aa r X i v : . [ qu a n t - ph ] O c t On von Neumann and Bell theorems applied to quantumness testsRobert Ali kiInstitute of Theoreti al Physi s and Astrophysi s, University of Gda«sk, Wita Stwosza 57, PL 80-952 Gda«sk, PolandNovember 6, 2018Abstra tThe issues, raised in arXiv:0809.011, on erning the relevan e of the von Neumann theorem for the single-system's quantumness test proposed in arXiv:0704.1962 and performed for the ase of single photon polarizationin arXiv:0804.1646, and the usefulness of Bell's inequality for testing the idea of ma ros opi quantum systems aredis ussed in some details. Finally, the proper quantum me hani al des ription of the experiment with polarizedphoton beams is presented.1 Introdu tionThe re ent paper of Marek ›ukowski [1℄ is aimed as a riti ism of the paper by myself and Van Ryn [2℄ devoted to" a simple test of quantumness for a single system" whi h was experimentally realized in [3℄. The ideas of [2℄ werefurther developed in [4℄ whi h appeared after [1℄ has been (cid:28)nished.The three main thesis of [1℄ are the following:I) "... simple test of quantumness for a single system....has exa tly the same relation to the dis ussion...as the vonNeumann theorem"II) "As far as a dire t dete tion of non existen e of any lassi al probabilisti models is on erned we are left withthe two theorems of Bell..."III) "..., the example given in [2℄ and realized in [3℄ does have a lassi al model, like every se ond order (in termsof (cid:28)elds) photoni interferen e e(cid:27)e t. The observed phenomena an be interpreted as non- lassi al only due to thestatisti al properties of the parametri down onversion pro ess".Although this kind of problems attra ted attention for several de ades and stimulated many authors to produ eenormous number of publi ations it seems that the serious onfusions and mis on eptions are still alive and needfurther debate and lari(cid:28) ation (see an ex ellent dis ussion in [5℄). Re ently, the problem of "quantumness" be ameimportant for more pra ti al reason, namely, the question of (non)existen e of ma ros opi quantum systems whi h ould be used as implementations of qubits in quantum information pro essing.2 Two theorems and related quantumness testsTo eliminate the main sour e of onfusion in the debate one should make the distin tion between two di(cid:27)erent lasses of problems:Q1) Impossibility of des ription of a given set of experimental data by a lassi al probabilisti model.Q2) Impossibility of a lo al realisti interpretation of quantum me hani s.The papers [2, 4℄ on ern the problem Q1 while the arguments of [1℄ are based on the ri h family of results knownunder olle tive name of Bell's Theorem [6℄ and on erning Q2. Firstly, Q1 and Q2 are independent from the logi alpoint of view. Se ondly, in Q1 the lassi al probabilisti model means a theory like statisti al lassi al me hani swith observables forming an algebra of fun tions and states being all probability distributions on a ertain "phasespa e". On the other hand, in Q2 the notion of lo al realisti model is often formulated in philosophi al languageleaving a spa e for di(cid:27)erent interpretations in pre ise mathemati al terms [9, 5℄. Nevertheless, both Q1 and Q2have something in ommon. This is the sear h for the parti ular features of lassi al probabilisti models whi h arenot present in the quantum formalism. For the purpose of further dis ussions I present two theorems formulatedin the algebrai language and on erning this type of features. In the following A denotes a C ∗ -algebra with theidentity I and provides a model for all bounded observables of the physi al systems as self-adjoint elements of A .The set of all linear, positive and normalized fun tionals on A denoted by S ( A ) is identi(cid:28)ed with all physi al statesof the system. The mean value of the observable A in the state ρ is given by ρ ( A ) . The spe trum Sp( A ) onsists1f all numbers a (real for self-adjoint A ) for whi h the element A − aI possesses no inverse in the algebra A . Oneinterprets the spe trum Sp( A ) as possible out omes of the measurement for the observable A [10℄.Theorem A For any self-adjoint elements A, B, C ∈ A the following impli ation ρ ( A ) + ρ ( B ) = ρ ( C ) for all ρ ∈ S ( A ) ⇒ Sp( C ) ⊂ Sp( A ) + Sp( B ) (1)holds if and only if A is ommutative and hen e isomorphi to an algebra of ontinuous fun tions on a ertain ompa t set.Theorem B For any self-adjoint elements A, B ∈ A the following impli ation ≤ ρ ( A ) ≤ ρ ( B ) for all ρ ∈ S ( A ) ⇒ ρ ( A ) ≤ ρ ( B ) (2)holds if and only if A is ommutative and hen e isomorphi to an algebra of ontinuous fun tions on a ertain ompa t set.Both theorems follow from the known results in the theory of operator algebras, the equivalen e of the assump-tions (2) and (1) is also dis ussed in [1℄. They an be used as a motivation for two quantumness tests whi h an beapplied to a olle tion of experimental data [11℄.QTest A Find three observables A, B, C whi h averaged values satisfy the equality ρ ( A ) + ρ ( B ) = ρ ( C ) for all states ρ (3)but the possible out omes of C are not given by the sums of the out omes of A and B .QTest B Find two observables A, B whi h averaged values satisfy the inequality ≤ ρ ( A ) ≤ ρ ( B ) for all states ρ (4)but for a ertain state σ the se ond moments ful(cid:28)ll σ ( A ) > σ ( B ) .Although no experiment an be on lusive in the philosophi al sense (even the existen e of the World annotbe proved [12℄) one an dis uss (always a bit subje tively) the level of on(cid:28)den e for the QTestA and QTestB. Inboth ases the di(cid:30) ulty is hidden in the words "for all states" as one an never perform experiments for all possiblestates. If the set of experimentally a essible states is restri ted too mu h, it an happen that for the QTestA thereexists another observable C ′ whi h yields the same mean values as C for the all a essible states but neverthelesspossesses as out omes the sums of out omes for A and B . Similarly, for the QTestB there may exist non a essiblestates whi h violate the inequality (4). To remove the later possibility an additional assumption - minimality of themodel - was introdu ed in [2, 4℄:If for any pair of experimentally a essible observables A, B the inequality (4) is on(cid:28)rmed for all experimentallya essible states, then the same inequality holds for all states in the model.Now, the positive result of the QTestB ex ludes the minimal lassi al model for the experimental data.The pra ti al and ommon sense justi(cid:28) ation of the minimality assumption was already proposed in [2℄ anddis ussed in some details using toy models in [4℄. It is based on a general observation that there is a ertainsymmetry between state preparation and measurement. Measuring apparatus involves some sele tion ((cid:28)ltering)pro edures whi h are used for state preparation as well. Therefore, for any (cid:28)xed te hnologi al implementation we an assume that the resolution on the side of state preparation is similar to the resolution on the side of measurement[13℄. On the other hand the existen e of a essible lassi al observables
A, B for whi h B − A possesses negativeout omes whi h are always averages out by all a essible states means that the resolution of the prepared states ismu h lower than the resolution of the measurable observables. The similar argument applies to the test A be ausefor lassi al systems one has C ′ = A + B and its experimental indistinguishability from C means again that thestates are mu h more oarse-grained than the observables.However, the test B has important pra ti al advantages in omparison with the test A. Namely, it is basedon two measurement's settings instead of three and employs inequalities instead of equalities and hen e does notrequire a (cid:28)ne tuning.3 Relation to von Neumann theoremThe main problem with the (in)famous no-go von Neumann theorem is that one annot (cid:28)nd it. The statement fromthe von Neumann book [14℄ whi h is referred in [1℄ after [7, 8℄ as von Neumann theorem is in fa t a ombination ofthe real theorem about the nonexisten e of dispersion-free states in quantum me hani s with rather loose remarkson the nonexisten e of hidden variable models (HVM) reprodu ing quantum me hani al predi tions. The mentioned2heorem is proved using rather expli it properties of the Hilbert spa e formalism instead of general axioms dis ussedalso in the von Neumann book [15℄. Unfortunately, von Neumann did not de(cid:28)ne pre isely what was his meaning ofHVM what prohibits a simple transformation of his remarks into a theorem.To ontinue dis ussion I an formulate the following hypothesis based on my understanding of [1, 7, 8, 14℄:Theorem A is the best approximation to what is alled in the literature von Neumann no-go theorem.Under this hypothesis I an agree that the on(cid:28)den e levels of the test based on the "von Neumann theorem" andthe test based on the Theorem B (used in [2℄) are similar ( ompare statement(I)), although important pra ti aladvantages of the later should be a knowledged.4 Counterexamples to no-go theorems for HVMIt is instru tive to dis uss brie(cid:29)y the stru ture of some models whi h might be onsidered as " ounterexamples tono-go theorems for HVM". They should provide "embeddings" of quantum theory into ertain lassi al ones andwere mentioned in [1℄.Bell's HVM for a qubitAny pure qubit's state ρ = ρ = 1 / I + ~k · ~σ ) , | ~k | = 1 , and any qubit's observable A = a I + ~a · ~σ an be representedby the following probability distribution p and the fun tion F on the phase spa e { ~m, ~n } whi h is a Cartesianprodu t of two unit spheres: ρ ≡ p ( ~m, ~n ) = δ ( ~n − ~k ) , A ≡ F ( ~m, ~n ) = (cid:8) a + | ~a | if ( ~m + ~n ) · ~a > a − | ~a | otherwise (cid:9) , (5)where a ± | ~a | are eigenvalues of A . Indeed, one an he k that Z d ~md~n p ( ~m, ~n ) F ( ~m, ~n ) = a + ~k · ~a = Tr( ρA ) . (6)This model is not minimal and the dis ussed in the Se tion 2 symmetry between states and observables is stronglyviolated. The allowed probability distributions are perfe tly lo alized in ~n and uniform with respe t to ~m while theallowed observables are equally sensitive to both variables.HVM of everythingAs the set of hidden variables one takes a Cartesian produ t of the out omes sets for all relevant observables,disregarding any algebrai relations between them, and as the probability the produ t of individual probabilitymeasures omputed from any theory [1℄. This "model" has predi tive power equal to zero and an be treated onlyas a meaningless "interpretation" .Phase spa e modelUsing an over omplete set of oherent ve tors {| α i} one an represent any density matrix ρ by a probability distri-bution on the phase spa e (Q-representation) and any observable A by a phase spa e fun tion (P-representation)[16℄ ρ ≡ p ( α ) = h α | ρ | α i , A ≡ F ( α ) such that A = Z d α F ( α ) | α ih α | . (7)Although, the lassi al-like formula holds Tr( ρA ) = Z d α p ( α ) F ( α ) , (8)this representation is not a HVM be ause the values of the fun tion F ( α ) do not oin ide with the eigenvalues of A orresponding to measurement out omes. One should noti e again the asymmetry between states and observablesin this representation. Probability distributions p ( α ) are "fuzzy" (Heisenberg relations) while F ( α ) ould be evena distribution more singular then Dira delta.All those " ounterexamples" are not minimal lassi al models, in the sense of Se tion 2, moreover they possessesother serious (cid:29)aws in their mathemati al and logi al stru ture.5 Di(cid:27)erent fa es of Bell's inequalityBell's theorem is a olle tive name for the vast family of results, some of them meeting the standards of mathemati altheorem another ones an be treated as philosophi al statements only. The main ingredient is always one of the many3orms of Bell's inequality. In the following I use always the so- alled Bell-Clauser-Horne-Shimony-Holt inequality.One onsiders four observables A i , B i , i = 1 , with the out omes or − under the assumption that any pair A i , B j is simultaneously measurable i.e. the orrelation observables A i B j make sense and yield outputs whi h are produ tof the outputs of A i and B j . Repeating the measurements for di(cid:27)erent pairs and a single initial state ρ one an ompute the following fun tion of state F ( ρ ) = ρ ( A B ) + ρ ( A B ) + ρ ( A B ) − ρ ( A B ) . (9)The famous BCHSH inequality reads | F ( ρ ) | ≤ . (10)I dis uss now three examples of assumptions whi h lead to (10):1) Bell's inequality for quantum separable statesConsider a quantum model of bipartite system with { A i } and { B i } being observables of two di(cid:27)erent subsystems.Then for all separable states, i.e. states of the form ρ = P α p α ρ Aα ⊗ ρ Bα , the inequality (10) holds.2) Bell's inequality for lassi al probabilisti modelsFor lassi al systems (de(cid:28)ned as in the Se tion 2) the inequality (10) holds for all observables with out omes or − [17, 5, 4℄.3)Bell's inequality for lo al realisti modelsHere the notions of lo ality and realism may have di(cid:27)erent mathemati al representations [18℄ and sometimes theassumption of free will is added [6, 19℄.The ase 1) is the least ontroversial and the most useful in quantum information. The se ond one suggestsa quantumness test in the spirit of the examples in Se tion 2 and for the lass of problems Q1. At the (cid:28)rstsight, it seems that su h a test is mu h stronger the the test B as it is enough to show violation of the inequality(10) for a single state only. However, one should remember that even in the lassi al theory the observables arejointly measurable only as abstra t idealized obje ts. In real experiments, on erning for example solid state oratomi implementations of qubits, one should he k whether on rete measuring devi es a ting simultaneously donot interfere with ea h other introdu ing unwanted orrelations. To ex lude this, one needs additional tests withdi(cid:27)erent initial states and observable settings. Therefore, for pra ti al appli ations su h quantumness tests basedon Bell's inequalities are mu h more involved than single parti le tests.Finally, one an assume that the ase 3) is what is really meant by "...any lassi al probabilisti model..." in[1℄. As stated , for example in [19℄ lo ality means that "...events and a tions in Ali e's lab annot in(cid:29)uen e dire tlysimultaneous events in Bob's lab and his a ts...". This is not the ase for the most interesting experimental situations on erning ontroversial ma ros opi quantum systems (super ondu ting qubits, oupled BEC's , Rydberg atoms,)where Ali e and Bob must share the same lab (see dis ussion of the previous ase). To deal with these importantand urgent questions one should (cid:28)rst apply more feasible single system tests like those proposed in [2, 4℄( omparestatement II).6 ExampleIn [3℄ the experiment realizing partially [20℄ test B for a single photon is des ribed. The author of [1℄ proposed touse instead of a single photon sour e a ma ros opi lassi al beam of light to show that for a ma ros opi systemthe "quantumness e(cid:27)e t" an be also observed. Unfortunately, the presented on lusions of his analysis are not orre t. A beam is a physi al system des ribed by the formalism of se ond quantisation. For any single-photonobservable A = P j α j | φ j ih φ j | there exists a se ond quantization observable Γ( A ) a ting on the Fo k spa e andgiven by Γ( A ) = X j α j a † ( φ j ) a ( φ j ) (11)where a ( φ j ) , ( a † ( φ j )) is an annihilation ( reation) operator orresponding to a mode of radiation (equivalently,a single photon wave fun tion normalized to 1) φ j . The out omes of the observables (11) are given by α j n j , n j = 0 , , , ... . On should remember that only the additive observables (11) and their fun tions (e.q. moments) an be measured in linear opti s experiments.The se ond quantization map A → Γ( A ) preserves the order i.e.: ≤ A ≤ B, A (cid:2) B ⇒ ≤ Γ( A ) ≤ Γ( B ) , Γ( A ) (cid:2) Γ( B ) . (12)4ut not the algebrai relations, e.g. Γ( A ) = (cid:0) Γ( A ) (cid:1) .The quantumnes test B involves squares of the observables like (cid:0) Γ( A ) (cid:1) with the out omes ( α j n j ) but not Γ( A ) with the out omes α j n j . Noti e that (cid:0) Γ( A ) (cid:1) oin ides with Γ( A ) only on the va uum and 1-photon se tor of theFo k spa e. Using the formula (cid:0) Γ( A ) (cid:1) = X i,j α i α j a † ( φ i ) a ( φ i ) a † ( φ j ) a ( φ j ) = X i,j α i α j a † ( φ i ) a † ( φ j ) a ( φ i )) a ( φ j ) + Γ( A ) . (13)one an ompute the mean value of the relevant observables in a oherent state Φ( ξ ) . Here ξ is interpreted as asingle-photon wave fun tion normalized to the averaged number of photons , h ξ | ξ i = N . For large N the oherentstate des ribes a quantum state of a ma ros opi light beam determined by ξ , interpreted now as the lassi alele tromagneti (cid:28)eld. One obtains the formulas h Φ( ξ ) , Γ( A )Φ( ξ ) i = h ξ, Aξ i (14)and h Φ( ξ ) , (cid:0) Γ( A ) (cid:1) Φ( ξ ) i = ( h ξ, Aξ i ) + h ξ, A ξ i . (15)The (cid:28)rst term on the RHS of (15) is of the order of N while the se ond is of the order of N . Therefore, only fora weak beam, i.e. N << the se ond term dominates and with the hoi e of single-photon observables ≤ A ≤ B and A (cid:2) B yields the "violation of lassi ality". For ma ros opi beams ( N >> the (cid:28)rst term dominates andthe lassi al order relations are preserved ( ompare with the dis ussion of "ma ros opi entanglement" in [21℄ or"additive observables" in [4℄). Therefore, only experiments with single photons an show dire tly deviations from lassi al probabilisti model. Of ourse, in su h experiments any single photon sour e is (cid:28)ne [22℄ and the results hasnothing to do with the "statisti al properties of the parametri down onversion" ( ompare statement III) appliedin the experimental setting of [3℄.The interesting aspe t of linear opti s experiments with light beams is that ompletely deterministi ma ro-s opi experiments on the ma ros opi obje ts an provide indire tly information about the quantum nature of theunderlying mi ros opi onstituents. It is possible under the hypothesis whi h an be alled Newton's model of light:A light beam onsists of nonintera ting indistinguishable parti les whi h intera t independently with the measuringapparatus.The unique nature of photons, in parti ular the ombination of bosoni statisti s, mass and harge equal to zeroand strong intera tion with matter allows to prepare states satisfying Newton's hypothesis and yielding high valuesof the out omes α j n j whi h produ e ma ros opi e(cid:27)e ts [23℄. Hen e, we an interpret a measurement performedon a single system - a light beam - as equivalent to a sequen e of many independent measurements performed on asingle photon prepared always in a (cid:28)xed state .As a onsequen e sir George Gabriel Stokes, who showed in 1852 that a state of light beam polarization isdes ribed by only four parameters, should be re ognized as the dis overer of quantum me hani s [24℄. Indeed,under the Newton's hypothesis the Stokes result implies that the polarization's state of a single photon must bedes ribed by three parameters represented by a ve tor in the interior of the qubit's Blo h sphere ( alled Poin arésphere in polarization opti s). If polarization would be a lassi al system its states should be represented by anin(cid:28)nitely dimensional simplex of probability measures instead of the 3-dimensional ball.A knowledgements. The author thanks Mar o Piani, Mi haªHorode ki and Ni k Van Ryn for dis ussions. Thework is supported by the Polish resear h network LFPPI.Referen es[1℄ M. ›ukowski, arXiv:0809.0115v1[2℄ R. Ali ki and N. Van Ryn, J. Phys. A: Math. Theor. 41 062001 (2008)[3℄ G. Brida, I. Degiovanni, M. Genovese, V. S hettini, S. Polyakov, and A. Migdall, (cid:16)Experimental test of non- lassi ality for a single parti le(cid:17), arXiv:0804.1646.[4℄ R. Ali ki, M. Piani and N. Van Ryn, arXiv:0807.2615[5℄ R.F. Streater, Lost Causes in and beyond Physi s, Springer, Berlin (2007)[6℄ Bell's Theorem in Stanford En y lopedia of Philosophy, http://plato,stanford.edu/entries/bell-theorem/57℄ J.S. Bell, Rev. Mod. Phys.38, 447 (1966)[8℄ N.D. Mermin, Rev. Mod. Phys.65, 803 (1993)[9℄ The relations between lassi al probabilisti models, hidden variable models, and lo al realisti models seemto be quite subtle and are usually dis ussed in philosophi al and not mathemati al language using the wordslike "reality", " ontextual", "non- ontextual" and even " ontingen y" [6℄. An often ignored di(cid:27)eren e between"model" and "interpretation" is another sour e of onfusions.[10℄ The reader unfamiliar with the algebrai approa h an always onsider two models: the lassi al one with thealgebra of fun tions and states as probability measures and the quantum one with the algebra of all boundedoperators on the Hilbert spa e and states being density operators (matri es).[11℄ To avoid a ompli ated notation I use the same symbol and the same name for an "observable" ("state") as aphysi al, operational notion and an "observable" ("state") as its mathemati al representation.[12℄ R. Ingarden, Der Streit um die Existenz der Welt vol.I, II, III, Niemeyer , Tuebingen (1963, 1965, 1974).[13℄ The symmetry between states and observables appears in the mathemati al framework in the form of dualitybetween states and observables. The states an be treated as linear fun tionals of observables and vi e versa.The alternative- C ∗ versus von Neumann algebras - re(cid:29)e ts this feature on a very abstra t level.[14℄ J. von Neumann, Mathematis he Grundlagen der Quantenme hanik, Springer, Berlin , 1932[15℄ The des ription of "von Neumann theorem" in [7℄ and [8℄ is not fair. For example, J.S. Bell wrote :"Theproblem is posed in the prefa e and on p.209. The formal proof o upies essentially pp. 305-324 and is followedby several pages of ommentary". A tually, the proof of absen e of dispersion-free states o upies 12 lines anda rather informal dis ussion of its onsequen es for the existen e of hidden variables - half of the page (see[14℄).[16℄ Noti e the reversed role of P(Q)-representations in omparison with [1, 4℄.[17℄ J.L. Landau, Phys.Lett.A 120, 54, (1987)[18℄ In all derivations of Bell's CHSH inequality the mathemati al assumption about joint probabilities, usuallyattributed to "lo ality", reads: p ( A i , B j ) = R µ ( dx ) p ( A i | x ) p ( B j | x ) , where µ ( dx ) is a probability distribution ofHV-s. This assumption is equivalent to the existen e of joint probability distribution p ( A , A , B , B ) . Indeedhaving p ( A i | x ) , p ( B j | x ) and µ ( dx ) one an de(cid:28)ne p ( A , A , B , B ) = R µ ( dx ) p ( A | x ) p ( A | x ) p ( B | x ) p ( B | x ) .Starting with p ( A , A , B , B ) one an treat A i , B j as random variables A i ( x ) , B j ( x ) and de(cid:28)ne onditionalprobability distributions p ( A i | x ) , p ( B j | x ) as atomi measures on entrated on the values A i ( x ) , B j ( x ) .[19℄ M. ›ukowski, arXiV:quant-ph/0605034v1[20℄ The missing part of the quantumness test in the experimental setting of [3℄ is the veri(cid:28) ation of the inequality ρ ( A ))
A, B for whi h B − A possesses negativeout omes whi h are always averages out by all a essible states means that the resolution of the prepared states ismu h lower than the resolution of the measurable observables. The similar argument applies to the test A be ausefor lassi al systems one has C ′ = A + B and its experimental indistinguishability from C means again that thestates are mu h more oarse-grained than the observables.However, the test B has important pra ti al advantages in omparison with the test A. Namely, it is basedon two measurement's settings instead of three and employs inequalities instead of equalities and hen e does notrequire a (cid:28)ne tuning.3 Relation to von Neumann theoremThe main problem with the (in)famous no-go von Neumann theorem is that one annot (cid:28)nd it. The statement fromthe von Neumann book [14℄ whi h is referred in [1℄ after [7, 8℄ as von Neumann theorem is in fa t a ombination ofthe real theorem about the nonexisten e of dispersion-free states in quantum me hani s with rather loose remarkson the nonexisten e of hidden variable models (HVM) reprodu ing quantum me hani al predi tions. The mentioned2heorem is proved using rather expli it properties of the Hilbert spa e formalism instead of general axioms dis ussedalso in the von Neumann book [15℄. Unfortunately, von Neumann did not de(cid:28)ne pre isely what was his meaning ofHVM what prohibits a simple transformation of his remarks into a theorem.To ontinue dis ussion I an formulate the following hypothesis based on my understanding of [1, 7, 8, 14℄:Theorem A is the best approximation to what is alled in the literature von Neumann no-go theorem.Under this hypothesis I an agree that the on(cid:28)den e levels of the test based on the "von Neumann theorem" andthe test based on the Theorem B (used in [2℄) are similar ( ompare statement(I)), although important pra ti aladvantages of the later should be a knowledged.4 Counterexamples to no-go theorems for HVMIt is instru tive to dis uss brie(cid:29)y the stru ture of some models whi h might be onsidered as " ounterexamples tono-go theorems for HVM". They should provide "embeddings" of quantum theory into ertain lassi al ones andwere mentioned in [1℄.Bell's HVM for a qubitAny pure qubit's state ρ = ρ = 1 / I + ~k · ~σ ) , | ~k | = 1 , and any qubit's observable A = a I + ~a · ~σ an be representedby the following probability distribution p and the fun tion F on the phase spa e { ~m, ~n } whi h is a Cartesianprodu t of two unit spheres: ρ ≡ p ( ~m, ~n ) = δ ( ~n − ~k ) , A ≡ F ( ~m, ~n ) = (cid:8) a + | ~a | if ( ~m + ~n ) · ~a > a − | ~a | otherwise (cid:9) , (5)where a ± | ~a | are eigenvalues of A . Indeed, one an he k that Z d ~md~n p ( ~m, ~n ) F ( ~m, ~n ) = a + ~k · ~a = Tr( ρA ) . (6)This model is not minimal and the dis ussed in the Se tion 2 symmetry between states and observables is stronglyviolated. The allowed probability distributions are perfe tly lo alized in ~n and uniform with respe t to ~m while theallowed observables are equally sensitive to both variables.HVM of everythingAs the set of hidden variables one takes a Cartesian produ t of the out omes sets for all relevant observables,disregarding any algebrai relations between them, and as the probability the produ t of individual probabilitymeasures omputed from any theory [1℄. This "model" has predi tive power equal to zero and an be treated onlyas a meaningless "interpretation" .Phase spa e modelUsing an over omplete set of oherent ve tors {| α i} one an represent any density matrix ρ by a probability distri-bution on the phase spa e (Q-representation) and any observable A by a phase spa e fun tion (P-representation)[16℄ ρ ≡ p ( α ) = h α | ρ | α i , A ≡ F ( α ) such that A = Z d α F ( α ) | α ih α | . (7)Although, the lassi al-like formula holds Tr( ρA ) = Z d α p ( α ) F ( α ) , (8)this representation is not a HVM be ause the values of the fun tion F ( α ) do not oin ide with the eigenvalues of A orresponding to measurement out omes. One should noti e again the asymmetry between states and observablesin this representation. Probability distributions p ( α ) are "fuzzy" (Heisenberg relations) while F ( α ) ould be evena distribution more singular then Dira delta.All those " ounterexamples" are not minimal lassi al models, in the sense of Se tion 2, moreover they possessesother serious (cid:29)aws in their mathemati al and logi al stru ture.5 Di(cid:27)erent fa es of Bell's inequalityBell's theorem is a olle tive name for the vast family of results, some of them meeting the standards of mathemati altheorem another ones an be treated as philosophi al statements only. The main ingredient is always one of the many3orms of Bell's inequality. In the following I use always the so- alled Bell-Clauser-Horne-Shimony-Holt inequality.One onsiders four observables A i , B i , i = 1 , with the out omes or − under the assumption that any pair A i , B j is simultaneously measurable i.e. the orrelation observables A i B j make sense and yield outputs whi h are produ tof the outputs of A i and B j . Repeating the measurements for di(cid:27)erent pairs and a single initial state ρ one an ompute the following fun tion of state F ( ρ ) = ρ ( A B ) + ρ ( A B ) + ρ ( A B ) − ρ ( A B ) . (9)The famous BCHSH inequality reads | F ( ρ ) | ≤ . (10)I dis uss now three examples of assumptions whi h lead to (10):1) Bell's inequality for quantum separable statesConsider a quantum model of bipartite system with { A i } and { B i } being observables of two di(cid:27)erent subsystems.Then for all separable states, i.e. states of the form ρ = P α p α ρ Aα ⊗ ρ Bα , the inequality (10) holds.2) Bell's inequality for lassi al probabilisti modelsFor lassi al systems (de(cid:28)ned as in the Se tion 2) the inequality (10) holds for all observables with out omes or − [17, 5, 4℄.3)Bell's inequality for lo al realisti modelsHere the notions of lo ality and realism may have di(cid:27)erent mathemati al representations [18℄ and sometimes theassumption of free will is added [6, 19℄.The ase 1) is the least ontroversial and the most useful in quantum information. The se ond one suggestsa quantumness test in the spirit of the examples in Se tion 2 and for the lass of problems Q1. At the (cid:28)rstsight, it seems that su h a test is mu h stronger the the test B as it is enough to show violation of the inequality(10) for a single state only. However, one should remember that even in the lassi al theory the observables arejointly measurable only as abstra t idealized obje ts. In real experiments, on erning for example solid state oratomi implementations of qubits, one should he k whether on rete measuring devi es a ting simultaneously donot interfere with ea h other introdu ing unwanted orrelations. To ex lude this, one needs additional tests withdi(cid:27)erent initial states and observable settings. Therefore, for pra ti al appli ations su h quantumness tests basedon Bell's inequalities are mu h more involved than single parti le tests.Finally, one an assume that the ase 3) is what is really meant by "...any lassi al probabilisti model..." in[1℄. As stated , for example in [19℄ lo ality means that "...events and a tions in Ali e's lab annot in(cid:29)uen e dire tlysimultaneous events in Bob's lab and his a ts...". This is not the ase for the most interesting experimental situations on erning ontroversial ma ros opi quantum systems (super ondu ting qubits, oupled BEC's , Rydberg atoms,)where Ali e and Bob must share the same lab (see dis ussion of the previous ase). To deal with these importantand urgent questions one should (cid:28)rst apply more feasible single system tests like those proposed in [2, 4℄( omparestatement II).6 ExampleIn [3℄ the experiment realizing partially [20℄ test B for a single photon is des ribed. The author of [1℄ proposed touse instead of a single photon sour e a ma ros opi lassi al beam of light to show that for a ma ros opi systemthe "quantumness e(cid:27)e t" an be also observed. Unfortunately, the presented on lusions of his analysis are not orre t. A beam is a physi al system des ribed by the formalism of se ond quantisation. For any single-photonobservable A = P j α j | φ j ih φ j | there exists a se ond quantization observable Γ( A ) a ting on the Fo k spa e andgiven by Γ( A ) = X j α j a † ( φ j ) a ( φ j ) (11)where a ( φ j ) , ( a † ( φ j )) is an annihilation ( reation) operator orresponding to a mode of radiation (equivalently,a single photon wave fun tion normalized to 1) φ j . The out omes of the observables (11) are given by α j n j , n j = 0 , , , ... . On should remember that only the additive observables (11) and their fun tions (e.q. moments) an be measured in linear opti s experiments.The se ond quantization map A → Γ( A ) preserves the order i.e.: ≤ A ≤ B, A (cid:2) B ⇒ ≤ Γ( A ) ≤ Γ( B ) , Γ( A ) (cid:2) Γ( B ) . (12)4ut not the algebrai relations, e.g. Γ( A ) = (cid:0) Γ( A ) (cid:1) .The quantumnes test B involves squares of the observables like (cid:0) Γ( A ) (cid:1) with the out omes ( α j n j ) but not Γ( A ) with the out omes α j n j . Noti e that (cid:0) Γ( A ) (cid:1) oin ides with Γ( A ) only on the va uum and 1-photon se tor of theFo k spa e. Using the formula (cid:0) Γ( A ) (cid:1) = X i,j α i α j a † ( φ i ) a ( φ i ) a † ( φ j ) a ( φ j ) = X i,j α i α j a † ( φ i ) a † ( φ j ) a ( φ i )) a ( φ j ) + Γ( A ) . (13)one an ompute the mean value of the relevant observables in a oherent state Φ( ξ ) . Here ξ is interpreted as asingle-photon wave fun tion normalized to the averaged number of photons , h ξ | ξ i = N . For large N the oherentstate des ribes a quantum state of a ma ros opi light beam determined by ξ , interpreted now as the lassi alele tromagneti (cid:28)eld. One obtains the formulas h Φ( ξ ) , Γ( A )Φ( ξ ) i = h ξ, Aξ i (14)and h Φ( ξ ) , (cid:0) Γ( A ) (cid:1) Φ( ξ ) i = ( h ξ, Aξ i ) + h ξ, A ξ i . (15)The (cid:28)rst term on the RHS of (15) is of the order of N while the se ond is of the order of N . Therefore, only fora weak beam, i.e. N << the se ond term dominates and with the hoi e of single-photon observables ≤ A ≤ B and A (cid:2) B yields the "violation of lassi ality". For ma ros opi beams ( N >> the (cid:28)rst term dominates andthe lassi al order relations are preserved ( ompare with the dis ussion of "ma ros opi entanglement" in [21℄ or"additive observables" in [4℄). Therefore, only experiments with single photons an show dire tly deviations from lassi al probabilisti model. Of ourse, in su h experiments any single photon sour e is (cid:28)ne [22℄ and the results hasnothing to do with the "statisti al properties of the parametri down onversion" ( ompare statement III) appliedin the experimental setting of [3℄.The interesting aspe t of linear opti s experiments with light beams is that ompletely deterministi ma ro-s opi experiments on the ma ros opi obje ts an provide indire tly information about the quantum nature of theunderlying mi ros opi onstituents. It is possible under the hypothesis whi h an be alled Newton's model of light:A light beam onsists of nonintera ting indistinguishable parti les whi h intera t independently with the measuringapparatus.The unique nature of photons, in parti ular the ombination of bosoni statisti s, mass and harge equal to zeroand strong intera tion with matter allows to prepare states satisfying Newton's hypothesis and yielding high valuesof the out omes α j n j whi h produ e ma ros opi e(cid:27)e ts [23℄. Hen e, we an interpret a measurement performedon a single system - a light beam - as equivalent to a sequen e of many independent measurements performed on asingle photon prepared always in a (cid:28)xed state .As a onsequen e sir George Gabriel Stokes, who showed in 1852 that a state of light beam polarization isdes ribed by only four parameters, should be re ognized as the dis overer of quantum me hani s [24℄. Indeed,under the Newton's hypothesis the Stokes result implies that the polarization's state of a single photon must bedes ribed by three parameters represented by a ve tor in the interior of the qubit's Blo h sphere ( alled Poin arésphere in polarization opti s). If polarization would be a lassi al system its states should be represented by anin(cid:28)nitely dimensional simplex of probability measures instead of the 3-dimensional ball.A knowledgements. The author thanks Mar o Piani, Mi haªHorode ki and Ni k Van Ryn for dis ussions. Thework is supported by the Polish resear h network LFPPI.Referen es[1℄ M. ›ukowski, arXiv:0809.0115v1[2℄ R. Ali ki and N. Van Ryn, J. Phys. A: Math. Theor. 41 062001 (2008)[3℄ G. Brida, I. Degiovanni, M. Genovese, V. S hettini, S. Polyakov, and A. Migdall, (cid:16)Experimental test of non- lassi ality for a single parti le(cid:17), arXiv:0804.1646.[4℄ R. Ali ki, M. Piani and N. Van Ryn, arXiv:0807.2615[5℄ R.F. Streater, Lost Causes in and beyond Physi s, Springer, Berlin (2007)[6℄ Bell's Theorem in Stanford En y lopedia of Philosophy, http://plato,stanford.edu/entries/bell-theorem/57℄ J.S. Bell, Rev. Mod. Phys.38, 447 (1966)[8℄ N.D. Mermin, Rev. Mod. Phys.65, 803 (1993)[9℄ The relations between lassi al probabilisti models, hidden variable models, and lo al realisti models seemto be quite subtle and are usually dis ussed in philosophi al and not mathemati al language using the wordslike "reality", " ontextual", "non- ontextual" and even " ontingen y" [6℄. An often ignored di(cid:27)eren e between"model" and "interpretation" is another sour e of onfusions.[10℄ The reader unfamiliar with the algebrai approa h an always onsider two models: the lassi al one with thealgebra of fun tions and states as probability measures and the quantum one with the algebra of all boundedoperators on the Hilbert spa e and states being density operators (matri es).[11℄ To avoid a ompli ated notation I use the same symbol and the same name for an "observable" ("state") as aphysi al, operational notion and an "observable" ("state") as its mathemati al representation.[12℄ R. Ingarden, Der Streit um die Existenz der Welt vol.I, II, III, Niemeyer , Tuebingen (1963, 1965, 1974).[13℄ The symmetry between states and observables appears in the mathemati al framework in the form of dualitybetween states and observables. The states an be treated as linear fun tionals of observables and vi e versa.The alternative- C ∗ versus von Neumann algebras - re(cid:29)e ts this feature on a very abstra t level.[14℄ J. von Neumann, Mathematis he Grundlagen der Quantenme hanik, Springer, Berlin , 1932[15℄ The des ription of "von Neumann theorem" in [7℄ and [8℄ is not fair. For example, J.S. Bell wrote :"Theproblem is posed in the prefa e and on p.209. The formal proof o upies essentially pp. 305-324 and is followedby several pages of ommentary". A tually, the proof of absen e of dispersion-free states o upies 12 lines anda rather informal dis ussion of its onsequen es for the existen e of hidden variables - half of the page (see[14℄).[16℄ Noti e the reversed role of P(Q)-representations in omparison with [1, 4℄.[17℄ J.L. Landau, Phys.Lett.A 120, 54, (1987)[18℄ In all derivations of Bell's CHSH inequality the mathemati al assumption about joint probabilities, usuallyattributed to "lo ality", reads: p ( A i , B j ) = R µ ( dx ) p ( A i | x ) p ( B j | x ) , where µ ( dx ) is a probability distribution ofHV-s. This assumption is equivalent to the existen e of joint probability distribution p ( A , A , B , B ) . Indeedhaving p ( A i | x ) , p ( B j | x ) and µ ( dx ) one an de(cid:28)ne p ( A , A , B , B ) = R µ ( dx ) p ( A | x ) p ( A | x ) p ( B | x ) p ( B | x ) .Starting with p ( A , A , B , B ) one an treat A i , B j as random variables A i ( x ) , B j ( x ) and de(cid:28)ne onditionalprobability distributions p ( A i | x ) , p ( B j | x ) as atomi measures on entrated on the values A i ( x ) , B j ( x ) .[19℄ M. ›ukowski, arXiV:quant-ph/0605034v1[20℄ The missing part of the quantumness test in the experimental setting of [3℄ is the veri(cid:28) ation of the inequality ρ ( A )) ≤ ρ ( B ))