Wigner-PDC description of photon entanglement as a local-realistic theory
aa r X i v : . [ m a t h - ph ] M a r Wigner-PDC description of photon entanglement as a local-realistic theory
David Rodr´ıguez Departamento de F´ısica Aplicada III, Universidad de Sevilla, E-41092 Sevilla, Spain ∗ (Dated: November 21, 2018) (Revised, additions: existence of f () , Γ() now guaranteed for more stringent restrictions)
The Wigner picture of Parametric Down Conversion (works by Casado et al ) can be interpretedas a local-realistic formalism, without the need to depart from quantum mechanical predictions atany step, at least for the relevant subset of QED-states. This involves reinterpreting the expressionsfor the detection probabilities, by means of an additional mathematical manipulation; though suchmanipulation seemingly provides enough freedom to guarantee consistency with the expectable,experimentally testable behavior of detectors, this is, in any case, irrelevant in relation to our mainresult, of a purely mathematical nature. We also include an overview of the consequences of thisframework in relation to: (i) typical Bell experiments; (ii) perhaps the most relevant recent relatedtest (Phys.Rev.Lett. 108, 2012). Additionally, we also propose an interpretation on that apparentlyawkward “subtraction” of the average ZPF intensity at the detection process.
PACS numbers: 03.65.Ud, 03.67.Mn, 42.25, 42.50.Xa
Previous comments
Before anything else, the tests by Brida et al [17] donot disprove the core of the Wigner-PDC approach asdeveloped in [2–8]; it only does so with some detectionmodels proposed to be able to interpret the expressionsin consistency with local-realism (LR). Precisely whatwe prove here is that such (local-realist) interpretationdoes not need to depart from QM at any step (at leastfor the relevant subset of quantum mechanical states), aproperty that former proposals like [16] did not satisfy(not to mention other problems of their own).This said, neither is ours here the ultimate proof thateverything in regard to the Wigner-PDC approach worksperfectly fine. The one and only value of this work isto show that, even still in need to determine furtherchoices both at the physical and purely mathematicallevels, there is room to accommodate all necessary re-strictions. Once more, without any need to depart fromquantum mechanical predictions, provided we stay insidethe subset of QED-states compatible with LR . Further ex-tensions of this formalism, giving rise to new predictionssuch as the “Spontaneous Parametric Up Conversion”proposed in [11] (and apparently also disproved by exper-imental work in [18], only to regain credit again follow-ing some recent reports from two different experimentalgroups [51]) are irrelevant here.I am still convinced that no conclusive proof of the vi-olation of local-realism (LR) has ever been obtained: wehave the absence of proper space-like separation or “local-ity loophole” in experiments with massive particles, andthe so-called “detection loophole” in experiments withphotons, and something related to this last in the latestimportant development ([34], see Secs.V). ∗ Electronic address: [email protected]
Each of these “loopholes” places the corresponding ex-perimental observations within the frontiers of LR, andonce there nothing is happening that cannot (and shouldnot try to) be understood from a purely classical frame-work, or at least from an intermediate one such as theone we deal with here. I use “intermediate” in the sensethat what we have is a quantum formalism, but we arealso incorporating, in an implicit way by considering thevacuum state which admits a well defined joint probabil-ity density as follows from the positivity of the Wignerfunction, the limits imposed by LR.Indeed, those latest developments in [34] are, in a way,even more compelling from the point of view of my claimsthan the usual Bell experiments are: while these last areusually interpreted as proof of the “non-local” nature ofQM, which is a way out to respect pure realism (“some-thing is there even if we do not look at it”), results in[34] would suggest the non-existence of a well definedjoint probability density for the results of a set of mea-surements performed locally upon a system, which is alsothat (the non-existence) of realism itself.After all, non-localities, such as action-at-a-distance,are also present in other branches of physics like for in-stance electromagnetism, but then always reconciled withLR once the full problem is considered (in regard to e.m.,the apparition of retarded potentials), and at least show-ing reasonable properties such a decrease with growingdistance between the parties. In view of [34], it is notany longer a matter of just some instantaneous interac-tion that does not fade away with distance, but some-thing clearly more fundamental: “things are not thereuntil we look at them”.The widespread willingness to accept such a view ofphysical reality before exhausting other alternatives isdifficult to understand: those alternatives have been pro-posed [2–8], including basic elements (a random back-ground of vacuum field fluctuations) that do not lookat all alien from the perspective of classical electromag-netism, and that even some recent works within the or-thodox approach to the field now acknowledge [35]. Be-sides, and to my knowledge, [34] is the only amongstother recent related tests [36, 37] which does not ac-knowledge, explicitly, one or other loophole leading tocompatibility with the local-realistic interpretation.In arxiv:1111.4092
I have already commented uponother recent developments; to the best of my knowledgethe situation remains the same as when this was written:there is no loophole-free evidence of non-locality.But moreover: even if the locality loophole was over-come, in order to be convincing violations (of one or otherinequalities) should be high enough to exclude other loop-holes (local coincidences, see arXiv:1111.4092) or possiblesystematic errors. If it is actually true that local-realism(LR) does not impose any limit on the quantum mechan-ical states that can be prepared in a lab, then there isno reason to expect any special “resistance” of detectionrates to go over the critical values (which is a determinedby LR), neither for technological limitations (detectors)to become so important at precisely that region of thespectrum (detection rates could saturate at any otherpoint well beyond the LR-frontier, and the issue wouldbe more than clear by now).Perhaps it is time to consider the possibility that notall states allowed by QM may have a physical counter-part, in particular those that would yield correlationsdefying LR. Perhaps is also time to start giving creditto models that, even though being quantum-mechanical,may include that restriction built within their formula-tion; here I propose a step in this direction.
I. INTRODUCTION
The Wigner picture of Quantum Optics of photon-entanglement generated by Parametric Down Conversion[1] was developed some years ago in a series of papers [2–8]; recently, the approach has been revitalized producinganother stream of very interesting results [9, 10]. Startingfrom an stochastic electrodynamical description (hence,based on continuous variables: electromagnetic fields de-fined at each and every point of space), by use of theso-called Wigner transformation [12] this model acquiresa form where all expectation values depend on a proba-bility distribution (hence one objectively defined) for thevalue of those fields in the vacuum.Such a picture clearly differs from the usual one inQuantum Information (QInf), based on a discrete de-scription of the photon (a particle) and not on a set offields in a continuum space. For instance, in the Wigner-PDC picture empty polarization channels at either of theexits of a polarizing beam splitter (PBS) become filledwith random components of the vacuum Zero-Point field(ZPF); these random components can for instance giverise to the enhancement of the detection probability forcertain realizations of the state of the fields (certain pho-tons, in the QED language): see note [13]. Phenomena such as this last come as a big difference with the cus-tomary model of the set polarizer-detector is treated justas a “black box”, able to extract polarization informationin principle without the (explicit) intervention of any ad-ditional noise.The Wigner-PDC framework provides an alternative,apparently local-realistic explanation of the results ofmany typically quantum experiments, the result of ameasurement depending on (and only on) the set of hid-den variables (HV) inside its light cone: in this case,both the signal propagating from the source and the ad-ditional noise introduced by the ZPF at intermediate de-vices and detectors. Within the first series of papers[2–8] those experiments included: frustrated two photoncreation via interference [2]; induced coherence and indis-tinguishableness in two-photon interference [2, 3]; Rarityand Tapster’s 1990 experiment with phase-momentumentanglement [3]; Franson’s (original, 1989) experiment[3]; quantum dispersion cancellation and Kwiat, Stein-berg and Chiao’s quantum eraser [4].From the most recent one [9, 10] we can also add, onone side, amongst quantum cryptography experimentsbased on PDC: two-qubit entanglement and cryptogra-phy [9] and quantum key distribution and eavesdropping[9]; on the other, an interpretation for the experimentalpartial measurement of the Bell states generated from asingle degree of freedom (polarization) in [10]. Recentlythere had been other promising developments, which Ido not know in detail though [14].Perhaps needless to say, this ”local realistic” pictureof the quantum experiments takes place within the lim-its (detection efficiencies) where it is already well ac-knowledged that such an explanation may exist; more-over, those limits arise as natural consequences of thetheory: they are simply limiting values of detection ratesfor states of light that are ultimately generated from thevacuum. This last feature is what makes the Wigner ap-proach so interesting; on the other hand, above we haveused “apparently” because that local-realistic interpre-tation, even within the corresponding limits of detectionrates, was until now not devoid of difficulties, in particu-lar related to the detection model (so far merely a one-to-one counterpart with Glauber’s original expressions [15]):as a result of the normal order of operators, there averageintensity due to vacuum fluctuations is “subtracted”, asubtraction that seems to introduce problems related tothe appearance of what could be interpreted as “negativeprobabilities”.That last is the issue that interests us here, one that, asto be expected, did motivate the proposal of several mod-ifications upon the expressions for the detection probabil-ities: for instance, as early as in [5], “our theory is also inalmost perfect one-to-one correspondence with the stan-dard Hilbert-space theory, the only difference being themodification in the detection probability that we pro-posed in relation...”.Such modifications ranged from the mere inclusion oftemporal and spatial integration [5] to the proposal ofmuch more complicated functional dependencies [8, 16,19], all of them seeming to pose their own problems; forinstance in [16], a departure from the quantum predic-tions at low or high intensities, experimentally disprovedfor instance in [17].Our route here is a different one however: we do notpropose any modification of the initial expressions forthe detection probabilities (in one-to-one correspondencewith the initial quantum electrodynamical model), butjust explore the possibility of performing some conve-nient mathematical manipulation that casts them in aform consistent with the axioms of probability, hence oneconsistent with local-realism. Following for instance [5](see also [15]), quantum mechanical detection probabili-ties, single and joint, can respectively be expressed as P i ∝ h I i − I ,i i = Z α,α ∗ ( I i ( α, α ∗ ) − I ,i ) W ( α, α ∗ ) dαdα ∗ , (1) P i,j ∝ Z α,α ∗ ( I i ( α, α ∗ ) − I ,i ) · ( I j ( α, α ∗ ) − I ,j ) × W ( α, α ∗ ) dαdα ∗ , (2)where α, α ∗ are vacuum amplitudes of the (relevant setof) frequency modes [20] at the entrance of the crystal, W ( α, α ∗ ) is obtained as the Wigner transform of the vac-uum state, I i ( α, α ∗ ) is the field intensity (for that mode)and I ,i the mean intensity due to the vacuum ampli-tudes, both at the entrance of the i -th detector (see [20]): I = Z α,α ∗ I ( α, α ∗ ) W ( α, α ∗ ) dαdα ∗ . (3)The last three expressions would in principle allow us toidentify the vacuum amplitudes with a vector of hiddenvariables λ ∈ Λ (Λ is the space of events or probabilisticspace), with an associated density function ρ ( λ ), λ ≡ α, α ∗ , (4) ρ ( λ ) ≡ W ( α, α ∗ ) . (5)Now, for instance in [5] it is already acknowledged that(1)–(2) cannot, because of the possible negativity of thedifference I i ( α, α ∗ ) − I ,i , be written as P i = Z Λ P i ( det | λ ) ρ ( λ ) dλ, (6) P i,j = Z Λ P i,j ( det | λ ) ρ ( λ ) dλ, (7)where naturally P i ( det | λ ) , P i,j ( det | λ ) should stay posi-tive (or zero) always. This last is our point of departure:in this paper we propose a reinterpretation of the for-mer marginal and joint detection probabilities based ona certain manipulation of expressions (1)–(2). The paper is organized as follows. In Sec.II we will con-sider a setup with just a source and two detectors; oncethat is understood, the interposition of other devices be-tween the source and the detectors poses no additionalconceptual difficulty though it is nevertheless convenientto address it in some detail: this will be done in Sec.III. The calculations in these two sections find supporton the proofs provided in Appendix 1, and stand for ourmain result in this paper. Sec. IV explores the ques-tion of “ α -factorability” in the model, not only from themathematical point of view but also providing some morephysical insights on its implications.Up to that point the novel points of the paper are madeand its results are self-contained; it is nevertheless natu-ral to extend our analysis to some of their further impli-cations (amongst these, the consequences for Bell testsof local-realism, and also a recent related proposal), inSec. V, where we also include a preliminary approachto questions regarding the physics of the real detectors.Finally, overall conclusions are presented in Sec. VI, andsome supplementary material is provided in Appendix 2,which may not only help make the paper self-containedbut perhaps also contribute to clarify some of the ques-tions addressed, in particular the non-factorability issue. FIG. 1: Wigner-PDC scheme: photon pair generation, po-larizing beam splitters (PBS) and detectors. Only relevant in-puts of Zero-Point vacuum field (ZPF) are represented in thepicture: “relevant” can be understood, in a classical wavelikeapproach, as “necessary to satisfy energy-momentum conser-vation” for the (set of) frequency modes of interest; in a purelyquantum electrodynamical framework we would be talkingabout conservation of the commutation relations at the emptyexit channels of the devices. Besides, those new ZPF compo-nents introduced at the empty exit channels of PBS’s 1 and2 can alter the detection probability for the signal arriving tothe detectors (for instance giving rise to its “enhancement”);that signal is on the other hand determined (amongst otherhidden variables) by the ZPF components entering the crys-tal. Figure: courtesy of A. Casado.
II. REINTERPRETING DETECTIONPROBABILITIES
Our aim is to adopt here an approach that is delib-erately as abstract as it can be chosen to be, becausewhat we are concerned about is an (apparent) prob-lem of the mathematical structure of the theory, ratherthan other details regarding its connection with physi-cal reality which should in principle be addressed at an-other stage of the investigation; nevertheless, of coursefor the sake of credibility some of these details need tobe brought up, and so we will do when necessary.We now propose a reinterpretation of expressions (1)–(2) which is based on the idea that it is legitimate to playwith internal degrees of freedom of a theory, as far as itsobservable predictions remain all of them invariant tothese transformations. We intend to make a progressiveexploration of the possible consistency conditions, fromless to more demanding, but we advance that in no waythis threatens the validity of our main result here, aswe will be able to provide proof of the existence of therequired solution for all of them.
A. Single detections
Knowing that the following equality holds (from hereon we drop detector indexes when unnecessary), for somereal constant K ( m ) (“marginal”), K ( m ) Z α,α ∗ ( I ( α, α ∗ ) − I ) W ( α, α ∗ ) dαdα ∗ = Z Λ P ( det | λ ) ρ ( λ ) dλ, (8)we realize we do not need to assume P ( det | λ ) ≡ K ( m ) · ( I ( α, α ∗ ) − I ) , (9)as a necessary, compulsory choice; it would be enough tofind some f ( x ) ≥
0, satisfying K ( m ) Z α,α ∗ ( I ( α, α ∗ ) − I ) W ( α, α ∗ ) dαdα ∗ = Z α,α ∗ f ( I ( α, α ∗ )) W ( α, α ∗ ) dαdα ∗ , (10)so we can then safely identify P ( det | λ ) ≡ f ( I ( α, α ∗ )) , (11)with f ( I ( α, α ∗ )) ≥ ∀ I ( α, α ∗ ). This last is nothingbut solving a linear system with only one restriction andan infinite number of free parameters, whose subspaceof solutions intersects the region 0 ≤ f ( I ( α, α ∗ )) ≤ ∀ α, α ∗ : see Appendix 1. B. Joint detections
From the perspective of our approach and regardingjoint detections, it is not very clear which are really theminimum conditions of consistence that one would haveto enforce; in order to proceed with the maximum gener-ality, let us simply define, for detectors i, j and a constant K ( j ) (“joint”), a new function Γ( x, y ), so that K ( j ) Z ( I i ( α, α ∗ ) − I ,i ) · ( I j ( α, α ∗ ) − I ,j ) × W ( α, α ∗ ) dαdα ∗ = Z Γ( I i ( α, α ∗ ) , I j ( α, α ∗ )) W ( α, α ∗ ) dαdα ∗ , (12)which we will now attempt to interpret as P i,j ( det | λ ) ≡ Γ( I i ( α, α ∗ ) , I j ( α, α ∗ )) . (13)Of course, at a first look it seems very desirable to guar-antee that the detection probabilities depend solely onthe amount of intensity at the entrance of each detector;hence it would seem natural to add the conditionΓ( I i ( α, α ∗ ) , I j ( α, α ∗ )) = f ( I i ( α, α ∗ )) · f ( I j ( α, α ∗ )) , (14)i.e., some “factorability” on the incoming intensities. Again, while such an additional condition seems neces-sary in order to preserve the “physical interpretability”of the “inner structure” of the theory, it is not at allsomething necessary from the point of view of its observ-able predictions, as long as these remain invariant; aswe will see later, I j ( α, α ∗ ) may not only be unobservableas corresponds to a particular realization of the randomfields... it may also happens that it does not actually rep-resent a real intensity, but just an average promediatedover another relevant random variable. In Appendix 1 we have shown that it is always possibleto find some suitable Γ satisfying all necessary conditionsto be interpreted as a probability distribution and con-sistent with the observable prediction of the theory re-garding joint detections, eq. (2); and, furthermore, thata solution can be found satisfying also (14).As said, our choice here is to proceed without loss ofgenerality: for this purpose, it is convenient to redefinenow α ≡ α, α ∗ , as well asˆ f i ( α ) ≡ f ( I i ( α )) , (15)ˆΓ i,j ( α ) ≡ Γ( I i ( α ) , I j ( α )) . (16)where we now assume that in general,Γ i,j ( α ) = f i ( α ) · f j ( α ) , (17)The absence of factorability on the α ’s may come perhapsas a surprise to some, given than Clauser-Horne factora-bility [21] on the hidden variable λ is usually taken forgranted; this is a mistake [22], that we have tried to clar-ify in Appendix 2. III. ADDING INTERMEDIATE DEVICES:POLARIZERS, PBS’S...
Once we place one or more devices between the crystaland the detectors, typically polarizers, polarizing beamsplitters (PBS) or other devices to allow polarizationmeasurements (such as in Fig. 1), in general we cannotany longer describe the fields between both with only oneset { α } of mode-amplitudes; we need to redefine our α ’sas now associated to a particular position r (they do notany longer determine a frequency mode for all space [20]).Hence, we will now have α ( r ) ≡ { α k ,γ ( r ) , α ∗ k ,γ ( r ) } , (18)and, letting r s be the position of the source (the crystal),and r i the position of the i -th polarizer or PBS (or anyother intermediate device), we will also redefine α s ≡ α ( r s ) , α i ≡ α ( r i ) , (19)with α i including the relevant amplitudes at the (empty)exit channels of that i -th intermediate device. With α s , α i corresponding each to (a set of) modes with dif-ferent (sets of relevant) wavevectors { k s } and { k i } [23],we can then regard them as two (sets of) statisticallyindependent random variables, with all generality. Theintensity at the entrance of detector i -th will thereforedepend now not only on α s but also on α i : P i ∝ h I i ( α s , α i ) − I ,i i = Z α s Z α i ( I i ( α s , α i ) − I ,i ) W ( α s ) W ( α i ) dα s dα i . (20)Once more, something like that can always be rewrit-ten (see former section), for some suitable and positivelydefined f ′ ( x ), as P i = Z α s Z α i f ′ ( I i ( α s , α i )) W ( α s ) W ( α i ) dα s dα i . (21)Integrating on α i we would obtain P i = Z α s ˆ f ′ i ( α s ) W ( α s ) dα s , (22)from where we define a new function ˆ f ′ i ( α s ). On the otherhand, for joint detections we would have P i,j ∝ Z α s Z α i Z α j ( I i ( α s , α i ) − I ,i ) · ( I j ( α s , α j ) − I ,j ) × W ( α s ) W ( α i ) W ( α j ) dα s dα i dα j , (23)which again can always be rewritten (again see formersection), for some positively defined Γ ′ ( x, y ), as P i,j = Z α s Z α i Z α j Γ ′ ( I i ( α s , α i ) , I j ( α s , α j )) × W ( α s ) W ( α i ) W ( α j ) dα s dα i dα j , (24) and integrating on α i , α j we would obtain P i,j = Z α s ˆΓ ′ i,j ( α s ) W ( α s ) dα s , (25)from where we can again define yet another new proba-bility density function ˆΓ ′ i,j ( α s ).It is interesting for the sake of clarity to compare thetwo situations: with (primed functions) and without po-larizers (unprimed). It is easy to see that, because thedetector only sees the intensity at its entrance channel,clearly (we drop the “s” subscript for simplicity), f ′ ( x ) = f ( x ) , ∀ x, (26)Γ ′ ( x, y ) = Γ( x, y ) , ∀ x, y. (27)while, in consistency with our approach, in generalˆ f ′ i ( α ) = ˆ f i ( α ), as well as ˆΓ ′ i,j ( α ) = ˆΓ i,j ( α ). IV. ON NON-FACTORABILITYA. Mathematical analysis
Let us go back to the case with just the source and thedetectors; we will soon see the following does neverthelessalso apply when polarizers or other devices are added tothe setup, just the same. According to our reasonings inApp. 2, and using (15)–(16), we now realize that thereis no way to avoidˆΓ i,j ( α ) = ˆ f i ( α ) · ˆ f j ( α ) , (28)unless we introduce some additional dependence of thekind ˆ f ( α ) → ˆˆ f ( α, µ ), so that thenˆˆΓ i,j ( α, µ ) = ˆˆ f i ( α, µ ) · ˆˆ f j ( α, µ ) , (29)where we add a second “hat” to avoid an abuse of nota-tion, and where µ stands for a new set of random vari-ables. This ˆˆ f ( α, µ ) should be interpreted as a detectionprobability conditioned to the new vector of random vari-ables µ , i.e, ˆˆ f ( α, µ ) ≡ P ( det | α, µ ) . (30)We will impose further demands on ˆˆ f ( α, µ ), definingˆˆ f ( α, µ ) ≡ f ( I ( α, µ )) , (31)something forced by strictly physical arguments: thechoice f ( I ( α, µ )) must prevail over other possible ones- for instance f ( I ( α ) , µ ) - due to the need to respect thedependence of the probabilities of detection (conditionedto α or not) alone on the intensity that arrives to thedetector, and nothing else.Now, with the density function ρ µ ( µ ), we could write P ( det | α ) = Z µ P ( det | α, µ ) ρ µ ( µ ) dµ, (32) P i,j ( det | α ) = Z µ P i,j ( det | α, µ ) ρ µ ( µ ) dµ, (33)allowing us to recover our former definitions (15)–(16):ˆ f ( α ) ≡ P ( det | α ) , (34)ˆΓ i,j ( α ) ≡ P i,j ( det | α ) . (35)For joint detections, the additional variable µ is particu-larly relevant because, we will always have that while P i,j ( det | α, µ ) = P i ( det | α, µ ) · P j ( det | α, µ ) , (36)in general P i,j ( det | α ) = P i ( det | α ) · P j ( det | α ) , (37)or we could equivalently say that while necessarilyˆˆΓ i,j ( α, µ ) = ˆˆ f i ( α, µ ) · ˆˆ f j ( α, µ ) , (38)in general ˆΓ i,j ( α ) = ˆ f i ( α ) · ˆ f j ( α ) , (39)where of courseˆΓ i,j ( α ) = Z µ ˆˆ f i ( α, µ ) · ˆˆ f j ( α, µ ) ρ µ ( µ ) dµ. (40)To conclude this section, we recover the case with in-termediate devices: due to α i , α j being, as defined, inde-pendent from one another and also from α s , our hypo-thetical “flag” µ cannot be associated with none of them.Therefore, in general, and in principle, not onlyˆΓ ′ i,j ( α ) = ˆ f ′ i ( α ) · ˆ f ′ j ( α ) , (41)but also ˆΓ i,j ( α ) = ˆ f i ( α ) · ˆ f j ( α ) either. We use “in prin-ciple” because this question is not yet analyzed in de-tail; we now see clear, though, that this possible non-factorability on α ’s is nothing more than an internal fea-ture of the model’s mathematical structure, bearing norelevance in regard to its double-sided compatibility (orabsence of it) both with local-realism and the quantumpredicted correlations. A possible candidate for that ad-ditional hidden variable would be, at least in my opinion,the phase of the laser µ [25]. V. COMPLEMENTARY QUESTIONSA. Wigner-PDC’s local realism vs. quantumcorrelations
Though former mathematical developments are fullymeaningful and self-contained on their own, yet it would be convenient to give some hints on how a local-realist(LR) model can account for typically quantum correla-tions, which are known to defy that very same local re-alism (LR). In the first place and as a general answer,what the Wigner-PDC picture proves is that LR is re-spected by a certain subset of all the possible quantumstates, specifically the ones that can be generated from anon-linear mix of the QED-vacuum (which therefore actsas an “input” for the model) with a quasi-classical (ahigh-intensity coherent state), highly directional signal,the laser “pump” (which indeed enters in the model asa non-quantized, external potential [2]). Moreover, sucha restriction is clearly not arbitrary at all, since it arisesfrom a very simple quantum electrodynamical model ofthe process of generation of polarization-entangled pairsof photons from Parametric Down Conversion (PDC): seefor instance eq. (4.2) in [2].
B. Detection rates and “efficiencies”
Aside from subscripts, we will now also drop “hats”and “primes” for simplicity; of course the fact thatΓ i,j ( α ) may not be in general α -factorisable,Γ i,j ( α ) = f i ( α ) · f j ( α ) , (42)does not at all mean that it cannot well satisfy Z Γ i,j ( α ) W ( α ) dα = (cid:20)Z f i ( α ) W ( α ) dα (cid:21) · (cid:20)Z f j ( α ′ ) W ( α ′ ) dα ′ (cid:21) , (43)i.e. (let us from now use superscripts “W” and “exp” todenote, respectively, theoretical and experimental detec-tion rates): P ( W ) i,j ( det ) = P ( W ) i ( det ) · P ( W ) j ( det ) , (44)which is indeed the sense in which the hypothesis of “er-ror independence” is introduced, to our knowledge, inevery work on LHV models [27]. This sort of conditionsover “average” probabilities (average in the sense thatthey are integrated in the hidden variable, may that be α alone or also some other one) are the only ones thatcan be tested in the actual experiment; there, we canjust rely on the number of counts registered on a certaintime-window ∆ T , and the corresponding estimates of thetype P ( exp ) i ( det ) ≈ n. joint det. ( i, j ) in ∆ Tn. marg. det. ( j ) in ∆ T . (45)Now, if we wish to include some additional uncertaintyelement reflecting the technological limitations (a “de-tection efficiency” parameter), what we have to do is toredefine the overall detection probabilities as P ( exp ) i ( det ) ≡ ˆ η i · P ( W ) i ( det ) , (46) P ( exp ) i,j ( det ) ≡ ˆ η i ˆ η j · P ( W ) i,j ( det ) , (47)as well as P ( exp ) i ( det | α ) ≡ ˆ η i · P ( W ) i ( det | α ) , (48) P ( exp ) i,j ( det | α ) ≡ ˆ η i ˆ η j · P ( W ) i,j ( det | α ) , (49)where 0 ≤ ˆ η i ≤ η ’s but also the non-technological contribution. From the point of view ofthe experimenter it is very difficult to isolate both com-ponents (Glauber’s theory [15] does not predict a unitdetection probability even for high intensity signals); weshould perhaps then confine ourselves to the term “ob-servable detection rate” instead of using the clearly mis-leading one of “detector inefficiency”. C. Consequences on Bell tests, theirsupplementary assumptions and critical efficiencies
That said and going to a lowest level of detail, statesin such an (LR) subset of QED can still indeed exhibitcorrelations of the class that is believed to collide withLR, yet the procedure through which they are extractedfrom the experimental set of data does not meet one ofthe basic assumptions required by every test of a Bellinequality: they do not keep statistical significance withrespect to the physical set of “states” or hidden instruc-tions [27]). To guarantee that statistical significance wemust introduce some of the following two hypothesis:(i) all coincidence detection probabilities are independentof the polarizers’ orientations φ i , φ j (this is what we call“fair-sampling” [28], for a test of an homogeneous in-equality [29]), which implies P ij ( det | φ i , φ j , α ) = P ij ( det | α s ) , (50)where of course (see Sec. III) α ≡ α s ⊕ α i ⊕ α j , andwhere we recall that φ i , φ j would determine whichvacuum modes amongst the sets α i , α j would intervenein the detection process,(ii) the interposition of an element between the sourceand the detector cannot in any case enhance the probabil-ity of detection (the “no-enhancement” hypothesis [30],needed to test the Clauser-Horne inequality [21], and pre-sumably every other inhomogeneous one [31]), P i ( det | φ i , α ) ≤ P i ( det |∞ , α s ) . (51)with ∞ denoting the absence of polarizer and with α ≡ α s ⊕ α i this time. Following our developments in Sec. III one can easilysee that (i) is not in general true, and according to [13]neither is (ii). I.e., whenever states of light are preparedso as to produce the sort of quantum correlations that areknown to defy LR, these last come supplemented with thenecessary features that prevent it from happening... howcould it not?Yet, a mere breach of (i) or (ii) is not enough to assertthe existence of a Local Hidden Variables (LHV) model,which is an equivalent way of saying that the results ofthe experiment respect LR: it is more than well knownthat this can only happen for certain values of the ob-served detection rates [27, 32]. However, from the pointof view of (48)–(49), and given the fact that, as proven inSecs II and III, the Wigner-PDC is in all circumstancesin accordance with expressions (71)–(72), and hence toall possible Bell inequalities whether our ˆ η ’s are equal orless than unity, the so-called “critical efficiencies” wouldmerely stand, at least as far as PDC-generated photonsare concerned, for bounds on the detection rates thatwe can experimentally observe (these last in turn con-strained by the only subset of quantum states that wecan physically prepare). D. A recent related test
The recent test in [34] would seem formally equiva-lent to a Bell test of an homogeneous inequality [29],but for two questions: (i) they do not require remoteobservers; (ii) the use of analogical measurements, thatwould seem to exclude the so-called “detection loophole”that is widely recognized in other tests. The purpose ofthis section is to show that such loophole still remains,though making it evident requires a subtler approach, ac-cording to our approach here one perhaps more realisticthan the usual assumption of “random errors”. Besides,and to my knowledge, [34] is the only amongst other re-cent related tests [36, 37] which does not acknowledge,explicitly, one or other loophole leading to compatibilitywith the local-realistic interpretation.Basically, Kot et al ’s proposal in [34] rests on theprobing of some “test function” F ≡ F ( Q , Q , . . . Q N ),where Q m is an outcome obtained when an observableˆ Q m is measured, and where { ˆ Q m } is a set of mutually ex-clusive observables (more precisely, according to eq. (8)also in [34], F involves a set of powers { ( Q m ) n , m, n =1 , . . . N } ). Later, they will be concerned with an “aver-age” β = h F i , which, if a local hidden variables (LHV)model turned out to exist, would have to be expressibleas h F i Λ = Z Λ F ( { Q m } ) · P ( { Q m }| λ ) ρ ( λ ) dλ, (52)with λ as usual a vector of hidden variables and Λ theoverall space of events. The quantity P ( { Q m }| λ ) = f ( Q , . . . Q N | λ ) would stand for a joint probability den-sity function (conditioned to the state λ ) for any set of(predetermined) results for any set of possible measure-ments upon the observables { ˆ Q m } .Each ˆ Q m will be then sampled on a different subsetΛ m ⊂ Λ... which would be no problem as long as all { Λ m } are statistically faithful to Λ; however, the fact thatmeasurements may not always give a result (or that thesemeasurements are effectively completed at different time-stamps) may destroy that statistical significance, henceinvalidating that of the estimate of β . To show that thismay happen even in view of (ii), we must go to Glauber’sexpression in (87). The key is that, for a given time-stamp, in general P ( t ) <
1, though of course after acertain interval ∆ t the accumulated probability, P det (∆ t ) = Z t +∆ tt P det ( τ ) dτ, (53)may approach unity, something irrelevant for our argu-ment: the hypothetical 0-instructions in a hypotheticalLHV model (see for instance [27]) would no longer haveanything to do with some “detection inefficiency”, butsimply express the fact that for some given time-stampand observable, P ( t ) may be less than unity.Other models of “detection” may involve a time-integral of the intensity at the entrance of the detector,or other sophistications; they would not affect our argu-ment as long as we recognize (87) as a time-dependentquantity, in general satisfying P ( r , t ) <
1. Once here,the physical connection with the test in [34] can be es-tablished by assigning to each time stamp t a set M t of hidden instructions of the usual kind (i.e., a differentLHV model M t for each t ): again see [27].In particular, following [38], a detection on the “signal”arm, at a time-stamp t , prompts the analysis of the signal(coming from the homodyne setup and entering a highfrequency oscilloscope) at the “idler” one, over a fixedtime window. Yet, a correlation between the detectiontime-stamp at the signal arm and the choice of observ-able at the idler would seem to require communicationor “signaling” between the two measurement setups.This last difficulty, which would burden our argumentwith the need for an additional “locality loophole”, can,however, be overcome too: consider an observable ˆ Q that(always) produces two possible results Q = { q , q } . Forsimplicity let us this time denote P Q ( q | λ ) the probabilityof an outcome q when Q is measured on the state λ .Then, h Q i Λ = q · P Q ( q | Λ) + q · P Q ( q | Λ) P Q ( det | Λ) ; (54)however, if each result is associated to a detection at adifferent set of time-stamps, { t i, } , { t i, } , then the exper-imentally accessible quantity is h Q i ob = q · P Q ( q | Λ ( Q )1 ) + q · P Q ( q | Λ ( Q )2 ) P Q ( det | Λ) . (55)In this last expression, Λ ( Q )1 , Λ ( Q )2 ⊂ Λ are again two dif-ferent sets selected by the correlation between the time- stamp at one arm and the result of the measurement atthe other when ˆ Q is measured.According to the approach in [2–8], the set of relevantvacuum electromagnetic modes inserted at the source arestill contained in the fields arriving at each detector, andthis will indeed induce some correlation between detec-tion time-stamp at one side and a measurement’s out-come at the other, and in general between any two eventswhich are related to the intensity of the incoming signal.This induced correlation is also at the core of other re-cent results in [9, 10]. A more detailed argument can beobtained by e-mail from the author. E. An approach to realistic detectors and averageZPF subtraction
This section approaches some physical considerationswith the aim of showing that there is plenty of room for asuitable physical interpretation of the model, even whenthat is not strictly necessary for the coherence of ourresults, at least from the purely mathematical point ofview. Indeed, we have already descended to the physicallevel when we established, in former sections, the depen-dence of our f ’s and Γ’s solely on the intensity arriving atthe detector. Expectable behavior for a physical devicewould typically include a “dead-zone”, an approximatelylinear range and a “saturation” at high intensities (thisis indeed the kind of behavior suggested for instance in[16]); amongst other restrictions this would imply, for in-stance, f ( I ( α )) = 0, when I ( α ) ≤ ¯ I (this last a thresh-old that may even surpass the expectation value of theZPF intensity), as well as Γ( I i ( α ) , I j ( α )) = 0 either for I i ( α ) or I j ( α ) below ¯ I .Neither these restrictions nor other similar ones wouldin principle invalidate our proofs in Appendix 1, whichseem to provide room enough to simulate a wide range ofpossible behaviors; however, we must remark that none ofour f ’s and Γ’s can ultimately be considered as fully phys-ical models, due to the fact that they represent point-likedetectors (the implications of such an over-simplificationmay become clearer in a moment).In close relation to the former, we also propose here asimple physical interpretation of the term − I appearingin the expressions for the detection probabilities. Fromthe mathematical point of view such subtraction arisesfrom a mere manipulation of Glauber’s original expres-sion [15]. From the physical one, a realistic interpretationwould be more than desirable, as that subtraction of ZPFintensity is for instance crucial to explain the absence ofan observable contribution of the vacuum field on the de-tectors’ rates [33]; of course we mean “explain in physicalterms”; from the mathematical point of view our modelhere already predicts a vanishing detection probabilityfor the ZPF alone.My suggestion is that the − I term must be (at least)related to the average flux of energy going through thesurface of the detector in the opposite direction to thesignal (therefore leaving the detector). That interpreta-tion fits the picture of a detector as a physical systemproducing a signal that depends (with more or less pro-portionality on some range) on the total energy (intensitytimes surface times time) that it accumulates. I am justsaying “at least related”, bearing in mind that to estab-lish such association we would first have to refine thepoint-like model of a detector which stems directly fromthe original Glauber’s expression [15]. VI. CONCLUSIONS
We have shown that the Wigner description ofPDC-generated (Parametric Down Conversion) photon-entanglement, so enthusiastically developed in the latenineties [2–8] but then ignored in recent years, can bereformulated as entirely local-realistic (LR). A formal-ism that is one-to-one with a quantum (field-theoretical)model of the experimental setup can be cast, thanks toan additional manipulation (also one-to-one), into a formthat respects all axiomatic laws of probability [39], andtherefore LR, as defined for instance in Sec. 2 a. Theoriginal quantum electrodynamical model takes as an in-put the vacuum state, which accepts a well defined prob-abilistic description through the so-called Wigner trans-form: this is the fact that the analogy with a local-realistic theory is conditioned to. What we call Wigner-PDC accounts, then, for a certain subset within the spaceof all possible QED-states, determined by a particular setof initial conditions and a certain Hamiltonian governingthe time-evolution [48], restrictions that seem to guaran-tee (according to us here) the compatibility with LR.Aiming for the maximum generality, as well as toavoid some possible (still under examination) difficultieswith factorisable expressions, we have renounced to whatwe call α - factorability of the joint detection expressions.Such a choice is not only perfectly legitimate [22], butmay also be supported by a well feasible interpretation(see Appendix IV); nevertheless, further implications ofthat non-factorability on α will also be left to be exam-ined elsewhere. Again, whatever they finally turn out tobe, they are also irrelevant for our main result in thispaper: the Wigner-PDC formalism can be cast into aform that respects all axiomatic laws of probability forspace-like separated events.Neither does the explicit distinction between the caseswhere polarizers (or other devices placed between thesource and the detectors) are or not included in the setupintroduce any conceptual difference from the point ofview of our main result; however, the question opensroom to remark some of the main differences of theWigner-PDC model (actually, also its Hilbert-space ana-logue) with the customary description used in the fieldof Quantum Information (QInf): here, each new deviceintroduces noise, new vacuum amplitudes that fill eachof its empty polarization channels at each of its exists, incontrast to the usual QInf “black box”, able to extract polarization information from a photon without any in-deterministic component.As a matter of fact, those additional random compo-nents may hold the key to explain the variability of thedetection probability (see note [13]) that is necessary,from the point of view of Bell inequalities, to reconcilequantum predictions and LR (see Sec. V A). In particu-lar, the immediacy with which the phenomenon of “de-tection probability enhancement” arises in the Wigner-PDC framework would suggest that this may be after allthe right track to understand why after several decadesthe minimum detection rates (or, in QInf terminology,critical detection efficiencies) that would lead to obtainconclusive evidence of non-locality are still out of reach(see an explanation of what these critical rates are in[32], for comments on the current state of the questionsee somewhere else).In Sec.V D we have also addressed a recent no-gotest that I consider particularly relevant, amongst otherthings because it does not explicitly acknowledge anyloophole. In Sec. V E we have done a first, general ap-proach on the question of whether our reinterpretationof the detection probabilities is consistent with the ac-tual physical behavior of detectors. A closer look to thisquestion is out of the scope of the paper; former proposals[16] in regard to this issue aimed perhaps too straight-forwardly to the physical level, while they did not evenguarantee consistency with the framework we have set-tled here (proof of this is that it introduces divergences inrelation to the purely quantum predictions, divergenceslater experimentally disproved for instance in [17]). Be-sides, we have also suggested a possible simple physicalinterpretation of I -subtraction taking place in the ex-pressions for the detection probabilities: work in any ofthese directions would anyway seem to require a depar-ture from the point-like model of a detector.Summarizing, we have shown that a whole family ofdetection models M det ≡ { f ( I ( α, µ )) , Γ( I i ( α, µ ) , I j ( α, µ ) } , (56)can be found, consistent both with the quantum mechan-ical expressions from the Wigner-PDC model and LR.A close examination of the constraints coming from thephysical behavior of the detectors and other experimen-tally testable features is left as a necessary step for thefuture, with the aim of establishing a subset physicallyfeasible ones; nevertheless, we have the guarantee thatall of them produce suitable predictions, as so does theirquantum electrodynamical counterpart.Yet, even at the purely theoretical level some otherfeatures remain open too: as a fundamental one, towhat extent the model requires what we have called non-factorability on α ’s. Once more and after all, QM isjust a theory, a theory that provides a formalism uponwhich to build models, models than can (and should) berefined based on experimental evidence, something that(again) also applies just as well to the one we are dealingwith here [49]. Perhaps the particle properties of light0are not enough to assume that the current model of “aphoton” is the best representation (and most complete)that we can achieve of light; many of those properties canbe understood, I believe, from entirely classical models[50], as well. Quantum entanglement seems to manifestin many of its “reasonable” features but at least as (thisparticular model of) Parametric Down Conversion is con-cerned and so far, whenever local-realism would seem tobe challenged new phenomena can be invoked so as toprevent, at least potentially, that possibility. Such are“unfair sampling”, “enhancement” (as a particular caseof variable detection probability) and over all detectionrates low enough to open room for the former two.Those phenomena find theoretical support in theWigner-PDC picture, but definitely not in the usual,based merely on the correspondence between the -spinalgebra for massive particles and the polarization statesof a plane wave (a photon then looks just as a -spinparticle, what some like to call a two-level system, butfor the magnitude of the angular momentum it carries,and its statistics of course). To explore, and exhaust ifthat is the case, alternative routes such as the one hereis not only sensible but also necessary. Acknowledgments
I thank R. Risco-Delgado, J. Mart´ınez and A. Casadofor very useful discussions, though most of them at amuch earlier stage of the manuscript. They may or maynot agree with either the whole or some part of my con-clusions and results, of course.
Appendix1. Auxiliary proofs
Lemma 1 : There always exists f ( x ) satisfying (10), with ≤ f ( x ) ≤ , for x = I ( α, α ∗ ) over the space of allpossible pairs { α, α ∗ } .Proof: We have a linear system with one restriction andinfinite variables: the set of values { f ( I ( α, α ∗ )) } ; coeffi-cients are given by the W ( α, α ∗ )’s, and the independentterm is the left-side term of (10), 0 ≤ P s ≤ V wherewe associate each value f ( I ( α, α ∗ )) per coordinate: wecan assume dim( V ) = N ( N → ∞ ) if we considera direct dependence on the α ’s, or we can also saydim( V ) = N (again N → ∞ ) if the dependence is definedupon the intensities, in principle a real number (dependson whether we choose to formulate the model upon in-stantaneous values, therefore real, or as a complex am-plitude, this is not essential here). Both options, f ’sdepending either on { α } or intensities { I ( α, α ∗ ) } serveright for our purposes, the second being perhaps more appropriate from the point of view of physical interpre-tation. If they do exist, compatible solutions f ( I ( α, α ∗ ))for the (unique) restriction will conform a linear manifold M ⊂ V . Now we have to see:(i) Due to W ( α, α ∗ ) > α, α ∗ [40], and P s ≥ M cannot be parallel with any of the coor-dinate hyper-planes in V ; i.e., M intersects all of them,defined each (each one for a pair α, α ∗ ) by the equation f ( I ( α, α ∗ )) = 0.(ii) Moreover, for P s > M has a non-trivial intersec-tion (more than one point, the origin) with all coordinateplanes inside the first hyper-quadrant V q (the subregion V q ⊂ V given by restricting V to f ( I ( α, α ∗ )) ≥ f ’s zero except one (alwayspossible because all the W ’s are strictly above zero), wecan see the crossings always take place at the positivehalf of the corresponding coordinate axis, therefore atthe boundary of V q . On the other hand, for P s = 0 thesolution for the system is trivial.With (i) and (ii), it is clear that under the restriction(under the set of inequalities) f ( I ( α, α ∗ )) ≤ ∀ α, α ∗ , theset of admissible solutions is still not empty, as guaran-teed by Z α,α ∗ W ( α, α ∗ ) dαdα ∗ = 1 , (57)and P s ≤
1, as we will now show. Here we give an induc-tive reasoning; let us consider the equivalent problem in3 dimensions, with ax + by + cz = d ; (58)clearly, for a, b, c, d ≥ x = y = z at a point of coordinates x = y = z = d/ ( a + b + c ) = d, (59)which is obviously in the first quadrant ( x, y, z ≥ a + b + c = 1 and 0 ≤ d ≤
1, clearlyalso inside of the region { ≤ x ≤ , ≤ y ≤ , ≤ z ≤ } . The extension to infinite dimensions, topologicalabnormalities all absent, is direct, with x ≡ α, α ∗ (oralternatively x ≡ I ( α, α ∗ ), a, b, c ≡ W ′ s and d ≡ P s ≤ Lemma 2 : There always exists Γ( x, y ) satisfying (12),with ≤ Γ( x, y ) ≤ , for x = I i ( α, α ∗ ) , y = I j ( α, α ∗ ) over the space of { α, α ∗ } .Proof: formally identical with Lemma 1. Lemma 3 : There always exists f ( x ) satisfying simultane-ously (10) for a finite collection of i = 1 , . . . , N detectors,with ≤ f ( x ) ≤ , for x = I i ( α, α ∗ ) , and satisfying, forany two pair of detectors, condition (14) as well.Proof: First we notice that condition (10) for each addi-tional detector stands for a new linear restriction on theproblem already treated in
Lemma 1 ; due to the system1being under-determined with an infinite number of freeparameters this poses no problem.However, condition (14) is not a linear but a non-linearrestriction. Consider first the case N = 2 (number of de-tectors), and let us solve the system under the 2 restric-tions (one per detector) of the kind (10), obtaining a solu-tion in accordance with Lemma 1 ; such solution, as seen,depends on an infinite set of free parameters. Let us givevalues to all these free parameters (from Lemma 1,suchvalues can be chosen so as to guarantee 0 ≤ f ( x ) ≤ , ∀ x )but one, which we will define as f ( I Θ ) = Θ , (60)for some particular (randomly chosen) value I ( α ) = I Θ (the same value of I ( α ) can be produced by several α ’s, soit is clearer to associate “labels” to I instead of α , thoughconceptually botch choices are equivalent); condition (14)between the two detectors stands now for a quadraticequation, A · Θ + B ( I Θ ) · Θ + C ( I Θ ) = D ; (61)before we make the former terms explicit, let Ω be theset configured by all possible values of α , and define thesubsets ˆ¯Ω( I Θ ) ≡ { α ; I ( α ) = I ( α ) = I Θ } , (62)¯Ω( I Θ ) ≡ { α ; I ( α ) = I Θ , I ( α ) = I Θ } , (63)ˆΩ( I Θ ) ≡ Ω − ˆ¯Ω( I Θ ) − ¯Ω( I Θ ) , (64)so that now we can write A ( I Θ ) = Z ˆ¯Ω( I Θ ) dα W ( α ) , (65) B ( I Θ ) = Z ˆΩ( I Θ ) dα W ( α ) · [ f ( I ( α )) + f ( I ( α ))] , (66) C ( I Θ ) = Z ¯Ω( I Θ ) dα W ( α ) · f ( I i ( α ) · f ( I j ( α )) . (67)Taking now into account that intensities are continuousfunctions of α it is reasonable to assume A ( I Θ ) ≈
0; (68)and now with 0 ≤ B, C, D ≤ C ≤ D (justlook at eq.61), and finally the equivalence of B to a meresingle detection probability (eq. 10 for instance), we cannow write B ( I Θ ) ≥ D ≥ D − C ( I Θ ) ≥ , (69)from where it is not difficult to see that eq.(61) alwaysadmits a solution 0 ≤ Θ ≤
1, where we recall the defini-tion of the free parameter as Θ ≡ f ( I Θ ).Extension to N detectors is inmediate by considering N −
2. Basic concepts revisited
We first briefly revisit the concepts of locality, deter-minism and factorability; a good understanding on theseconcepts is crucial for the main results of the paper, whatmakes this review not only convenient but almost un-avoidable, especially given the presence of some confusionin the literature. In any case, it shall be clearly under-stood that
Clauser and Horne’s factorability [21] is nota requisite for local-realism. a. Locality and realism
A theory predicting the results of two measurements A and B that take place under causal disconnection (rel-ativistic space-like separation) can be defined as local ifand only if we can write A = A ( λ, φ A ) , B = B ( λ, φ B ) , (70)where λ is a (set of) hidden (or explicit) variables definedinside the intersection of both light cones, and φ A , φ B are another two other sets of variables (amongst themthe configurable parameters of the measuring devices)defined locally at A and B , respectively, and causallydisconnected from each other, i.e., P ( A = a | λ, φ A , φ B ) = P ( A = a | λ, φ A ) , (71) P ( B = b | λ, φ A , φ B ) = P ( B = b | λ, φ B ) . (72)These last two expressions are usually taken as a defini-tion of local causality [41].Now, realism simply stands for λ (and φ A , φ B as well)having a well defined probability distribution. A set ofphysical observables corresponding to a particular quan-tum state can be sometimes described by a well definedjoint probability density (such is the case of field ampli-tudes in any point of space for the vacuum state in QED);for other quantum states that is not possible though. b. Determinism A measurement M upon a certain physical system,with k possible outcomes m k , is deterministic on a hiddenvariable (HV) λ (summarizing the state of that system),if (and only if) P ( M = m k | λ ) ∈ { , } , ∀ k, λ, (73)which allows us to write M ≡ M ( λ ) , (74)and indeterministic iff, for some λ , some k ′ , P ( M = m k ′ | λ ) = { , } , (75)2i.e., at least for some (at least two) of the results for atleast one (physically meaningful) value of λ .Now, indeterminism can be turn into determinism, i.e.,(75) can into (73), by defining a new hidden variable µ ,so that now, with γ ≡ λ ⊕ µ : P ( M = m k | γ ) ∈ { , } , ∀ k, γ, (76)which means we can write, M ≡ M ( λ, µ ) , (77)a proof that such a new hidden variable µ can always befound (or built) given in [42].So far, then, our determinism and indeterminism areconceptually equivalent, though of course they may cor-respond to different physical situations, depending for in-stance on whether γ is experimentally accessible or not. c. Factorability Let now M = { M i } be a set of possible measurements,each with a set { m i,k } of possible outcomes, not neces-sarily isolated from each other by a space-like interval.We will introduce the following distinction: we will say M is(a) λ -factorisable , iff we can find a set { ξ i } of randomvariables, independent from each other and from λ too ,such that µ = M i ξ i , (78)and (73) holds again for each M i on γ i ≡ λ ⊕ ξ i : P ( M i = m i,k | γ i ) ∈ { , } , ∀ i, ∀ k, γ i , (79)this last expression meaning of course that we can write,again for any of the M i ’s, M i ≡ M i ( λ, ξ i ) . (80)(b) non λ -factorisable , iff (79) is not possible for any setof statistically independent ξ i ’s.We will restrict, for simplicity, our reasonings to justtwo possible measurements A, B ∈ M , with two possibleoutcomes,
A, B ∈ { +1 , − } , all without loss of gener-ality. We have seen that, as the more general formula-tion, we can always write something like A = A ( λ, ξ A ), B = B ( λ, ξ B ). Lemma 1 –(i) If A and B are deterministic on λ , i.e., (73) holds for A and B , then they are also λ -factorisable, i.e., P ( A = a, B = b | λ ) = P ( A = a | λ ) · P ( B = b | λ ) , (81)for any a, b ∈ { +1 , − } . Eq. (81) is nothing but theso-called Clauser-Horne factorability condition [21]. (ii) If A and B are indeterministic on λ , i.e., if (73) doesnot hold for λ , then : for some µ (always possible to find[42]) such that now (76) holds for γ ≡ λ ⊕ µ , A, B are γ -factorisable, P ( A = a, B = b | γ ) = P ( A = a | γ ) · P ( B = b | γ ) , (82)i.e., (81) holds for γ , this time not necessarily for λ .(iii) Let (79) hold for A, B , on λ, ξ A , ξ B : if λ, ξ A , ξ B arestatistically independent, (hence, A and B are what wehave called λ -factorisable), then (81) holds for λ , notnecessarily on the contrary. Proof –(i) When (73) holds, for any λ and any a, b ∈ { +1 , − } , P ( A = a | λ ) , P ( B = b | λ ) ∈ { , } , from where we can,trivially, get to (81).(ii) It is also trivial that, if (76) holds, (81) can be recov-ered for γ . That the same is not necessary for λ can beseen with the following counterexample: suppose, for in-stance, that for λ = λ , either A = B = 1 or A = B = − P ( A = B = 1 | λ ) = P ( A = 1 | λ ) · P ( B = 1 | λ ) , (83)numerically: = .(iii) We have, from independence of λ, ξ A , ξ B , and work-ing with probability densities ρ ’s: ρ λ ( λ, ξ A , ξ B ) = ρ λ ( λ ) · ρ A ( ξ A ) · ρ B ( ξ B ), which we can use to write P ( A = a, B = b | λ ) = Z P ( A = a, B = b | λ, ξ A , ξ B ) × ρ A ( ξ A ) · ρ B ( ξ B ) dξ A dξ B . (84)and now with the fact that we can recover (76) for A ( B )on γ A = λ ⊕ ξ A ( γ B = λ ⊕ ξ B ), P ( A = a, B = b | λ )= Z P ( A = a | λ, ξ A ) · P ( B = b | λ, ξ B ) × ρ A ( ξ A ) · ρ B ( ξ B ) dξ A dξ B = Z P ( A = a | λ, ξ A ) · ρ A ( ξ A ) dξ A × Z P ( B = b | λ, ξ B ) · ρ B ( ξ B ) dξ B = P ( A = a | λ ) · P ( B = b | λ ) . (85)On the other hand, let λ, ξ A , ξ B be not statistically in-dependent: we can set for instance, as a particular case, ξ i ≡ µ , ∀ i , therefore reducing our case to that of (76).Once this is done, our previous counterexample in (ii) isalso valid to show that factorability is not necessary for λ here.3 [1] Parametric Down Conversion (PDC): a pair of entangledphotons is obtained by pumping a laser beam into a non-linear crystal.[2] A. Casado, T.W. Marshall and E. Santos. J. Opt. Soc.Am. B , 494 (1997).[3] A. Casado, A. Fern´andez-Rueda, T.W. Marshall, R.Risco-Delgado, E. Santos. Phys. Rev. A , , 3879 (1997).[4] A. Casado, A. Fern´andez-Rueda, T.W. Marshall, R.Risco-Delgado, E. Santos. Phys. Rev. A , , 2477 (1997).[5] A. Casado, T.W. Marshall, E. Santos. J. Opt. Soc. Am.B , 1572 (1998).[6] A. Casado, A. Fern´andez-Rueda, T.W. Marshall, J.Mart´ınez, R. Risco-Delgado, E. Santos. Eur. Phys. J. D , 465 (2000),[7] A. Casado, T.W. Marshall, R. Risco-Delgado, E. Santos. Eur. Phys. J. D , 109 (2001).[8] A. Casado, R. Risco-Delgado, E. Santos. Z. Naturforsch. , 178 (2001).[9] A. Casado, S. Guerra, J. Pl´acido.
J. Phys. B: At. Mol.Opt. Phys. , 045501 (2008).[10] A. Casado, S. Guerra, J. Pl´acido. Advances in Mathe-matical Physics
J. mod. Optics , 1273 (2000).[12] The Wigner transform applies a quantum state ρ in areal, mulitivariable function: W [ ρ ] : ρ → W ( { x, p } H ) ∈ R , (86)where the set { x, p } H of variables upon which W ( · ) takesvalues depends on the structure of H , the (quantum me-chanical) Hilbert space of the system. Under certain re-strictions (for a subset of all possible quantum states), W [ ρ ] can be interpreted as a (joint) probability densityfunction. For instance see:Y.S. Kim, M.E. Noz. “Phase space picture of QuantumMechanics (Group theoretical approach)”, Lecture Notesin Physics Series - Vol , ed. World Scientific.[13] For instance, from E. Santos, “Photons Are Fluctuationsof a Random (Zeropoint) Radiation Filling the WholeSpace”, in The Nature of Light: What Is a Photon? ,p.163, edited by Taylor & Francis Group, LLC (2008):“one of the additional hypotheses used, introduced byClauser and Horne with the name of “no-enhancement”,is naturally violated in SO because a light beam crossinga polarizer may increase its intensity, due to the insertionof ZPF in the fourth channel (...), which is the possibilityexcluded by the no-enhancement assumption”.[14] See in arxiv, Casado et al .[15] The expressions for the detection probabilities in theWigner-PDC picture can be obtained by mere manipu-lation of Glauber’s original expressions for marginal andjoint detections, respectively: P i ∝ h ψ | ˆ E ( − ) i ˆ E (+) i | ψ i , (87) P i,j ∝ h ψ | ˆ E ( − ) i ˆ E ( − ) j ˆ E (+) j ˆ E (+) i | ψ i . (88)with | ψ i representing the state of the fields in all space,the field operator ˆ E ( − ) i ( ˆ E (+) i ) defined at the position ofthe i -th detector r i and containing only creation (anni-hilation) operators, therefore giving rise to expressions in the so-called normal order of quantum operators. See[2, 5] for details.[16] A. Casado, T.W. Marshall, R. Risco-Delgado, E. Santos.“A local hidden variables model for experiments involv-ing photon pairs produced in parametric Down Conver-sion”, arXiv:quant-ph/0202097v1 (2002).Roughly speaking, their proposal can be understood asa substitution of our function f ( I ( α )) ≡ f ( I ( α ) − I ) bya new one f trial ( I ( α ) − I ( α )) ≡ ˆ f trial ( α ) ≡ (1 − e − ζ · [ ¯ I ( α ) − ¯ I ( α ) ]) · Θ (cid:2) ¯ I ( α ) − I m (cid:3) , (89)where ζ, I m are parameters, Θ is the Heaviside func-tion, I m > I represents a “threshold” intensity and¯ I ( α ) , ¯ I ( α ) correspond to our intensities I ( α ) , I ( α ), ex-cept for the fact that they are calculated performing aprevious spatial and temporal integration (on the respec-tive angular and temporal windows of observation) of thefield complex amplitude.The differences between the former proposal and our ap-proach are two. First, the inclusion of spatial/temporalintegration; regardless of its relevance in relation to thephysical behavior of detectors [33] (we insist that thatis not our focus here), the (quantum electrodynamical)model from where we depart (for instance see eq. 4.2 in[2]) involves indeed a “point-like” model of a detector: itis therefore at that stage were proper refinements shouldbe done, and we justified in leaving the question asidefor further works. As a second one, f trial deviates fromthe quantum predictions both at low and high intensi-ties [24], consistently with the fact that neither f trial northe models in [19] belong to our class of acceptable func-tions f (guaranteeing a one-to-one correspondence withthe initial quantum electrodynamical model).Indeed, we are left to wonder how our much less am-bitious but certainly necessary step was not properly at-tacked before; it is our view that a project which involvesunsolved problems both at the purely mathematical andphysical levels should be approached (at least) in twosteps. Here we have taken the first of them (regardingthe purely mathematical issues); the second would standfor applying all the necessary constraints to reproducethe actual physical behavior of detectors.[17] G. Brida, M.Genovese, M. Gramegna, C. Novero, E.Predazzi, arxiv.org/abs/quant-ph/0203048 (2002).[18] G. Brida, M.Genovese, J. mod. Optics t, r , one particular mode of the ZPF, with k thewave-vector and γ the two possible orthogonal polariza-tion states, is always expressible as E ( α, α ∗ ) = X k ,γ h α k ,γ e − i ( ωt − k · r ) + α ∗ k ,γ e i ( ωt − k · r ) i , (90)where the (stochastic) amplitude α k ,γ determines, for allspace, a mode ( k , γ ). Of course, we can always define α k ,γ ( r ) = α k ,γ ( r = 0) e i k · r , (91) so that now, for a certain mode ω and at a particularpoint of space: E ( r ) = X k ,γ h α k ,γ ( r ) e − iωt + α ∗ k ,γ ( r ) e iωt i , (92)where it is obvious that the amplitudes α k ,γ ( r ) followexactly the same probability distribution as α k ,γ , i.e.,their distribution is governed by W ( α, α ∗ ). This all saidand for the sake of simplicity, we shall abuse notation (asis done customarily in [2–8]) with α, α ∗ ≡ { α k ,γ , α ∗ k ,γ } (93)i.e., α, α ∗ denotes a set of mode amplitudes (therefore inprinciple defined for all space) for a given ω , (one or a setof) relevant wave-vectors k and (the relevant) polariza-tions components γ . Finally, of course the overall vacuumfield is obtained as a sum of all modes.[21] J.F. Clauser, M.A. Horne. Phys. Rev. D , 526 (1974).[22] The so-called factorability condition [21], which for what-ever two space-like separated observables A, B wouldread, in our notation, P ( A = a, B = b | λ ) = P ( A = a | λ ) · P ( B = b | λ ) , (94)is not, as sometimes assumed, a necessary one ( ∀ a, b, λ ),unless the outcomes of the measurements (either numeri-cal ones or simply whether there is going to be a detectionof not) are completely deterministic on λ .For instance, suppose that for λ = λ , either A = B = 1or A = B = 0 with equal probability; it is easy to seethat P ( A = B = 1 | λ ) = P ( A = 1 | λ ) · P ( B = 1 | λ )( = ).We have addressed thoroughly this question in App. 2;anyway and as shown there, the assumption of (94) inthe case of [21] is nevertheless not criticizable, since onecan always find (or define) a new hidden variable thatguarantees it.[23] In the present framework, each wave-vector k defines aplane wave propagating through the entire coordinatespace, which obliges us to take some precautions. Let { k s } and { k i } be the sets of relevant wavevectors for thesets of relevant amplitudes { α s } and { α i } , respectively,then we will suppose that { k s } ∩ { k i } = Ø; in the (re-ally very unlikely) case that this did not happen, we canalways redefine α i so the independence with { α s } stillholds.[24] For low intensities, the departure from the quantum me-chanical model stands for a certain “dead-zone” and amuch higher dark count rate.At first order in perturbation theory, the expectationvalue of the number of photons is zero and therefore suchrate vanishes: experimentally observed dark counts areusually associated to thermal effects or electronic noise.Besides, a saturation effect is introduced, which on theother hand is a feature of all real detectors, but in thiscase it appears at much lower intensities (for energieslower than one photon). Consistency with experimentaldata demands compliance with certain relations providedin that same paper [16].[25] Indeed, the laser is described by a coherent state, whichfor a high intensity signal means that it can be regardedas quasi-classical wave with well defined amplitude andphase (and introduced as a non-quantized external po-tential, see [2] for instance). The phase of this complex amplitude is for instance rele-vant in expressions (4.10) from [2], and has the potentialto interfere constructive or destructively with the α ’s, in-creasing or decreasing the overall intensity of the signal,therefore modifying the detection probabilities. From thispoint of view, such phase cannot be at all excluded fromthe vector of relevant hidden variables and therefore non-factorability (of detection probabilities) in α is the mostnatural feature to expect (though not strictly necessary),see our detailed analysis in Sec. IV.[26] R. Risco-Delgado, private communication.[27] For reference on how, and under which hypothesis LHVmodels are built:(i) A. Cabello, D. Rodr´ıguez, I. Villanueva. Phys. Rev.Lett. , 120402 (2008),(ii) A. Cabello, J. -˚A. Larsson, D. Rodr´ıguez. Phys.Rev. A , 062109 (2009).[28] The fair sampling assumption is already formulated in[44]: “given a pair of photons emerges from the polariz-ers, the probability of their joint detection is independentof the polarizer orientations”.[29] We believe the distinction between homogeneous and inhomogeneous inequalities was first introduced byM.Horne, then later updated by E.Santos ( ref ). We willsupersede previous definitions with our own here: we willcall inhomogeneous inequalities all the ones involving co-incidence rates of different order (for instance, marginaland joint ones), homogeneous on the contrary.Of course, if for instance a certain subset of the observ-ables involved in the inequality does not have any un-certainty associated to its detection, the inclusion of in-homogeneous terms may not lead to the requirement ofsupplementary assumptions; this is not the case of exper-iments with photons, anyway.[30] The no-enhancement assumption [21]: “for every emis-sion λ , the probability of a count with a polarizer in placeis less or equal to the probability with the polarizer re-moved”, where λ is the (hypothetical) hidden variableexpressing the state (at least the initial one) of the pairof particles. We remark: “for every emission λ ”.[31] An inhomogeneous Bell inequality [29] requires the es-timation of coincidence rates of different order (for in-stance single and double, or double and triple); due tothe fact that there is no way to identify if the whole setof particles have been simultaneously emitted and thengone undetected, or they were simply never emitted atall, the test would always require supplementary assump-tions; of course from a wave-like perspective, the issue iseven more evident. In any case, marginal detection rateswould not be experimentally accessible unless we assume“independent errors” at the detector for every and eachof the “states” in the model (i.e., detection probabilitiescannot depend on the hidden variables: α ’s, other suchas the phase of the pump. etc).[32] Let us consider a Bell experiment and let η be what peo-ple in QInf define as “detection efficiency” [27] (from ourpoint of view this is clearly a misleading term, we shouldcall it simply “detection rate”, whether is reduced or notby technological imperfections, see Sec. V A). A possibleLocal Hidden Variables (LHV) model for the experimentis then composed by a set of states { s } with probabilis-tic weights (or frequencies) ρ s : all restrictions the modelmust satisfy (either regarding the behavior of detectors orthe quantum mechanical predictions) are linear in { ρ s } and the trivial solution η = 0 (all ρ s = 0 but for theone that predetermines no detections at all) is alwaysadmissible, implying there is always some η crit ≥ η ≤ η crit the desired LHV can be built. Thereasoning applies whether if the LHV just simulates a vi-olation of a particular Bell inequality or also every otherquantum prediction for a given state and set of observ-ables (the addition of more restrictions would in principlelower η crit ).[33] Though from the mathematical point of view our modelhere already predicts a vanishing detection probabilityfor the ZPF alone, a physical interpretation of theabsence of an observable detection rate as a result ofthe vacuum fluctuations, or at least that of a significantone, is still an open question. Following Santos’s work,the absence of the ZPF from the observational spectrumcould be justified, for realistic detectors, on the combi-nation of the following properties:(i) the already commented subtraction of the averageZPF intensity as a result of Glauber’s expression in thenormal order [15],(ii) spatial and temporal integration, involving theautocorrelation properties of the ZPF,(iii) a “low band pass” frequency response.See, for instance: Emilio Santos, “How pho-ton detectors remove vacuum fluctuations”,http://arxiv.org/abs/quant-ph/0207073v2 (2008).We remark again that this issue is in any case irrelevantfor our main results here: it concerns models builtoutside of the quantum framework (which is not ourcase). Nevertheless, we conjecture that our proposedinterpretation of (i) in Sec. VI may be of use to makeprogress on the question.[34] E. Kot, N. Grønbech-Jensen, B.M. Nielsen, J.S.Neergaard-Nielsen, E.S. Polzik, A.S. Sørensen.Phys.Rev.Lett. , 233601 (2012).[35] V. Handchen et al , NATURE PHOTONICS (2012).[36] Peruzzo et al , SCIENCE, Reports, (2012).[37] Kaiser et al , SCIENCE, Reports, (2012).[38] J.S. Neergard-Nielsen et al , OPTIC EXPRESS , 7940(2007).[39] Let Λ be a space of events, all possible probability assig-nations within the formalism, F ( · ) : λ ∈ Λ → F ( λ ),satisfy 0 ≤ F ( λ ) ≤ ∀ λ and, for any partition { Λ i } ofΛ (assuming of course additivity on disjoint subsets ofevents), P i F (Λ i ) = 1.All Bell inequalities (at least those written in terms ofprobabilities) can be obtained as a derivation from theselaws, plus some very simple supplementary assumptions:statistical independence of variables defined at distantlocations. Of course, Bell inequalities written in terms ofcorrelations can be also written in terms of probabilities.Many do no even need such supplementary assumptions(space-like separation ones or of another kind): such isthe case of both the CH [21] and the CHSH [44] inequali-ties. To illustrate this see for instance [45]’s derivation ofthe CH inequality; the CHSH inequality can on the otherhand also be interpreted as a mere algebraic inequalityon whatever four quantities taking values ± W ( α, α ∗ ) is a Gaussian [2]; hence W ( α, α ∗ ) > ∀ α, α ∗ . [41] For a recent work reviewing these concepts see for in-stance: T. Norsen, “J.S. Bell’s Concept of Local Causal-ity”, http://arxiv.org/abs/0707.0401.[42] A possible (not unique) procedure to build µ is this: foreach M i , we define a new random variable σ i and assignvalues for each pair ( λ, m i,k ): σ i ≡ σ i ( λ, m i,k ), and nowsimply do µ ≡ L i σ i . As built, σ i ’s are not necessarilyindependent from one another, nor are they necessarilyindependent from λ .[43] A. Casado, private communication.[44] J.F. Clauser, M.A. Horne, A. Shimony, R.A. Holt. Phys.Rev. Lett. , 880 (1969).[45] J.-˚A. Larsson and J. Semitecolos, Phys. Rev. A ,022117 (2001).[46] R. Risco-Delgado. PHD Thesis. Universidad de Sevilla(1997).[47] A.Casado, PHD Thesis: “Estudio de los experimentos deconversi´on param´etrica a la baja con el formalismo de lafunci´on de Wigner” (Universidad de Sevilla, 1998).[48] The PDC model is by construction restricted to a certainsubset of QED-states, obtained directly from a mix ofthe vacuum state (hence one with positive Wigner func-tion) with a quasi-classical signal (the laser), and a time-evolution governed by a quadratic Hamiltonian (henceone that preserves the positivity of the Wigner function);for this last point, see [12] or [47].[49] For instance, the description of the “pump” (laser beam)as a non-quantized, external potential is just an ap-proximation (one that allows for a Hamiltonian that isquadratic in creation/annihilation operators, and hencepreserves positivity of the Wigner function); in any case,we should not forget that further refinements may in-clude not only quantization of the laser but also addi-tional terms expressing the interaction with further ZPFmodes neglected in the present formulation.[50] For instance, the inclusion of particle sub-structure couldalso be the way forward to explain other peculiarities ofthe quantum world that seem so far alien to a purely clas-sical framework. In particular, two very crucial points:(i) the appearance of a discrete spectrum of observableenergy exchanges between matter and the electromag-netic field, due to the presence of meta-stable states (“at-tractors”) in the classical phase space of the system (forinstance the Hydrogen atom), once observations are as-sumed as some time-average that smears out transients,being therefore reduced to a discrete spectrum of statesand transitions between them;(ii) the possibility of highly directional radiation patterns(emission of energy) and resonances (absorption of en-ergy), in the same way as they arise in macroscopic sys-tems with rich spatial structure.Point (i) can potentially accommodate the famous“quanta” E = hω relation, attempts at which we haveleft at an intermediate step elsewhere; point (ii) wouldmake possible the transfer of energy through long dis-tances with no spread.[51] See for instance “Can we celebrate defeatfor the photon by Maxwell-Planck theory?”“Can we celebrate defeatfor the photon by Maxwell-Planck theory?”