Finite-time collapse process and non-local correlations are incompatible with non-signaling theories
aa r X i v : . [ qu a n t - ph ] S e p Finite-time collapse process and non-local correlations are incompatible withnon-signaling theories
M. G. M. Moreno ∗ and Alejandro Fonseca Departamento de F´ısica, Universidade Federal de Pernambuco. Recife 50670-901, PE, Brazil.
M´arcio M. Cunha
Departamento de Matem´atica, Universidade Federal de Pernambuco. Recife 50670-901, PE, Brazil.Departamento de F´ısica, Universidade Federal Rural de Pernambuco. Recife 52171-900, PE, Brazil. (Dated: September 5, 2018)We propose a hidden variable analysis of collapse dynamics in which the state’s reduction processmay take a finite time δt . A full characterization of the model is given for the case of black boxes. Byintroducing nonlocal perfect correlations to a two black-boxes scenario, it is shown that in order toavoid faster than light communication, the reduction time associated to the system must be strictlynull. Furthermore we prove that the result above holds even when there is a time window betweenthe choice of both part’s inputs. Our results represent a new evidence of the instantaneous natureof the wave function collapse process which could have implications in foundations of quantummechanics and information science. I. INTRODUCTION
After almost a century of Quantum Theory, one of itsfundamental postulates represents yet an issue: the mea-surement problem. This widely discussed phenomenon,in principle forbidden by the linearity of Schr¨odinger’sequation, motivated many works aiming to present anexplanation on the mechanism leading the system, in aprobabilistic way to the collapsed state, in perfect har-mony with Born’s rule and L¨uders postulate. From theperspective of the Copenhagen interpretation those prob-lems are solved by taking them as fundamental postu-lates [1]. Following the
Many World interpretation, allpossible outcomes from a measurement coexist in differ-ent universes [2], avoiding the necessity of L¨uders postu-late nevertheless it doesn’t explain the mechanism behindBorn’s rule. Decoherence [3] approaches the problem byshowing that a quantum system in contact with a en-vironment should swiftly reach a classic statistical dis-tribution, nonetheless closed systems remain a problem.Among the attempts to solve the measurement problem,the most plausible explanation is that Quantum Mechan-ics is an approximated theory, in the sense that theremust a nonlinear equation that rules the dynamics of allsystems which in the microscopic limit becomes approx-imately linear in accordance with Schr¨odinger equation,after all no microscopic experiment ever indicated Quan-tum Theory to be wrong. Such a theory should also bestochastic, since any nonlinear deterministic extra termin Schr¨odinger’s equation leads to signaling [4]. Thatapproach motivated many collapse models introduced inthe last few decades, as well as several experiments wereproposed to investigate the features of the wave functioncollapse [5–12]. For a complete review on the currentstate of the area see [13]. ∗ [email protected] If collapse comes from a dynamical process, a funda-mental question should be addressed: once initiated, howlong, in average, does the process of collapse take to beaccomplished?
The mean time of the collapse may bringa deep insight on the dynamics that rules it. This prob-lem, which has already been considered and experimentaltests to verify the duration of the collapse have been pro-posed [14–16], will be the central issue of this work.In order to investigate the mean time associated withthe collapse process, the present work brings into play aquite general approach: a hidden variables model, in aDevice Independent (DI) scenario. A hidden variablesmodel allows for the contemplation of possible conse-quences and effects of unknown parameters, and its dy-namics, that may be playing some rule on a given prob-lem. Under this approach, we are particularly interestedinto hypotheses on the set of hidden variable which cangenerate appreciable differences in the outputs of someexperiment. This method is well know by the seminalwork of J. Bell [17]. Recently Bedingham employed thisconcept to perform a link between the collapse dynam-ics and the Bohmian mechanics [18] (see also [19]). Onthe other hand, the DI certification program provides ro-bust results, for it only relies on the statistics of a givenexperiment and it is not necessary to make any extra as-sumption on the system to be tested. A system underthe DI procedure is treated as an assortment of boxesequipped with buttons which after being pushed produceone out of an array of outcomes, in general different ineach run of the experiment. Associated with hidden vari-able models, the DI idea has been of remarkable impor-tance on protocols of certification. For an introductionto the subject, we refer the reader to [20]. At the end ofthe day, from the set of frequencies it is possible to inferunderlying properties of the whole system. In this paper,by using nonlocally correlated systems and applying theideas above we obtain general results that are indepen-dent of Quantum Mechanics, however, they lead to veryremarkable consequences in its context.The paper is organized as follows: In section II we showthe usual hidden variable (HV) model in the context ofnon-local correlations. After in section III we introduce aHV model to treat the problem of a single system subjectto an arbitrary collapse dynamics. Section IV is devotedto make a connection between nonlocal correlations andcollapse dynamics under the HV models overlook, andwe present some implications of the collapse dynamics.In section V we expose our main conclusions.
II. HIDDEN VARIABLES AND NON-LOCALCORRELATIONS
The most known application of hidden variables mod-els is perhaps the definition of local correlations, as-sociated with the derivation of Bell’s inequalities. Inthis problem it is investigated the statistical behavior P ( a, b | x, y ) of two separated parts in which inputs areperformed, x in one and y in the other side, generatingoutputs a and b respectively. We can always write: P ( a, b | x, y ) = Z Λ dλζ ( λ | x, y ) P ( a, b | x, y, λ ) . (1)In the above expression, λ represents variable(s) whichare sampled from a set Λ, following a distribution ζ ( λ | x, y ), responsible for the probability of the system,and yet out of the reach for the experimenter.Equation (1) represents the basic assumption behindhidden variable models, however one can always add ex-tra hypothesis on them. For instance, one can assumethat superluminal communication between parts is for-bidden, i. e. the non-signaling assumption, in this casethe marginal probabilities of each part should be inde-pendent of what happens in the other: P ( a | x, y, λ ) = P ( a | x, λ ) P ( b | x, y, λ ) = P ( b | y, λ ) , then: P ( a, b | x, y, λ ) = P ( a | x, λ ) · P ( b | y, λ ) . (2)We can go further and assume that all correlations comefrom the λ variables and that their distribution is well de-fined despite the inputs x and y , i. e., ζ ( λ | x, y ) = ζ ( λ ),which is known as the measurement independence as-sumption. Moreover note that the inputs x and y do notdepend on the set of hidden variables (free will assump-tion). Considering this, correlations described by a localhidden variables model may be written as: P ( a, b | x, y ) = Z Λ dλ ζ ( λ ) P ( a | x, λ ) P ( b | y, λ ) , (3)attainable by any classically correlated composed system.This example illustrates the power of hidden variablesassumptions and how to handle them in order to get valu-able information on the system under consideration. III. HIDDEN VARIABLES AND COLLAPSE
Imagine Alice receives a closed box containing a flippedcoin. Right before Alice looks inside the box, she wouldsay that the probability of getting either heads or tails isone half (assuming a faithful coin). If Alice finds that theoutput was tails (heads) any further observation of thesame coin will yield the output tails (heads) with proba-bility one. We may say that Alice’s system collapses afterit is measured. So far the problem seems very trivial: theposition of the coin is well defined from the moment thesystem was created, and the act of looking to the coin justmeans to learn the value of some unknown well definedvariable. This is not always the case, for instance onemay consider that instead of a coin, there is an electroninside the box prepared in a spin state ( | i + | i ) / √ | i and | i representing the eigenstates of ˆ σ z . WhenAlice carries out a spin measurement in z direction, quan-tum theory states that there are not a priori establishedvariables hidden from Alice defining the correspondingoutcomes. In contrast with the former case, now a phys-ical process is expected to take place, leading to the finaloutputs. This is what we mean by collapse .Following the above scenario, now we consider Alice re-ceiving a box, on which she can provide an input x froma set of inputs X that generates some output a ∈ A , witha well defined probability. Before making any further as-sumption, it is useful to introduce a formal distinctionbetween the two classes of inputs that Alice may pro-vide to her box. On one hand, we have inputs leading tooutcomes in a non-deterministic way -hereafter collapsetriggering operations (CT). On the other hand, opera-tions conducting to deterministic outputs, defined hereas non collapse triggering operations (NCT). Hence itis possible to divide the set of inputs X in two parts: X = X CT ∪ X NCT .In order to contemplate any possible collapse dynam-ics (consequence of a CT input) and its effect on thedescription of an arbitrary system, we propose a hiddenvariable approach similar to that introduced in previoussection. Let state some assumptions: (i) the collapse istriggered by an input x ∈ X CT on the box at an instant τ which returns some output a ∈ A , (ii) the system takesa time δt a to collapse (i.e. to generate an output), (iii)the probability of obtaining the output “ a ” as a conse-quence of the first input x is known to be P ( a | x ), and(iv) the collapse time δt a depends in a non trivial way onthe probability of its output P ( a | x ). Considering that,the most general expression describing the probability ofan output a ′ given a second input x in a posterior time τ ≤ t ≤ τ + δt , is: P ( a ′ | x ; t > τ ) = Z Γ dγ ( t ) χ ( γ ( t ) | x ) P ( a ′ | x ; γ ( t )) , (4)here Γ, χ and γ play the same role as Λ, ζ and λ inequation 1, respectively. This model encompasses anypossible description behind the phenomenon of collapse.In fact we could make χ ( γ ( t ) | x ) = δ [ γ ( t ) − ˆ ρ ( t )], where δ [ . ] is the Dirac’s delta and ˆ ρ ( t ) is a density operator,and considering the POVM ˆ M x = { ˆ E xa | a ∈ A} , and P ( a ′ | x ; γ ( t )) = tr (cid:16) γ ( t ) · ˆ E xa ′ (cid:17) , then we have: P ( a ′ | x ; t > τ ) = tr (cid:16) ˆ ρ ( t ) · ˆ E xa ′ (cid:17) , (5)which corresponds to the standard formulation in quan-tum mechanics [20].After the largest among the collapse times ( t ≥ τ + δt ∗ ),where δt ∗ = max { δt a } a ∈A , we expect the box to evolvein such a way that P ( a ′ | x ; t ≥ τ + δt ) = δ a,a ′ , where δ a,a ′ is the Kronecker’s delta, and a represents the firstoutput. Without loss of generality we can divide the setof variables Γ into subsets Γ a , each containing all possible γ ( t ) leading to every output “ a ”. As we know a priori that the result “ a ” should be obtained with probability P ( a | x ), then we can write: P ( a ′ | x ; t ) = X a ∈A P ( a | x ) Z Γ a dγ ( t ) χ a ( γ ( t ) | x ) P ( a ′ | x ; γ ( t )) , for τ ≤ t ≤ τ + δt ∗ . To gain some insight on this particu-lar step, it is possible to consider the one-dimensionalrandom walker, which after n steps has a probability P ( j | n ) of being found in the position j . There may existseveral possible paths leading to this configuration, thusone can assemble all these paths together in the set Γ j and argue that with probability P ( j | n ) a path from thisset is sorted out. Also notice that no knowledge from theoutputs to be obtained is required to conceive the exis-tence of this partition, only the assumption that one ofthe possible results will happen.We can simplify the above equation, by defining thefunctions f aa ′ ( t ): f aa ′ ( t ) = Z Γ a dγχ ( γ ( t ) | x ) P ( a ′ | x ; γ ( t )) , (6)which should respect the following bounds: f aa ′ ( τ ) = P ( a ′ | x ) ,f aa ′ ( t ′ ≥ τ + δt a ) = δ a,a ′ , (7) P a ′ f aa ′ ( t ′ ≥ τ ) = 1 . The first condition is related to the initial probabilitydistribution of the box, the second sets the final config-uration after the collapse and the last one ensures thatnormalization is satisfied.Thus using these new functions, we have that: P ( a ′ | x ; t ≥ τ ) = X a f aa ′ ( t ) P ( a | x ) . (8)Note that we have not considered any specific dynamicsso that this result remains as general as possible. Fur-thermore, the collapse time intervals δt a can be taken aszero or finite without loss of generality.Equation (8) and relations (7) lead to the followingresult: if the second input x is performed at an instant t ≥ δt ∗ , the output a will occur with probability P ( a | x ),however if t ≤ δ ˜ t , for δ ˜ t = min { δt a } a ∈A , then the proba-bility distributions will depend on the functions f aa ′ ( t ).In particular, we can consider the quantity: P ( a | x ; τ ≤ t ≤ δ ˜ t ) − P ( a | x ) , (9)which can be experimentally assessed. For a dichotomicsystem, the only condition that allows it to be zero wouldbe P ( a | x ) = , otherwise this quantity is non-vanishing. IV. HIDDEN VARIABLES, NON-LOCALITYAND COLLAPSE
Following the previous reasoning, an extension to thecase of two separated boxes sharing some correlation ispresented. Two balls, one black and the other white, arerandomly placed into Alice and Bob’s boxes respectively.If Alice looks inside her box and learns the colors of herball, then due to the correlation they shared, she alsolearns that of Bob. Setting a = { , } as the color of theball and x = 1 to the act of measuring it, the first timethe input x = 1 is provided, the output a is obtainedwith probability P ( a | x = 1), however any further “colormeasurement” will yield the output a ′ with probability P ( a ′ | x = 1) = δ a,a ′ , given that we are leading with alocally-correlated system. The same behavior is observedin Bob’s box. Like the first example, in this case thereare variables that could give a complete description ofthe box at first, but are not revealed to the parts. Thusthe collapse represents only the knowledge of some hid-den variable. No dynamical process is expected here, forthere is no evidence of physical changes. To observe somecollapse dynamics as discussed above, one must look forcorrelations that cannot be represented by a local hiddenvariables model (eq. 3), where the collapse represents aphysical transformation in both parts.With this in mind, assume we have two arbitrarily sep-arated parts, Alice and Bob as usual, both in inertialreference frames, possessing nonlocally correlated boxes.Alice can provide either an input or x = 0 ∈ X NCT or x = 1 ∈ X CT , such that P ( a |
0) = δ a, , obtaining someoutput a ∈ A , and Bob also gives either an input y =0 ∈ Y NCT or y = 1 ∈ Y CT where P ( b |
0) = δ b, returningsome output b ∈ B . Furthermore, assume that the corre-lation is such that given x = 1 and y = 1, then a = b , andthat the probability of the first measurement in the sys-tem P ( a, b | x = 1 , y = 1) = P ( a | x = 1) = P ( b | y = 1) isknown, where the equalities hold due to the correlationwhich forbids results where a = b . Notice that the inputpairs ( x = 0 , y = 1), ( x = 1 , y = 0), ( x = 1 , y = 1) arecollapse triggering.Suppose Alice and Bob agree that at an instant τ inBob’s watch, she decides the value for x , while Bob sets y = 0. And at an instant t ′ ≥ τ , Alice provides the input x = 0, and Bob y = 1. In this point the natural step is tocompare both scenarios x = 0 and x = 1, given by Alice’sinitial choice. This is a crucial aspect in our approach,for usually when the subject of collapse is tested withrespect to whether it is signaling or not, only collapsetriggering inputs are considered [21].Whenever Alice chooses x = 0, Bob providing y = 1in t ′ will be able to observe that the outputs b follow theknown distribution: P ( b | , t ′ ) = P ( b | . (10)Nevertheless, when she supplies x = 1, the system as awhole starts to collapse, and according to equation (4)the probability in t ′ can be described by: P ( a, b | , t ) = Z Γ dγ ( t ) χ ( γ ( t ) | , P ( a, b | , γ ( t )) . Following the treatment for the single box, it is a fact thatthe system collapses to some output ( a, b ) with probabil-ity P ( a, b | ,
1) = δ a,b P ( b | , P ( b ′ | , t ) = X b ∈B P ( b | Z Γ b dγ ( t ) χ b ( γ ( t ) | P ( b | , γ ( t )) . Here we define: f bb ′ ( t ) = Z Γ b dγ ( t ) χ ( γ ( t ) | P ( b | , γ ( t )) , (11)such that: f bb ′ ( τ ) = P ( b ′ | y ) ,f bb ′ ( t ′ ≥ τ + δt ) = δ b,b ′ , P b ′ f bb ′ ( t ′ ≥ τ ) = 1 , so we can write: P ( b | , t ′ ) = X a f bb ′ ( t ′ ) P ( b | . (12)It is always possible to find distributions for which equa-tions (10) and (12) are in agreement with non-signalingconditions if and only if f ( t ′ ) = f ( t ′ ) = 1 and f ( t ′ ) = f ( t ′ ) = 0 for t ′ > τ , representing an instan-taneous collapse. Otherwise Alice’s choice affects Bob’sstatistics. This analysis suggests that only instantaneouscollapses are compatible with non-signaling theories.The results shown above are quite general in the sensethat no assumptions are made on the dynamics behindthe process. However this scheme does not contemplateall possible scenarios yet. Particularly, we have assumedthat Alice and Bob can choose the specific time in whichthe inputs are delivered, which is not necessarily feasi-ble. For instance, following Quantum Theory one cannotchoose exactly the instant in which a photon is emittednor when it will hit the detector. Now, we repeat ouranalysis by taking into account that Alice and Bob canonly decide the time window ∆ t > δ ˜ t in which the inputacts happen. In addition, we consider that each of themcan only perform one input: Alice deciding x and Bobapplying y = 1. When the time window starts we assume that an in-put takes place according to some probability distribu-tion g ( t ). Thus, the probability of it to happen at aninstant t ′ , τ ≤ t ′ ≤ τ + ∆ t , is: P ( t ′ ) = Z t ′ τ g ( t ) dt, (13)where R τ +∆ tτ g ( t ) dt = 1.Now we are interested in what happens if the timeinterval between the instant in which Alice’s input hap-pens, t A , and the moment Bob’s input occur, (hereafter t B ), are smaller than δ ˜ t . This is because if the time inter-val is larger, then the collapse process will be completedand the observed effect in the previous case will not playany role here. In general, we have: P ( | t B − t A | < δ ˜ t ) = Z τ +∆ tτ g ( t A ) Z t A + δ ˜ tt A − δ ˜ t g ( t B ) dt B dt A , hereafter Θ = P ( t B − t A ≤ δ ˜ t ).Let us investigate the partial probability distributionfor Bob. Once again, when Alice decides x = 0, Bobobserves that the statistics associated to his outputs isequal to the already know distribution P ( b | y ). Alterna-tively, when she chooses x = 1, triggering the collapse,then Bob’s probability can be described in the followingway: P ( b ′ | y ) = (1 − Θ) P ( b ′ | y )++ ΘΩ ( X b ∈B P ( b | y ) Z δ ˜ t f bb ′ ( t ′ ) g ( t ′ ) dt ′ ) , (14)where Ω = R δ ˜ t p ( t ) dt .The positivity of the probability distributions guaran-tees that equation (14) may be different from the a pri-ori known distribution P ( b | y ), unless f ab ( t ′ ) = δ a,b for t ′ ≥ τ . Hence the only way to avoid signaling for anydistribution is an instantaneous collapse dynamics. V. CONCLUSION
We have addressed the question of the finiteness ofcollapse time for two different scenarios, on one hand asingle system and on the other, a bipartite correlated one,treated as boxes in analogy to Bell scenarios. Based onour idea of collapse, the inputs are divided in two sets:collapse and non-collapse triggering. A hidden variablesapproach is employed to model an arbitrary collapse dy-namics. For the first case we demonstrate that in princi-ple it is possible to distinguish finite time from instanta-neous collapse dynamics.For the case of two nonlocally correlated parts, we de-rive some conditions necessary to such correlations do notviolate non-signaling constraints during the collapse pro-cess. The obtained result is quite general, and suggeststhat any collapse dynamics with finite time is incom-patible with non-signaling constraints. Our results areparticularly relevant in the context of Quantum Foun-dations, because it can can bring some insights on themeasurement problem, still an open question nowadays.
VI. ACKNOWLEDGEMENTS
We thank Rafael Chaves and Barbara Amaral forthe comments and suggestions. M´arcio M. Cunha issupported by FACEPE-FULBRIGHT BCT 0060-1.05/18grant. Financial support from Conselho Nacional de De-senvolvimento Cient´ıfico e Tecnol´ogico (CNPq) throughits program INCT-IQ, Coordena¸c˜ao de Aperfei¸coamentode Pessoal de N´ıvel Superior (CAPES), and Funda¸c˜ao deAmparo `a Ciˆencia e Tecnologia do Estado de Pernam-buco (FACEPE) is acknowledged. [1] N. Bohr, “The quantum postulate and the recent devel-opment of atomic theory,” (1928).[2] H. Everett III, Reviews of modern physics , 454 (1957).[3] W. H. Zurek, Reviews of modern physics , 715 (2003).[4] N. Gisin, Physics Letters A , 1 (1990).[5] M. Bahrami, M. Paternostro, A. Bassi, and H. Ulbricht,Physical Review Letters , 210404 (2014).[6] L. Di´osi, Physical review letters , 050403 (2015).[7] D. Goldwater, M. Paternostro, and P. Barker, PhysicalReview A , 010104 (2016).[8] S. Nimmrichter, K. Hornberger, and K. Hammerer,Physical review letters , 020405 (2014).[9] M. G. Genoni, O. Duarte, and A. Serafini, New Journalof Physics , 103040 (2016).[10] M. Bilardello, A. Trombettoni, and A. Bassi, PhysicalReview A , 032134 (2017).[11] A. Vinante, M. Bahrami, A. Bassi, O. Usenko, G. Wijts,and T. Oosterkamp, Physical review letters , 090402 (2016).[12] A. Vinante, R. Mezzena, P. Falferi, M. Carlesso, andA. Bassi, Physical Review Letters , 110401 (2017).[13] A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ul-bricht, Reviews of Modern Physics , 471 (2013).[14] F. Parisio, Physical Review A , 062108 (2011).[15] M. Moreno and F. Parisio, Physical Review A , 012118(2013).[16] R. Moreira, A. Carvalho, M. Mendes, M. Moreno, J. Fer-raz, F. Parisio, L. Acioli, and D. Felinto, Optics Com-munications , 212 (2018).[17] J. S. Bell, Physics , 195 (1964).[18] D. J. Bedingham, Journal of Physics A: Mathematicaland Theoretical , 275303 (2011).[19] R. Tumulka, Journal of Physics A: Mathematical andTheoretical , 478001 (2011).[20] V. Scarani, Acta Physica Slovaca , 347 (2012).[21] D. J. Bedingham, Journal of Physics A: Mathematicaland Theoretical42