Counterfactuality, Definiteness, and Bell's Theorem

Home , Branching (linguistics), Conjugate variables, Superdeterminism

arXiv:1909.06608v4 [quant-ph] 14 Jan 2020 n oeueu olfrdaigwt counterfactuals with to dealing hope for world. we tool our this, useful in Through more have. a these find effects remove the to in- and various us it, extent require the mechanics evaluating quantum by of this exclusion terpretations do this We strong be. how needs just counterfactual seeing excluding by before universe definiteness, quantum a of sense acted we if be counterfactual would requires un- world indefiniteness. mechanics are the quantum we how - so about differently and talk results, to no able unper- have origi- claims experiments mechanics, the formed quantum Interpretation, us of mea- Copenhagen interpretation makes but small The nal mechanics small, a quantum this. a However, imagining doubt in This universe life; way. the day-to-day surable, differently. affect our done would of change would be world part to the how forms something about were talk to change, meaningful is it that nnidpnet aeucin( the wavefunction make (non-independent) you uncertain. particles, particle correlated other the two for for of variables variable conjugate pair conjugate of a set a para- of of a one one to measuring came by can- - Einstein dox they independently), correlated written so entangle- be being and not states [1]) particles’ decreases (two other’s ment the as increase must uigteposition, the suring r eae by related are st t momentum, its to as ∗ [email protected] eivsiaeteagmn htw a nymake only can we that argument the investigate We idea commonsensical the is definiteness Counterfactual osdrtoprils ahdsrbdb hi own their by described each particles, two Consider uncertainty variable’s one (where conjugacy Between I P N ELSTHEOREM BELL’S AND EPR II. 2 unu niern ehooyLbrtre,Department Laboratories, Technology Engineering Quantum t eaint o onefculpsiiiisinteract done. possibilities soph counterfactual yet how more not to a far relation in a its variables us multi-valued allows of it interpretation indefi definiteness, counterfactual counterfactual full cal than definit rather (the event), semi-definiteness measurement interpr counterfactual various by require prohibited it just what comparing By Theorem. 1 eso onefculdfiieessprtscasclfr classical separates definiteness counterfactual show We ..WlsPyisLbrtr,Uiest fBitl Tynd Bristol, of University Laboratory, Physics Wills H.H. .TeERParadox EPR The A. .INTRODUCTION I. x 1 p fpril ,w eoeuncertain become we 1, particle of , 1 onefcult,Dfiiees n elsTheorem Bell’s and Definiteness, Counterfactuality, steucraniso h two the of uncertainties the as , nvriyo rso,Woln od rso,B81S UK 1US, BS8 Bristol, Road, Woodland Bristol, of University ψ 1 and ψ 2 .B mea- By ). ot .Hance R. Jonte ta,i t lc,i oehn neemnt n beto able and indeterminate something is In- place, transmission. its physics, information in classical stead, for speed of maximum universe a deterministic with stable, the is instantly another, information to mechanism. - without place distance” and one a from at passing he action supposedly when is “spooky about This talking to was referred to state. Einstein said paradox corresponding be the the can exactly electron into second This ’pre-prepared’ the way random. be is no into is Further, collapses there electron means first mechanism. instan- the direct state both the any the - pushing without spin and state, spins corresponding taneously overall its electrons’ into the the electron collapses of other it one when measured, one means, is of This independently spins another. electrons’ two these describe h on pnsaeo w nage electrons: of entangled two example of the state gave spin He prod- joint tensor the states)[2]. the single-particle as of written on be uct cannot focusing (which and states doing observables joint this, non-commuting simplified with Bohm away variables. conjugate ducing the incomplete. was to quantum reality the corresponding of showing reality treatment - mechanical of momentum particle’s element second the the indef- lose being - to definite inite being supposedly something from state. 2’s particle changed we contact, making derived direct without any instantaneously, and Therefore, momentum, we its 2’s. position, particle measured 1’s freely particle have measuring con- could before initial system’s But, otherwise the using as ditions. 1’s momentum, particle derive 2’s could particle we of uncertain come . hsi noptbewt isenswrdiw-gone - worldview Einstein’s with incompatible is This cannot we state, product a as written be cannot this As oee,hseapeoe-ovltstig yintro- by things over-convolutes example his However, go could you that nonsense a seemed it Einstein, To be- also we 1, particle of position the taking However, ,2, 1, ns fpsil pin vial fe a after available options possible of eness iees hl esdfiieta classi- than definite less While niteness. mqatmpyis yaayigBell’s analysing by physics, quantum om okn rmti,w ute consider further we this, from Working ∗ fEetia n lcrncEngineering, Electronic and Electrical of siae olt osdrtephysical the consider to tool isticated ttos eso otinterpretations most show we etations, l vne rso,B81L UK 1TL, BS8 Bristol, Avenue, all | ψ i = √ 1 2 ∆ ( ↑ |↓i ⊗ |↑i x ∆ p ≥ ~ 2 + ↓ |↑i ⊗ |↓i (2) ) (1) 2 send information instantaneously. To reconcile this with problem was rendered into quantum-informational form, the world around us, one of three options previously as- as is valid if quantum mechanics provides a complete pic- sumed impossible must be allowed. ture of reality. This was Bell’s Theorem - that these Bell The first option is to remove locality - to assume, in Inequalities, which act as limits for local-hidden-variable quantum cases, an instantaneous signal can be sent, with- models, can be violated using quantum mechanics. out any visible mechanism, to allow the second particle The most famous of these is the CHSH Inequality [4]. to align with the first. This didn’t appeal to Einstein, Here, we imagine a source that emits pairs of particles, given, on the face of it, it appears this means nature each of which goes into a test unit. For the left-hand violates Special Relativity. particle, it either is subjected to test a or a′, and for the The second option is to say, in all previous experi- right, to test b or b′. All these tests have result either ments, any attempt to remove pre-existing correlations +1 or 1 (e.g. tests of spin, where a and a′ (b and b′) has failed, and that whatever causes the first particle to are along− different axes). While the two tests on each prefer one option forces the second into its correspond- side have no requirement to be orthogonal to each other, ing state - the result was never random, but one of two the result is most visible if they do (e.g. a and b testing similar-looking but ultimately superdetermined options. x-spin, and a′ and b′ y-spin). This involves either claiming all evidence so far for quan- We then derive the correlations, E(x, y), for x being tum theory is the result of undetectably small correla- one of the tests on the left-hand particle (a or a′), and y tions (which grows less and less likely the more data we being one of the tests on the right (b or b′). We do this collect), or that our universe is fully superdetermined. by taking the four coincidence counts for our particular The third option is to deny that the state of the par- choice of x and y (N++, when both detectors register ticles is counterfactually definite - to say the universe is +1, N−−, when both register 1, N+−, when the left − indeterminate before measurement, and these simultane- registers +1 and the right 1, and N−+, when the left ous joint probability states (e.g. Eq.2) truly exist. While registers 1 and the right +1),− and get a weighted mea- the previous two options violate physical laws or experi- sure of the− correlation of x and y, mental evidence, this one, while mathematically more al- lowable, seems to strike at the heart of the assumptions (N++ + N−−) (N+− N−+) of science - repeatability, causality, and determinacy. E(x, y)= − − (3) To that end, Einstein advocated that, rather than ac- (N++ + N−−) + (N+− + N−+) cept any of these three seemingly nonsensical options, quantum mechanics must be incomplete, and so there From this, we then generate S, must be some local hidden variable governing quantum phenomena in a way that at least respects the basic tenets ′ ′ ′ ′ S = E(a,b) E(a,b )+ E(a ,b)+ E(a ,b ) 2 (4) of our understanding of reality (locality/no superluminal- − ≤ ity, counterfactual definiteness, and regular causality/no This gives our Bell Inequality - if a quantum version superdeterminism). This question - whether quantum of the system can get a value of S greater than 2, then theory was incomplete, and a local hidden variable model we have a test for which of these two forms of logic the could complete it - was the basis of the debate between universe obeys. This quantum case we compare it to Bohr and Einstein; but little could be said to prove one (Tsirelson’s Bound [5]) is where we take the sum of the side over the other. expectation values of the products of the observables that correspond to the tests (Aˆ for a, Aˆ′ for a′, etc...) to get B. Bell’s Theorem S = AˆBˆ + AˆBˆ′ + Aˆ′Bˆ Aˆ′Bˆ′ 2√2 (5) For nearly thirty years, the question of which of these h i h i h i − h i≤ options were correct remained a point of interpretation - This shows the Bell Inequality for the CHSH set-up can there was no physical difference ascribable to either quan- be violated in the quantum case - local-hidden-variable tum mechanics being an incomplete local hidden variable models fail to account for quantum correlations between theory, or being complete but meaning one of a number the two particles. This means an experiment giving an S- of peculiar conclusions for the universe. Bell, however, value of greater than 2, as was just shown possible, proves changed this by showing, in certain circumstances, a dif- Bell’s Theorem, and shows that the Universe doesn’t ference between predictions resulting from local-hidden- obey a local-hidden-variable model. variable models and fully quantum-mechanical models of A large body of work has been done to experimen- the world [3]. tally test Bell’s Theorem, given the difficulty in ensuring Specifically, he proposed there would be a set of ex- apparent quantum correlations that allow the Bell in- periments you could undertake, where there would be an equality violation aren’t due to preexisting correlations upper bound on the correlations you would get classically governing the choice of test. Of these, the most notable (say, if the universe could be fully described by a local- is Aspect et al’s, where the test photons were subjected hidden-variable model), which could be beaten if the to was chosen while they were in flight [6, 7]. 3

Since then, there have been many more tests, each with the particles collapse into is still random, and so there is ever more ingenious ways of reducing the likelihood that still arguably no action at a distance [16]. the results are due to loopholes. The most signiﬁcant re- However, while not usable to send meaningful infor- cent one, by Hensen et al, closes this correlation loophole mation between observers, non-locality does still allow entirely, meaning there can only be prior correlation gov- information to pass between the two particles, which in- erning the choice of test if superdeterminism is true [8, 9]. tuitively seems a violation of special relativity. Given no This proves once and for all quantum mechanics isn’t the other violation of special relativity has been found as of incomplete form of a local hidden variable model. yet [17, 18], we conclude it is unlikely quantum systems communicate nonlocally [19].

III. COUNTERFACTUALITY AND DEFINITENESS 2. Superdeterminsim

A. The Trilemma Revisited The second possibility is the universe is superdetermined: each and every event is uniquely and independently caused separate to any other event. Each parti- Given experimental proofs of Bell’s Theorem mean cle ‘knows’ in advance which state it will be in. While quantum mechanics can’t be the incomplete form of a lo- explaining the supposed instantaneous, mechanism-free cal hidden-variable theory, we have to return to the other collapse of one particle based on the other (due to it options which allow a quantum-mechanical description of instead having been separately pre-ordained into which reality to match the universe. These are that either su- states the particles will collapse), it is both epistemically perluminal communication is possible, superdeterminism difficult, and presents physical issues. is in effect in our universe, or there is no counterfactual Superdeterminism is epistemically difficult as it makes definiteness. These all have issues. it impossible to establish causal relationships between any two things, given everything is caused independently and separately from anything else. This makes it use- 1. Superluminal Communication less to form laws governing causal interactions, as there are no causal interactions. Admittedly, this implies The first option allowing Bell Inequality violation is all events are superdetermined, but, given all quantum that, on the collapse of one particle’s state, it can instan- events would need to be, this isn’t too different. taneously send a message to its partner, allowing collapse Further, it makes us ask what predetermines these into the corresponding state. This respects standard defi- events? While initially popular with metaphysicians, nitions of causality and definiteness, but it has one major such as Bishop Berkeley, divinely-inspired superdeter- flaw - it violates special relativity. minism has fallen out of favour with even the majority According to Special Relativity, the fastest information of religious philosophers, who view the sheer sum of evi- can propagate is c, the vacuum speed of light [10]. This dence that events causally determine one another seems has been repeatedly demonstrated [11–15], and is a cor- far more parsimonious than everything being caused in- nerstone of modern physics. Entanglement circumvent- dependently of everything else. ing this limit and allowing one particle to change another, Aside from this, there is also the more physical issue distant from it, simultaneously - is implausible. Further that we have observed no mechanism by which such a full Lorentz invariance makes it akin to saying the collapse of predestination could be ‘remembered’ by every single ob- the particle’s wavefunction can cause something to occur ject in the universe. This choreographing of events would which happened before it - retrocausality. lead to a number of absurdities, such as being able to in- However, the way in which this collapse would occur terrogate the future of anything in the universe from any- would not be immediately observable - it could only be thing else, allowing effective superluminal classical com- seen once both parts of the entangled state were mea- munication, and rendering meaningless all of the conclu- sured, and then would require classical communication sions of special relativity. This makes superdeterminism between the measurers of these two to confirm. Further, even less palatable than non-locality, as at least for that, the state the two particles collapse into would still be ran- only quantum correlations need be superluminal. dom, and so couldn’t be modulated to send information. That is, with this non-local collapse, no object or meaningful information would be propagating faster than the 3. Counterfactual Definiteness speed of light - it is a case of passion, rather than action, at a distance. The only way this could be made action The final option is the one Einstein was most critical at a distance is if there were some way that, depending of - that the universe, at least for quantum interactions, on the basis one state was measured in, the state of the lacks counterfactual definiteness. second particle could be altered. While this can be done Counterfactual definiteness is the idea that, were some- through steering, the states within the chosen basis that thing done differently, there is a matter of the fact as to 4 how the universe would be. While this is commonly con- 2. Relative State Interpretation fused with determinacy - that the universe evolves deterministically, from a set of initial conditions - it is perfectly Everett’s Relative State (‘Many-Worlds’) Interpreta- valid to say that the universe can be probabilistic, but tion, posits that different possible eigenstates simultane- counterfactually definite; so long as there is some mat- ously co-exist within the state of the world - we merely ter of fact about how the world would be if conditions observe one of these eigenstates. This is analogised as a changed, the universe is counterfactually definite. variety of alternate worlds, one for each eigenstate, with While the other options allow some conditions around decoherence acting as branching points from which they a measurement to change in a way the measurer cannot split and can no longer interact. Counterfactual definite- control (either by instantaneous signalling, or vastly in- ness is avoided by having not one definite counterfactual creasing the number of conditions), the universe being option, but many simultaneously-acting possibilities [23]. counterfactually indefinite removes this control entirely - Further, the Relative State interpretation argues a lack of we become unable to say, if we changed what was mea- counterfactual definiteness is different to indeterminism - sured, how the universe would be. physical quantities can be multi-valued (and so not fully This makes us ask just how counterfactually indefinite counterfactually definite), but the overall state evolves quantum theory actually is - given it appears, except at deterministically. This is very different from what is typ- measurement, that the wavefunction evolves determinis- ically expected from counterfactual indefiniteness. tically, in a way entirely determined by initial conditions - and if weakening this indefiniteness can help it look more commonsensical to an outside observer. 3. Bohmian Interpretation

In Bohm’s Interpretation, particles have a definite value for position, momentum, and all other observables, as in classical physics. The difference, which makes it a B. Interpretations and Definiteness valid interpretation of quantum mechanics, is that each particle is subject to an expressly non-local quantum po- The easiest way to evaluate the extent we need to re- tential. This perturbs the particle into precisely the state move counterfactual definiteness is to look at interpreta- expected by quantum mechanics [24]. tions in which it is removed. Given quantum mechanics However, this still leaves particles with objectively- demands the existence of conjugate variables, and so a existing values for all observables (in Bohm’s theory, the variable’s ability to be unknown (to allow its conjugate to uncertainty relation is a limit on our ability to know the be precisely known), all interpretations must permit un- value of conjugate variables, rather than uncertainty on certainty about certain variables. How they implement their existence), and so counterfactual definiteness exists this is where they differ. - the element of the trilemma lost here is locality. Bohm theory shows how we can keep counterfactual definiteness by weakening of one of the other facets - given Bohm’s non-locality prohibits faster-than-light signalling, it has been argued it still preserves special relativity, despite 1. Copenhagen Interpretation being non-local [25].

Originally the standard interpretation, the Copen- hagen Interpretation rejects counterfactual definiteness 4. Collapse-Based Interpretations - we cannot say, if we were to measure a conjugate variable of one we had already measured, what the result There are many Collapse-Based Interpretations of would be. Bohr, the interpretation’s founder, believed quantum mechanics, but they all have decoherence as a in complementarity - that truth conditions of sentences physical process. Whether based on consciousness, grav- giving an observable’s value depend on the apparatus in- ity, or some other objective phenomenon, at some point, volved, insofar as these truth conditions have to reference the wavefunction decays from a superposition of many the experimental setup as well as the experiment’s actual eigenstates to just one [26]. In these interpretations, the outcome [20]. In this interpretation, questions like “what other eigenstates, previously part of this superposition, would the spin of an electron be, if it was measured in are lost entirely - however, counterfactual definiteness the x-direction rather than the y?” are meaningless. This still exists for these options probabilistically. Further, almost-quietist interpretation bans us from even consid- similar to the Relative State Interpretation, before deco- ering counterfactual cases [21, 22]. Therefore, it, along- herence, possible values exist as a multi-valued variable - side other minimalist interpretations (e.g. the Ensem- both of these are different to full counterfactual definite- ble and the Statistical Interpretation), provides us with ness, but are not counterfactual indefiniteness either. an account of what quantum mechanics looks like when However, these theories also have the issue that col- completely counterfactually indefinite. lapse is difficult to observe, and we currently have no rea- 5 son for preferring one over the others. This shows just how possible options can interfere to create classically- removing counterfactual definiteness without any addi- impossible effects, which is even more useful than having tional loss from one of the other possible options may one counterfactually definite option. not be as useful as thought for producing a valid interpretation of quantum theory. D. Shifted Counterfactuality

C. Weakened Definiteness Looking at counterfactuality less as possible results of actions, and more as possible ways measurement could From our analysis of other interpretations, we can now have collapsed a state, we see some similarity, especially if see how much counterfactual indefiniteness is necessary. we attribute counterfactual semi-definiteness to all these possible options. Indeed, prior to collapse, the density matrix formulation shows different modes (representing 1. Uncertainty of Conjugate Variables different ways the system can collapse) interact, which seems impossible classically [26]. The first thing the above analysis blocks is the simul- These interactions allow wondrous things, like observ- taneous reality of conjugate variables. The only inter- ing something indirectly, as with Elitzur and Vaidman’s pretation proposing the definiteness of both variables in ‘Bomb Detector’ [28], or communicating without sending a pair is Bohm’s theory, and that is one of its key weak- anything between two parties [29], as with Salih et al’s nesses. This is for two reasons. Firstly, experiments counterfactual communication protocol [30–32] - things show the indeterminacy isn’t due to experimental issues, which, without considering these semi-definite options, but conjugate variables really not being simultaneously would never have been imagined. measurable. The indeterminacy is built into the world, This shows how useful it is to consider counterfactual rather than being due to experimental disturbances [27]. options, more than the Copenhagen interpretation al- Secondly, this indeterminacy is key to how quantum ob- lows [33]. By making Einstein’s challenge to quantum jects behave (e.g. electrons taking up all of their shell, mechanics testable, even though it disproved his local- rather than just difficult-to-locate points, causing Pauli hidden-variable theory, researchers once again investi- exclusion). While you could slide this indeterminacy to gated the more philosophical side of quantum mechan- Bohm’s quantum potential, as modern Bohmians tend ics, long-neglected by followers of Heisenberg and Dirac - to do, that seems to make the potential the real phe- and we hope consideration of counterfactuality will lead nomenon, and the point-like particle solely a marker we to similar advances. use for our own comprehension - bringing in indeterminacy by the backdoor. This means the uncertainty relation between two conjugate variables, and the indetermi- IV. CONCLUSION nacy caused by this, must be included. By looking at Bell’s Theorem, and the trilemma it causes, we showed a lack of counterfactual definiteness 2. Counterfactual Semi-Definiteness is necessary to allow quantum phenomena to occur. We then evaluated the minimal amount of indefiniteness re- Comparing the Copenhagen Interpretation to the quired for us to maintain the theoretical underpinnings Relative State and collapse-based interpretations, we of the experimental evidence we observe. have come across a new concept: counterfactual semi- From this, we reached the conclusion that, while the definiteness. While being unable to attribute a single uncertainty relation prevents us from having the strong result to potential measurement as we can do with full counterfactual definiteness of classical physics, we can counterfactual definiteness, we can determine the result still have some form of counterfactual semi-definiteness. each of them would bring (for a finite number of possible This definiteness of possible options available after states), and weigh these up using the Born rule - getting measurement, while not as solid as the counterfactual multi-valued observables. This is not the strict single- definiteness the EPR paradox and Bell’s Theorem pro- valued counterfactual definiteness we expect, it also isn’t hibit, allows us to start investigating the physical inter- the complete counterfactual indefiniteness of the Copen- pretation of possible states in a way that, until Bell, had hagen Interpretation; being deterministic, we can de- been neglected since the early days of quantum theory. scribe how this multi-valued function would change if the Working from this insight, the myriad possible areas of evolution were altered. development in counterfactuality seem breathtaking, and This returns some common sense to the discussion. likely to underpin developments in quantum theory and Unlike the Copenhagen interpretation, where counterfac- application for years to come. tual possibilities are treated as non-existent, now we can Acknowledgements - Originally written while studying properly evaluate them. This is useful not just for after for the MSci Physics and Philosophy degree at the Uni- measurement, but before - as we see below, it explains versity of Bristol, under the supervision of James La- 6 dyman, who I thank for his valuable comments. I also thank John Rarity, and the University of Bristol’s Quan- tum Engineering Technology Laboratories.

[1] A. Einstein, B. Podolsky, and N. Rosen, mann, K. Andeen, T. Anderson, et al., arXiv preprint Phys. Rev. 47, 777 (1935). arXiv:1709.03434 (2017). [2] D. Bohm, Quantum Theory, Dover Books on Physics Se- [18] P. Delva, J. Lodewyck, S. Bilicki, E. Bookjans, G. Vallet, ries (Dover Publications, 1951). R. Le Targat, P.-E. Pottie, C. Guerlin, F. Meynadier, [3] J. S. Bell, Physics Physique Fizika 1, 195 (1964). C. Le Poncin-Lafitte, et al., Physical review letters 118, [4] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, 221102 (2017). Phys. Rev. Lett. 23, 880 (1969). [19] W. De Muynck, W. De Baere, and H. Martens, Founda- [5] B. S. Cirel’son, Letters in Mathematical Physics 4, 93 (1980). tions of Physics 24, 1589 (1994). [6] S. J. Freedman and J. F. Clauser, [20] J. Faye, in The Stanford Encyclopedia of Philosophy, Phys. Rev. Lett. 28, 938 (1972). edited by E. N. Zalta (Metaphysics Research Lab, Stan- [7] A. Aspect, P. Grangier, and G. Roger, ford University, 2019) winter 2019 ed. Phys. Rev. Lett. 49, 91 (1982). [21] D. R. Murdoch, Niels Bohr’s philosophy of physics (Cam- [8] M. Giustina, M. A. M. Versteegh, S. Wengerowsky, bridge University Press, 1987). J. Handsteiner, A. Hochrainer, K. Phelan, F. Steinlech- [22] J. Faye, Niels Bohr: his heritage and legacy: an anti- ner, J. Kofler, J.-A. Larsson, C. Abellán, W. Amaya, realist view of quantum mechanics (Kluwer Academic V. Pruneri, M. W. Mitchell, J. Beyer, T. Ger- Publisher, 1991). rits, A. E. Lita, L. K. Shalm, S. W. Nam, [23] G. Blaylock, American Journal of Physics 78, 111 (2010). T. Scheidl, R. Ursin, B. Wittmann, and A. Zeilinger, [24] D. Bohm, Phys. Rev. 85, 166 (1952). Phys. Rev. Lett. 115, 250401 (2015). [25] T. Maudlin, Quantum non-locality and relativity: Meta- [9] B. Hensen, H. Bernien, A. E. Dréau, A. Reiserer, N. Kalb, physical intimations of modern physics (John Wiley & M. S. Blok, J. Ruitenberg, R. F. Vermeulen, R. N. Sons, 2011). Schouten, C. Abellán, et al., Nature 526, 682 (2015). [26] J. Von Neumann, Mathematical Foundations of Quantum [10] A. Einstein et al., Annalen der Physik 17, 50 (1905). Mechanics: New Edition (Princeton university press, [11] Z. G. Guiragossian, G. Rothbart, M. Year- 2018). ian, R. Gearhart, and J. Murray, [27] M. Ozawa, Physics Letters A 320, 367 (2004). Physical Review Letters 34, 335 (1975). [28] A. C. Elitzur and L. Vaidman, [12] J. Alspector, G. Kalbfleisch, N. Baggett, E. C. Fowler, Foundations of Physics 23, 987 (1993). B. Barish, A. Bodek, D. Buchholz, F. Sciulli, E. Siskind, [29] J. R. Hance, J. Ladyman, and J. Rarity, L. Stutte, et al., Physical Review Letters 36, 837 (1976). arXiv preprint arXiv:1909.07530 (2019). [13] G. R. Kalbfleisch, N. Baggett, E. C. Fowler, and J. Al- [30] H. Salih, Z.-H. Li, M. Al-Amri, and M. S. Zubairy, spector, Physical Review Letters 43, 1361 (1979). Physical review letters 110, 170502 (2013). [14] L. Stodolsky, Physics Letters B 201, 353 (1988). [31] H. Salih, W. McCutcheon, J. Hance, P. Skrzypczyk, and [15] G. L. Greene, M. S. Dewey, E. Kessler Jr, and E. Fis- J. Rarity, arXiv preprint , arXiv:1806.01257 (2018). chbach, Physical Review D 44, R2216 (1991). [32] J. Hance, W. McCutcheon, P. Yard, and J. Rarity, in [16] H. M. Wiseman, S. J. Jones, and A. C. Doherty, Physical Quantum Information and Measurement (QIM) V: Quantum Technologies review letters 98, 140402 (2007). (Optical Society of America, 2019) p. T5A.50. [17] M. G. Aartsen, M. Ackermann, J. Adams, J. Aguilar, [33] J. F. Clauser, in Quantum (Un)speakables: From Bell to M. Ahlers, M. Ahrens, I. A. Samarai, D. Alt- Quantum Information (Springer, 2002) pp. 61–98.