Table Of Content

Is single-particle interference spooky? Pawel Blasiak ∗ Institute of Nuclear Physics Polish Academy of Sciences, PL-31342 Kraków, Poland Itissaidaboutquantuminterferencethat"Inreality,itcontainstheonlymystery". Indeed,together withnon-localityitisoftenconsideredasthecharacteristicfeatureofquantumtheorywhichcannot beexplainedinanyclassicalway. Inthisworkweareconcernedwitharestrictedsettingofasingle particle propagating in multi-path interferometric circuits, that is physical realisation of a qudit. It is shown that this framework, including collapse of the wave function, can be simulated with classicalresourceswithoutviolatingthelocalityprinciple. Wepresentalocalontologicalmodelwhose predictionsareindistinguishablefromthequantumcase. ’Non-locality’inthemodelappearsmerely asanepistemiceffectarisingonthelevelofdescriptionbyagentswhoseknowledgeisincomplete. Thisresultsuggeststhattherealquantummysteryshouldbesoughtinthemulti-particlebehaviour, 7 sincesingle-particleinterferometricphenomenaareexplicableinaclassicalmanner. 1 0 PACSnumbers:03.65.Ta,03.65.Ud 2 n In the Feynman Lectures on Physics quantum inter- at all clear to what extent these features are unique to a ference is described as "a phenomenon which is impossi- thequantumrealm. Ontheonehand,therearevarious J ble, absolutely impossible, to explain in any classical way, models indicating analogies on the grounds of classi- 0 1 and which has in it the heart of quantum mechanics" [1]. cal probabilistic theories, see e.g. [28–38]. On the other Broadlyspeaking,thephenomenonconcernsbehaviour hand, none of these results fully reconstruct quantum ] ofaparticleintheinterferometriccircuitsandtheprob- predictionsforgeneralsingle-particlescenarios. Allthis h p lem consists in reconciling wave and particle character makes the question about the distinctive quantum fea- - of the phenomenon. Another difficulty is a common- turesaninterestingproblem. Inparticular,itisnotclear t n sense explanation of the collapse of the wave function whethernon-localityinthesingle-particleframeworkis a uponmeasurement. Insomemysteriouswaybehaviour onaparwiththemulti-particlecase,i.e. doesnotadmit u of the quantum particle depends on the knowledge of explanationvialocalhiddenvariablemodels[10–13]. A q whatishappeninginthedistantpartsoftheexperimen- decisive answer would require either a rigorous no-go [ talsetup. Notably,non-localityofthecollapsemanifests proof, like the Bell’s theorem is for two particles, or 1 itself already in the single-particle scenarios, as first a counterexample encompassing all relevant aspects of v pointedoutbyA.EinsteinduringtheFifthSolvayCon- quantuminterferometricsetups. 2 5 ference[2]whometaphoricallycalledsuchaninfluence In this paper we are concerned with a single particle 5 "spookyaction-at-a-distance"[3,4]. Afullyfledgedargu- propagating in general multi-path interferometric cir- 2 ment against local realism in quantum theory is due cuits – that is physical realisation of finite dimensional 0 to profound insight of J. S. Bell [5, 6]. It requires two Hilbert space =CN (qudit) [39] – and explicitly con- . H 1 particles to show non-local correlations between mea- struct local ontological model which faithfully imitates 0 surements in distant arms of the interferometric setup. allquantummechanicalpredictions. Thissuggestscau- 7 Remarkably,allfurtherrefinementsoftheargumentex- tion against statements to the effect of non-locality of 1 ploitpropertiesofentangledstatesinmulti-particlesce- thecollapseofthewavefunctionorabsoluteimpossibil- : v narios, see e.g. [7–9]. This leaves open the question of ity of classical explanation of single-particle interfero- i X possible local explanation of quantum interferometric metric phenomena. The model shows that local expla- phenomenainthesingle-particlecase,cf.[10–13]. nation is conceivable and the real mystery lays in the r a multi-particlebehaviour[5–9]. Analysisofthemodelil- Quantum mechanics of single-particle phenomena lustratestheroleofepistemicconstraintsindescription is a rich source of paradoxes and surprising effects of the system by agents with limited resources, which whichchallengeourclassicalintuitionabouttheworld. Apart from quantum interference [14], they include leadtoallkindsofweirdquantum-likeeffects. e.g.: interaction-free measurements [15–17], quantum Zeno effect [16–19], Wheeler’s delayed-choice experiment[20,21],violationofLeggett-Garginequalities[22, RESULTS 23],pre-andpost-selectionparadoxes[24,25]andcon- textuality [26, 27]. These phenomena are often consid- We begin with a brief account of quantum interfer- ered as strictly quantum mechanical effects and some ometric circuits. It is meant to introduce the notation of them, like contextuality or Leggett-Garg inequali- and provide a basis for comparison of the model con- ties,aresometimestreatedassignaturesofthequantum structed in this paper with the standard quantum me- regime. However, as suggestive it might look it is not chanicaldescription. 2 in spite of the fact that detector D is localised only in j Quantum interferometry in a nutshell. In the follow- j-thpath. Explanationofthisbehaviourleadstotheno- ingweconsiderstandardinterferometricframeworkfor torious problem concerning ontological status of quan- asingleparticlepropagatingthroughanetworkofspa- tum states and the issue of non-locality of the collapse tially separated paths. Evolution of the system is im- ofthewavefunction. plemented by gates attached to the paths which repre- These rules provide mathematical description of a sentnon-trivialtransformations(withemptypathscor- single particle the interferometric circuit. It was shown responding to free evolution). See Fig. 1 (on the right) in Ref. [39] that any unitary and projective measure- for illustration. It is enough to consider only a few mentin =CN canbeexperimentallyrealisedinacir- H kinds of gates which form a basis for construction of cuitcomposedofNpathsasasequenceofinterferomet- complex interferometric circuits [39]. These gates in- ric gates defined above. Thus it provides a convenient clude phase shifters S and detectors D which are at- physicalframeworkforfoundationalexplorations. j j tached to individual paths and beam splitters B on Our main goal in this paper is explicit construction st which two paths cross, with j and s,t indicating the ofaclassicalanaloguewiththesamestructuralcompo- respective paths. A special role of detectors is to pro- nents (comprised of paths and gates arranged into cir- vide an outcome ’Click’/’No Click’ which attests to cuits)whichmimicsquantumbehaviourofaparticlein thepresence/absenceoftheparticleinagivenpath. the interferometric circuits described above. The crux Quantumdescriptionofasingleparticleintheinter- of the matter is to provide a model with well-defined ferometric circuit which consist of N paths associates underlyingontologywhichdoesnotviolatethelocality position of the particle with the vectors of computa- principle, and yet on the operational level its predic- tional basis 1 ,..., N , where j represents the fact tionsareindistinguishableformthequantumcase. | (cid:105) | (cid:105) | (cid:105) of particle being in j-th path. In general, state of the system is a superposition with complex coefficients ψ Ontology of the model. Let us consider circuits com- j definingavector(ray)in =CN,i.e. posed of N paths labelled with index j = 1,...,N. H Definingthemodelweassumethatinthecircuitprop- |ψ(cid:105) = ∑N ψj|j(cid:105) = (cid:32)ψ...1 (cid:33) = ψ(cid:126), (1) aqga=tes1,a..s.i,nNgl.e pAardtidciletiownhailclyh, hwaes pwoesltludlaetfienethdatpoasliotinogn j=1 ψN each path propagates a local field characterised by two wdiiftfherninogrmbealaisnatoiovner(cid:107)al|lψp(cid:105)(cid:107)h2as=e b∑ejin|ψgj|e2qu=iv1alaenndt. vEevcotolurs- d|uejg|r(cid:54)ee1saonfdfr(ereedalo)mst:re(ncgotmhpτjlesxu)chamthplaittu0de(cid:54)uτjj s(cid:54)uc1h. Tthhaist tionimplementedbygatescorrespondstoasequenceof means that at each time the system of N paths is fully unitaryandprojectivetransformationsdescribedasfol- specifiedbyapoint (q,(cid:126)u,(cid:126)τ) intheonticstatespace lows. Free evolution in j-th path acts trivially and phase Λ = q : q =1,...,N shifter Sj introducesphase eiω intherelevantpath,i.e. { }× (cid:126)u CN : u (cid:54)1 (5) { ∈ | j| }× ψj free (cid:47)(cid:47) ψj and ψj Sj (cid:47)(cid:47) eiωψj. (2) {(cid:126)τ ∈RN :0(cid:54)τj (cid:54)1}, Beamsplitter B located at the crossing of paths s and t whereu andτ describethefieldinj-thpath. SeeFig.1. st j j implements a unitary in the subspace spanned by kets In the following we will be interested in stochastic s and t givenby evolution which requires probabilistic description and | (cid:105) | (cid:105) hence consider the set of all possible probability distri- (cid:18)ψs(cid:19) Bst (cid:47)(cid:47) (cid:18)ψs(cid:48)(cid:19) = (cid:18)i√R √T(cid:19)(cid:18)ψs(cid:19) , (3) butionsovertheonticstates ψt ψt(cid:48) √T i√R ψt (cid:110) (cid:90) (cid:111) (Λ) = p : Λ [0,1] : p(λ)dλ =1 , (6) where R,T are reflectivity and transitivity coefficients. P −→ Λ Finally, according to the measurement postulate (von which will be called epistemic state space. A general Neumann–Lüders rule) detector D is described by the PVM P , 1 P whereP j jj ,i.e. dependingon stochastic transformation (or gate) is defined as a map- { j − j} j ≡ | (cid:105)(cid:104) | pingT : Λ (Λ),whereT(λ)specifiesdistribution theoutcomeiteffectstheprojection −→ P of final states given the system was in state λ Λ. In  ∈ |ψ(cid:105) Dj (cid:47)(cid:47)  (cid:107)((11−−|PPj(cid:105)jj))||ψψ(cid:105)(cid:105)(cid:107) ’’CNloicCkl’i,ck’, (4) tohfetrmanosdfoelrmwaetiwonilsl(bgeatceosn)cwerhnicehdawreithdeasclriimbeitdedbeclhoowi.ce Localinterferometricgates. Forsuchdefinedontology with probability that detector D ’Clicks’ given by the j we need to define stochastic counterparts of the inter- Born rule Pr(Dj|ψ) = |(cid:104)j|ψ(cid:105)|2 = |ψj|2. Note that pro- ferometricgates. Notethatinordertoobeythelocality jectionpostulateEq.(4)affectsthewholespace =CN principle action of the gates should be restricted to the H 3 ! ! ! Ontology Interferometric circuit FIG.1. Ontologyofthemodelandinterferometriccircuits. Onetheleft,ontologyofthemodelconsistsofasingleparticleand localfieldspropagatingineachpathofthecircuit. Ateachtimetheparticlehaswell-definedpositionq=1,...,N andthefields arecharacterisedbyamplitudeu andstrengthτ withj=1,...,Nlabellingthepaths. Ontheright,circuitsdescribepropagation j j of a particle through a network of (spatially separated) paths and gates which represent a sequence of transformations. Basic interferometric toolkit consists of freeevolution (empty path), phaseshifters Sj, beamsplitters Bst (on which two paths meet) and detectorsD whichinform(’Click’/’NoClick’)aboutthepresence/absenceofaparticleinagivenpath. Thisselectionofgates j isgeneralenoughtoprovidephysicalrealisationofanyunitaryandprojectivemeasurementdescribedbyquantumformalism in = CN [39]. In this paper we show that the outlined ontology (on the left) completed with appropriately defined local H stochasticgatesfullyreconstructsquantummechanicalpredictionsforasingleparticleintheinterferometriccircuits. paths they are attached to, i.e. modify degrees of free- Beam splitter B is a gate which brings paths s and t st domonlyintherespectivepathsandtheeffectedtrans- togetherandimplementsthefollowingtransformation. formationbeingnotdependentonthesituation(config- Amplitude and strength of the fields are modified ac- urationofgates,outcomesorfields)intheotherpaths. cordingtotherecipe: whWicehsctoarrrteswpiotnhddteoscfrreiepteivoonlutoifonp.aItthws iwllibtheoaustsugmateeds (cid:18)us(cid:19) Bst (cid:18)u(cid:48)s(cid:19) = (cid:18)i√R √T(cid:19)(cid:18)δτsτ(st) 0 (cid:19)(cid:18)uj(cid:19), (10) thatthefieldinsuchapathissubjectto’naturalageing’, ut −→ u(cid:48)t √T i√R 0 δτtτ(st) uk namely at each step its strength decreases and ampli- and tude remains unchanged. We make the following defi- τs,τt Bst τ(st)/2, where τ(st) =max τs,τt . (11) nitionoffreeevolutionin j-thpath: −→ { } In plain words, the role of δ’s in the diagonal matrix in uj −fr→ee uj & τj −fr→ee τj/2. (7) Esoq.th(1a0t)itisdtooessunpoptrceosnsetrsibthuetefiteoldthweittrhanwsfeoarkmeredstraemnpgltih- Phaseshifter Sj isadeterministicgatewhichactsin j- tudesattheoutput. Notethatstrengthsoftheoutgoing thpathbyrotatingphaseofthefieldbyeiω andstrength fieldsaresubsequentlylevelleduptoτ(st)/2;seeEq.(11). ’ageingnaturally’,i.e. wehave: Additionally, if the particle happens to be in one of the crossing paths, i.e. q = s or q = t, then it may change uj −S→j eiωuj & τj −S→j τj/2, (8) itspositionfollowingtheprobabilisticrule: paDthet(eic.eto.rdDetjecchtoerck’Cslfiocrkps’reosnelnyceifoqf=thej).pFaurtritchleerimnoj-rteh, q Bst (cid:40) q(cid:48) = s withprobability |u(cid:48)s||2u+(cid:48)s||2u(cid:48)t|2 , (12) w(’Celpiocskt’u/l’aNteotChalticthke’)dmetoedcitfiioens dameppelnitduidnegaonndthsetrerensgutlht −→ q(cid:48) = t withprobability |u(cid:48)s||2u+(cid:48)t||2u(cid:48)t|2 . andotherwise,for q = s and q = t,itremainsoutside. ofthefieldin j-thpathinthefollowingway: (cid:54) (cid:54) Allgatesdefinedabovearelocal(withtheinteraction uj −D→j (cid:26) 1u & τj −→ (cid:26) 10 iiff qq == jj . (9) biseptwlaeceendtahtethpeatchrsososnintghepobienatm). Wspeliattlesroanlolotwetehdatsitnrcaensi-t j (cid:54) formations effected by free evolution, phase shifters S j In the above definitions it is implicitly assumed that anddetectors D aredeterministic,whilethebeamsplit- j the particle can not jump between the paths. In other ters B arenon-trivialstochasticgates. st words, if the particle happens to be in path q = j, then Observe that the structure of circuits constructed in itstaysthereq q,andotherwiseforq = jitremains themodelisanalogoustothoseinthequantuminterfer- −→ (cid:54) outside q q = j. The particle may change its loca- ometricframework. The differencelaysintheunderly- −→ (cid:54) tiononlyatthecrossingpoints,i.e. wherethethebeam ingontologywhichinthepresentedmodelisgivenex- splittersareplaced. plicitly with locality being built in from the outset. We 4 willshowthatstatisticalpredictionsforanyexperimen- In the following, we show how to construct such an talcircuitinthemodelarethesameasforitsquantum- operational account of the model which makes no ref- mechanicalcounterpart. erencetotheunderlyingontology. Operational desideratum. Imagine agent without any Reconstruction of quantum predictions. Closer anal- prior knowledge of the model making an effort to un- ysis of the model reveals significance of special classes derstand how it works only by analysing results of ex- of distributions [(cid:126)z] (Λ) which can be labeled with perimentsthatshecanperform. Clearly,herconception complexvectors(ray⊂s)P(cid:126)z Cn,thatis ∈ oscfrtibheedmaobdoevle,msainycdeihveerrgcehofriocemotfhgea’ttersuein’ ocnotnosltorguyctidneg- (cid:126)z = ∑N zjej = (cid:32)z...1 (cid:33), (13) experimental circuits is constrained. In the following j=1 zN we are interested in forming minimal account of the modelasseenbytheagentavoidinganyunfoundedin- with normalisation (cid:126)z = ∑ z 2 = 1 and equivalence (cid:107) (cid:107) j| j| terpretational commitments. It is thus appropriate to up to the overall phase. These classes can be shown to adopt operational approach and restrict attention solely formdisjointfamilyofsubsetsin (Λ),i.e. wehave P todescriptionofexperimentalpredictionsinthecircuits builtaccordingtotherulesofthemodel. [(cid:126)z]∩[(cid:126)z(cid:48)] (cid:54)=∅ ⇔ (cid:126)z =(cid:126)z(cid:48) (uptophase). (14) For this purpose we need to identify what informa- See Section Methods for explicit definitions and Fig. 2 tionisactuallyavailabletotheagentsubjecttothiskind forillustration. of constraints. The following questions provide guid- Interest in these very special classes of distributions anceinthisprocess: [(cid:126)z] (Λ) is due to their behaviour under action of ⊂ P thegatesdefinedinthemodel. Letussummarisemain (i) Which distributions in (Λ) can be prepared by the P resultsrelevantforthediscussionoftheoperationalac- agentwithlimitedresourcesathand? count of the model (see Section Methods). Firstly, it In general, it may be the case that the agent explores can be shown that free evolution, phase shifters, beam only a restricted range of distributions in (Λ), mean- splitters and detectors (with post-selection) act congru- P ingthatsomedistributionsarebeyondherreach. Then ently on such defined family of classes, i.e. all distribu- itisnaturaltoask: tions in a given class are mapped into distributions in some other class [(cid:126)z] p p(cid:48) [(cid:126)z(cid:48)]; see Theorem 1. (ii) How do these distributions transform under action of (cid:51) −→ ∈ Secondly, one observes that available preparation pro- thegatesinthemodel? ceduresmaketheagentstartoffwithdistributionscon- Whatremainsistoabstractawayredundantontological tained in one of the initial classes [e1],...,[eN], where structure. Hereisthekeytotheoperationalaccount: ej = (0,...,1,...,0)T hassingle1inj-thpositionwhich indicates position of the particle (’Click’) ascertained Operational indifference principle: Distributions bytheinitialpreparation;seeEq.(25). that are not distinguishable by means available to the Combining these facts together provides answer to agent, that is give the same probabilistic predictions questions(i)and(ii)fromtheoperationaldesideratum foranyconceivableexperiment(circuit),areequivalent discussed above. Since the family of classes is closed fromtheoperationalpointofview. under available transformation, we infer that the agent withalimitedchoiceofgatesatcommandremainscon- It allows to discard ontological details which are irrel- finedinherexplorationstoarestrictedsubsetofdistri- evant (or inaccessible) to the agent by treating all in- butionsin (Λ) givenbytheunionofallclasses,i.e. distinguishable distributions as a single entity. At this P point one should be able to identify the underlying (cid:91)(cid:110)[(cid:126)z] : (cid:126)z CN, (cid:126)z =1(cid:111)(cid:40) (Λ). (15) mathematicalframeworkandanswerthequestion: ∈ (cid:107) (cid:107) P Notethatthissethasnaturalcoarse-graining(partition- (iii) What is the minimal operational account which cor- ing)intoclasses[(cid:126)z]whichhavethepropertythataction rectlydescribespredictionsofthemodel? ofthegatesinthemodelisconciselydescribedastrans- In short, we seek for the bare-bone description with- formation of the labelling vectors(cid:126)z (cid:126)z(cid:48). A crucial −→ outpreferencetoanyparticularinterpretation,withthe observationisthatonthelevelofclassesthesetransfor- only purpose to provide a tool for prediction of exper- mation rules are exactly the same as for the quantum imental results. Such an account should specify the set interferometric gates; cf. Eqs. (2)–(4) and Eqs. (27)–(30) of possible operational states which correspond to in- inTheorem1. equivalent preparation procedures and provide trans- Such a coarse-grained description is just enough for formation rules describing evolution in conceivable ex- ourpurposes. Thisisbecausedistributionsinthesame perimentalcircuits(includingmeasurementoutcomes). class [(cid:126)z] give identical measurement predictions, i.e. 5 probability of a ’Click’ in detector D is equal to z 2. seeemergentgeometryoftheprojectivespace =CN Moreover, since classes transform asja whole the|rej|is andquantummechanicalaccountofaqudit[40H]. no way to differentiate by the agent between two dis- Thisresultisanexplicitcounterexampleshowingim- tributions in the same class by arranging any compli- possibility of proving non-locality for a single particle catedcircuitfromthegatesavailableinthemodel. This intheinterferometricsetups. Forthesakeofclarity,we allows to make use of the operational indifference prin- addressthequestionof(non-)localityinadifferentcon- ciple and observe that all information relevant for pre- text than the Bell-type scenario; the latter is concerned dicting behaviour of the system is held by the class it- with correlations between measurements on a pair of self, that is knowledge of a particular distribution in quantumparticles,whereashereweareconcernedwith [(cid:126)z] is redundant. It means that label(cid:126)z plays the role a single quantum particle interacting with classical ap- of operational state which encodes complete informa- paratus(phaseshifters,beam-splittersanddetectors)as tion available to the agent, thereby answering question describedbyquantumtheoryin =CN [39]. H (iii)fromtheoperationaldesideratumdiscussedabove. At first sight our conclusion seems to contradict Notice that we get full analogy with the quantum de- proofsclaimingnon-localityofasingleparticle[10–13]. scription of interferometric circuits, i.e. we have the We note that these arguments exploit additional quan- same structure (geometry) of states which are complex tumresource,namelycoherentstateswhoseproperties vectors (rays) in CN with identical transformation and rely on superposition of multi-particle states. This re- measurement rules given in Eqs. (2)–(4) and Eqs. (27)– quirespresenceofotherparticlesinthesystemmaking (30) respectively. All things considered, both descrip- the claim of single-particle character of the considered tionsareequivalentandthuswecanidentify phenomena open to question [12]. A similar objection applies to recent demonstration of the collapse of the ψ e(cid:33)quiv. [(cid:126)z] (or ψ(cid:126) e(cid:33)quiv. (cid:126)z). (16) wave function using homodyne detection [41]. In view | (cid:105) Inconclusion,operationalaccountofthemodelboils of the presented model, these proofs seem to illustrate downtospecificationofastategivenbyacomplexvec- non-trivialaspectof’almost’classicalresourceprovided tor (ray) (cid:126)z CN with the transformation rules and by local oscillators (coherent states), as compared with statistics of o∈utcomes (’Clicks’) being the same as for ’clean’single-particlescenariosconsideredinthispaper. the quantum gates. This means that from the perspec- We note that the single-particle framework is a rich tive of an agent unaware or indifferent to the underly- source of paradoxes and weird phenomena which are ingontologythebehaviourofquantum-interferometric often considered as typically quantum effects with- circuits and their counterparts in the presented model out classical explanation [14–27]. The latter assertion areforallpracticalpurposesindistinguishable. shouldbetreatedwithcaution,sinceanyargumentfor non-classicality of an effect always depends on additional assumptions whose plausibility should be prop- DISCUSSION erly assessed. For example, interaction-free measurements assume null effect of negative measurement re- In summary, we have constructed local ontological sults [15–17], Leggett-Garg inequalities require non- model which faithfully imitates quantum predictions invasive measurements, pre- and post-selection para- forasingleparticleintheinterferometriccircuits. Cru- doxes rest upon contextual effects [42–44], etc. Our cialfortheanalysisofthemodelisdistinctionbetween model illustrates non-trivial aspect of these assump- two levels of description. On the one hand, we have tions and shows that mere local state disturbance by ontologicaldescription by an omniscient observer having detectors ushers in a possibility for classical-like expla- accesstoalldetailsofthemodel,i.e. seeingstructureof nation of single-particle phenomena. A strong point of the ontic state space and familiar with construction of themodelisthatthepresentedontologyismadeready the gates. On the other hand, we have epistemicdescrip- for any kind of circuit with arbitrary number of paths. tion concerned only with the information which is ac- Assuch,itprovidesexhaustivereconstructionofsingle- tually available. The latter adopts operationalperspective particlephenomenainaunifiedframeworkasopposed ofanagentunawareoftheunderlyingontologyandin- toseparatemodelsdevisedforsimulationofparticular vestigating the system only with the tools at hand, i.e. effects,cf.[28–38]. building interferometric circuits and analysing experi- Let us remark that in the classification of Harrighan- mental results (statistics of ’Clicks’). We have shown Spekkens [45] our construction is ψ-ontic, that is dis- that operational predictions of the constructed model tributionscorrespondingdifferentquantumstateshave areindistinguishablefromthequantummechanicalbe- non-overlapping supports. We should also point out haviour. This illustrates that properly chosen con- that the model allows for different representations of straints on gaining knowledge can modify the picture the the same quantum state, that is any distribution ontheepistemiclevel. Inourcase,fromthelocalontol- p [(cid:126)z] is a valid representation of the same state ψ ogy with classical probabilistic description in (Λ) we (w∈ith the identification ψ(cid:126) (cid:33)(cid:126)z). This variety is nec|es(cid:105)- P 6 sary to accommodate contextual effects which abound parison with the single-particle phenomena which are inthequantumregime[26,27,46,47]. less problematic in this respect. In particular, we have To give a broader perspective we hasten to note that shown that single-particle framework is not enough to thereisonlyahandfulofontologicalmodelswhichre- establish non-locality, since in this case ’spooky action- construct a qudit. One of them is the ψ-ontic model at-a-disstance’ can be understood as merely an effect of by Beltrametti-Bugajski [48] which is essentially re- descriptionontheepistemiclevel. statement of the standard Copenhagen interpretation (wherenon-localityofthecollapseofthewavefunction METHODS is built in from the outset). There is also an interesting proposal by Lewis-Jennings-Barrett-Rudolph [49] built within the framework of ψ-epistemic models. It is ex- Here we give all necessary definitions and state our plicitlynon-localand,inaddition,violatesthesocalled main result which describes structure of distributions preparation independence principle – the latter seems within the reach of an agent exploring the model (see to be a generic feature of any successful ψ-epistemic SupplementaryInformationfortheproof). approach, see [50, 51]. One should also mention the de Broglie-Bohm interpretation of quantum mechan- ClassesofdistributionsinP(Λ). Crucialfortheanal- ics [52, 53] which postulates local guidance of particles ysisofthemodelisthefollowingconstructionofdistin- by a quantum potential. For a single particle quantum guished classes of distributions in (Λ); see Fig. 2 for P potential (directlyrelated to thewave function)lives in illustration. a 3D space and its dependence on the configuration of Step1: Letusconsiderspecialsubsetsoftheonticstate theapparatusisasourceofnon-localeffects. Addition- space Λi(cid:126)z ⊂ Λ labeled by integers i ∈ {1,...,N} and ally, the de Broglie-Bohm model has many weird fea- complexvectors(cid:126)z CN definedasfollows: ∈ tures, such as complicated spatial description, ’surreal-  istic’trajectories[54]orexcessivecontextualeffects[55],  a) q =i wwhhiocshepreerlesvisatnetvdenegirneethseosfimfrepeldeoinmterrefedruocmeettoricasqeutudpits. (q,(cid:126)u,(cid:126)τ) ∈ Λi(cid:126)z ⇐d⇒f  bc)) τ∆iτ=(cid:126)uτ >(cid:126)z0 (17) ∼ In summary, all these models have built in non-local effects in the description and therefore do not make a where∆τ(cid:126)uisavectorobtainedfrom(cid:126)ubyretainingfield amplitudes corresponding to the highest field strength case against non-locality of single-particle interferome- τ:=max τ ,...,τ andtheremainingonesputequal trydiscussedinthispaper. { 1 N} tozero. Hencetheroleofthediagonalmatrix Toconclude,letusquoteE.T.Jaynes[56]onthecur- rent understanding of quantum mechanical formalism: ∆τ:=diag(δτ1τ,..., δτNτ), (18) "But our present QM formalism is not purely epistemolog- which picks out those entries of (cid:126)u which correspond ical; it is a peculiar mixture describing in part realities of to the highest strength τ. In our notation symbol ’ ’ Nature,inpartincompletehumaninformationaboutNature ∼ stands for proportionality, i.e. (cid:126)z (cid:126)z(cid:48) iff(cid:126)z = α(cid:126)z(cid:48) for – all scrambled up by Heisenberg and Bohr into an omelette some α C, α = 0. In plain wo∼rds, these conditions that nobody has seen how to unscramble. Yet we think that ∈ (cid:54) expressthefollowingrequirements: theunscramblingisaprerequisiteforanyfurtheradvancein a) particleispresentinpathi, basicphysicaltheory. For,ifwecannotseparatethesubjective andobjectiveaspectsoftheformalism,wecannotknowwhat b) fieldinpathihashigheststrength(non-vanishing; we are talking about; it is just that simple." In this spirit, withpossibilityofequalstrengthsinotherpaths), ourmodelisanillustrationoftheideathatcarefuldis- c) vector of field amplitudes with highest strengths tinctionbetweentheepistemicaspectofthedescription ∆τ(cid:126)u isproportionalto(cid:126)z. andtheunderlyingontologicalaccountprovidesaway Clearly, for different labels i and(cid:126)z (up to proportional- of understanding weird quantum phenomena as an ef- ity)thesesubsetsaredisjoint,i.e. wehave fect of incomplete knowledge – it is tenable at least for sTihnigsleg-ipvaerstisculeppfroarmt teowtohrekbaesliethfethmatoudneslcrdaemmbolinnsgtroafttehse. Λi(cid:126)z ∩Λ(cid:126)jz(cid:48) (cid:54)=∅ ⇔ i = j & (cid:126)z ∼(cid:126)z(cid:48). (19) quantum omelette should be in principle possible, albeit Step 2: Then, we introduce auxiliary classes of proba- it is not evident at the moment how to construct such bilitydistributionswithsupportin Λi(cid:126)z anddenote a theory. It seems that non-local effects should play (cid:110) (cid:111) a role in the full reconstruction – as the Bell’s theo- [(cid:126)z]i := p ∈ P(Λ) : suppp ⊂ Λi(cid:126)z ⊂ P(Λ).(20) rem suggests – however, it is not clear to what extent ByvirtueofEq.(19)theseclassesformadisjointfamily and in what form (see [57] for some hints). The pre- ofsubsetsin (Λ),i.e. wehave sentedmodelpointsouttothemulti-particlebehaviour P as the real source of the quantum mystery in com- [(cid:126)z]i∩[(cid:126)z(cid:48)]j (cid:54)=∅ ⇔ i = j & (cid:126)z ∼(cid:126)z(cid:48). (21) 7 |z1|2 t |z2|2 t |z3|2 t Lpejj00t2[ej] t |z10||2z20t|2 t |z30|2 t [ej]t Ontic state space Epistemic state space FIG.2. Constructionofclassesofinterestin (Λ). Ontheleft,distributions p [(cid:126)z]aredefinedtohavesupportin (cid:83)N Λi P ∈ i=1 (cid:126)z with cumulative probability over the respective subsets Λi equal to z 2. Since all Λi are disjoint, distributions in different (cid:126)z | i| (cid:126)z colfadssiestsrihbauvteionnosns-uopveprolartpepdiningasuspinpgolretssu(sbeseetpΛ∈jej[(cid:126)z(s]eaenpd(cid:48)(cid:48)p∈(cid:48) ∈[e[j(cid:126)z])(cid:48)]a)n.dCldaessscersib[eesj]infoitriajl=pr1e,p.a.r.a,tNionasrewsipthectihale:peaacrthicilsecionmapgriivseedn path. Ontheright, illustrationofthewholespaceofprobabilitydistributionsover Λ, denotedby (Λ), withdisjointsubsets P representingclassesofinterest[(cid:126)z]. Theseclassestransformcongruently(asawhole)underactionofthegatesinthemodelas explainedinTheorem1(inthepictureinitialclassundergoessequenceoftransformations[ej]→[(cid:126)z]→[(cid:126)z(cid:48)]→[(cid:126)z(cid:48)(cid:48)]→...). Step3: Now,wearereadytodefineclassesofdistribu- array of detectors D ,D ,...,D placed in each path 1 2 N tionswhichplayacentralroleinanalysisofthemodel. and retaining only those cases when a single detection occurred. In this way the agent carries out an effec- Definition 1. With each normalised vector(cid:126)z CN, such ∈ tive initial preparation which attests to the presence of that (cid:107)(cid:126)z(cid:107) := ∑iN=1|zi|2 = 1, we associate the following class a single particle (’Click’) in a given path. Note that ofprobabilitydistributions: ontheontologicallevelselectionofeventswithasingle (cid:110) N (cid:111) ’Click’indetectorDjresultsinanensembledistributed [(cid:126)z] := ∑|zi|2pi : pi ∈ [(cid:126)z]i ⊂ P(Λ). (22) over the ontic states (q,(cid:126)u,(cid:126)τ) ∈ Λ subject to the follow- i=1 ingconditions: See Fig. 2 for illustration. This means that distribu- q = j , u =1 , τ =1 , tions in [(cid:126)z] have support in (cid:83)N Λi with cumulative j j (24) i=1 (cid:126)z u =? , τ =0 , for k = j , probabilityovertherespectivesubsetsΛi equalto z 2 k k (cid:54) (cid:126)z | i| (and otherwise the shape of distributions being arbi- where u ’s depend on the unknown source; see Eq.(9). k trary). Anotherwaytocharacteriseclassesofinterestis A quick look at definitions in Eqs. (17), (20) and (22) to write [(cid:126)z] = ∑iN=1|zi|2[(cid:126)z]i, which means that its ele- revealsthatsuchdistributionshavesupportinΛjej,and mentsareconvexcombinationsofdistributionsin[(cid:126)z]’s henceareincludedinclass i with weights z 2. As a consequence of Eq.(21) we ob- | i| [e ] (Λ) (if D ’Clicks’), (25) serve that such defined classes are disjoint subsets in j ⊂ P j (Λ),i.e. wehave where e = (0,...,1,...,0) hassingle1in j-thposition. P j [(cid:126)z] [(cid:126)z(cid:48)] =∅ (cid:126)z (cid:126)z(cid:48). (23) In conclusion, the agent starts off in one of the classes ∩ (cid:54) ⇔ ∼ [e ],...,[e ] which correspond to initial preparation 1 N of the system with a single particle (’Click’) in a given Initialpreparation. Anypredictionofexperimentalbe- path. haviour rests upon knowledge of initial preparation of Intheabovewehaveassumednopriorknowledgeof the system. In general, it is an intrinsic characteristic the source and hence the need of initial filtering of the of the source which provides an ensemble of systems unknown ensemble. We note that it could have been with a given distribution of the ontic states. However, bypassed if the agent was granted access to a single ifnosuchinformationisavailable,thentheagentgiven particlesourcewithallpathsblockedexceptone(likeit some unknown (possibly random) source has to pre- isusuallyassumedinthequantumscenarios). Thiscan pare initial ensembles of systems by herself. Here is a be easily realised within the model by postulating that genericschemehowtoproceedinsuchacase. the source injects particles (with non-vanishing ampli- Sinceweareinterestedinsingle-particlescenarios,in tudes and strengths) into a given path, and the blocks the first place presence of a single particle (’Click’) in remove particles resetting strength of the field to zero. thesystemshouldbeverified. Thispropertycanbecon- We observe that it boils down to preparation of distri- firmed by sieving an unknown ensemble through the butionsinoneoftheclassesinEq.(25)again. 8 [4] A.Einstein,“LettertoMaxBorn,3rdMarch1947,”inThe Geometry of classes. It appears that the structure of Born-EinsteinLetters(Macmillan,1971). classes [(cid:126)z] (Λ) defined in Eq.(22) is closed un- [5] J.S.Bell,“OntheProblemofHiddenVariablesinQuan- ⊂ P tumMechanics,”Rev.Mod.Phys.38,447–452(1966). der transformations (circuits) considered in the model. [6] J. S. Bell, Speakable and unspeakable in quantum mechanics Here is the key result describing behaviour of classes (CambridgeUniversityPress,1993). underactionofthegatesinthemodel,cf. Fig.2(onthe [7] D. M. Greenberger, M. A. Horne, A. Shimony, and right). (FortheproofseeSupplementaryInformation). A. Zeilinger, “Bell’s theorem without inequalities,” Am. J.Phys.58,1131–1143(1990). Theorem1. Transformationsimplementedbyphaseshifters [8] L. Hardy, “Quantum mechanics, local realistic theories, Sj,detectorsDjandbeamsplittersBstactcongruentlyonthe and lorentz-invariant realistic theories,” Phys. Rev. Lett. family of classes (cid:8)[(cid:126)z] (Λ) :(cid:126)z Cn, (cid:126)z = 1(cid:9) defined 68,2981–2984(1992). in Eq.(22). This mean⊂s Pthat classe∈s trans(cid:107)fo(cid:107)rm as a whole, [9] N.D.Mermin,“Hiddenvariablesandthetwotheorems ofJohnBell,”Rev.Mod.Phys.65,803–815(1993). i.e. all distributions in a given class map into distributions [10] S.M.Tan,D.F.Walls, andM.J.Collett,“Nonlocalityof insomeotherclass asinglephoton,”Phys.Rev.Lett.66,252–255(1991). [(cid:126)z] (cid:51) p (cid:47)(cid:47) p(cid:48) ∈ [(cid:126)z(cid:48)], (26) [11] LPh.yHs.arRdeyv,.L“Netot.n7l3o,ca2l2i7ty9–o2f28a3S(1in99g4le).Photon Revisited,” where mapping(cid:126)z (cid:126)z(cid:48) is determined by the gates imple- [12] D.M.Greenberger,M.A.Horne, andA.Zeilinger,“Non- −→ locality of a Single Photon?” Phys. Rev. Lett. 75, 2064– mentedinthecircuitaccordingtothefollowingrules. 2064(1995). •FreeevolutionactstriviallyandphaseshifterSj introduces [13] J. Dunningham and V. Vedral, “Nonlocality of a Single phaseintherelevantcomponentofvector(cid:126)z Particle,”Phys.Rev.Lett.99,180404(2007). [14] V. Scarani and A. Suarez, “Introducing quantum me- zj free (cid:47)(cid:47) zj and zj Sj (cid:47)(cid:47) eiωzj. (27) 7ch18a–n7ic2s1:(1O99n8e)-.particle interferences,” Am. J. Phys. 66, •Detector Dj placedin j-thpath’Clicks’withprobability [15] AIn.terEalcittziounr-FarneedMLe.asVuariedmmeannts,,”“QFouuanndtu.mPhyMs.e2ch3,an9i8c7a–l 997(1993). Pr(Dj|(cid:126)z) = |zj|2, (28) [16] P. Kwiat, H. Weinfurter, T. Herzog, A. Zeilinger, and M. A. Kasevich, “Interaction-Free Measurement,” Phys. anddependingontheoutcomeeffectsprojectionofvector(cid:126)z Rev.Lett.74,4763–4766(1995). (cid:126)z Dj (cid:47)(cid:47)  (cid:107)((11−−ePPjjj))(cid:126)(cid:126)zz(cid:107) ’’NCloicCkl’i,ck’. (29) [[[111789]]] ABAPquh...MayJPns.ie.tsDur7rema0es,W,a2tneh“7deZ2ero–eEd2rn.,y7oC,5“”.Ip(nJG2.at0.eMr0aSr2adau)ctod.htxia.orPnsinhh-fyarqnes.ue,1a“m8nT,theu7ae5msZ6u–ert7neh6moe3’oes(rn1pyt9,a,”7”r7aAA)d.momx..iJJn.. Beam splitter Bst at the crossing of two paths s and t im- Phys.48,931–932(1980). • plementsthefollowingunitaryonthecorrespondingcompo- [20] J. A. Wheeler, “The ’Past’ and the ’Delayed-Choice nentsofvector(cid:126)z Double-Slit Experiment’,” in Mathematical Foundations of Quantum Theory, edited by A. R. Marlow (Academic (cid:18)zzst(cid:19) Bst (cid:47)(cid:47) (cid:18)i√√TR i√√TR(cid:19)(cid:18)zzst(cid:19). (30) [21] VP.reJassc,q1u9e7s8,)E..Wu,F.Grosshans,F.Treussart,P.Grangier, A. Aspect, and J.-F. Roch, “Experimental Realization of We note that Theorem 1 is also valid for parallel trans- Wheeler’s Delayed-Choice Gedanken Experiment,” Sci- formations(gates)implementedindifferentpathsatthe ence315,966–968(2007). same time. In such a case, it is implied that evolution [22] A. J. Leggett and A. Garg, “Quantum mechanics ver- susmacroscopicrealism: Isthefluxtherewhennobody is given by joint transformation of the respective com- looks?”Phys.Rev.Lett.54,857–860(1985). ponents of vector(cid:126)z (see Theorem 2 in Supplementary [23] C. Emary, N. Lambert, and F. Nori, “Leggett–Garg in- Informationfordetailedformulation). equalities,”Rep.Prog.Phys.77,016001(2014). [24] Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the resultofameasurementofacomponentofthespinofa spin-1/2particlecanturnouttobe100,”Phys.Rev.Lett. 60,1351–1354(1988). ∗ [email protected] [25] R. E. George, L. M. Robledo, O. J. E. Maroney, M. S. [1] R.P.Feynman,R.B.Leighton, andM.Sands,TheFeyn- Blok,H.Bernien,M.L.Markham,D.J.Twitchen,J.J.L. manLecturesonPhysics,Vol.III(Addison-Wesley,1965). Morton, G. A. D. D. Briggs, and R. Hanson, “Opening [2] G.BacciagaluppiandA.Valentini,QuantumTheoryatthe up three quantum boxes causes classically undetectable Crossroads: Reconsideringthe1927SolvayConference(Cam- wavefunction collapse,” Proc. Natl. Acad. Sci. USA 110, bridgeUniversityPress,2009). 3777–3781(2013). [3] A.Einstein,B.Podolsky, andN.Rosen,“CanQuantum- [26] S.KochenandE.Specker,“TheProblemofHiddenVari- Mechanical Description of Physical Reality Be Consid- ables in Quantum Mechanics,” J. Math. Mech. 17, 59–87 eredComplete?”Phys.Rev.47,777–780(1935). (1967). 9 [27] A.A.Klyachko,M.A.Can,S.Biniciog˘lu, andA.S.Shu- [49] P.G.Lewis,D.Jennings,J.Barrett, andT.Rudolph,“Dis- movsky,“SimpleTestforHiddenVariablesinSpin-1sys- tinct Quantum States Can Be Compatible with a Single tems,”Phys.Rev.Lett.101,020403(2008). StateofReality,”Phys.Rev.Lett.109,150404(2012). [28] L. Hardy, “Disentangling Nonlocality and Teleporta- [50] M. F. Pusey, J. Barrett, and T. Rudolph, “On the reality tion,”arXiv:quant-ph/9906123 (1999). ofthequantumstate,”NaturePhys.8,476–479(2012). [29] A. Daffertshofer, A. R. Plastino, and A. Plastino, “Clas- [51] M. S. Leifer, “Is the Quantum State Real? An Ex- sical No-Cloning Theorem,” Phys. Rev. Lett. 88, 210601 tendedReviewof ψ-ontologyTheorems,”Quanta3,67– (2002). 155(2014). [30] R. W. Spekkens, “Evidence for the epistemic view of [52] D. Bohm, “A Suggested Interpretation of the Quantum quantum states: A toy theory,” Phys. Rev. A 75, 032110 Theory in Terms of "Hidden" Variables. Part I and II,” (2007). Phys.Rev.85,166–193(1952). [31] M. Kleinmann, O. Gühne, J. R. Portillo, J.-A. Larsson, [53] J. T. Cushing, A. Fine, and S. Goldstein, eds., Bohmian and A. Cabello, “Memory cost of quantum contextual- Mechanics and Quantum Theory: An Appraisal, Boston ity,”NewJ.Phys.13,113011(2011). Studies in the Philosophy of Science, Vol. Volume 184 [32] S.D.Bartlett,T.Rudolph, andR.W.Spekkens,“Recon- (KluwerAcademicPublishers,1996). structionofGaussianquantummechanicsfromLiouville [54] B.-G. Englert, M. O. Scully, G. Sussmann, and mechanics with an epistemic restriction,” Phys. Rev. A H. Walther, “Surrealistic Bohm Trajectories,” Z. Natur- 86,012103(2012). forsch.47a,1175–1186(1992). [33] J. J. Wallman and S. D. Bartlett, “Non-negative subthe- [55] L. Hardy, “Contextuality in bohmian mechanics,” in ories and quasiprobability representations of qubits,” Bohmian Mechanics and Quantum Theory: An Appraisal, Phys.Rev.A85,062121(2012). Boston Studies in the Philosophy of Science, Vol. 184, [34] P. Blasiak, “Quantum cube: A toy model of a qubit,” edited by J. T. Cushing, A. Fine, and S. Goldstein Phys.Lett.A377,847–850(2013). (Springer,1996)pp.67–76. [35] C. Ferrie and J. Combes, “How the Result of a Single [56] E. T. Jaynes, “Probability in Quantum Theory,” in Com- Coin Toss Can Turn Out to be 100 Heads,” Phys. Rev. plexity, Entropy, and the Physics of Information, edited by Lett.113,120404(2014). W.H.Zurek(Addison-Wesley,1990)p.381. [36] P.Blasiak,“Classicalsystemscanbecontextualtoo: Ana- [57] B.F.TonerandD.Bacon,“CommunicationCostofSim- logueoftheMermin–Peressquare,”Ann.Phys.353,326– ulating Bell Correlations,” Phys. Rev. Lett. 91, 187904 339(2015). (2003). [37] P.Blasiak,“Localmodelofaqubitintheinterferometric setup,”NewJ.Phys.17,113043(2015). [38] A. Karanjai, E. G. Cavalcanti, S. D. Bartlett, and T. Rudolph, “Weak values in a classical theory with an epistemicrestriction,”NewJ.Phys.17,073015(2015). [39] M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental Realization of Any Discrete Unitary Op- erator,”Phys.Rev.Lett.73,58–61(1994). [40] I. Bengtsson and K. Z˙yczkowski, Geometry of Quantum States: An Introduction to Quantum Entanglement (Cam- bridgeUniversityPress,2006). [41] M. Fuwa, S. Takeda, M. Zwierz, H. M. Wiseman, and A.Furusawa,“Experimentalproofofnonlocalwavefunc- tion collapse for a single particle using homodyne measurements,”Nat.Commun.6,6665(2015). [42] M.S.LeiferandR.W.Spekkens,“Pre-andPost-Selection Paradoxes and Contextuality in Quantum Mechanics,” Phys.Rev.Lett.95,200405(2005). [43] O. J. E. Maroney, “Detectability, Invasiveness and the Quantum Three Box Paradox,” arXiv:1207.3114 [quant- ph] (2012). [44] M. F. Pusey, “Anomalous Weak Values Are Proofs of Contextuality,”Phys.Rev.Lett.113,200401(2014). [45] N. Harrigan and R. W. Spekkens, “Einstein, Incom- pleteness, and the Epistemic View of Quantum States,” Found.Phys.40,125–157(2010). [46] R. W. Spekkens, “Contextuality for preparations, transformations, and unsharp measurements,” Phys. Rev. A 71,052108(2005). [47] N. Harrigan and T. Rudolph, “Ontological models and the interpretation of contextuality,” arXiv:0709.4266 [quant-ph] (2007). [48] E. G. Beltrametti and S. Bugajski, “A classical extension ofquantummechanics,”J.Phys.A:Math.Gen.28,3329 (1995). 10 Is single-particle interference spooky? Supplementary Information Proof of Theorem 1 PawelBlasiak InstituteofNuclearPhysicsPolishAcademyofSciences,PL-31342Kraków,Poland PRELIMINARIES Now, we give a detailed version of Theorem 1 which inthematrixnotationtakesthefollowingform. For the proof we switch to the matrix notation and Theorem2. Parallelconfigurationofgatesactscongruently denote(cid:126)z CN asacolumnvector on the family of classes (cid:8)[(cid:126)z] (Λ): (cid:126)z CN, (cid:126)z =1(cid:9) ∈N (cid:32)z.1 (cid:33) defined in Eq.(22). This mean⊂sPthat classe∈s trans(cid:107)for(cid:107)m as a (cid:126)z = ∑ zjej = .. . (31) whole, i.e. all distributions in a given class map into distri- j=1 zN butionsinsomeotherclass Lloewtiunsgwanriatelyosiust. eWxeplwiciiltllyusmeadtriaicgeosnraellemvaatnrticfeosr the fol- [(cid:126)z] (cid:51) p (cid:47)(cid:47) p(cid:48) ∈ [(cid:126)z(cid:48)], (35) 0 ...  1 ...  wgahteerse m,app,in,g(cid:126)zan−d→me(cid:126)za(cid:48)sudreepmenendtsoountctohmeecso(n’fiCgluicraktsio’)n. oItf Pj =  1 ... , Sk =  eiω... , (32) isspecFifieDdbSytBhefollowingrules: 0 1 For the system described by a distribution in class [(cid:126)z] de- wherePj isaprojectoron j-thcomponentandSk intro- t•ector Dj ’Clicks’withprobability duces phase eiω in k-th component leaving the remain- ingonesunchanged. Anotherusefulmatrixwhichacts Pr(Dj|(cid:126)z) = |zj|2, (36) nontriviallyonlyincomponents s,t hastheform and conditioning (post-selecting) on a ’Click’ in D leaves { } j 1 ...  thesysteminstatedescribedbyadistribution p(cid:48) ∈ [ej],i.e. Bst =  i√...R...... √...T , (33) (cid:126)z Dj (cid:47)(cid:47) (cid:126)z(cid:48) = ej ((cid:48)Click(cid:48)). (37)    √T ... i√R  Ateachrunofexperimenteitheroneofthedetectors’Click’  ...  or all detectors remain silent (negative measurement result), 1 withthelatterhappeningwithprobability1 ∑ z 2. whereRandT aresomeconstantssuchthatR+T =1. − j∈D | j| Clearly,matricesSj andBst areunitary. i•nIanlltdheeteccatsoersofDnefgoartijve mea)sourrenmoemntearseusurelmt (e’nNtoatCalllic(nko’ In the following we consider evolution of the system detectors = ∅)j transf∈ormDation implemented by the gates implementedbyaparallelconfigurationofgatesacting D isgivenby indifferentpathsatthesametime–thiscorrespondsto a single step in the circuit model. It will be convenient (cid:126)z (cid:47)(cid:47) (cid:126)z(cid:48) ∼ ∏(1−Pj)∏Sk ∏ Bst (cid:126)z, (38) to group paths with the same kind of gates and define j k s,t ∈D ∈S { }∈B thecorrespondingsubsetsas 2 withtheorderofmatricesintheproductbeingirrelevant. pathswithoutgates(emptypaths), F − Thisformulationdealsexplicitlywiththecaseofpar- pathswithdetectors, D − (34) alleltransformationsindifferentpaths. Itisstraightfor- pathswithphaseshifters, ward to convince oneself that Theorem 1 follows from S − pairsofpathscrossingonbeamsplitters. Theorem 2 (both are actually equivalent with the latter B − (cid:83) one being a more rigorous version). It is thus enough Notethat , , and aredisjointandexhaustive; F D S B 1 toprovethematrixversiongivenabove. henceitisapartitionofthesetofallpaths 1,...,N . { } 1TWp{1ahe,ier.shs.ea.Bv,sNee=t}sF.(cid:8)a,Br{Deys,1dSe,ixts1h⊂j}oa,iun{{st1st2,bi,.vet.ec2.a},wu,Ne.s.}em.,g(cid:9)aeanawtndeesFsgainec∪ttceiD(cid:83)nBB∪diifSs=fear∪{enss(cid:83)e1tt,Bpto1af=,ths(2us{,.n1t2o,,.r..d..e.,r}Ned⊂}). 2totDhhfuaesenudtbe1oss.cenHrqoiupenent-inucotennritteohanferoyqprumpraornaopltjoiuesrcmatttiioioomnnnaes(cid:126)lazilste(cid:48)yun(cid:32)rgseytmh(cid:126)zm(cid:48)e/obn(cid:107)fo(cid:126)tz(cid:126)zl(cid:48)i(cid:107)n(cid:48)’∼(iEcn’fq.eE.xa(qp4n.r)a()e3l.so8sg)inomguastyhisebseuneleeeisdns