No-go for device independent protocols with Tan-Walls-Collett `nonlocality of a single photon'
Tamoghna Das, Marcin Karczewski, Antonio Mandarino, Marcin Markiewicz, Bianka Woloncewicz, Marek Żukowski
NNo-go for device independent protocols with Tan-Walls-Collett ‘nonlocality of a singlephoton’
Tamoghna Das, Marcin Karczewski, Antonio Mandarino, Marcin Markiewicz, Bianka Woloncewicz, and Marek Żukowski International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-308 Gdaśk, Poland
We investigate the interferometric scheme put forward by Tan, Walls and Collett [Phys. Rev.Lett. , 256 (1991)] that aims to reveal Bell non-classicality of a single photon. By providinga local hidden variable model that reproduces their results, we decisively refute this claim. Inparticular, this means that the scheme cannot be used in device-independent protocols. ‘Nonlocality of a single photon’ is a controversial andlong debated subject. It was first addressed by Tan,Walls and Collett (TWC) in [1]. The authors aimed todemonstrate in a most striking way an effect that cannotbe duplicated in any classical theory , namely a violationof local realism with a single particle. TWC consideredthe state | ψ (cid:105) b ,b = 1 √ (cid:104) | (cid:105) b ,b + i | (cid:105) b ,b (cid:105) , (1)obtained by casting a single photon on a balanced beam-splitter, where e.g. | (cid:105) b ,b , indicates one photon excita-tion in the Fock space of exit mode b and the vacuum ofthe Fock space relative to exit mode b , see Fig.(1). Theform of such state appears to be similar to the singletstate of two level systems, which is known to maximallyviolate a Bell’s inequality. The two states are howeverintrinsically different in terms of the number of particlesinvolved and | ψ (cid:105) b ,b can be thought of as a plain super-position of the photon in either of the beams.As a result of these two opposing views, it is not intu-itively obvious if it is possible to violate a Bell inequal-ity using the single-photon state | ψ (cid:105) b ,b . Apart fromthe fundamental theoretical significance of this problem,it is also important from a practical point of view. Asthis state is very easy to obtain, we would like to knowwhether it could be treated as a resource in quantum in-formation. In this Brief Report we will argue that its usein the device independent protocols, which rely on Bellnon-classicality to work regardless of the internal func-tioning of the apparatus implementing them, cannot bebased on the original scheme presented by TWC.In the experimental proposal of TWC shown in Fig.(1),the state | ψ (cid:105) b ,b is distributed between Alice (control-ling the mode b ) and Bob ( b ). They both performhomodyne measurements on their parts. To his end, thestate in mode b j and the auxiliary coherent fields in state | α j (cid:105) a j = (cid:12)(cid:12) αe iθ j (cid:11) a j are cast into the different input portsof 50-50 beamsplitters BS j , j = 1 , and end up in photonnumber measuring detectors D c j and D d j , which registerthe intensities in the output ports.To benchmark the violation of local realism, TWC used s t b (cid:2869) (cid:2870) a a c d c d (cid:3031) (cid:3117) (cid:3030) (cid:3117) (cid:3030) (cid:3118) (cid:3031) (cid:3118) b BS BS BS FIG. 1. Experimental configuration proposed by Tan, Wallsand Collett in [1]. A single photon impinges on a 50-50 beam-splitter via input s , along with the vacuum in the input t . As aresult we get state | ψ (cid:105) b ,b , which propagates to the laborato-ries of Alice and Bob, who perform homodyne measurementsinvolving weak coherent local oscillator fields (their ampli-tudes satisfy | α | = | α | ). the correlation function E ( θ , θ ) = (cid:104) ( I c ( θ ) − I d ( θ ))( I c ( θ ) − I d ( θ )) (cid:105) LHV (cid:104) ( I c ( θ ) + I d ( θ ))( I c ( θ ) + I d ( θ )) (cid:105) LHV , (2)where I x j ( θ j ) is the intensity at output x = c, d mea-sured by the observer j = 1 , and the averaging is doneover local hidden variables (LHV). Then, they consideredthe inequality | E ( θ , θ ) + E ( θ (cid:48) , θ ) + E ( θ , θ (cid:48) ) − E ( θ (cid:48) , θ (cid:48) ) | ≤ , (3)in which the settings are defined by the local phases θ j and θ (cid:48) j . For the amplitudes of the local oscillators sat-isfying α < √ − , they observed a violation of theinequality in formula (3) and concluded that the singlephoton state in Eq. (1) is ‘nonlocal’.However, the inequality in formula (3), derivedin [2], rests on the assumption that the total intensity a r X i v : . [ qu a n t - ph ] F e b I c j ( θ j , λ )+ I d j ( θ j , λ ) registered by each observer does not depend on θ j . As observed in [3] and [4], we cannot becertain that this condition holds. Therefore the TWCattempt to demonstrate the Bell non-classicality of thesingle-photon state | ψ (cid:105) b ,b has to be re-examined.Since its appearance, the TWC letter [1] attractedmuch attention, and stirred a lot of controversy. San-tos suggested that the intensity correlations in the TWCscheme can be explained with local hidden variables [4]and cannot be used to convincingly demonstrate non-classicality of a single photon. However, his LHV modelreproduced only the correlation functions, and not thefull quantum predictions concerning, for instance, regis-tered photon numbers. Other works challenge the single-photon nature of the effect [5, 6], or suggest modificationsof the experiment which would allow provable violationsof local realism [7–9]. Thus far, no definite answer wasgiven to the problem whether the TWC interference ef-fect, which seemingly violates local realism, admits a pre-cise local realistic model, or not.Papers describing the experimental realizations of vari-ants of this scheme [10, 11] report violations of a Bellinequality. However these claims were presented withcaution, e.g in [11], where it is effectively stated thatthe results are not better than for conventional Bell testswhich involve the efficiency loophole.In this Brief Report we introduce an LHV modelthat reproduces precisely all quantum predictions for theTWC setup. Its applicability is limited by the strength ofthe local oscillators, but covers the range reported in [1]as revealing the ‘nonlocality’ of the single-photon state.This result definitely closes the case and precludes anyattempt to implement device-independent protocols re-lying on TWC correlations. On the other hand, in aforthcoming publication ( in preparation ) we present amodification of the TWC setup that would allow for agenuine violation of local realism. Its main idea is that,in contrast to the TWC case, the strengths of local os-cillators need to depend on settings. This means thatno definite initial state can be ascribed to all quantumoptical fields involved in this scheme. Explicit LHV model of TWC correlations.
Quantumpredictions for the TWC setup are fully characterizedby the probabilities p ( n ) of events consisting in register-ing a specific numbers of photons in the output modes: n = ( k c , l d , r c , s d ) ∈ N (for readability, we omit theindices indicating the modes in further parts of this re-port). They read (see Appendix A for the derivation) p ( n ) = A ( α, n ) (cid:104) ( k − l ) + ( r − s ) + 2( k − l )( r − s ) sin( θ ) (cid:105) , (4)where θ = θ − θ and A ( α, n ) = e − α k ! l ! r ! s ! (cid:16) α (cid:17) k + l + r + s α . (5)These probabilities have some features, which hint athow one could reproduce them with an LHV model. First, whenever both detectors of Alice or Bob registerthe same number of photons, the probability does notdepend on θ . Let us denote the set of these events N := { n : k = l or r = s } . We will cover them by a fam-ily of trivial LHV submodels assigning fixed outcomes toAlice and Bob, see further.Next, notice that all the probabilities that do dependon θ are of the form p ( n ) = B ( α, n ) (1 + V ( n ) sin( θ )) , (6)where B ( α, n ) = A ( α, n ) (cid:104) ( k − l ) + ( r − s ) (cid:105) and V ( n ) = 2( k − l )( r − s )( k − l ) + ( r − s ) . (7)To reproduce them, we will adapt a model by Larsson[12], which reproduces all the quantum predictions for atwo-qubit singlet state, provided that the detection ineffi-ciency is lower than /π . A variant of the Larsson modelis also reproducing two-qubit Franson-type interference[13].Our model M is a convex combination of submodels M n , each chosen with probability P ( M n ) . The submod-els belong to two infinite families: the trivial {M n } n ∈N and Larsson-like one {M n } n ∈ ˜ N , where ˜ N := { n : k >l and r > s } . We shall focus on the latter first.The approach that we use differs from the one of Lars-son in that we do not exploit the detection loophole. In-stead, we group the probabilities that depend on the localsettings θ j with the ones that correspond to the eventsin which (perfect) detectors of either Alice or Bob do notregister any photons. We shall denote the set of suchevents by O := { n ∈ N : k = l = 0 or r = s = 0 } .Each Larsson-like submodel {M ( k,l,r,s ) } is going topredict eight events resulting from applying (or not) theswaps k ↔ l and r ↔ s to (0 , , r, s ) , ( k, l, , and ( k, l, r, s ) . Notice that only one of them matches the in-dex ( k, l, r, s ) ∈ ˜ N of the model. To construct it, wetake a uniformly distributed continuous hidden variable λ ∈ [0 , π ] and a coin toss one x ∈ { , } , which char-acterize the LHV probabilities P A n and P B n , assigned tooutcomes obtained by Alice and Bob in each submodel M n . Their form stems from the model presented in [12],generalizing it to the case of < |V ( n ) | ≤ .Specifically, for x = 0 Alice can register the event ( c, d ) ∈ { ( k, l ) , ( l, k ) } with probability P A n ( c, d | θ , λ,
0) = R n ( c, d | θ , λ ) = −V ( n ) π + V ( n ) | sin ( θ − λ ) | H (( c − d ) sin( θ − λ )) , (8)where H is the Heaviside function. Otherwise, she de-tects no photons at all P A n ( c = 0 , d = 0 | θ , λ,
0) = R n (0 , | θ , λ )= 1 − (cid:88) ( e,f ) ∈{ ( k,l ) , ( l,k ) } R n ( e, f | θ , λ ) . (9)Bob detects ( c (cid:48) , d (cid:48) ) ∈ { ( r, s ) , ( s, r ) } with probabilities P B n ( c (cid:48) , d (cid:48) | θ , λ, Q n ( c (cid:48) , d (cid:48) | θ , λ ) = H (cid:0) ( c (cid:48) − d (cid:48) ) cos( θ − λ ) (cid:1) . (10)For x = 1 we swap the forms of the functions defin-ing the hidden probabilities. This symmetrizes the sub-model, as now P A n ( c, d | θ , λ,
1) = Q n ( c, d | θ , λ ) , and P B n ( c (cid:48) , d (cid:48) | θ , λ,
1) = R n ( c (cid:48) , d (cid:48) | θ , λ ) . Having established the local probabilities, let us nowturn our attention to the joint ones. For each submodelthey are given by P AB n ( c, d, c (cid:48) , d (cid:48) | θ , θ )= π (cid:80) x =0 (cid:82) π dλ P A n ( c, d | θ , λ, x ) P B n ( c (cid:48) , d (cid:48) | θ , λ, x ) . (11)For instance, the probability that the submodel M n pre-dicts the event ( k, l, r, s ) in the simplest case of π/ >θ > θ > , k > l and r > s , can be calculated asfollows P AB n ( n | θ , θ )= (cid:80) x =0 (cid:82) π dλ P A n ( k,l | θ ,λ,x ) P B n ( r,s | θ ,λ,x )4 π = −V ( n )2 π + V ( n )4 π (cid:82) θ π − θ dλ sin ( θ − λ ) = V ( n ) sin( θ )2 π . (12)All other predictions of the submodel can be obtainedsimilarly. For events ( k, l, r, s ) , ( l, k, r, s ) , ( k, l, s, r ) and ( l, k, s, r ) we get P AB n ( c, d, c (cid:48) , d (cid:48) | θ , θ )= V ( n ) sign ( ( c − d )( c (cid:48) − d (cid:48) ) ) sin( θ )2 π . (13)In the case of the O -events (0 , , r, s ) , (0 , , s, r ) , ( k, l, , and ( l, k, , , the probability is flat and reads − π ,which follows directly from the normalisation conditionin Eq. (9). Comparing Eq. (13) with the correspond-ing quantum probabilities, we see that each Larsson-likesubmodel M n must appear in the full model M withprobability P ( M n ) = 2 πB ( α, n ) . (14)In the Appendix B we show that our formulas for P ( M n ) lead to a properly normalized probability distribution,with the proviso described below.The presented model definitely reproduces all proba-bilities which reveal interference. However a sine quanon condition for consistency of the full model is toproperly describe also the events of the O class. Theabove construction ascribes all the probabilities ( π − B ( α, ( k, l, c (cid:48) , d (cid:48) )) to the event ( k, l, , in the fullmodel M . They result from the submodels M ( k,l,c (cid:48) ,d (cid:48) ) ,each drawn with the probability (14), giving constantprobabilities − π for the events from O . The sum of allsuch contributions cannot be greater than the quantumprobability for the event, p ( k, l, , , but can be lower since the difference can be compensated by the trivialmodels. This gives the following consistency conditions ∆ ( k,l, , = p ( k, l, , − (cid:16) π − (cid:17) (cid:88) c (cid:48) >d (cid:48) B ( α, ( k, l, c (cid:48) , d (cid:48) )) ≥ , (15)which must hold for any k (cid:54) = l . Obviously, due to thesymmetrization an analogous condition can be writtenfor events of (0 , , r, s ) type.In the Appendix C we show that the condition in Eq.(15) is satisfied for any ( k, l ) and ( r, s ) , whenever α < . . The threshold value is given by the case | k − l | = 1 ,as the larger this difference, the higher the α for whichthe condition holds.The model can be completed using a family of trivialsubmodels M n for events n ∈ N . They predict fixedoutcomes for Alice and Bob, P A n ( k, l ) = P B n ( r, s ) = 1 ,which lead to P AB n (( k, l, r, s )) = 1 . Obviously, for events n ∈ N \ O , we choose each corresponding trivial model M n with probability p ( n ) . Finally, for events n ∈ O we might need to compensate the potential difference ∆ ( k,l, , > between the quantum predictions for the O -events and the predictions specified by the Larsson-like models. To do that, we use an additional trivialsubmodel for event ( k, l, , , which appears in the fullmodel with probability P ( M ( k,l, , ) = ∆ ( k,l, , . Thecase of ∆ (0 , ,r,s ) > is treated the same way.One can easily build a better version of the modelwhich would hold for slightly higher values of α . How-ever, we were not able to find a model which has anunconstrained validity, and one can conjecture that theLarsson-like approach cannot lead to such. Still, ourmodel fully covers the range of α for which TWC pre-dicted a violation of local realism. Thus, this claim isfully revoked. Closing remarks.
The above result closes the ambigu-ity of the relation of TWC correlations with violations oflocal realism. Similar form of correlations for coincidentcounts stemmed from several other experimental schemesproposed in the early times of photonic entanglement-interferometry. Their functional sine-like dependence v sin ( θ ± θ ) was considered to violate a Bell inequalitywhen the visibility v , was higher than √ . Such depen-dence appears, for instance, in the standard two qubitBell experiments, and v > √ indeed blocks any possi-bility of having a local realistic model in the idealizedscenario (perfect efficiency, etc.). The most importantclaims of this character were put in [14], for a differentsituation in [15], and in a still different experimental con-text in the TWC paper. Experiments of [14] were shownto be, in their idealized versions, proper Bell experiments[16]. However Franson interferometry, even in the idealform, was shown to have a local realistic model repro-ducing the correlations for all values of the local phases[13]. Modifications of the Franson scheme were shown tobe a necessity. Here we showed that the idealized TWCcorrelations have also a simple model in the region inwhich they were supposed to violate local realism. Onehas to modify the experiment in order to see a genuineviolation of local realism, e.g., like in the proposal of [8]or [7]. However this is not a minor modification. In aforthcoming paper we shall analyse modifications whichhave some of the traits of the one of [7], but still involvejust initial photon in mode s , Fig.(1), like it is in the caseof [1]. They cannot have a local realistic model.We stress that the results so far presented have atwofold value. We answer the vexata quaestio about the‘nonlocality of a single photon’ as presented in [1], but wealso put a warning on its possible exploitation for futurequantum technologies application. In fact, the latter is arising field with a flourishing literature and our findingsshould serve as a caveat lector for any attempt to use this scheme in any protocol requiring as the main resource theviolation of local realism. ACKNOWLEDGEMENTS
We acknowledge support by the Foundation forPolish Science (IRAP project, ICTQT, contract no.2018/MAB/5, co-financed by EU within Smart GrowthOperational Programme). MK acknowledges support bythe Foundation for Polish Science through the STARTscholarship. AM acknowledges support by National Re-search Center through the grant MINIATURA DEC-2020/04/X/ST2/01794. [1] S. M. Tan, D. F. Walls, and M. J. Collett, Nonlocality ofa single photon, Phys. Rev. Lett. , 252 (1991).[2] M. D. Reid and D. F. Walls, Violations of classical in-equalities in quantum optics, Phys. Rev. A , 1260(1986).[3] M. Żukowski, M. Wieśniak, and W. Laskowski, Bell in-equalities for quantum optical fields, Phys. Rev. A ,020102 (2016).[4] E. Santos, Comment on “nonlocality of a single photon”,Phys. Rev. Lett. , 894 (1992).[5] D. M. Greenberger, M. A. Horne, and A. Zeilinger, Non-locality of a single photon?, Phys. Rev. Lett. , 2064(1995).[6] A. Peres, Nonlocal effects in Fock space, Phys. Rev. Lett. , 4571 (1995).[7] L. Hardy, Nonlocality of a single photon revisited, Phys.Rev. Lett. , 2279 (1994).[8] K. Banaszek and K. Wódkiewicz, Testing quantum non-locality in phase space, Phys. Rev. Lett. , 2009 (1999). [9] S. J. van Enk, Single-particle entanglement, Phys. Rev.A , 064306 (2005).[10] B. Hessmo, P. Usachev, H. Heydari, and G. Björk, Ex-perimental demonstration of single photon nonlocality,Phys. Rev. Lett. , 180401 (2004).[11] S. A. Babichev, J. Appel, and A. I. Lvovsky, Homodynetomography characterization and nonlocality of a dual-mode optical qubit, Phys. Rev. Lett. , 193601 (2004).[12] J. Åke Larsson, Modeling the singlet state with local vari-ables, Physics Letters A , 245 (1999).[13] S. Aerts, P. Kwiat, J.-A. Larsson, and M. Żukowski,Two-photon Franson-type experiments and local realism,Phys. Rev. Lett. , 2872 (1999).[14] Z. Y. Ou and L. Mandel, Violation of Bell’s inequalityand classical probability in a two-photon correlation ex-periment, Phys. Rev. Lett. , 50 (1988).[15] J. D. Franson, Bell inequality for position and time, Phys.Rev. Lett. , 2205 (1989).[16] S. Popescu, L. Hardy, and M. Żukowski, Revisiting Bell’stheorem for a class of down-conversion experiments,Phys. Rev. A , R4353 (1997). APPENDIX A – QUANTUM PHOTODETECTION PROBABILITIES
In this section, we are going to calculate the probability of detecting the event n = ( k c , l d , r c , s d ) , consisting inregistering specific numbers of photons in the output modes of the Tan-Walls-Collett setup.The initial state, obtained by transforming a single photon with a balanced beamsplitter and adding two coherentstates as ancillas reads | Ψ (cid:105) = | αe iθ (cid:105) a √ | (cid:105) + i | (cid:105) ) b b | αe iθ (cid:105) a . (16)We will show how the state (16) transforms on balanced beamsplitters U BSj , j = 1 , which link the output andinput modes via ˆ c j = 1 √ a j + i ˆ b j ) and ˆ d j = 1 √ i ˆ a j + ˆ b j ) . (17)Applying (17) to the state (16) we get | Ψ (cid:105) = e − α ∞ (cid:88) j =0 ( αe iθ ) j j ! (ˆ a † ) j √ i ˆ b † + ˆ b † ) ∞ (cid:88) k =0 ( αe iθ ) k k ! (ˆ a † ) k = e − α ∞ (cid:88) j,k =0 − j + k ( αe iθ ) j j ! ( αe iθ ) k k ! (cid:16) ˆ c † + i ˆ d † (cid:17) j (cid:16) − ˆ c † + i ˆ d † + i ˆ c † + ˆ d † (cid:17)(cid:16) ˆ c † + i ˆ d † (cid:17) k | Ω (cid:105) = e − α ∞ (cid:88) j,k =0 − j + k ( αe iθ ) j j ! ( αe iθ ) k k ! j (cid:88) p =0 (cid:18) jp (cid:19) (ˆ c † ) j − p ( i ˆ d † ) p (cid:16) − ˆ c † + i ˆ d † + i ˆ c † + ˆ d † (cid:17) k (cid:88) q =0 (cid:18) kq (cid:19) (ˆ c † ) k − q ( i ˆ d † ) q | Ω (cid:105) , = ∞ (cid:88) j,k =0 j (cid:88) p =0 k (cid:88) q =0 f ( j, p, k, q )(ˆ c † ) j − p ( ˆ d † ) p (cid:16) − ˆ c † + i ˆ d † + i ˆ c † + ˆ d † (cid:17) (ˆ c † ) k − q ( ˆ d † ) q | Ω (cid:105) , (18) = ∞ (cid:88) j,k =0 j (cid:88) p k (cid:88) q =0 f ( j, p, k, q ) (cid:20) − (cid:112) ( j − p + 1)! p !( k − q )! q ! | j − p + 1 (cid:105) c | p (cid:105) d | k − q (cid:105) c | q (cid:105) d + i (cid:112) ( j − p )!( p + 1)!( k − q )! q ! | j − p (cid:105) c | p + 1 (cid:105) d | k − q (cid:105) c | q (cid:105) d + i (cid:112) ( j − p )! p !( k − q + 1)! q ! | j − p (cid:105) c | p (cid:105) d | k − q + 1 (cid:105) c | q (cid:105) d + (cid:112) ( j − p )! p !( k − q )!( q + 1)! | j − p (cid:105) c | p (cid:105) d | k − q (cid:105) c | q + 1 (cid:105) d (cid:21) (19)where f ( j, p, k, q ) = e − α − j + k − ( αe iθ ) j j ! ( αe iθ ) k k ! (cid:18) jp (cid:19)(cid:18) kq (cid:19) ( i ) p + q , ∀ p ≤ j, q ≤ k. (20)Now, Pr ( k, l ; r, s ) = | (cid:104) k, l, r, s | Ψ (cid:105) | = (cid:12)(cid:12)(cid:12)(cid:12) − f ( k + l − , l, r + s, s ) + if ( k + l − , l − , r + s, s )+ if ( k + l, l, r + s − , s ) + f ( k + l, l, r + s − , s − (cid:12)(cid:12)(cid:12)(cid:12) k ! l ! r ! s != e − α k ! l ! r ! s ! (cid:16) α (cid:17) k + l + r + s α (cid:104) ( k − l ) + ( r − s ) + 2( k − l )( r − s ) sin( θ − θ ) (cid:105) , (21) APPENDIX B – ON THE SUM OF PROBABILITIES OF ALL SUBMODELS M n In this section we prove that the probabilities P ( M n ) of choosing specific submodels are properly normalized. Wehave (cid:88) n ∈N ∩ ˜ N P ( M n ) = (cid:88) n ∈N \O B ( α, M n ) + (cid:88) n ∈ ˜ N πB ( α, M n ) + (cid:88) n ∈O ∆ n , (22)where (cid:88) n ∈O ∆ n = (cid:88) k (cid:54) = l (cid:32) p (( k , l , , )) − (cid:16) π − (cid:17) (cid:88) c (cid:48) >d (cid:48) B ( α, ( k, l, c (cid:48) , d (cid:48) )) (cid:33) + (cid:88) r (cid:54) = s (cid:32) p (( , , r , s )) − (cid:16) π − (cid:17) (cid:88) c>d B ( α, ( c, d, r, s )) (cid:33) = (cid:88) n ∈O B ( α, M n ) − (cid:16) π − (cid:17) (cid:88) n ∈ ˜ N B ( α, M n ) = (cid:88) n ∈O B ( α, M n ) − (2 π − (cid:88) n ∈ ˜ N B ( α, M n ) . (23)Moreover, notice that (cid:88) n ∈ ˜ N B ( α, M n ) = 14 (cid:88) n ∈ N (cid:52) c (cid:54) = d, c (cid:48) (cid:54) = d (cid:48) B ( α, M n ) . (24)Plugging Eqs.(23) and (24 )into Eq. (22) we get (cid:80) n ∈N ∩ ˜ N P ( M n ) = (cid:80) n ∈N \O B ( α, M n ) + 2 π (cid:80) n ∈ ˜ N B ( α, M n )+ (cid:80) n ∈O B ( α, M n ) − (2 π − (cid:80) n ∈ ˜ N B ( α, n ) = (cid:80) n ∈ N (cid:52) B ( α, n ) = 1 . (25) APPENDIX C – THRESHOLD INTENSITY FOR THE VALIDITY OF THE LHV MODEL
In this section we prove that if α < . , the probabilities of choosing a specific submodel P ( M n ) are non-negative.To do that, we only need to consider n ∈ O , for which P ( M n ) = ∆ n . Let us fix n = ( k, l, , , k (cid:54) = l , as thereasoning for n = (0 , , r, s ) is fully analogous. We need to check the conditions in which ∆ n = B ( α, ( k , l , , )) − (cid:16) π − (cid:17) (cid:88) c (cid:48) >d (cid:48) B ( α, ( k, l, c (cid:48) , d (cid:48) ) ≥ . (26)We plug the definition of the function B ( α, n ) from the main text into (26) and obtain, after some transformations, ∆ n = e − α − k − l − (cid:0) α (cid:1) k + l − (cid:16) − ( π − e α (cid:0) α + ( k − l ) (cid:1) + ( π − I (cid:0) α (cid:1) ( k − l ) + 4( k − l ) (cid:17) k ! l ! . (27)It is easy to see that the condition ∆ n ≥ is equivalent to − ( π − e α (cid:0) α + ( k − l ) (cid:1) + ( π − I (cid:0) α (cid:1) ( k − l ) + 4( k − l ) ≥ . (28)As the Bessel function I satisfies I (cid:0) α (cid:1) ≥ , the inequality (28) can be approximated by a slightly stricter − ( π − e α (cid:0) α + ( k − l ) (cid:1) + ( π − k − l ) + 4( k − l ) = (cid:16) ( π − (cid:16) − e α (cid:17) + π + 2 (cid:17) ( k − l ) − ( π − α e α ≥ . (29)For α < , the coefficient (cid:16) ( π − (cid:16) − e α (cid:17) + π + 2 (cid:17) standing in front of ( k − l ) is positive. This means that thecritical case we need to consider is ( k − l ) = 1 . Thus, we arrive at (cid:16) ( π − (cid:16) − e α (cid:17) + π + 2 (cid:17) − ( π − α e α ≥ . (30)It can be shown that the inequality 30 is satisfied for α ≤ W (cid:18) e + eππ − (cid:19) − ≈ . , (31)where W denotes the Lambert W function ( W ( z ) returns the principal solution for w in z = we ww