Unambiguous identification of coherent states II: Multiple resources

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by  Michal Sedlak

Unambiguous identification of coherent states II: Multiple resources
Michal Sedla´k1,3, Ma´rio Ziman1,2, Vladim´ır Buˇzek1,3, and Mark Hillery4
1Research Center for Quantum Information, Slovak Academy of Sciences, Du´bravsk´a cesta 9, 845 11 Bratislava, Slovakia
2Faculty of Informatics, Masaryk University, Botanick´a 68a, 602 00 Brno, Czech Republic
3Quniverse, L´ıˇsˇcie u´dolie 116, 841 04 Bratislava, Slovakia
4Department of Physics and Astronomy, Hunter College of the City
University of New York, 695 Park Avenue, New York, NY 10021, USA
Weconsiderunambiguousidentification ofcoherentstatesof electromagnetic field. Inparticular,
westudypossiblegeneralizationsofanopticalsetupproposedinM.Sedl´aketal.,Phys. Rev. A76,
022326 (2007). We show how the unambiguous identification of coherent states can be performed
in a general case when multiple copies of unknown and the reference states are available. We also
9 investigatewhetherreference statesafterthemeasurement canbe“recovered”andfurtherused for
0 subsequent unambiguous identification tasks. We show that in spite of the fact that the recovered
0 referencestatesaredisturbedbymeasurementstheycanberepeatedly usedforunambiguousiden-
2 tifications. We analyze the role of various imperfections in preparation of the unknown and the

n reference coherent states on theperformance of our unambiguous identification setup.
a
J PACSnumbers: 03.67.Lx,02.50.Ga

1
2
I. INTRODUCTION comes a standard need for a re-calibration of communi-

] cation channel induced by the changes in the medium.
h
In our previous paper [5] (to which we will refer as to
p Discriminationofquantumstates is achallengingtask
- whichdramaticallydiffersfromdiscriminationofstatesof Paper I) we showed that the UI for modes of an electro-
nt aclassicalsystem. Themaindifferencebetweenquantum magnetic field can be efficiently performed if we known
a andclassicaldiscriminatingproblemsisthatsinglecopies that states to be discriminated are coherent states with
u unknown amplitudes. We shown that this prior knowl-
ofnonorthogonalquantumstatescannotbedistinguished
q perfectly. There exist severalstrategieshow to approach edge (or, a reduction of a set of considered states of an
[
the problem of quantum-state discrimination which are electromagneticmode)significantlyimprovesthesuccess

1 related to different choices of a figure of merit that de- of a discrimination process. Furthermore, we proposed
v a simple optical setup consisting of a set of beam split-
fines what is considered to be the best measurement. In
6 ters and photodetectors for implementation of optimal
particular,the strategythatminimizes a mean probabil-
0
UI measurement. This setup was also recently experi-
ity oferroris calledthemin-errorapproach[1]. Another
2
mentally realized by L. Bart˚uˇskov´aet al. [6].
3 equally well motivated strategy is the so-called unam-
. biguous discrimination [2, 3, 4]. In this strategy errors In the present paper we study possible generalizations
1
inconclusionsarenotpermitted, whichimplies that also of an optical setup proposed in Paper I and we investi-
0
9 inconclusive results are acceptable. Nevertheless, when- gate modifications and generalizations of the unambigu-
0 ever we obtain a conclusive result we know for sure that ous identification of coherent states. More specifically,
: it is correct. Various discrimination tasks may differ by in Sec. II we show how the UI of coherent states can
v
a prior information about the quantum system we are be performed in a general case when multiple copies of
i
X measuring. The discrimination among two known pure unknown and reference states are available. This inves-
r states represents a situation, where we have the maxi- tigation is motivated by the observation that with the
a
mum knowledge about the possible preparations of the increase of the number of identically prepared particles
system. On the other hand one can consider a situation we can better identify the preparator,so we are ”closer”
when we have no classical information available about to a classical domain. In Sec. III we investigate the
the preparation. Specifically, instead of being given only possibilities for a recovery of reference states after the
one unknown quantum system, we are given additional measurement is performed. This recovery process might
quantum systems, denoted as reference states, that rep- seem to be prohibited by the rules of quantum mechan-
resent possible preparations. Our task is to determine ics (due to irreversible disturbance of a quantum state
unambiguouslywith which reference states the unknown by a measurement). Nevertheless, we show that in spite
quantum system match. This problem is coined as the of the fact that the recovered reference states are “de-
UnambiguousIdentification(UI)andcanbeused,forex- graded”(disturbed),neverthelesstheycanbeusedinthe
ample, in a communication of parties that do not share next round of UI - this is true under the condition that
common reference frame. In such case reference states an undisturbed (new) copy of an unknown state is pro-
can be seen as an “alphabet” that is always send along vided. Suchascenariocanbeseenasarepeatedsearchin
with the classical information carried by the unknown a quantum database, where the data, i.e. the reference
state. If parties are communicating over medium unsta- states, degrade with their repeated use. In Sec IV we
ble in time (e.g. optical fiber, air,...) then the UI over- investigate the influence of various imperfections and a
2
noise on the performance of our UI setups. Our findings where χ(α ,...,α ) is the probability distribution de-
1 M
are summarized in Sec. V while the Appendix contains scribing our knowledge about the choice of reference
some technical details of our calculations and proofs. states. The optimality of a UI measurement is defined
withrespectto the averageperformanceP. However,we
will see that in most of the optical setups we propose it
II. GENERAL UI MEASUREMENT STRATEGY suffices to optimize P(α ,..., α ), because the opti-
1 M
WITH BEAM SPLITTERS mal value of the trans|mititivitie|s inithe setup does not
depend on specific reference states. In Eq. (2.5) we in-
We start our investigation of unambiguous identifica- tegrate over multiple infinite (complex) planes of com-
tion tasks with posing the problem within a sufficiently plex amplitudes. Unfortunately, a uniform distribution
wide framework. Specifically, we shall consider a set on an infinite plane can not be properly defined. Thus,
of modes of quantum electromagnetic field (linear har- χ(α ,...,α ) can not be uniform, but instead should
1 M
monic oscillators) each of which with a semi-infinite- be “regularized”, i.e. it should satisfy some reasonable
dimensional Hilbert space . In addition, we shall physical requirements. For example, the probability of
consider that each of the mHod∞e is prepared in a coher- having reference states with very big amplitudes, i.e. of
ent state of a specific amplitude. All together we will very high energy, should be vanishing. In the present
considerasetofM+1groupsofelectromagneticmodes. paper we will calculate P(α ,..., α ), because most
1 M
| i | i
The groupA of n modes A ,...,A carriesn copies of the features that the averagedprobability P will have
A 1 nA A
of an unknown coherent state α . Each of the remain- are already apparent in P(α ,..., α ).
? 1 M
| i | i | i
ing M groupsB,C,D,...containsn ,n ,n ,...copies In Paper I we focused our attention mainly on the UI
B C D
of the reference states α , α , α ,..., α , respec- problem with a single copy of an unknown state and a
1 2 3 M
| i | i | i | i
tively. Moreover, we are guaranteed that the unknown single copy of each of two reference states (M = 2,n =
A
state α is the same as one of the reference states α n = n = 1). We proposed a simple efficient UI mea-
| ?i | ki B C
k = 1,...,M. Our task is to find out unambiguously surementutilizingthreebeamsplittersandtwophotode-
with which reference state the unknown state matches. tectors. The whole setup is supposed to operate as fol-
Thus,ingeneral,thediscriminationproblemcorresponds lows: The unknown state α is split by the first 50/50
?
| i
to a selection among one of M possible types of states: beam splitter. As a result we obtain two equally “di-
luted” copies of the unknown state [15] described by a
|ΨiiABC... ≡ .|α..ii⊗A⊗n|αA1⊗iN⊗|nαN1,iB⊗nB ⊗|α2iC⊗nC ⊗..(.2.1) ivse,cttoogre|t√h1e2rαw?iit⊗h|o√n12eαo?fi.thEearcehfeorefntcheesseta“tceos,pfieeds”in|√t1o2αth?ei
second (respectively the third) unbalanced beam split-
with i = 1,2,...,M. The most general measurement
ter. The second (respectively the third) beam splitter
strategy unambiguously discriminating among these M
performs the comparison of the diluted unknown state
possibilitiescanbedescribedbyapositiveoperatorvalue
1 α with the first (respectively the second) reference
measure (POVM) consisting of M +1 elements Ei with |√2 ?i
i=0,1,...,M. The elementE correspondsto a correct state. Detectionofphotonsattheoutputmodesofthese
i
identification of Ψ while E corresponds to an incon- quantum-state comparison measurement setups [7, 8, 9]
i 0
clusive result, i.e.| thie failure of the measurement. These unambiguously indicates that α? differs from α1 or
| i | i
elementsmustobey“no-error”conditions[Eq. (2.2)]and α2 . This enables us to conclude that α? = α2 or
| i | i | i
they have to constitute a proper POVM [Eq. (2.3)]: α? = α1 , respectively. We show in Appendix A, that
| i | i
this measurement is optimal if we restrict ourselves to
i=j Tr[Eiρj]=0; ρi = Ψi Ψi ; (2.2) discrimination of coherent states with the use of linear
∀ 6 | ih |
M optical elements, number resolving photodetectors.
Ei 0,E0 0; E0+ Ei =11. (2.3) Naturally, coherent states encode complex numbers.
≥ ≥ Xi=1 From this point of view the state |√12α?i carries for-
mally the whole information about the complex ampli-
We assume that the states of the type Ψ appear with
i
| i tude. This is due to the fact that we know the factor
anequalpriorprobabilityη =1/M. Theperformanceof
i
λ=1/√2 by whichα is rescaled. If the complex ampli-
this UI measurement can be quantified by a probability ?
tude α is encoded in the state λα then for 0 λ<1
ofidentificationforaparticularchoiceofreferencestates ? ?
| i ≤
wewillspeakabouta“diluted”unknownstatewhile the
M case λ > 1 will be referred to as a “concentrated” un-
P(|α1i,...,|αMi)= ηiTr[Eiρi]. (2.4) known state |α?i. These terms come from the fact that
i=1 the “diluted” state can be obtained by mixing a coher-
X
However, a more adequate figure of merit is its average ent state and a vacuum at an input of a beam split-
value, i.e. ter. As a result of the beam splitter transformation two
modes at the output of the beam splitter are in the di-
P = P(α ,..., α )χ(α ,...,α ) dα ...dα , luted states. On the contrary, the “concentrated” state
1 M 1 M 1 M
ZCM | i | i can be prepared by launching two copies of the same
(2.5) coherentstate into the beam splitter. As a result we ob-
3
FIG. 1: The beamsplitter setup designed for constructive in-
terference of thesame input coherent states. Out of k copies
of acoherent state α we obtain at theoutputof a sequence FIG. 2: The beamsplitter setup designed for an unambigu-
of k 1 beamspitte|rsione mode in the coherent state √2α ous identification of multiple copies of two types of coherent
and k− 1 modes in a vacuum state vert0 . | i states.
− | i
M = 2 and n ,n ,n are arbitrary. The above men-
tainoneoftheoutputmodesinthe“concentrated”state A B C
tioned idea of “concentration” of quantum information
while the second mode in the vacuum state. Using a se-
implies that we first feed all provided copies of the un-
quenceofbeamsplittersandcorrespondingresourcesone
knownstateinton 1beamsplitterstoobtainthefirst
canprepare“diluted’or“concentrated”stateswitharbi- A−
state √n α in the mode A (for brevity later called
trary value of the scaling factor λ. Actually, preparation | A ?i 1
onlyA). TheothermodesA ...A endupinavacuum
of “concentrated”states is the main idea we will employ 2 nA
state, therefore we will not consider them further. Simi-
inourinvestigationoftheUImeasurementwithmultiple
larly,n 1(respectively,n 1)beamsplittersareused
copiesofunknownandreferencestates. Atthebeginning B− C−
to prepare the state √n α (respectively, √n α ) in
of the UI measurement we will, for each kind of a state, | B 1i | C 2i
the modes B (C ). Next, we feed these concentrated
concentratetheinformationencodedinitskcopiesintoa 1 1
statesintoessentiallythe sameschemeasinPaperI(see
single quantum system. This can be done by a sequence
Fig.2). Thus,altogetherwearegoingtousen +n +n
of k 1 beam splitters (see Fig. 1) with transmitivities A B C
− beam-splitters. The analysis of the setup presented in
chosensothatthe inputstate β k constructivelyinter-
| i⊗ Fig. 2 is analogous to Paper I, therefore we comment on
feres to produce the state √kβ 0 k 1. More details
⊗ − it only briefly.
| i⊗| i
about this transformation can be found in Sec. III.A
A beamsplitter transforms two input modes prepared
of [9]. The result of these preliminary transformations
incoherentstates α and β ,respectively,as α β
is a mapping of possible types of states Ψ into states | i | i | i⊗| i7→
|√nAαiiA1⊗|√nBα1iB1⊗|√nCα2iC1⊗.|..i⊗i |0it,where |f√orTaαr+efle√cRtivβiity⊗a|n−d a√tRraαn+sm√itTtivβiitywchoeerffiecRie,nTtssotfanthdes
t = n 1+n 1+...+n 1. As a next step
A − B − M − beamsplitter. ThesetupinFig.2employsoneadditional
we will use the setup proposed in Sec. III of Paper I for
input mode D initially prepared in a vacuum state, i.e.
a single copy of the unknown state and single copies of
the state vector describing four input modes reads
M reference states. Of course, as we will see below the
transmitivities ofall beam splitters in the setup must be
Φ = √n α √n α √n α 0 ,
modified according to the number of copies of the un- | ini | A ?iA⊗| B 1iB ⊗| C 2iC ⊗| iD
known and the reference states we are given.
whereα isguaranteedtobe eitherα orα . Theaction
? 1 2
of the three beamsplitters in the setup is described by a
unitary transformation
A. Two types of reference states
Φ Φ =(U(2) U(3))(U(1) I )Φ ,
| ini7→| outi AB⊗ CD AD⊗ BC | ini
The unambiguous identification of two types of coher-
entreferencestatesisthefirstnaturalstepingeneralizing where U(j) is associated with the j-th beamsplitter B
XY j
the scenariowith single copies ofunknownandreference actingonthemodesX andY. Sincebeamsplittersdonot
statesinvestigatedinPaperI.Inthissectionweconsider entangle coherent states it follows that the output state
4
Φ remains factorized. In the first step the beam- Ifα =α , thenthe probabilityofa correctidentifica-
out ? 1
| i
splitterB withtransmittivityT preparestwo“diluted” tion is given as the probability of detecting at least one
1 1
copies of the state √n α , i.e. photon in the mode C
A ?
| i
|0iD⊗|√nAα?iA 7→| R1nAα?iD⊗| T1nAα?iA(.2.6) P1 =1−|h0| T3nC(α2−α1)i|2 =1−e−nnCC+nnAA((11−−TT11))|α1−α2|2.
(2.11)
In the second step thepbeamsplitters Bp,B perform the p
2 3 Analogously, in the case α = α the probability of a
? 2
transformation such that the output state reads
correct identification reads
|Φouti=|outiA⊗|outiB ⊗|outiC ⊗|outiD, (2.7) P2 =1 0 R2nB(α2 α1) 2 =1 e−nnBB+nnAATT11|α1−α2|2.
−|h | − i| −
with (2.12)
p
Thus the total probability of the identification of refer-
|outiA = |− R2nBα1+ T2T1nAα?iA, ence states |α1i and |α2i is equal to
out B = Tp2nBα1+ Rp2T1nAα? B, 1
| i | i P(α , α )=η P +η P = (P +P ). (2.13)
1 2 1 1 2 2 1 2
out C = p R3R1nApα?+ T3nCα2 C, | i | i 2
| i |− i
out D = Tp3R1nAα?+ Rp3nCα2 D. Nextwewilloptimize the performanceofthe setupby
| i | i
choosing an appropriate value of transmittivity T . The
p p 1
A crucial observation is that the parameters T ,R =
j j definition of the uniform distribution on the set of co-
1 T can be adjusted so that either the mode A, or
j herent states is problematic, therefore we first focus our
−
the mode C, ends up in a vacuum state providing that
attentiononthe probabilityofidentificationforapartic-
α = α , or α = α , respectively. In particular, setting
? 1 ? 2 ularchoiceofreferencestates α and α ,respectively,
1 2
the transmittivities to | i | i
expressed by Eq. (2.13).
T2 = 1+1nnBAT1 ; T3 = nnCA1+−1T−1T1 , (2.8) vtheeaTlhsreetfheinraevtnetschteiegsatoatpitoteinsmoaαfltchh,eofiαicrestoodfneTlryi1vaidftoinvees n∂=Po(tn|α∂d1Tei.1,p|αAe2nsid)oronene-
1 2 B C
| i | i
we find expects,becauseofsymmetryarguments,T1 isoptimally
set to 1/2 if n = n . In such a case, P(α , α ) can
B C 1 2
| i | i
out A = R2nB(α? α1) A; be simplified to take the following form:
| i | − i
|outiB = |pT2nBα1+ R2T1nAα?iB; P(α1 , α2 )=1 e−nnAA+n2nBB|α1−α2|2. (2.14)
out C = pT3nC(α2 pα?) C; | i | i −
| i | − i
|outiD = |pT3R1nAα?+ R3nCα2iD. (2.9) pLreotbuasbinliotyteηth=at1if nηB 6=fornwChtichhenthteheorpetiemxaisltschaoipcerioorf
1 2
p p −
Finally, we perform photodetection in output the T does not depend on the reference states. However, as
1
modes A and C by photodetectors D and D , respec- already mentioned, we focus on the η = 1/M case and
2 1 i
tively. By detecting a photon in one of the two modes we will assume that we are given the same number of
we can unambiguously identify the unknown state. In copies of each reference state.
particular, for these two modes we have
α? =α1 0 A T3nC(α2 α1) C; 1. Trade-off of resources
↔| i ⊗| − i
α? =α2 R2nBp(α2 α1) A 0 C. (2.10)
↔| − i ⊗| i The number of copies of an unknown state or of a ref-
p
We note that due to the fact that at least one of the erence state we have can be seen as a measure of some
modesisinavacuumstatebothdetectorscannot“click” resource. Fromthispointofview aninterestingquestion
(detectthephotons)atthesametime. Therefore,ineach immediately arises. Which type of resource is more use-
single run of the experiment only three situations can ful for an unambiguous identification of coherentstates?
happen: Are unknown states more useful than reference states or
i) none of the detectors click, vice versa? To answer these questions we consider the
ii) only the detector D clicks, following situation. Imagine we will get altogether N
1
iii) only the detector D clicks. quantum systems (modes of electromagnetic field) but
2
If only the detector D clicks that following Eqs. (2.10) we have a liberty to specify whether the specific mode is
1
we unambiguously conclude that α = α . Similarly, if prepared in in the unknown state or in one of the two
? 1
only the detector D clicks we unambiguously conclude reference states. Thus, if we ask for n copies of the
2 A
that α = α . If none of the detectors click we cannot unknown state we will obtain n = n = (N n )/2
? 2 B C A
−
determine which mode was not in the vacuum state and copiesper a referencestate. Let usfor simplicity assume
therefore this situation represents an inconclusive result. that N and n have the same parity. The probability of
A
5
identification for a reference states α , α then reads this result is easily obtained by taking the limit of Eq.
1 2
| i | i
(2.14).
P(α1 , α2 )=1 e−nA(N2N−nA)|α1−α2|2 (2.15)
| i | i −
and it is maximized for nA = N/2 , because the terms 3. Weak implementation of UI measurement
⌊ ⌋
in the exponent are nonnegative. Hence, from the point
of view of the resources, it is optimal to ask for a prepa-
Let us consider a basic version of UI of coherent
rationof N/2 unknownstatesandthe equalnumberof
states (M = 2, n = n = n = 1). We will
⌊ ⌋ A B C
copies per a reference state (specifically, N/4 ).
describe a measurement, which in the case of suc-
⌊ ⌋
cess, leaves all the input states nearly unperturbed and
achieves the same probability of identification as the
2. Infinite number of copies of reference states
original setup from Paper I. The measurement proce-
dure goes as follows. We first equally split each of
Unambiguous identification is a discrimination task our resource states into N parts. Thus, we have N
in which we have very limited prior knowledge about copies of states 1 α , 1 α , 1 α . We use the
the possible preparations of the quantum system we are |√N ?i |√N 1i |√N 2i
beam-splitter setup from Paper I for each of these N
given. The amount of information about the possible
triples. TheUImeasurementperformedonthefirsttriple
ponfuremtphbaerearretofioefrnetshniecsmeessttsahetneetspiarwleleypaogrbiavtteainoinnb.yoIfnthrteehfeenruleimnmcbietersotofafitncefiosnpbiiteees- 1will seu−cc31Nee|dα1−wαit2h|2.probIfabwilietyfi1nd− eα−?31|√=1Nα1α−1√1Nwαe2|2ca=n
−
comeknowntousandthustheUIisbecomingequivalent combine the unmeasured 3N 3 modes into states
−
to discrimination among known states. The unambigu- 2N 2α , N 1α . For α = α we operate anal-
| N− 1i | N− 2i ? 2
ous discrimination among pair of known pure states (for
oqgously obtaiqning N 1α , 2N 2α . If UI mea-
equal prior probabilities) was solved by Ivanovic, Dieks | N− 1i | N− 2i
and Peres [2, 3, 4] in 1987. Their optimal measurement surement of the firsqt triples faqils we continue by mea-
succeedswithaprobability1 ϕ ϕ ,where ϕ , ϕ suring the other triples until we find a conclusive out-
1 2 1 2
−|h | i| | i | i
are the known states in which the system can be pre- come or use all the triples. In case of k-th triple lead-
pared. Inwhatfollowswewillshowthatintheaforemen- ing to conclusive result we concentrate the remaining
tsieotnuepdalcimhiietve(sMth=e s2a,mnBe o=ptnimCa→l p∞erf)oromuranbceea.mI-nspolirtdteerr resources to obtain states | 2(NN−k)α1i,| NN−kα2i or
to provethis we have to evaluate the limit of Eq. (2.14): | NN−kα1i,| 2(NN−k)α2i depqending on α?qbeing α1 or
P(α , α ,n = n ) αq2. Wedonoqtgetaconclusiveresultonlyifthemeasure-
| 1i | 2i B C →∞ ments of all N triples yield inconclusive results. Hence,
= lim 1 e−nnAA+n2nBB|α1−α2|2 the overall probability of successful identification of the
= n1B→e∞−n2A−|α1−α2|2 uanndkneoqwunalssttahteastiosf1t−he(eo−pt31Nim|αa1l−bαe2a|2m)N-sp=lit1t−eres−et13u|αp1−frαo2m|2
−
= 1 α α nA. (2.16) Paper I. However, in contrast to the setup from Paper
1 2
−|h | i| I, if a conclusive result is obtained before measuring the
In the last equality we have used the expression for N-th triple we still have “diluted” input states at our
the modulus of the overlap of the two coherent states disposal.
α1 α2 2 = e−|α1−α2|2. In the limit nB = nC
|h | i| → ∞
the two known states that could be unambiguously dis-
criminatedbytheIvanovic-Dieks-Peresmeasurementare B. More types of reference states
ϕ1 = α1 ⊗nA, ϕ2 = α2 ⊗nA. Thus, we see that
| i | i | i | i
Eq. (2.16) is equal to 1 ϕ1 ϕ2 and so our beam- Intheprevioussectiontheoptimalvaluestransmittiv-
−|h | i|
splitter setup performs optimally in this limit. Let us ities in our beam-splitter setup were state-independent
note that for nA =1 our setup is in this limit equivalent only in the case of equal number of copies per refer-
to the setup proposed by K. Banaszek [10] for unam- ence state. Thus, for more than two types of reference
biguous discrimination between a pair of known coher- stateswewilldiscussonlycaseswiththesamenumberof
ent states. For nB = nC our T2 1, T3 0, i.e. copies of each reference state. Unfortunately, we will see
→ ∞ → →
the “concentrated” reference states are nearly reflected, that even in this restricted scenario, the optimal choice
which induces a displacement of the “diluted” unknown of transmittivities in the setup we propose will depend
state |√12α?i. In the same way K. Banaszek uses very on the reference states.
unbalanced beam-splitters to cause the displacement of The generalization of the beam-splitter unambiguous
the outputs of the beam-splitter. identification scheme from the previous subsection is
For the limiting case M = 2, n = n = n straightforward. We start by preparing the “concen-
A B C
→ ∞
it is natural to expect a classical behavior, i.e. a unit trated”states √n α , √n α ,..., √n α . Weuse
A ? B 1 B M
| i | i | i
probabilityofidentification. Forunequalreferencestates M 1 beam-splitters to sequentially split the “concen-
−
6
trated” unknown state √n α into M states. Each can be useful for creating a quantum database, which
A ?
| i
of these M states is then merged with one of the “con- would not be completely destroyed by the search per-
centrated” reference states √n α ,..., √n α on formed on it. Instead, the data i.e. the reference states
B 1 B M
| i | i
beam-splitter C ,...,C . The transmittivity T (of would degrade gradually with repeated use. First, we
1 M k
beam-splitter C ) is chosen so that destructive interfer- showthatreferencestatescannotbe”recreated”without
k
ence yields vacuum on the second output port of C for additional resources if the first unambiguous identifica-
k
α =α . These output ports are monitoredby photode- tion yields an inconclusive outcome. Although, this may
? k
tectors D ,...,D . Detection of at leastone photon by seemdisappointing,weshowthatthe unmeasuredstates
1 M
photodetector D unambiguously indicates α = α . If still can be used efficiently for UI if the same unknown
k ? k
6
all photodetectors except the k-th fire, then we conclude state is expected. Next, we examine the situation of the
that α = α . For M = 2 we had freedom in choos- first UI producing a conclusive result known to us. In
? k
ing the ratioT withwhichthe “concentrated”unknown that case “diluted” reference states can be created, and
1
state √n α was split into two parts used for the two they can be used for another independent unambiguous
A ?
| i
comparisons. In order to maximize the probability of identification.
identification we can tune M 1 transmittivities of the Letusconsiderthebasicversionofunambiguousiden-
−
beam-splittersthatresultinsplittingthe“concentrated” tificationofcoherentstates(M =2,n =n =n =1).
A B C
unknownstate. The optimal choiceofthese transmittiv- The beam-splitter setup for this scenario was originally
itiesevenforequalpriorprobabilitiesη =1/M depends proposed in Paper I and coincides with the setup de-
j
on the choice of the reference states. Once we consider picted in Fig.2. Modes B and D are not entangled with
n = n = ... then let us consider equal splitting of other modes, therefore their state does not depend on
B C
the “concentrated” unknown state into M parts, even the measurement performed by the two photodetectors.
though it is not necessarily the optimal choice. In such The states of the modes B,D is given by the Eqs. (2.8),
a case the beam-splitters C ,...,C are performing the (2.9), where T is setto 1/2(for details see sectionIIA).
1 M 1
following transformation:
2 1
out = α + α
n B 1 ? B
A | i | 3 6 i
Ck : α? √nBαk out1 out2 ; r r
| M i⊗| i7→| i⊗| i
r 1 2
out = α + α (3.1)
TknA | iD | 6 ? 3 2iD
out1 = α + R n α ; (2.17) r r
? k B k
| i | M i
r Using beam-splitters, phase shifters and knowncoherent
p
RknA states we can produce out of states Eq. (3.1) a coherent
out2 = α + T n α .
? k B k
| i |− M i state of the form
r
p
The condition of out2 being a vacuum for α? = αk 2 1 1 2
forces us to set th|e trainsmittivity to T = n /(n + a( α1+ α?)+b( α?+ α2)+γ , (3.2)
k A A | 3 6 6 3 i
r r r r
Mn ). The probability of observing at least one photon
B
in |out2i if α? =αj is 1−e−nAn+AMnBnB|αj−αk|2. The corre- rwehfeerreenace,bs,tγat∈e.CH.enIcmea,gwineewwanetwthanetsttaoterefcroovmerEtqh.e(fi3r.2s)t
sponding probability of identification therefore reads:
to be λα . Even though we know that either α = α ,
1 ? 1
| i
P(α1 ,..., αM )= M 1 (1 e−nAn+AMnBnB|αj−αk|2). osirbαili?ti=esαp2r,oadusuceitsa“bjluenckh”oiicne tohfea,obthfoerr ocnaseeo.fAthneasleogpoouss-
| i | i M −
reasoning works for the second reference state. For the
j=1 k=j
X Y6
inconclusive result of UI measurement we do not know,
(2.18)
whichpossibilitytookplace,andthusthereferencestates
Let us note that for a single copy of an unknown state can not be recovered.
and a single copy of a reference state (n =n =n =
A B C
... = 1) the scenario is the same as in the Sec. IV.E of
Paper I. In that special case both proposed setups coin- A. Repetition of UI for same unknown state
cide, and therefore also the previous expression reduces
to Eq. (4.30) from Paper I. Although, the unmeasuredmodes of the beam-splitter
setup seem useless they can be exploited in the UI of
the same unknown state α . Namely, we can feed
?
III. RECOVERY OF REFERENCE STATES them instead of reference s|taties into the beam-splitter
AFTER THE MEASUREMENT scheme shown in Fig. 2. The concatenation is illus-
trated in Fig. 3. The transmittivity of the beamsplit-
In this section we examine the information that re- ter B (respectively, B ) can be set so that its mea-
2 3
mains in the unmeasured modes of our beam-splitter sured output is in a vacuum if α = α (respectively,
? 1
UI setups. In particular, we focus on the possibility of ifα =α ). Ifwechose(forsymmetryreasons)T =1/2
? 2 1
“recreating”thereferencestatesoutofthosemodes. This thenthetransmittivitiesT ,T shouldbesettoT =3/4,
2 3 2
7
FIG. 3: The beam-splitter setup designed for a subsequent
unambiguous identification of multiple copies of an unknown
coherent state.
T =1/4. This implies that the photodetectors measure
3
the states (α α )/√6 , (α α )/√6 . Thus, for
? 1 2 ?
| − i | − i
both cases α =α ,α =α we can observe a photon in
? 1 ? 2
only one of the photodetectors and with the probability
1 e−16|α1−α2|2 unambiguously conclude which possibil-
ity−took place. Hence, the probability 1 e−61|α1−α2|2
is a conditional UI probability after a first−identification FIG. 4: The beam-splitter setups designed for the recovery
measurement returned an inconclusive result. The over- of unmeasured modes from Fig. 2. In the case α? = α1 the
setup a) is used, howeverfor α =α thesetup b) is used.
all probability of an unambiguous identification for this ? 2
two-round measurement is 1 e−21|α1−α2|2. This is due
−
to the fact that the measurement fails only if both mea-
surement rounds yield an inconclusive outcome. statesfor anUI with adifferent, independently prepared
The two-round measurement is essentially an UI unknown state β (either β =α or β =α ).
? ? 1 ? 2
scheme for M = 2, nA = 2, nB = nC = 1, so we can When the α?|=iα1 result is found in the first round
compare its performance with the corresponding beam- of the UI, the unmeasured modes B,D are in states
splitter scheme Eq. (2.14) analyzed in Sec. IIA. Indeed,
3α and 1α + 2α , respectively. Thus, we
theperformanceisthesame,butthetworoundmeasure- | 2 1iB | 6 1 3 2iD
ment has one possible advantage. If the first roundgives hqave the “conceqntrated”qfirst reference state 3α in
aconclusiveresultthenwestillhaveanunmeasuredcopy | 2 1i
the mode B. Let us now examine whether theqreference
ofthe unknownstate (i.e. copyof α respectively α )
| 1i | 2i state α2 canbe “recreated”outofthe modesB andD.
atourdisposal. Thisisasimilaradvantageasinthecase | i
The natural idea is to use the mode B to shift the mode
ofweakimplementationoftheUImeasurementdiscussed
D via a beam-splitter so that the α part of the ampli-
1
in Sec. IIA3.
tude in 1α + 2α is canceled. This happens for
| 6 1 3 2iD
the transqmittivityqof the beam-splitter equal to 9/10:
B. Repetition of UI with different unknown state
3 1 2
α α + α
1 1 2
| 2 i⊗| 6 3 i7→
As we illustrated in the beginning of Sec. III it is not r r r
possibleto“recreate”thereferencestatesbylinearoptics 27 1 1 3
+ α + α α . (3.3)
after an inconclusive result of an UI measurement is ob- 7→| r20 r15! 1 r60 2i⊗|r5 2i
tained. Onthecontrary,wewillshowthatwhenaconclu-
sive resultis registeredthen both referencestates canbe Hence, we know how to recover separately either the
“recreated”. Although, thisrecreationisnotperfect,the first or the second reference state. If solely such a single
recreated reference states are bit “diluted”. Neverthe- stateisusedinthe subsequentUImeasurementthenthe
less, subsequently, these states can be used as reference probabilityof success is bounded from aboveby 1/2,be-
8
causeonlyonetypeofareferencestatecanbeidentified.
Thus,wewanttofindasetup,whichextractsbothtypes
of reference states simultaneously and allows for a sub-
sequent round of the unambiguous identification of β .
?
| i
SuchaschemeispresentedinFig. 4a. Thebeam-splitter
B splitsthe“concentrated”firstreferencestateintotwo
1
parts. One part can be directly used for the next round
of UI, the second partcancels the α contribution in the
1
amplitude of the coherent state in the mode D via the
beam-splitter B . If we set transmittivity of the beam-
2
splitterB tobeTR,thentherequirementofcancelation
1 1
of the α contribution of the amplitude of the coherent
1
state in the mode D constrains the transmittivity of B
2
to be TR = (9 9TR)/(10 9TR). The corresponding
2 − 1 − 1
“recreated” reference states then read
3 6 6TR
TRα , − 1 α . (3.4)
|r2 1 1i |s10−9T1R 2i
We want to use these two states instead of the refer-
ence states α1 , α2 in the next round of UI. Both FIG.5: Thebeam-splittersetupdesignedforrepetitionofUI
| i | i
possible preparations β = α , β = α will be withdifferentunknownstate,whichcanbeseenasarepeated
? 1 ? 2
| i | i | i | i
equally likely, therefore we chose TR = (7 √13)/9 so search in a quantum database. The gray beam-splitters in
1 − recovery steps are used if the unknown state from previous
that equally diluted reference states
round of UI matches the second reference state otherwise
7 √13 black ones are used.
λ α , λ α , λ − (3.5)
2 1 2 2 2
| i | i ≡ 6
p p
enterthenextroundofUI.Ifaconclusiveresultα =α
? 2 n = 1, n = n = (7 √13)/6. It turns out
isobtainedinthefirstroundofUI,thenafterexchanging A B C −
that we can perform infinitely many additional rounds
therolesofthemodesBandD,analogousrecoverysetup
of UI, where in each round the unknown state is inde-
(see Fig.4b) can be used to produce the “diluted” refer-
pendently chosen to be either α or α . It suffices
ence states Eq. (3.5). Thus, for both conclusive results | 1i | 2i
to use the beam-splitter setup from Fig. 2 followed by
from the first round of UI, one type of UI measurement
the setup from Fig. 4 recovering the reference states in
using the recovered reference states can be used in the
each round of UI. However, the transmittivities of the
second round. Actually, the beam-splitter setup from
beam-splitters used in those setups must be set as fol-
Fig. 2 can be used (see Fig. 5) if we take into account
lows. Let us denote by √λ the factor by which the
thatforourinputstatesn =1,n =n =(7 √13)/6. k
A B C − reference states are suppressed at the beginning of the
Upon making this substitution the performance of the
k-th round (e.g. λ =1). In k-th round of UI we should
setup is the same as in Sec. IIA, and all the formulas 1
set T = 1/2, T = 2λ /(1+2λ ), T = 1/(1+2λ ) in
derived there remain valid. The aforementioned setup 1 2 k k 3 k
the scheme from Fig. 2 and
succeeds in UI with the probability given by Eq. (2.14).
However, the second round of UI will be possible only if
2λ2 + 4λ4 +(1+2λ )2
the first UI succeeded, which implies the following prob- T = 1 k k k ;
ability of UI in the second round 1 − p(1+2λk)2
(1 T )(1+2λ )2
1 k
P(|α1i,|α2i) = (1−e−13|α1−α2|2) T2 = 1+(−1−T1)(1+2λk)2, (3.7)
× (1−e−2(170−−√√1313)|α1−α2|2). (3.6) intheschemepresentedinFig. 4. Thesuppressionofthe
amplitude of reference states is given by λ λ =
It is interesting that UI with nearlyorthogonalreference k 7→ k+1
f(λ ), where
states canbe done alsoin the secondroundwith a prob- k
ability of success approaching unity.
(1+2x)2 2x2 4x4+(1+2x)2
Let us now see, whether further rounds of UI are still f(x)= − − .
possible. The first round of UI can be seen as use of the 2(1+p2x)
beam-splitter setupfromFig. 2 withn =n =n =1 (3.8)
A B C
followed by the setup from Fig. 4 recovering the refer-
ence states. In the second round we have used again The probability of successfully performing the UI in the
the beam-splitter setup from Fig. 2 this time with k-th round is P(k)(α , α ) = P(k 1)(α , α )(1
1 2 − 1 2
| i | i | i | i −
9
1.0 0.7
N=30
L> 0.6 N=25
0.8 Α2 N=20
L> ÈΑ>,10.5 N=10N=15
È>Α,20.6 HLNHÈPS0.4
Α10.4 L>- 0.3 N=5
HÈP È>Α,2
0.2 Α10.2
LNHÈ
0.0 HP 0.1
0 5 10 15 20 25 30 0.0
0 5 10 15 20
ÈΑ -Α È
1 2 ÈΑ1-Α2È
FIG.6: (Coloronline)Theperformanceoftherecoverysetup. FIG.7: Thedifferencebetweentheperformanceoftherecov-
TheprobabilityofidentificationP(α1 , α2 )asafunctionof eryandthesplittingstrategiesfordifferentnumberofidenti-
| i | i
the scalar product (given by α1 α2 ) depicted for various fication rounds N as a function of the scalar product (given
| − |
numbers of measurement rounds. Starting from the left the by α α ).
1 2
curves correspond to the probability of identification in the | − |
first, 20th, 40th, 60th, 80th round of theUI.
IV. INFLUENCE OF NOISE ON RELIABILITY
OF UI SETUPS
e−1+λ2kλk|α1−α2|2),becausethek-throundoftheUIispos-
sible only if all previous UI succeeded [11]. The depen- In this section we investigate how noise (uncertainty)
denceoftheprobabilityofidentificationonthedifference inthestatepreparationaffectsthereliabilityofthemea-
oftheamplitudesofthereferencestatesandonthenum- surementresults. TheUIsetupswehavepresentedabove
ber of measurement rounds is depicted on Fig. 6. are designed specifically for coherent states and ideally
Let us now discuss an alternative approach to the re- they are 100% reliable, i.e. whenever we obtain a con-
covery of reference states. Imagine that our task is to clusive result Ei then we can be completely sure that
identify N independent unknown states with reference the possibility xi (i.e., α? = αi) took place.However, it
states. Instead of recovering reference states after iden- mightbe that the unknownand referencestates aresent
tifying each of the unknown states we can first split the tousviaanoisychannelorsimplythattheirpreparation
reference states into N parts and then performthe iden- is noisy. We assume that this disturbance has the form
tifications independently. We are goingto illustrate that of a technical noise [12], and therefore the unknown and
even though we know value N ahead of time, the split- thereferencestatesarenotpurecoherentstates αi ,but
| i
ting strategy does not outperform the strategy based on rather their mixtures ωi:
recovery of reference states.
1 β2
forTmhaetisopnlitintintghesttrwaotegreyfebreegnicnessbtaytedsisitnrtibouNtingcotphieesino-f ωi = 2πσ2 ZCdβe−|2σ|2|αi+βihαi+β|; (4.1)
the states |√1Nα1i,|√1Nα2i. These two states are then ρi(α) = (ωi)⊗nA ⊗(ω1)⊗nB ⊗(ω2)⊗nC ⊗...,(4.2)
puttogetherwith oneofthe unknownstatesandareun-
with σ defining the strength of the noise [13] and α in-
ambiguously identified by the scheme for M = 2,n =
A
dicates the dependence on α . In such a case conclu-
1,n =n =1/N. The probability of a successful iden- i
B C
sive results ofour UI setups willno longerbe unambigu-
tification the unknown state depends only on the refer-
ous. More precisely, there will be a certain probability
encestates,henceforeachoftheN UImeasurementswe
thhaavteaPl(l|αof1it,h|eαm2i)su=cc1ee−dei−sNt1h+2e|rαe1f−orαe2|P2.S(NT)h(e|αp1rio,b|αa2bii)lit=y Pspur(roxebima|Ebeniil)titwyisiitshthcwaelhlceiocdnhtstheheqeuroeelbniatcaebiinloieftydtohofeuttphcoeomsosuiebtEciloiitmoyfextEhi.ei.mTTehhaies-
(1 e−N1+2|α1−α2|2)N. On the other hand in the scheme corresponding mathematical definition reads:
−
with the recoveryof the reference states the N-th round
η P(E x )
can succeed only if all the previous identification rounds i i i
R(E )=P(x E )= | , (4.3)
were successful. This means that the probability of suc- i i| i M η P(E x )
cess of the N-th round P(N)(α , α ) is the same as j=1 j i| j
1 2
the probability that all the N|roiun|dsiof the identifica- where η is the a priori probPability of the possibility x
i i
tion task were successful. The difference between the and P(E x ) is the probability that the measurement
i j
|
performance of the recovery and the splitting strategies of the system prepared in the possibility x will give
j
for different N is depicted in Fig. 7. a result E . Let us note that under the possibility x
i i
10
we understandallsituations inwhich the unknownstate UI setupactsonstates α +ν , α +β , α +γ , ̺ fed
? 1 2
| i | i | i | i
is the same as the i-th reference state. Thus x stands intomodesA,B,C,D(seeFig.II)andfinallyweperform
i
for the whole set of situations, which differ by complex integration over ν,β,γ,̺.
amplitudes αk of the “centers” of the reference states For multiple copies of the unknown and the refer-
ωk. How those “center points” of all reference states are ence states we assume that the noise is acting indepen-
chosen in xi is described by the probability distribution dently on each of the copies, i.e. we would analyze
χi(α1,...,αM). The support of χi is Cm corresponding nB copies of the first reference state entering as states
to aninfinite plane. Thereforeauniformprobabilitydis- α +β ,..., α +β . The first partof the UI setup,
| 1 1i | 1 nBi
tribution can not be defined on it. Nevertheless, we can which “concentrates” copies of the same species, gener-
express the reliability as: ates the state |√nBα1+ √n1B(β1+...+βnB)i and sim-
ilarly, the state √n α + 1 (γ +...+γ ) for the
R(Ei)= Mjη=i1RηCjMCdMαχdαi(χαi)(Tαr)(TEri(ρEi(iαρj)()α)), (4.4) fsoercotnhdeurenfkerneonwcne|ssttaattCee,.a2Tnhde|√√bnenCaAmα1s?p+li√ttn1eAr(tνr1a+nnCs.f.oi.r+mνantAio)ni
wheredα dαP...dαR . Inthelimitσ 0statesω be- for coherent input states does not entangle its outputs,
≡ 1 M → i thus we can, in the same way as in section IIA, derive
come α α . Because of the no-error conditions (2.2),
| iih i| expressionsforthestatesofthe modesthatthe photode-
which for coherent states are satisfied by our UI setups,
tectorsD ,D measure. Consequently,the finalstatesof
onlythei-thtermofthesuminEq. (4.3)survives. Thus, 1 2
the modes A and C read:
without noise the reliability is equal to unity. For σ >0
also other terms in Eq. (4.3) will contribute and hence
n n 1 1 1
reliability will be less than one. Moreover, the precise A B α α ̺+ ν β
? 1
value of R(Ei) will depend on the probability distribu- (cid:12)rnA+2nB (cid:20) − −rnA nA − nB (cid:21)EA
tions χi(α). (cid:12)(cid:12) ≡|µ1iA
Intheremainingpartofthissectionwewillinvestigate (4.6)
ascenario,whichmightbecalledasthephasekeying. We
n n 1 1 1
assume that two reference states (M = 2) have always A C α α ̺ ν+ γ
2 ?
opposite phases, i.e. if ω1 is centered around the am- (cid:12)rnA+2nC (cid:20) − −rnA − nA nC (cid:21)EC
plitude α then ω2 is centered around the amplitude −α. (cid:12)(cid:12) ≡|µ2iC,
ValuesofαhaveaGaussiandistributioncenteredaround
0 (vacuum) with a dispersion ξ , so where ν ≡ nk=A1νk, β ≡ nk=B1βk, γ ≡ nk=C1γk. Now
we have to evaluate the probability of the projection of
χi(α1,α2)=δ(α1+α2)2π1ξ2e−|α1|2/(2ξ2), i=1,2. wtheesweislltaintetesPg|rµa1tieAt,h|µis2ipCarotniPatol rtehseuvltactuouomb.tPaSinubtsheequpernotblay-,
(4.5) bility P(Dk ρi(α)) that the photodetector Dk (k =1,2)
|
does not click. Probabilities P(D ρ (α)) are related to
k i
|
In order to calculate the reliability we must first eval- Tr(Eiρj(α)) in the following way:
uate Tr[E ρ (α)]. This means we have to derive the
i j
probabilitieswithwhichdetectorsD ,D clickif“fuzzy” Tr(E ρ ) = [1 P(D ρ )].P(D ρ );
1 2 1 1 − 1| 1 2| 1
states ω?,ω1,ω2 are fed into the UI setup instead of Tr(E1ρ2) = [1 P(D1 ρ2)].P(D2 ρ2);
α , α , α . Our UI setup uses an additional mode − | |
D| ?tih|at1sih|ou2lid be initially prepared in vacuum. We as- Tr(E2ρ1) = P(D1|ρ1).[1−P(D2|ρ1)];
sume that also this mode is noisy and initially in a state Tr(E2ρ2) = P(D1 ρ2).[1 P(D2 ρ2)], (4.7)
| − |
ω centered around 0 (vacuum).
i
To present our calculations concisely, we first derive where the argument of ρi(α) is omitted for brevity. Fi-
howthe setupactsoncoherentstates: We integrateover nally,weobtainthe quantitiesTr[Eiρj(α)] thatweneed
coherent states in Eq. (4.1) (e.g. α +β ) and then we for evaluating the reliability according to Eq. (4.4).
i
integrate those partialresults. Thu|s, for aisingle copy of Using the formula 0µi 2 = e−|µi|2 for the modulus
|h | i|
the unknown and the reference states we derive how the of the overlap of two coherent states we obtain: