General linear-optical quantum state generation scheme: Applications to maximally path-entangled states

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by  N. M. VanMeter

General linear-optical quantum stategeneration scheme:
Applications to maximallypath-entangled states
N.M.VanMeter1, P.Lougovski1, D.B. Uskov1,2, K.Kieling3,4, J. Eisert3,4, andJonathanP. Dowling1
1HearneInstituteforTheoretical Physics, LouisianaStateUniversity, BatonRouge, LA70803, USA
2Department ofPhysics, TulaneUniversity, 2001 PercivalSternHall, NewOrleans, LA70118, USA
3Institute for Mathematical Sciences, Imperial College London, London SW7 2PE, UK
4QOLS,BlackettLaboratory, ImperialCollegeLondon, PrinceConsort Road, LondonSW72BW,UK
We introduce schemes for linear-optical quantum state generation. A quantum state generator is a device
thatpreparesadesiredquantum stateusingproduct inputsfromphoton sources, linear-opticalnetworks, and
post-selectionusingphotoncounters. Weshowthatthisdevicecanbeconciselydescribedintermsofpolyno-
mialequationsandunitaryconstraints. WeillustratethepowerofthislanguagebyapplyingtheGro¨bner-basis
technique along with the notion of vacuum extensions to solve the problem of how to construct a quantum
8 stategenerator analyticallyfor anydesiredstate, and usemethods ofconvex optimizationtoidentifybounds
0 tosuccess probabilities. Inparticular, wedisprove aconjecture concerning thepreparation ofthemaximally
0 path-entangledNOON-statebyprovidingacounterexample usingthesemethods, andwederiveanewupper
2 boundontheresourcesrequiredforNOON-stategeneration.

n
a PACSnumbers:42.50.Dv,42.50.St,03.67.Lx,03.67.Mn
J

2
I. INTRODUCTION the problem of quantum state generation with linear optics

andprojectivemeasurements. WefirstformuIatethephysical
2
v Therearemanyquantumstatesoflight,whichareingreat problemofstategenerationintermsofpolynomialequations
4 demand in quantum technology [1]. Due to their high ro- inSectionII.Wethenarguethattheequationsobtainedcanbe
5 bustness to decoherence, and relatively simple manipulation solvedanalytically,byapplyingstandardtoolsofalgebraicge-
1 techniques, photons are often exploited as primary carriers ometry,whenaunitarityconstraintisrelaxed. Moreover,we
2
of quantum information. Recently, a number of schemes show how unitarity can be reintroduced into the solution by
1
6 have been suggested enablingquantuminformationprocess- additionofauxiliaryvacuummodes. InSection IIIweillus-
0 ingwithphotonsusingonlybeamsplitters,phaseshifters,and tratetheformalismbyconsideringanexampleofa“NOON”
/ photodetectors[2,3].Inadifferentapproachtoquantumcom- state generator. We demonstrate how an optimal “NOON”
h
puting,clusterstatesofphotonscanbeusedtoperformcom- stategeneratorcanbeconstructedusingconvexoptimization
p
- putation [4]. Moreover,exoticstatesofmanyphotons,such tools. Asimpleexamplefor5-photon“NOON”stategenera-
t as maximallypath-entangledNOON states, havefoundtheir tiondisprovesthe“No-Go”conjecture[3]. We drawconclu-
n
a placeinquantummetrology[5],lithography[6],andsensing sionsandsummarizeinSectionIV.
u [7].
q Anaturalquestionarises:howcanthesecomplicatedstates
v: oflightbeprepared?Oneapproachtotheproblemistomake II. LINEAR-OPTICALQUANTUMSTATEGENERATOR
i useofanopticalnonlinearity. However,duetotherelatively
X
small average number of photons involved, the overall non- Any linear-optical quantum state generator (LOQSG) can
r linear effect is extremely weak and typically of little practi- be thoughtof as consisting of two main blocks(see Fig. 1).
a
cal use [8, 9, 10]. An alternative way to enable an effective ThefirstblockisaN-portlinear-opticaldevice,describedby
photon-photoninteractionistouseancillamodesandprojec- a unitarymatrixU U(N)combiningN inputintoN out-
∈
tive measurements [2, 3]. In this way a quantumstate gen- put modes. The second block represents a projective mea-
erator can be realized utilizing only linear optical elements surement of some of the modes of the first block, in which
(beamsplittersandphaseshifters)andphotoncounters,atthe a certain pattern of photons measured in some M < N of
expense of the process becoming probabilistic [11]. Hybrid themodesisconsidereda“successfulmeasurement”,leading
schemes, combining weak nonlinearities and measurements, to a preparation of the desired state in the remaining modes
havebeenalsoproposedrecently[12]. (compare also Refs. [11, 13, 15, 16]). This measurement is
Despite many theoretical efforts, the problem of quantum probabilistic—butheralded—or“event-ready”.Clearly,the
statepreparationoflightwiththehelpofprojectivemeasure- outputisdeterminedbyaninterplaybetweenthenumbersof
ments has not been formalized and solved in a general set- inputphotons,theentriesofthematrixU,andthenumbersof
ting. Progress with respect to the problem of constructing photonsdetected.
anoptimalall-opticaltwo-qubitgatehasbeenachieved[13]. There are two types of problems that can be formulated
Moreover,aslightlydifferentproblemofattributinganeffec- around the conceptof LOQSG. The first problem is the fol-
tive physical nonlinearity to a given combination of linear- lowing: Given the matrix U and a known inputstate, which
opticaltransformationsandprojectivemeasurementshasbeen output states can be generated for different projective mea-
demonstratedtohaveadefiniteanswer[14]. surements? Thisiswhatcouldbereferredtoasthe“forward
In this paper we concentrate on formalizing and solving problem”. Thisquestionisequivalenttotheproblemoffind-
2
in tar the creation operatorsof the outputmodes. The outputstate
ψ 〉 ψ 〉
oftheLOQSGbeforeaprojectivemeasurementhencereads
L.O.
m
1
0〉 m02 |ψouti = F(a†1,out,...,a†N,out)|0i
N 1 N ni
:= √n !(cid:18) Ui,ja†j,out(cid:19) |0i. (1)
FIG.1:Thelinearopticalquantumstategenerationscheme. Yi=1 i Xj=1
Here, F(a†1,out,...,a†N,out) is a homogeneouspolynomial of
ingtheeffectivenonlinearitygeneratedbyagivenprojective degreen = N n in thecreationoperators. Coefficients
measurementandwasaddressedinRefs.[14,17].Thesecond k=1 k
ofthemonomPialsinF areinturnhomogeneouspolynomials
problem is that of state preparation: Given an input state, a ofdegreenbelongingtothepolynomialringC[U ].
i,j
projectivemeasurement,andatargetstate,isitpossibletode-
TheoutputoftheLOQSGafterasuccessfulprojectivemea-
termineaunitarymatrixU,ofappropriatedimension,involv- surement of (m ,...,m ) photons in modes N M +
1 M
ingpotentiallyfurtherauxiliarymodes,describingtheunitary 1,...,N isthengivenby −
blockoftheLOQSG?Givenacertaininputstate,thisimpor-
tant problem asks whether a device for the preparation of a Φ = m ,...,m ψout .
1 M
| i h | i
certain quantum state can be identified, and if so, what ele-
mentsitcontains.Oncetheunitaryisfound,thereisasimple, Thestatevector Φ istheresultoftheactionofapolynomial
| i
wmeelnl-taktnioonwnwpitrhesbceraipmti-osnplfiottrercsonanvedrtpinhgasiet-tsohiafnteorsp,tiectacl.im[1p8le]-. GG(ias†1c,oount,s.tr.u.c,tae†Nd−frMom,ouFt)obnytsheelevcaticnuguamll|t0eir.mTshwehpicohlycnoonmtaiianl
Findingsuchaunitarywecallthe“inverseproblem”. Inthis a†N M+1,out,...,a†N,out exactly to the powers m1,...,mM,
section, we provide a mathematical description of LOQSG resp−ectively,andreplacingthesecreationoperatorsby1after-
andillustratehowmethodsofalgebraicgeometrycanbeused wards. Thus, the polynomialG acting on vacuum describes
tosolvethisinverseproblem. the final state in the first N-M unmeasured modes, which
To be more specific, we start from a given input ψin formtheoutputofthedevice.
| i
in N optical modes obtained from photon sources. Then, Forfurtherconvenience,wehavesummarizedallsymbols
we investigate all state vectors that can be reached from used in our paper relevant to our description of the LOQSG
this using arbitrarynetworks of linear optical elements [18]. andtheircorrespondingmeaninginthetablecontainedinAp-
We hence investigate the orbit Ω = Ψ : Ψ = pendixA.
(U)ψin forsomeU U(N) of an i{n|puit state| viector ByconstructionGisahomogeneouspolynomialofdegree
Uψin .|Theiunitary (U)∈actingin}theHilbertspaceofquan- n m,m= M m ,withcoefficientsinC[U ].Similarly,
t|umistates is the stUandard Fock-Bargmann representation of th−e desired oPutkp=u1t ofkthe LOQSG, ψtar , is a si,tjate vector in
theunitarytransformationU actingonmodes. Ifwenowde- the modes 1,...,N-M, and can be| wriitten as a polynomial
notewith ψtar thedesiredoutputinN-M ofthemodesupon
aparticula|rproijectivemeasurementP = ψD ψD onM of Qfin(dai†1n,gouat,L..O.Q,aS†NG−iMst,houetn)eaqcutiinvgaloenntvtoacfiunudmin.gTahueniptaroryblmematroixf
| ih |
themodes,then
U suchthatG=αQ,forsomenonzeroα.
WenoticethatforthepolynomialequalityG=αQtohold
Θ={|Ψi:|Ψi=α|ψtari|ψDi+ βi|ψianyi|ψiD⊥i,α>0} true, the coefficients of like monomials in G and αQ must
Xi be equal. This leads to a system of polynomial equations
inthevariablesU ,theentriesofourtransformationmatrix
isthesetofallstatevectorswhichcanbeconvertedinto ψtar i,j
U. Thissystemofpolynomialequations,alongwiththecon-
by the measurement of ψD in auxiliary modes. The| α2i
straint that U be unitary, is the mathematical formulation of
represents a probability o|f suiccess, and each ψD den|ote|s
a state vector orthogonal to ψD . The solutio|nit⊥oithe state thestatepreparationproblem. Ingeneral,thereisnoefficient
solutiontechniqueforsolvingsuchasystem. Itisbecausethe
| i
preparationproblemisthentheintersectionofΩandΘ.
unitarityconstraintinvolvesthe operationof complexconju-
Since there exists a one-to-one correspondence between
gationandhencecannotbewritteninanalgebraicform,i.e.,
photonicstatesand(possiblyinfinite)polynomialsinthecre-
inthesamewayasthepolynomialequations. So,weinstead
ationoperatorsofthemodes[19],theproblemoffindingthe
proposeamethodforfindingthemodetransformationmatrix
intersectionofΩandΘcanberecastintermsofpolynomial foralargerLOQSGdescribedbyinput(n ,...,n ,0,...,0)
1 N
equalities.Wewillassumethroughoutthepaperthattheinput
and measurement (m ,...,m ,0,...,0). The method in-
1 M
hasbeenpreparedusingindividualsources,sotheinputstates
volves two steps: (1) For the original system, we relax the
are product states of Fock states with respect to N modes:
constraintthatthematrixU beunitary,andsolvethepolyno-
ψin = n1,...,nN . Theseinputmodesareassociatedwith mial system G = αQ for non-unitarymatrix A with entries
| i | i
N creation operators a† = (a†1,...,a†N). The mode trans- Ai,j; (2) For A as foundin step (1), we find a larger matrix
formation of the LOQSG is given by a N N unitary ma- U whichisunitaryandcontainsAasasubmatrix;thisU de-
× ′ ′
trix U. In other words, the creation operators transform as scribestheLOQSG,andwillgeneratethedesiredstate ψtar
(a†)T = U(a†out)T, wherea†out = (a†1,out,...,a†N,out)denotes intheunmeasuredmodes. | i
3
To realizethe step (1)andto solvethepolynomialsystem (m ,...,m ,0,...,0) reflecting success are then the solu-
1 M
G = αQ, weuse the Gro¨bnerbasistechnique[21]. We first tiontotheproblemoffindingaLOQSG.Thequestionarises:
find a Gro¨bner basis, g ,...,g C[A ], for the ideal for a given non-unitarymatrix A, when is it possible to find
1 l i,j
{ } ⊂
generatedbythecoefficientsofG αQ. ThisGro¨bnerbasis a matrix U which is unitary and contains some member of
′
−
istheminimalsetofpolynomialsthatdonotleavearemain- [A]asasubmatrix? Moreover,howcanwefindtheextension
derwhentheoriginalpolynomialsaredividedbythem. One whichoptimizesthesuccessprobabilityoftheLOQSG?
of the essential features of this approach is contained in the Itisnotdifficultto see thatanymatrixA canbe extended
eliminationtheorem,whichstatesthattheGro¨bnerbasiscon- ifλ = A 1,soifthelargestsingularvalueofAisnot
||| ||| ≤
sistsofpolynomialsofonlythefirstvariableintheordering, largerthanunity(comparealsoRef.[13]). HenceforanyA,
wecanalwaysextendA/λ [A]toaunitary.Letthesingular
g1 =g1(A1,1) C[A1,1], valuedecompositionofAb∈edenotedasA = VDW, where
⊂
thefirsttwovariables,g2 = g2(A1,1,A1,2) ⊂ C[A1,1,A1,2], tVh,eWmi∈nimUu(mN)diamreenusniiotanroyfatnhdeeDxt=enddeiadgu(ndi1t,a.ry..U,dcNo)n.taTinhienng,
and so forth. This property allows us to simply find the
A/λ is N +d, where d = rank(D/λ ). Thus, d < N
solution to the whole system by solving for each variable −1
additionalvacuummodesaresufficienttoextendamemberof
subsequently— a procedurethat is often regardedas exten-
[A] to a unitary. The minimum number of additionalmodes
sion of a solution. That means finding the solutions A0 of
1,1 neededtoextendatleastonememberof[A]remainsunsolved.
g (A ) = 0, A0 ofg (A0 ,A ) = 0, andso on. Thus,
1 1,1 1,2 2 1,1 1,2 However,wecansolvetheproblemofoptimizingthesuccess
the whole problem is reduced to finding roots of monovari-
probabilityoftheextendedLOQSGusing,forexample,ideas
ate polynomials. In this way, the Gro¨bnerbasis techniqueis
fromoptimizationtheory[23].
the generalizationto polynomialsystemsofthe techniqueof
For any solution of the polynomial equations the success
Gaussianeliminationforlinearsystems,andissimpletoim-
probabilityof correctstate preparationis by definition given
plementinalgebraicsoftware.
by α2. The allowed rescaling of rows and columnsrescale
For the case of product-state inputs, we can simplify the | |
theamplitudesintherespectiveinputandoutputmodessuch
polynomialsystemofequationsbyeliminatingsomearbitrary
thatwithinanequivalenceclass
factorsbeforeusingtheGro¨bnerbasistechnique. Weremark
that for any matrix A, which is a solution to the polynomial 2
system, arescalingoftherowsofA—correspondingtothe p = α xnk yml , (2)
s (cid:12) k l (cid:12)
inputmodesoftheproductstate—byarbitrarynonzerofac- (cid:12)(cid:12) Yk Yl (cid:12)(cid:12)
torsx andofthecolumnsofA—correspondingtothemea- (cid:12) (cid:12)
suredimodes — by arbitrary nonzero factors yj will gener- wherepsisthesuccess(cid:12)probabilityandxk(cid:12)andylarethearbi-
ate another solution to the system. Hence, for any solution traryrowandcolumnmultipliersaspresentedinthedefinition
A, we define the equivalenceclass of A by [A] = XAY : ofour equivalenceclasses. Note, thatw.l.o.g. itis sufficient
X = diag(x1,...,xN),Y = diag(y1,...,yN),y1 {= ... = to chose xk and yl real and positive, thus every quantity in
yN M = 1 . In any such equivalence class, there is ex- (2)isreal. Theproblemisthentomaximizeps subjecttothe
actl−y one me}mber A for which A = ... = A = constraintXAY2A†X . Forsimplicitywewillfocuson
1,1 N,1 ≤ 1
A1,N M+1 = ... = A1,N = 1. Hence, by setting the theimportantcasewhenni = nj foralli,j = 1,...,N and
aforem−entioned variables to 1 from the very beginning, one mi =mj foralli,j =1,...,M. Then,forgivenY,thecon-
cansolveamuchsimplerpolynomialsystemintheremaining straint can be written as a so-called semi-definite constraint.
variablesusingtheGro¨bnerbasistechnique;eachsolutionof Semi-definiteoptimizationproblemscanbeefficientlysolved
that system will correspond to an equivalence class of solu- andsolversarereadilyavailable[24].AsAY2A†isinvertible,
tionsfortheentiresystem. Wecallsuchasolutionanequiva- theconstraintisequivalentto
lenceclassrepresentative.
X
Foragiveninputandprojectivemeasurement,onewillfind, 1 0.
in general, many equivalence classes of solutions. In most (cid:20) X (AY2A†)−1 (cid:21)≥
cases, an equivalenceclass representativeA will notbe uni-
Theobjectivefunction,p = c( x )2n1 foraconstantc =
tary and hence will not conservethe canonicalcommutation s k k
relations,whichimpliesthatAandhenceanunderlyingLO- α2( lyl)2mM, is a monomial,Qand hence clearly not linear.
HowQever, this monomialcan be relaxedto again a hierarchy
QSGisnotphysicallyrealizable. However,foranon-unitary
of semi-definite constraints, without altering the optimalob-
A,onecanthinkofalargeropticalsystemdescribedbyauni-
jectivevalue: Letsbethesmallestintegersuchthat2s N.
tary matrix U′, and hence experimentallyimplementable, in Let us assume that 2s = N; if N is smaller, we can al≥ways
N = N +dmodeswhichinturncontainsamemberof[A]
′ pad with variables that we enforce to be unity by means of
asasubmatrix. Physically,thiscorrespondstoanadditionof
linear constraints. Then, let x(1) = x for k = 1,...,N,
auxiliary modes to the LOQSG. Then, if we input and mea- k k
surevacuumintheseadditionalmodes, theentriesofU cor- and x(kj)1x(kj) = (x(kj/+21))2 for k ∈ {2,4,...,2s+1−j} and
respondingtotheaddedmodesdonotaffectthefinalstateof j = 1,−...,s. Hence, we have introduced a number of new
theoriginalmodes.GivenaninputonN modesandadesired variables, according to a hierarchy. Each of the quadratic
outputonN M modes,the (i)totalnumberofmodesN , equalityconstraintsof thisformcan actuallybe enforced,as
′
−
the(ii)unitaryU U(N ),andthe(iii)measurementpattern is easy to see, see footnote [25]. In fact, given Y, we have
′ ′
∈
4
writtentheaboveproblemasasemi-definiteproblem.Inturn, andG. Theyread,
relaxationsgive efficientupperboundsto the success proba-
2
bilityfortheLOQSGwhensimultaneouslyvaryingX andY. 1 3 3
ThesolutiontothisproblemgivesrisetotheLOQSGoperat- F = √8  Ai,ja†j,out ,
ingwiththehighestprobabilityofsuccesswithintheequiva- iY=1 Xj=1
 
lenceclassofA.Then,foragiveninputandmeasurement,the
overall optimal LOQSG can be found by simply optimizing G = Coefficient[F(a†1,out,a†2,out,a†3,out),a†3,out], (4)
1
wanidthcinhoeoascihngeqtuhievbaleesntceeqculiavsaslebnycethcelamsse.thoddescribedabove, Q = √2 5!(cid:16)(a†1,out)5+(a†2,out)5(cid:17).
·
The equality G = αQ leads to a system of six polynomial
equations for the coefficients of (a†1,out)k(a†2,out)5−k, k =
0,...,5, with respect to ten complex variables: the nine el-
III. EXAMPLE:NOON-STATELOQSG ementsofAandα.
We nowsimplifythe systemto solveonlyforequivalence
Toillustratehowtosolvetheproblemoftheintersectionof classes by setting A1,1 = A2,1 = A3,1 = A1,3 = 1. Once
Ω and Θ using the language of polynomials, let us consider theGro¨bnerbasistechniqueisusedtosolvethesimplifiedsys-
thestategenerationproblemfortargetstatesoftheform tem,onefindsthattherearefinitelymanyequivalenceclasses.
Here,wewillillustrativelyshowtheworkforoneofthem(in
fact,foranoptimalone). Thefollowingmatrixistheoptimal
ψtar =(n,0 + 0,n )/√2,
| i | i | i equivalenceclassrepresentative:
1 1 1
the so-called path-entangled NOON state [20]. The NOON
state is importantin a number of applications such as quan- A=1 e−4π5i 3+2√5e−2π5i  (5)
tum metrology and lithography [6]. It cannot, however, be 1 e4π5i 3+√5e25πi
 2 
generatedusingonlylinearopticsfromproductsources,and,
Although the mode transformation in Eq. (5) generates the
therefore,theNOON-stategenerationproblemisbothimpor-
finaldesiredNOONstatevectoruptonormalization
tantandnontrivial.ForNOON-stategeneration,werequire
α(5,0 + 0,5 )/√2
| i | i
N
α
Coefficient[F,jY=3(a†j,out)mj]= √2n!((a†1,out)n+(a†2,out)n). nwoitnh-uαni=tar√ya3n0d(3m+us√tb5e),eixttiesnedaesdy.toseethatthematrixAis
(3) We now seek to optimize the success probability, ps =
Eq. (3) cannot be satisfied for input states for which ni > α2(x1x2x3)4(y3)2 of the extended unitary U over all
m+1foranyi 1,...,N . Toseethis,notethattheRHS x1,x2,x3, and y3 such that XAY is extendable. Let
ofEq.(3)hasau∈ni{quefactor}izationintodistinctlinearterms. us view x1,x2,x3 in spherical coordinates: ps =
Ifn >m+1foranymodeitheLHSwouldnecessarilyhave α2r12(sin2θcosθsinφcosφ)4y2. Foragivendirectionθ,φ,
i 3
multipleidenticallinearfactorsofina†1,out anda†2,out. Hence, the optimalps is reachedwhenr is chosenmaximally; thus,
for the equality to hold, we must have n m+1. Thus, we choose r so that the largest singular value of XAY is 1.
i
usingpolynomialproperties,wehaveshown≤thattheintersec- For a given y3, then, the optimal success probability of the
tion of Ω andΘ is emptyfor any ψin whichcontainsmore extension over all X can be found by varying θ and φ over
thanm+1photonsin anyofitsm|odeis. Inotherwords, we an entire sphere; call this ps,opt. Then, by evaluating ps,opt
havedemonstratedthatthelargestNOONstatethatmightbe overareasonablylargerangeofy3,theglobaloptimumofthe
generatedfromN modeswithatotalofmmeasuredphotons extensioncanbefound.
isN(m+1) m. Usingtheoptimizationproceduredescribedabove,wefind
− that the global maximal success probability is reached at a
Theargumentaboveisonlyusefulfordeterminingwhether
point (x ,x ,x ,y ) for which the matrix can be extended
an intersection is empty. However, the ultimate goal of this 1 2 3 3
minimally, i.e., by only 1 additional dimension. It remains
section is to present a constructive method to find all points
unclearwhetherthisresultwillholdtrueingeneral. Onecan
in the intersection when it is non-empty. To illustrate our
checkthattheunitarymatrix
method,wedemonstratehowtobuildaNOON-stateLOQSG.
LetusconsiderthefollowingLOQSGinput ψin = 2,2,2 . 0.5722 0.5722 0.1894 0.5561
| i | i
InthiscasethemodetransformationmatrixAisa3×3ma-  0.5257 0.5257e−4π5i 0.4556e−2π5i 0.4895e35πi 
tersitxedwiintha9prcoojmecptlievxe menetariseusreAmi,ejn.toWfeonareepphaorttoicnuilnarmlyoidnete3r-,  0.5257 0.5257e45πi 0.4556e25πi 0.4895e−3π5i 
0.3461eπi 0.3461eπi 0.7409 0.4599 
i.e., ψD = m = 1 . Thetotalnumberofphotonsinthe  
1
| i | i | i
remaining (unmeasured) modes is thus 5 and, therefore, the containsamemberof[A]asasubmatrixandgeneratesauni-
onlyNOONstatethatcanbecreatedistheonecorresponding tary transformation on four modes which results in a five-
to the state vector ψtar = (5,0 + 0,5 )/√2. Proceeding photon NOON state starting with input 2,2,2,0 and pro-
| i | i | i | i
asdiscussedinSectionIIwecalculatethepolynomialsF,Q, jectivemeasurementofonephotoninmode3andvacuumin
5
mode4withoptimalsuccessprobability 0.05639.Byusing V. APPENDIXA
≈
the well-known algorithm presented in Ref. [18], the above
unitarymatrixcaneasilybetransformedintoalinear-optical
Symbol Meaning
network consisting of phase-shifters and beam-splitters and,
N totalnumberofmodes
assuch,canbeconstructedinthelab.
M numberofancillarymodes
Remarkably, the example we have just considered also
serves as a counterexample to the “No-Go” conjecture [3]. U N–dim.unitaryrepresentingthelinearoptics
TheconjecturestatesthatthelargestNOONstatethatcanbe n numberofphotonsinputinmodei
i
producedfromproductinputswithonlyN modesisthatofN n N n –totalnumberofinputphotons
i=1 i
photons.However,wehaveseenthatthereisasituationwhen
m PnumberofphotonsmeasuredinmodeN M +i
theNOONstateoffivephotonscanbegeneratedinonlyfour i −
m N m –detectedphotonsnumber
modes. i=N M+1 i
ψin PN–dim−.inputproductstate
| i
ψout N–dim.stateafterlinearoptics
| i
IV. CONCLUSIONSANDOUTLOOK ψD M–dim.specifiedmeasurementstate
| i
Φ N M–dim.stateremainingaftermeasurement
Inthiswork,wehaveshownhowtheproblemofidentifying | i −
ψtar N M–dim.desiredoutputstate
alinearopticalstatepreparationdevicecanbeformulatedand | i −
F polynomialrepresenting ψout
solvedusingthelanguageofpolynomials. Inthisway,wedo | i
nothavetoincludeunitarityconditionsasconstraintsonour G polynomialrepresenting Φ
| i
system. Instead,wesolvethepolynomialequationsusingthe Q polynomialrepresenting ψtar
| i
methodsofalgebraicgeometryandlaterrestoreunitaritywith α probabilityamplitudeforsuccessfulmeasurement
asacrificeofatmostdoublingthenumberofmodes.Wehave
A N–dim.matrixsolutiontoG=αQ
introduced a general framework that allows for the system-
d numberofvacuummodesaddedtosystem
aticconstructionoflinearopticaldevicespreparingentangled
N N +d
quantumstatesoflight. ′
It is worth noting that in addition to solving the problem U′ N′–dim.unitarycontainingAasasubmatrix
ofstategenerationbylinear-opticalmeans,thetechniquewe
propose can also be used to construct optimal linear-optical
quantum gates. With care taken to formulate appropriate VI. AKNOWLEDGEMENT
equivalenceclasses,theproblemofconstructinga(probabilis-
tic) linear-opticalquantum gate involvesthe same system of NMV,PL,DU,andJPDacknowledgesupportfromthethe
equations,andhencecanbesolvedusingGro¨bnerbasesand Army Research Office and the Disruptive Technologies Of-
vacuum extensions. Applied to gate construction, our solu- fice. KK and JE acknowledge support from the DFG (SPP
tion scheme can be seen as the generalization of the proce- 1116),the EU(QAP),the EPSRC, the QIP-IRC,the EURYI
dure used to construct an optimal linear-optical NS gate in Award, and Microsoft Research through the European PhD
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[25] Tostartwith,wecanmaximizet,subjectto hierarchy
x(j) x(j+1)
»x1(ts) x(ts) –≥0. » x(kkj/+−211) xk/(kj2) –≥0
2
Then, each of the equality constraints can be enforced by the fork∈{2,4,...,2s+1−j}andj =1,...,s.