiffiffiffiffi
4πω
peiωðc þxÞeiω
μðc xÞ
aðωÞþh:c:(1)
whe e μ=(1 − /c)/(1 + /c) is he Dopple pa ame e . We choose
o es ima e he pa ame e μins ead o , as i na u ally a ises in
he Dopple e ec . The es ima ion e o o is ela ed o he one
o μ ia he e o p opaga ion o mula o he QFI
Jð Þ¼ð∂ μð ÞÞ2Jðμð ÞÞ. The Fou ie coe ficien s a(ω) and hei
complex conjuga es ge p omo ed by canonical quan iza ion o
annihila ion and c ea ion ope a o s, which we deno e by ^
aðωÞand
^
ayðωÞ. They sa is y he ela ions ½^
aðωÞ;^
að~
ωÞ ¼ ½^
ayðωÞ;^
ayð~
ωÞ ¼ 0
and ½^
aðωÞ;^
ayð~
ωÞ ¼ δðω~
ωÞ. The idle equency mode is
e e ed o as ^
bðωÞand sa isfies he same commu a ion ela ions.
I commu es wi h he signal mode as bo h beams a e spa ially
sepa a ed. The fi s e m in Eq. (1) in b acke s ep esen s he
incoming wa e, while he second e m is he ou going wa e
which is Dopple shi ed ω→ω/μ. Now, le us de i e he
Bogoliubo ans o ma ion ^
Uμwhich maps he incoming modes
^
aðωÞ o he Dopple eflec ed ou going modes, deno ed as ^
aðωÞ.
Fo his, a change o in eg a ion a iables is pe o med in he
second e m in Eq. (1), leading o
^
ϕðx; Þ¼Z1
0
dω
ffiffiffiffiffiffiffiffiffi
4πω
peiωðc þxÞ^
aðωÞþeiωðc xÞ^
aðωÞþh:c:
;
wi h he ope a o ^
aðωÞμ1=2^
aðμωÞ. Thus, he p ocess o
eflec ion is desc ibed by he uni a y ans o ma ion
^
Uμ^
aðωÞ^
Uy
μ¼μ1=2^
aðω=μÞ. The p e ac o μ
−1/2
ensu es a p ope
no maliza ion and he change o sign is he πphase shi ha
adia ion expe iences when eflec ed. The acuum s a e 0
ji
, which
sa isfies ^
aðωÞ0
ji¼^
bðωÞ0
ji¼0, emains unchanged a e Dopple
eflec ion, ha is Uμ0
ji¼0
ji
. In he mos gene al amewo k, he
ou going mode also picks up a phase ac o expði2ωxm=ðc ÞÞ
depending on he eloci y and loca ion x
m
o he objec .
The e o e, his phase could in p inciple also be used o es ima e
he eloci y, bu gene ally a he cos o an addi ional knowledge
abou he loca ion. Fu he mo e, in eal wo ld applica ions he
phase o en is andomized due o su ace p ope ies o he objec
and in o ma ion abou is los . Hence, as a fi s s ep, we will
neglec he in o ma ion om he phase and we will only conside
he in o ma ion abou he eloci y ha is encoded in he
equency spec um o he ligh beams. The QFI J
q
de i ed he e
is a lowe bound o he QFI in which phases a e also aken in o
accoun .
Fig. 1 Scheme o a quan um Dopple lida . A win-beam mul i-
mode squeezed- acuum s a e is p oduced by he ansmi e on he
bo om le . The signal beam is sen owa ds he mo ing a ge
whe e i is eflec ed and i s equency Dopple shi ed. The idle
beam does no in e ac wi h he mo ing a ge and is e ained. Bo h
he eflec ed signal beam and he idle beam a e measu ed a he
ecei e on he bo om igh .
M. Reiche e al.
2
npj Quan um In o ma ion (2022) 147 Published in pa ne ship wi h The Uni e si y o New Sou h Wales
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Classical p o ocol
In he classical p o ocol we ake a cohe en signal as he p obe s a e.
Fo a con inuum o equency modes, a cohe en s a e is defined as
ψ
ji
¼exp½αRdω ðωÞð^
aðωÞ^
ayðωÞÞ 0
ji
, whe e we ake he dis-
placemen cons an α o be a eal numbe o he sake o simplici y.
The spec al ampli ude (ω) shall be an a bi a y di e en iable and
no malized unc ion, i.e. ∫dω∣ (ω)∣
2
=1. We assume ha he ca ie
equency ω
c
=∫dω∣ (ω)∣
2
ωis much la ge han he bandwid h Δω
defined as Δω2¼Rdωj ðωÞj2ðωωcÞ2, he so-called na ow-
bandwid h app oxima ion. This allows us o change he limi s o
in eg a ion o ( −∞,∞). The eflec ed s a e is gi en by
^
Uμψ
ji
¼ψμ
¼exp½αRdωμ1=2 ðμωÞð^
ayðωÞ^
aðωÞÞ 0
ji
, whe e we
ha e used ^
Uμe^
A^
Uy
μ¼e^
Uμ^
A^
Uy
μ¼e^
Aμ,and^
Ais he exponen o he
cohe en s a e. Thus, he s a e is s ill a cohe en s a e a e he
eflec ion bu wi h an ampli ude (ω)→−μ
1/2
(μω). The mean
equency is shi ed o ω
c
/μand he spec al bandwid h is s e ched
o comp essed by a ac o o 1/μ, see Supplemen a y No e 2.
The e o e, es ima ing he equency and he a iance p o ides
in o ma ion abou he pa ame e μ. The calcula ion o he QFI is
s aigh o wa d, we need o compu e ∂μψμ
.As½∂μ^
Aμ;^
Aμ¼0, we
can w i e ∂μψμ
¼e^
Aμ∂μ^
Aμ0ji. As a consequence, i ollows ha
〈ψ
μ
∣∂
μ
ψ
μ
〉=0andh∂μψμj∂μψμi¼h0j∂μ^
Ay
μ∂μ^
Aμj0i.Thisleads o
he exp ession o he QFI (see Supplemen a y No e 2)
JcðμÞ¼4α2
μ2Zdω1
2 ðωÞþω∂ω ðωÞ
2
:(2)
This is he gene al exp ession o he QFI. Le us now make some
app oxima ions o gain physical insigh s. Fi s , we conside small
eloci ies compa ed o he speed o ligh /c≪1, o which he
equency Dopple shi is app oxima ely 2ω
c
/c. Le us now
in oduce he spec al ampli udes’Fou ie ans o m g( )=∫
dω (ω)e
iω
. The ime du a ion ΔTo he pulse is gi en by
ΔT2¼Rd
2jgð Þj2Rd jgð Þj2
2. Using he u he app ox-
ima ion ΔTΔω /c≪1, which is s anda d in he classical li e a-
u e
37
, we ob ain ( o mo e de ails see Supplemen a y No e 2)
JcðμÞ4
μ2ω2
cNcΔT2:(3)
We see ha he classical p o ocol ollows he SQL scaling
expec ed o a classical s a egy. Fu he mo e, we no e ha h ee
pa ame e s comple ely define he op imal pe o mance o a
classical lida , he pho on numbe α
2
=N
c
, he ca ie equency ω
c
and he ime du a ion ΔTo he pulse.
Quan um p o ocol
Fo he quan um p o ocol, we use a win-beam mul imode
squeezed- acuum s a e. This s a e can be p oduced in he
labo a o y by non-linea op ical p ocesses, such as spon aneous
pa ame ic down-con e sion (SPDC). In his p ocess o SPDC, a
pump beam, which is conside ed o be classical, in e ac s wi h a
χ
(2)
non-linea op ical medium. Pho ons o he pump field decay
in o signal and idle pho on pai s. The use o a wa eguide o
SPDC allows o educing he numbe o spa ial modes o one o
each beam
38–42
gi en by ^
aðωÞ(signal) and ^
bð~
ωÞ(idle ). The
e ec i e Hamil onian desc ibing he p ocess is gi en by
43
^
HI¼i_ξZdωZd~
ω ðω;~
ωÞ^
aðωÞ^
bð~
ωÞþh:c:; (4)
whe e he coupling cons an ξ, e e ed o as he squeezing
pa ame e , is chosen o be eal o simplici y, and p opo ional o
he in ensi y o he classical pump beam and he s eng h o he
in e ac ion. The no malized join -spec al ampli ude ðω;~
ωÞ
depends on he specifics o he non-linea p ocess and on he
pump beam. In he case o SPDC, he join -spec al ampli ude can
be in many cases app oxima ed as a double Gaussian
44
which also
simplifies analy ic calcula ions
ðω;~
ωÞ¼ ffiffiffiffiffiffiffiffi
2
πσϵ
exp ðωþ~
ωω0Þ2
2σ2
!
´exp ðω~
ωÞ2
2ϵ2
!
:
(5)
The fi s exponen ial unc ion in Eq. (5)wi ha gumen ωþ~
ω
comp ises ene gy conse a ion o he pho on decay p ocess and i is
inhe i ed by he equency mode spec um o he pump beam,
which is assumed o be Gaussian wi h mean equency ω
0
and
a iance σ
2
/2. The second exponen ial unc ion wi h a gumen ω~
ω
co esponds o he phase ma ching condi ion, i.e. momen um
conse a ion o he pho on decay p ocess, and depends on he
spa ial p ope ies o he pump beam and he non-linea medium.
Thus, by modi ying he pump beam, bo h unc ions composing
ðω;~
ωÞcan independen ly be ailo ed
42
.Weagainassume he
na ow-bandwid h app oxima ion ω
0
≫σand ω
0
≫ϵ. The double
Gaussian can be decomposed in o i s Schmid modes
45
as
ðω;~
ωÞ¼P1
n¼0 nψnðωω0=2Þψnð~
ωω0=2Þ,whe e{ψ
n
(ω)} is an
o hono mal se closely ela ed o he He mi e unc ions ( u he
de ails in Supplemen a y No e 3). The ela i e weigh 2
no each
indi idual mode is gi en by n¼2ffiffiffiffi
σϵ
p
σþϵðσϵ
σþϵÞnwi h P 2
n¼1. The
numbe o ac i e modes is gi en by he Schmid numbe
K¼ð
Pn 4
nÞ1¼σ2þϵ2
2σϵ, which we in e p e as a measu e o equency
en anglemen wi hin he signal and idle pho on pai . Fo K=1, only
one pai o modes is necessa y o desc ibe he s a e and he double
Gaussian ac o izes, ha is ðω;~
ωÞ¼ψ0ðωω0=2Þψ0ð~
ωω0=2Þ,
which implies no equency en anglemen . Fo K>1, he s a e is
equency en angled and he deg ee o en anglemen g ows
mono onically wi h K.In e s.
42,46
, echniques we e p oposed o
gene a e Schmid numbe s in he ange o K~400–5000, which
co esponds o an ex emely high- equency en anglemen o he
pho on pai . The Schmid modes cap u e he spec al s uc u e o
ðω;~
ωÞin a disc e e manne , and hus i is na u al o in oduce
disc e e annihila ion and c ea ion ope a o s ^
an¼Rdωψnðω
ω0=2Þ^
aðωÞand ^
bn¼Rdωψnðωω0=2Þ^
bðωÞwhich a e smea ed
ou e sions o ^
aðωÞand ^
bð~
ωÞ
47
. The modes sa is y he commu a ion
ela ions ½^
an;^
am¼½
^
bn;^
bm¼½^
an;^
by
m¼0and½^
an;^
ay
m¼
½^
bn;^
by
m¼δnm due o he o hono mali y o {ψ
n
(ω)}. The disc e e
desc ip ion o he p oblem subs an ially acili a es he calcula ion o
heQFI.TheHamil onianinEq.(4) is gi en in he disc e e desc ip ion
by
^
HI¼i_ξX
1
n¼0
n^
an^
bn^
ay
n
^
by
n
i_ξX
1
n¼0
^
Hn:(6)
As he Hamil onians o he indi idual modes commu e
½^
Hn;^
Hm¼0, he o al squeezing ope a o ^
S¼ei^
HI=_o he SPDC
p ocess can be w i en as a enso p oduc o squeezing ope a o s
o each indi idual mode ^
S¼N1
n¼0
^
Snwi h ^
Sn¼eξ^
Hn. The
squeezing pa ame e o he squeeze co esponding o he mode
nis gi en by ξ
n
. Finally, we a e able o exp ess he p obe s a e o
he quan um p o ocol using disc e e c ea ion ope a o s. Using he
no mal o de ed ep esen a ion o squeezing ope a o s
48
,wefind
(see Supplemen a y No e 4 o de ails)
^
S0ji¼O
1
n¼0
1
coshðξ nÞexp anhðξ nÞ^
ay
n
^
by
n
0ji:(7)
Thus, he win-beam mul imode squeezed- acuum s a e is jus he
p oduc s a e o independen wo-mode squeezed- acuum s a es.
Now, he eflec ed s a e ψμ
¼^
Uμ^
S0
jiis
ψμ
¼Nexp X
1
n¼0
anhðξ nÞ^
ay
nμ
^
by
n
!
0
ji
;(8)
M. Reiche e al.
3
Published in pa ne ship wi h The Uni e si y o New Sou h Wales npj Quan um In o ma ion (2022) 147
whe e we ha e ans o med he p oduc in Eq. (7) in o a sum in
he exponen and we ha e in oduced he no maliza ion cons an
N¼Qn1=coshðξ nÞ, which is independen o μ. The ope a o ^
ay
n
ans o ms in o ^
Uμ^
an^
Uy
μ¼^
ay
nμ¼Rdωμ1=2ψnðμω ω0=2Þ^
ayðωÞ,
picking up a phase shi and a μ-dependence, whe eas ξ,
n
, and
he idle modes ^
bn emain μ-independen . The mean equency o
he ans o med mode is gi en by ω0=2μ¼ω, as one would
expec om he Dopple e ec . The bandwid h o each mode is
p opo ional o ffiffiffiffiffiffiffiffiffiffi
σϵ=2
p, and i ans o ms in o ffiffiffiffiffiffiffiffiffiffiffiffiffi
σϵ=2μ
pσa e
he eflec ion. In he con inuous o malism, he join -spec al
ampli ude con e s in o ðω;~
ωÞ!μ1=2 ðμω;~
ωÞ. Now, in o de
o calcula e he QFI, we need o fi s e alua e he de i a i e
∂μψμ
. The only componen o he s a e ha depends on μis ^
ay
nμ.
The de i a i e can be calcula ed using he p ope ies o he
He mi e unc ions and we find ha ∂μ^
ay
nμis a linea combina ion
o c ea ion ope a o s ^
ay
nμ anging om modes n−2 on+2. As
he de i a i e o he exponen in Eq. (8) commu es wi h he
exponen i sel , we find ∂μψμ
¼
Pn anhðξ nÞð∂μ^
ay
nÞ^
by
nS0
ji
, see
Supplemen a y No e 5. By using he ans o ma ion ule ^
Sy^
anμ^
S¼
^
anμcoshðξ nÞ^
by
nsinhðξ nÞand he analogous ule o he idle
mode, whose de i a ion is discussed in Supplemen a y No e 4, we
finally find he analy ic exp ession o he QFI (see Supplemen a y
No e 5 o he ull de i a ion). This spli s up in o equency and
mode-bandwid h con ibu ions as JqðμÞ¼ð∂μωÞ2JqðωÞþ
ð∂μσÞ2JqðσÞwi h
JqðμÞ¼1
μ2
ω2
0
σϵZωþσϵ
ω2
0
Zσ
;(9)
wi h he equency e m defined as
Zω¼X
1
n¼0
sinh2ðξ nÞncosh2ðξ n1Þþðnþ1Þcosh2ðξ nþ1Þ
(10)
and he mode-bandwid h e m as
Zσ¼P
1
n¼0
sinh2ðξ nÞnðn1Þcosh2ðξ n2Þ
þðnþ1Þðnþ2Þcosh2ðξ nþ2Þ:
(11)
The bandwid h con ibu ion is supp essed by he ac o σϵ=ω2
0as
can be seen in Eq. (9), which is small due o he na ow-bandwid h
app oxima ion. Fo a ypical SPDC p ocess in po assium dihyd o-
gen phospha e c ys al pumped by a equency doubled i anium-
sapphi e lase , his ac o is app oxima ely ffiffiffiffiffiffiffiffiffiffiffiffiffi
σϵ=ω2
0
p0:01
49
.
A ai compa ison
Le us now compa e he pe o mance o he quan um and he
classical p o ocols and find ou unde which condi ions quan um
ad an age is achie ed. Fo ha , we examine he quan um
ad an age a io J
q
/J
c
, whe e we ha e omi ed he dependence
on μ o he sake o eadabili y. In he case J
q
/J
c
> 1, he quan um
s a egy ou pe o ms he classical one assuming ha an op imal
measu emen is pe o med and he C amé -Rao bound is a ained,
which is usually he case in he absence o he mal noise pho ons.
We al eady poin ed ou in Sec ion “Classical p o ocol”, ha he
classical lida is solely cha ac e ized by he h ee pa ame e s
pho on numbe , ca ie equency and ime du a ion. So o ai ly
compa e bo h p o ocols, we se hese h ee pa ame e s equal o
bo h signal beams. Fo pho on numbe and mean equency, his
co esponds o ω
c
=ω
0
/2 and α2¼Pnsinh2ðξ nÞ. Now, le us
calcula e he ime du a ion o he quan um signal beam. Fo ha
we in oduce he ime-domain e sion o he c ea ion and
annihila ion ope a o s ia ^
Eyð Þ¼Rdωeiω ^
ayðωÞ, which is he
ope a o c ea ing a pho on a ime a he ansmi e . The
no malized powe o he signal beam is defined as
jsð Þj2¼hψj^
Eyð Þ^
Eð Þjψi=NS. The ime du a ion can hen be
calcula ed and we find
ΔT2¼Zd
2jsð Þj2Zd jsð Þj2
2
(12)
¼2
σϵP1
n¼0sinh2ðξ nÞn
P1
m¼0sinh2ðξ mÞþ1
2
!
:(13)
The de ailed calcula ions can be ound in Supplemen a y No e 5.
As bo h J
q
and ΔTa e gi en by infini e se ies con aining
hype bolic igonome ic unc ions, we will in he ollowing s udy
pa ame e egimes in which simple analy ic exp ession o he
espec i e quan i ies can be ob ained, which helps o in e p e he
esul s.
No equency en anglemen
Le us fi s s udy he case in which no equency en anglemen is
p esen be ween signal and idle beams. In his case, we ha e
K=1, i.e. σ=ϵ. The s a e educes o he well-known wo-mode
squeezed- acuum s a e ψμ
¼expðξð^
a0μ^
b0^
ay
0μ
^
by
0ÞÞ 0ji wi h
signal pho on numbe NS¼sinh2ðξÞ.Wefind ha
Jq
Jc¼1 (14)
o all alues o he squeezing pa ame e ξ(Supplemen a y No e
6). Thus, no quan um ad an age is achie ed wi h a wo-mode
squeezed- acuum s a e. Bo h p o ocols obey he SQL J
q
,J
c
~N
S
.In
simila in e e ome ic phase es ima ion p o ocols, Heisenbe g
scaling is achie ed wi h he wo-mode squeezed s a e. Bu due o
ou igno ance o he a ge ’s posi ion x
m
, he in o ma ion abou
he eloci y con ained in he phase expði2ωxm=ðc ÞÞ canno be
accessed and hus Heisenbe g scaling is no achie able in ou
case. Thus, equency en anglemen K> 1 is necessa y o
quan um ad an age in ou p o ocol wi h pu e p obe s a es gi en
in Eq. (7).
High- equency-en anglemen egime
Le us now conside he case in which he equency en angle-
men is he dominan quan um esou ce. We speci y his egime
by he condi ion ξ≪K
1/2
, which allows us o app oxima e he
hype bolic unc ions as sinh2ðξ nÞξ2 2
nand cosh2ðξ nÞ1. The
numbe o pho ons in mode nis gi en by NSn ¼sinh2ðξ nÞ1
and he o al pho on numbe can be app oxima ed as N
S
≈ξ
2
,
whe e we ha e only aken he fi s e m o he app oxima ion in o
accoun (Supplemen a y No e 7). Wi h his, he a io o QFIs is
Jq
Jc1þσ2þϵ2
2ω2
0
1þ1
K2þ1
KðK2þKÞ
:(15)
The fi s e m is he equency con ibu ion and is equal o 1. The
emaining e ms co espond o he bandwid h con ibu ion which
is small due o he na ow-bandwid h app oxima ion. Thus, in he
high- equency en anglemen egime no quan um ad an age can
be ob ained, he classical and quan um p o ocol pe o m equally
well. Wi h he u he cons ain ξ≪1, he s a e becomes a
supe posi ion o he acuum and a wo-pho on s a e, he same
s a e used in e .
12,13
. E en hough hese s a es yield no quan um
ad an age in es ima ing he eloci y alone, hey yield ad an age
in join ly es ima ing he posi ion and eloci y o a a ge .
High-squeezing egime
Le us now conside a egime in which squeezing is he dominan
quan um esou ce and he equency en anglemen is ela i ely
M. Reiche e al.
4
npj Quan um In o ma ion (2022) 147 Published in pa ne ship wi h The Uni e si y o New Sou h Wales
weak. We speci y he pa ame e condi ions as ξ≫K
3/2
and K≳1.5.
These condi ions helps us o pu he QFI in o a concise analy ic
o m. Addi ionally, by equi ing K≳1.5, he high-squeezing egime
is su ficien ly dis inc om he no-en anglemen egime wi h
K=1. The de ails abou he calcula ions pe o med in his
Subsec ion can be ound in Supplemen a y No e 8. The ac ion
o pho ons in he mode nis gi en by N
Sn
/N
S
, whe e NSn ¼
sinh2ðξ nÞis he pho on numbe o he mode no he signal
beam. By inc easing ξ o a fixed K, he ela i e con ibu ion o
highe modes n> 0 dec eases. In he high-squeezing egime,
almos all he pho ons eside in he 0 mode, ha is N
S
≈N
S0
≫
N
S1
≫1, implying a high pho on numbe pe mode bu a low
numbe o ac i e modes, con a y o he high- equency
en anglemen egime. Wi h his, we a i e in he asymp o ic limi
a he esul
Jq
Jc1
34NS
ðÞ
ffiffiffiffiffi
K1
Kþ1
p;(16)
whe e i was used ha NSn ¼sinh2ðξ nÞcosh2ðξ nÞ o n=0, 1
and only e ms o o de N
S0
N
S1
in Eq. (9) con ibu e significan ly.
This is he eason why he bandwid h e ms a e negligible. In Fig.
2, bo h he no malized equency (solid lines) and bandwid h
(dashed lines) e ms a e plo ed agains ξ o Schmid numbe s
K=10 and K=20, confi ming ou analy ical esul s. In e es ingly,
he QFI in e ms o he mode pho on numbe s is J
q
~N
S0
N
S1
,
which indeed indica es a scaling be e han he SQL, and nea ly
eaches he HL o Kbig enough. As a conclusion, inc easing he
squeezing o a fixed Kalso inc eases he quan um ad an age, so
squeezing can be seen as a sensi i i y-enhancing esou ce o he
p o ocol. Up o his poin , pho on loss and noise has been
neglec ed. In sec ion “A loss analysis” o he high-squeezing
egime, he impac o losses, bu no he impac o noise, will be
examined on he high-squeezing egime.
The mixed egime
Now, le us s udy he in e media e pa ame e egime K
1/2
≪ξ≪
K
3/2
. Unde hese condi ions, mul iple modes a e ac i e like in he
high- equency en anglemen egime and he pho on numbe pe
mode is high N
Sn
≫1 like in he high-squeezing egime, hence he
name mixed egime. Using hese condi ions, we can de i e in he
asymp o ic limi an analy ic exp ession o he QFI
Jq
Jcξ
21=2K3=2þσϵ
4ω2
0
NS;(17)
whe e he fi s e m is again he equency con ibu ion and he
second e m he bandwid h con ibu ion, ollowing bo h a
Heisenbe g scaling JqN2
S. Fo de ails abou he calcula ions in
his Subsec ion, see Supplemen a y No e 9. The ac o ξ/2
1/2
K
3/2
is
smalle han 1, bu we s ill ha e (ξ/2
1/2
K
3/2
)N
S
≫1 hus gua an ee-
ing quan um ad an age. Since bo h e ms σϵ=4ω2
0and ξ/2
1/2
K
3/2
a e smalle han 1, we canno gene ally decide which con ibu ion
is dominan . Fo ins ance, in he expe imen al se up e e ed o in
sec ion “Quan um p o ocol”, we had ha ffiffiffiffiffiffiffiffi
σϵ=
pω00:01, he
bandwid h can be sa ely neglec ed in he mixed egime, a leas
o alues o ξand Kup o 100 as can be seen in Supplemen a y
No e 9. The e o e, we will neglec he bandwid h con ibu ion
om his poin on.
In Fig. 3, he a io 2μ2σϵ
ω2
0
Jq=N2
S2Zω=N2
Sis plo ed o bo h ξ
and Kup o he alues o 100. Th ee dis inc egions
co esponding o he h ee pa ame e egimes can be app e-
cia ed. In he whi e a ea, which co esponds o a alue o 1 o
he a io,weobse eabeha iou o JqN2
S o he QFI and hus
Heisenbe g scaling. The ed a ea is he high- equency en angle-
men egime and he blue a ea is he high-squeezing egime.
Cu iously, quan um ad an age is achie ed in he wo egimes
wi h high pho on numbe pe mode, and no in he high-
equency en anglemen egime wi h a low pho on numbe pe
mode. This in con as o he quan um illumina ion p o ocol,
whe e a small pho on numbe pe mode is necessa y o achie e
quan um ad an age. The pa ame e condi ions o he h ee
egimes and hei co esponding quan um ad an ages a e
summe ized in Table 1.
Fig. 2 The equency and bandwid h con ibu ion in he high-
squeezing egime. The no malized equency (solid lines) and
bandwid h (dashed lines) con ibu ions 8Zω;σ=ð4NSÞffiffiffiffiffi
K1
Kþ1
pa e plo ed
agains he squeezing pa ame e ξ o Schmid numbe s K=10 and
K=20. The no malized QFI app oaches 1, which indica es a scaling
abo e he SQL in he limi K
3/2
≪ξ. The no malized bandwid h
con ibu ion is much smalle and goes o 0 o K
3/2
≪ξ. I is e en
u he supp essed by he ac o σϵ=ω2
0.
Fig. 3 The h ee pa ame e egimes. We plo he no malized QFI
2μ2σϵ
ω2
0
Jq=N2
S, no o be con used wi h he quan um ad an age a io.
The plo shows he h ee pa ame e egimes and hei co espond-
ing bo de s gi en by he con ou s ξ=K
1/2
and ξ=K
3/2
. The mixed
egime is cha ac e ized by he alue o 1, depic ed as whi e, and
hus shows Heisenbe g scaling and alida es ou analy ical
exp ession o he QFI. The high-squeezing egime, in which
quan um ad an age abo e he SQL is achie ed, is depic ed as blue
wi h alues below 1. The ed a ea is he high-en anglemen egime,
whe e he alues ange a abo e 2 bu we e cu o . In his egime,
no quan um ad an age is achie ed.
M. Reiche e al.
5
Published in pa ne ship wi h The Uni e si y o New Sou h Wales npj Quan um In o ma ion (2022) 147
A loss analysis o he high-squeezing egime
So a we ha e conside ed he ideal scena io in which no pho ons a e
los and he e u ned s a e is pu e. In ealis ic scena ios, he p obe
s a e a he ecei e will be mixed due o pho on loss and he mal
noise. In e .
11
, ime-o -fligh es ima ion in he mic owa e egime was
s udied, in which he he mal pho on numbe pe mode is much
la ge han 1. A simila s a e was used, a con inuous wa e squeezed
s a e, and a conside able quan um ad an age was p o ed a a ce ain
h eshold o he signal- o-noise a io. This so-called h eshold e ec
a ises only in he p esence o he mal noise and equi es an analysis
ha goes beyond he calcula ion o he QFI. In ou p o ocol, we
assume ope a ion in he op ical egime, in which he mal noise can be
neglec ed and solely elying on he QFI su fices. Pho on loss, howe e ,
has o be conside ed o assess i he p o ocol shows quan um
ad an age in mo e ealis ic scena ios. Because pho on loss mixes he
s a e, he calcula ion o he QFI is significan ly mo e complica ed.
Thus, we will only s udy he high-squeezing egime in which he s a e
can be desc ibed su ficien lywellbyonlyacoupleo Schmid modes
and hus allows us o de i e analy ical exp essions. We assume no
losses in he idle beam. Pho on loss in he signal beam can occu on
he way o and om he a ge and/o du ing he in e ac ion wi h he
objec (which gene alizes he p o ocol o non-pe ec ly eflec ing
objec s). The p obabili y o losing a signal pho on is assumed o be
equency-independen and i is modelled by a beam spli e
^
UB^
aðωÞ^
Uy
B¼ffiffiffi
η
p^
aðωÞþ ffiffiffiffiffiffiffiffiffiffiffi
1η
p^
cðωÞ;(18)
whe e ^
cðωÞis an auxilia y mode which canno be accessed by he
expe imen e and will be aced ou a he end. In his amewo k,
he beam spli e commu es wi h he Dopple eflec ion ope a-
ion, so only one beam spli e wi h e ec i e ansmissi i y ηis
equi ed o he lida - o- a ge - o-lida ound ip. We choose o
apply his beam spli e ope a ion a e he eflec ion a he
ecei e le el. The final s a e is a Gaussian s a e. Gaussian s a es
a e ully desc ibed by hei fi s wo momen s dand Σ, he
defini ions and an in oduc ion o Gaussian s a es can be ound in
sec ion “Gaussian s a es”and e .
50
. As we discussed in sec ion
“High-squeezing egime”, only he fi s wo pai s o modes ^
a0μ;^
a1μ
and ^
b0;^
b1a e popula ed wi h a significan amoun o pho ons.
This allows us o omi he es o he modes by acing hem ou
and hus de i e a lowe bound o he QFI. Al e na i ely, we could
jus i y he neglec o he highe modes by ailo ing he join -
spec al ampli ude ðω;~
ωÞ, such ha only he fi s wo modes a e
ac i e. To use he o mula o he QFI o Gaussian s a es gi en in
e .
50
, we need o change he basis (i.e. he modes ^
a0μand ^
a1μ) o
make i pa ame e independen . To do so, we assume ha a p io
es ima e μ
0
o he pa ame e is known and we only wan o
es ima e he small de ia ion δwi h μ=μ
0
+δ, which is s anda d
in mos pa ame e es ima ion p o ocols. We expand he Schmid
modes a ound μ
0
up o he fi s o de and find
^
a0μ^
a0μ0δω0
2μffiffiffiffiffi
σϵ
p^
a1μ0(19)
^
a1μ^
a1μ0þδω0
2μffiffiffiffiffi
σϵ
p^
a0μ0ffiffiffi
2
p^
a2μ0
;(20)
whe e we ha e neglec ed he e ms co esponding o he
bandwid h con ibu ion, as hey a e small in his egime, which
we ha e p e iously es ablished in sec ion “High-squeezing egime”.
Now, we ha e he modes ^
a0μ0;^
a1μ0;^
a2μ0;^
b0;^
b1;^
c0μ0;^
c1μ0;^
c2μ0,
whe e ^
cnμ0¼Rdωμ1=2
0ψnðμ0ωω0=2Þ^
cðωÞa e he auxilia y
Schmid modes. The esul ing co a iance ma ix and QFI a e
calcula ed in Supplemen a y No e 10. We eco e he esul om
Eq. (16) o he lossless case η=1, which confi ms he alidi y o
ou app oach and ou app oxima ions. To make a ai compa ison,
we also ha e o conside he classical p o ocol unde he e ec o
pho on loss. The QFI o he classical s a egy is simply educed by
he ac o η, ha isJcηω2
0NSΔT2=μ2. We a i e a he a io
Jq
Jc1
1η;(21)
whe e we assumed N
S1
(1 −η)≫1 o ob ain a compac esul . This
assump ion causes he di e gence in Eq. (21) since, as η→1, we
ha e N
S1
→∞. Wi hou his assump ion, we eco e he esul o
he lossless scena io in he limi o η→1. The quan um ad an age
a io in Eq. (21) does no depend on he pho on numbe , and hus,
pho on loss des oys he nea HL scaling and b ings i down o he
SQL, ha is J
q
~N
S
. A cons an ac o quan um ad an age is
achie ed o all alues o η, bu his ac o becomes insignifican
o small pa h ansmissi i ies η≪1. Fo ansmissi i ies η≥50%,
he quan um ad an age ac o is J
q
/J
c
≥2≈3dB. This makes ou
p o ocol p omising o sho - ange applica ions whe e he pa h
losses a e small, such as Dopple mic oscopy o biologicals.
Op imal measu emen
As we a e es ima ing only he eloci y o he objec , he e
always exis s a leas one op imal measu emen sa u a ing he
QFI, bu i is no necessa ily unique. Quan um es ima ion
heo y p o ides echniques o cons uc some o hese
obse ables, in pa icula he one ela ed o he symme ic
loga i hmic de i a i e (SLD) ^
Oμ¼1μþ^
Lμ=JðμÞ. Howe e , i s
implemen a ion in a ealis ic expe imen al se up is a highly
non- i ial ask. In he case o a pu e-s a e mani old, he SLD ^
Lμ
can be w i en as ^
Lμ¼∂μψμ
ψμ
þψμ
∂μψμ
51
.Thus,only
∂μψμ
needs o be calcula ed, which has been done o he
calcula ion o he QFI and i can be ound in Supplemen a y
No e 5. Howe e , a cons uc ion o his obse able in a lab in
an op ical se up is a om i ial. Fu he mo e, i depends on
he pa ame e μi sel , and we would like o ha e a
measu emen wo king on he whole ange o eloci ies i
possible. O he wise, an adap i e measu emen s a egy could
be ollowed
52
. In he Gaussian o malism, he SLD can be
w i en as a sum o e ms ha a e a mos quad a ic in he
modes
50
. In Supplemen a y No e 10 we ha e gi en he explici
exp ession o he SLD de i ed in he limi o N
S
≫1in he
high-squeezing egimeunde pho onloss.
Le us analyze a measu emen based on equency- esol ed
pho on coun ing o signal and idle pho ons o he lossless
scena io, which is discussed in de ail in Supplemen a y No e 11.
This measu emen co esponds o a p ojec ion on o he equency
eigens a es ω;~
ωjiω1;¼;ωn;~
ω1;¼~
ωm
ji
1
ffiffiffiffiffiffiffi
n!m!
pn
i¼1m
j¼1ay
ðωiÞbyð~
ωjÞ0
ji
, whe e n;m2Na e he signal and idle pho on
numbe s and ωi;~
ωj2R>0a e he espec i e equencies o each
pho on. The co esponding se o POVM ope a o s a e
ω;~
ω
ji
ω;~
ω
hjj
n;m2N;ωi;~
ωj2R>0g. We calcula e he Fishe
in o ma ion (FI), ~
Fq, co esponding o his measu emen o a
gene aliza ion ~
ψμ
o he p obe s a e ψμ
gi en in Eq. (8). This
gene alized p obe s a e con ains phase ac o s depending on he
kine ic p ope ies o he a ge and a complex squeezing
pa ame e , which we e p e iously omi ed in ou analysis. We
can show ha he measu emen ou comes do no depend on
Table 1. Lis ed a e he quan um ad an ages o he di e en
pa ame e egimes.
Regime 1 Regime 2 Regime 3
ξ≪K
1/2
ξ≫K
3/2
K
1/2
≪ξ≪K
3/2
Jq
Jc1Jq
JcNffiffiffiffiffi
K1
Kþ1
p
S
Jq
Jcξ
K3=2NS
Regime 1, 2 and 3 co espond o he high- equency en anglemen , he
high-squeezing and he mixed egime, espec i ely. He e, he con ibu-
ions due o he bandwid h shi a e neglec ed.
M. Reiche e al.
6
npj Quan um In o ma ion (2022) 147 Published in pa ne ship wi h The Uni e si y o New Sou h Wales
hese phases. Indeed, bo h s a es ~
ψμ
and ψμ
gi e ise o he
same p obabili y dis ibu ion o measu emen ou comes. Thus, he
POVM ω;~
ω
ji
ω;~
ω
hjg
is ac ually phase insensi i e. Finally, we p o e
ha ~
Fq¼Jq, so his measu emen also sa u a es he QFI in sec ion
“Quan um p o ocol”. Le us ema k ha his measu emen does
no depend on he pa ame e μ, so i can be used o sa u a ing
he QFI o any eloci y. Also, i could in p inciple be expe imen ally
easible by using di ac ion g a ings ha map equency
componen s o dis inc loca ions whe e pho on coun e s a e
placed
53,54
. We no e, ha he POVM is a sepa a e measu emen o
he signal and idle beams, which u he acili a es he expe i-
men al implemen a ion. This also indica es ha he idle , and hus
he en anglemen , solely se es as a s a e p epa a ion ool. Fo
example, one can check ha he idle less Fock s a e
ð^
ay
0ÞNS0ð^
ay
1ÞNS10ji, which has no equency en anglemen and
could be app oxima ely he alded wi h ou p obe s a e, bea s he
SQL and shows he same beha iou unde loss as in Eq. (21) o he
limi N
S
≫1.
Fu he pe spec i es
Las ly, we wan o emphasize ha he p o ocol can be easily
adap ed o di e en equency and/o bandwid h es ima ing
p o ocols. Also, he a ge ’s ajec o y can be gene alized o an
accele a ing one ia a Bogoliubo ans o ma ion
55,56
, bu wi h an
addi ional complica ion due o he p esence o Casimi adia ion.
Fo s a iona y a ge s, he p o ocol can be adap ed o es ima e he
loca ion, which boils down o he es ima ion o a i al imes o he
signal beam. The p obe s a e w i en in he ime domain has
exac ly he same s uc u e as in he equency domain, whe e
he a iances o he double Gaussian change as σ
2
/2 →2σ
2
and
ϵ
2
/2 →2ϵ
2
. The Schmid numbe emains unal e ed unde his
ans o ma ion. Thus, he es ima ion o ime a i al o signal
pho ons is analogous o he es ima ion o mean equency o he
signal pho ons. Analogously, a measu emen ha a ains he
op imal pe o mance is he measu emen o pho on a i al imes.
DISCUSSION
We ha e p oposed a p o ocol o a quan um Dopple lida ha
es ima es he adial eloci y o a eflec ing mo ing a ge using a
win beam wi h equency en anglemen and squeezing as
quan um esou ces. This quan um p o ocol was benchma ked
agains a classical one by calcula ing he QFIs o bo h s a egies.
We ha e iden ified h ee di e en pa ame e egimes, achie ing
quan um ad an age in wo o hem. In he high-squeezing
egime, whe e he equency en anglemen becomes less ele an
compa ed o squeezing, he quan um p o ocol exceeds he
s anda d quan um limi . In he mixed egime, whe e bo h
quan um esou ces a e compa able, he quan um p o ocol ollows
he Heisenbe g limi . We ha e ound ha equency- esol ed
pho on coun ing o signal and idle beam is an op imal
measu emen in he lossless case. The e ec o losses on he
pe o mance o he p o ocol was s udied in he high-squeezing
egime by modelling he loss channel as a equency-independen
eflec i i y beam spli e . A cons an ac o quan um ad an age
≥3 dB in he a iance o he es ima o is achie ed gi en a pa h
ansmissi i y ≥50%.
METHODS
Quan um es ima ion heo y
The objec i e o quan um es ima ion heo y is o find he ul ima e
p ecision limi o he es ima ion o a pa ame e μ ha is encoded
in a quan um sys em. In ou scena io, he p obe s a e ρ ha is
emi ed by he lida acqui es in o ma ion abou μdu ing he
eflec ion o he mo ing a ge , which ans o ms he s a e as
ρ→ρ
μ
. The classical Fishe in o ma ion (FI) F(μ) is a measu e o he
in o ma ion abou he pa ame e μ ha can be ex ac ed by a
gi en measu emen co esponding o he posi i e ope a o -
alued measu e (POVM) {Π
z
} wi h RdzΠz¼1. The FI is gi en by
FðμÞ¼Zdz1
pμðzÞ∂μpμðzÞ
2;(22)
whe e pμðzÞ¼T ðΠzρμÞis he p obabili y o ha ing he measu e-
men ou come zgi en he pa ame e μ. The C amé -Rao bound is
gi en by
51
Va ðμÞ⩾1
MFðμÞ;(23)
whe e μis an unbiased es ima o ha maps he measu emen
da a o he Mexpe imen epe i ions o an es ima e o he
pa ame e μ. The bound can be sa u a ed using he maximum
likelihood es ima o in he limi o la ge M
57
. Maximizing he FI
o e all POVMs {Π
z
} yields he quan um Fishe in o ma ion
J(μ)⩾F(μ). The Eq. (23) o he QFI is called he quan um
C amé -Rao bound which se s he absolu e p ecision limi o
he es ima ion o μ. In he case o a pu e-s a e mani old, i.e. when
^
ρμ¼ψμ
ψμ
o any μ, he QFI is gi en by
51
JðμÞ¼4h∂μψμj∂μψμijhψμj∂μψμij2
:(24)
To p o e a quan um ad an age, we calcula e he QFIs J
q
and J
c
o
bo h he quan um and classical s a egy. A quan um ad an age is
achie ed, i he a io is J
q
/J
c
> 1 assuming bo h s a egies
illumina e he objec wi h he same ene gy and an op imal
measu emen is pe o med. An obse able co esponding o he
op imal measu emen is gi en by ^
Oμ¼1μþ^
Lμ=JðμÞ, whe e ^
Lμis
he symme ic loga i hmic de i a i e (SLD), which sa isfies
^
Lμ^
ρμþ^
ρμ^
Lμ¼2∂μ^
ρμ. As he op imal obse able gene ally
depends on he pa ame e i sel , a p io guess abou he
pa ame e is equi ed o cons uc he measu emen . The
measu emen can hen be adap i ely op imized
52
.
Gaussian s a es
A Gaussian s a e is ully defined by i s fi s wo momen s dand Σ.
Thei componen s a e defined as
dm¼ ^
ρ^
Rm
;(25)
and
Σnm ¼ ^
ρ Δ^
Rm;Δ^
Ry
ng
hi
;(26)
whe e ^
R¼ð^
a0;^
ay
0;^
a1;^
ay
1;¼ÞTand Δ^
R¼^
R^
d. Gaussian uni-
a ies ha ans o m he s a e as ^
ρ0¼^
U^
ρ^
Uy ans o m he fi s
momen s as
d0¼Gd þb;(27)
and
Σ0¼GΣGy;(28)
whe e Gis he co esponding symplec ic ma ix, see e .
50
o
mo e in o ma ion on how Gand b ela e o he Gaussian uni a y
^
U. A o mula o he QFI o a Gaussian s a e is gi en by
JðμÞ¼lim
κ!1
1
2 ec½∂μΣyM1
κ ec½∂μΣþ2∂μdyΣ1∂μd;(29)
whe e Mκ¼κΣyΣKKwi h he symplec ic o m K=
diag(1, −1, 1, −1, …). The ope a ion ec[⋅] u ns a ma ix in o a
ec o as
ec ab
cd
¼
a
b
c
d
0
B
B
B
@
1
C
C
C
A
:(30)
M. Reiche e al.
7
Published in pa ne ship wi h The Uni e si y o New Sou h Wales npj Quan um In o ma ion (2022) 147
We can also calcula e he SLD in his o malism. I is gi en by
^
Lμ¼Δ^
RyAμΔ^
R1
2 ½ΣAμþ2Δ^
RyΣ1∂μd;(31)
whe e ec½Aμ¼lim
κ!1M1
κ ec½∂μΣ.
DATA AVAILABILITY
The au ho s decla e ha all da a suppo ing he findings o his s udy a e a ailable
wi hin he a icle and i s Supplemen a y Ma e ial.
Recei ed: 7 July 2022; Accep ed: 30 No embe 2022;
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ACKNOWLEDGEMENTS
We hank Robe Jonsson, Gö an Johansson and Benjamin Hua d o insigh ul
discussions. We acknowledge financial suppo om Basque Go e nmen QUANTEK
p ojec om ELKARTEK p og am (KK-2021/00070) and he Basque Go e nmen
p ojec IT1470-22, Spanish Ramón y Cajal G an RYC-2020-030503-I and he p ojec
g an PID2021-125823NA-I00 unded by MCIN/AEI/10.13039/501100011033 and by
“ERDF A way o making Eu ope”and “ERDF In es in you Fu u e”, as well as om
QMiCS (820505) and OpenSupe Q (820363) p ojec s o he EU Flagship on Quan um
Technologies, and he EU FET-Open p ojec s Qu omo phic (828826) and EPIQUS
M. Reiche e al.
8
npj Quan um In o ma ion (2022) 147 Published in pa ne ship wi h The Uni e si y o New Sou h Wales
(899368). M.R. acknowledges suppo om UPV/EHU PhD G an PIF21/289. M.W.
acknowledges suppo om he Na ional Science Founda ion unde G an CCF-
1956211. R.D.C. acknowledges suppo om he Ma ie Skłodowska Cu ie ellowship
numbe 891517 (MSC-IF G een- MIQUEC), he Alexande on Humbold Founda ion,
he Knu and Alice Wallenbe g Founda ion h ough he Wallenbe g Cen e o
Quan um Technology (WACQT), and he Academy o Finland, g an s nos. 353832,
349199.
AUTHOR CONTRIBUTIONS
M.R. de eloped he heo e ical o malism and pe o med he analy ic calcula ions.
M.S. sugges ed he seminal idea and supe ised he p ojec h oughou all s ages.
M.R., M.S., R.D.C. and M.W. con ibu ed o he in e p e a ion and imp o emen o he
esul s. M.R. ook he lead in w i ing he manusc ip and all au ho s p o ided c i ical
eedback.
COMPETING INTERESTS
The au ho s decla e no compe ing in e es s.
ADDITIONAL INFORMATION
Supplemen a y in o ma ion The online e sion con ains supplemen a y ma e ial
a ailable a h ps://doi.o g/10.1038/s41534-022-00662-9.
Co espondence and eques s o ma e ials should be add essed o Maximilian
Reiche o Mikel Sanz.
Rep in s and pe mission in o ma ion is a ailable a h p://www.na u e.com/
ep in s
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M. Reiche e al.
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