A Very Common Fallacy in Quantum Mechanics: Superposition, Delayed Choice, Quantum Erasers, Retrocausality, and All That
AA Very Common Fallacy in Quantum Mechanics:Superposition, Delayed Choice, Quantum Erasers,Retrocausality, and All That
David EllermanUniversity of California at RiversideDecember 21, 2011
Abstract
There is a very common fallacy, here called the separation fallacy , that is involved in the inter-pretation of quantum experiments involving a certain type of separation such as the: double-slitexperiments, which-way interferometer experiments, polarization analyzer experiments, Stern-Gerlach experiments, and quantum eraser experiments. It is the separation fallacy that leadsnot only to flawed textbook accounts of these experiments but to flawed inferences about retro-causality in the context of ”delayed choice” versions of separation experiments.
Contents
There is a very common fallacy, here called the separation fallacy , that is involved in the interpre-tation of quantum experiments involving a certain type of separation such as the: • double-slit experiments, • which-way interferometer experiments, 1 a r X i v : . [ qu a n t - ph ] D ec polarization analyzer experiments, • Stern-Gerlach experiments, and • quantum eraser experiments.In each case, given an incoming quantum particle, the apparatus creates a certain labelled ortagged (i.e., entangled) superposition of certain eigenstates (the ”separation”). Detectors can beplaced in certain positions (determined by the tags) so that when the evolving superposition stateis finally projected or collapsed by the detectors, then only one of the eigenstates can register ateach detector. The separation fallacy mistakes the creation of a tagged or entangled superpositionfor a measurement. Thus it treats the particle as if it had already been projected or collapsed to aneigenstate at the separation apparatus rather than at the later detectors. But if the detectors weresuddenly removed while the particle was in the apparatus, then the superposition would continue toevolve and have distinctive effects (e.g., interference patterns in the two-slit experiment).Hence the separation fallacy makes it seem that by the delayed choice to insert or remove theappropriately positioned detectors, one can retro-cause either a collapse to an eigenstate or not atthe particle’s entrance into the separation apparatus.The separation fallacy is remedied by: • taking superposition seriously, i.e., by seeing that the separation apparatus created an entan-gled superposition state of the alternatives (regardless of what happens later) which evolvesuntil a measurement is taken, and • taking into account the role of detector placement (”contextuality”), i.e., by seeing that if asuitably positioned detector, as determined by the positional labels or tags, can only detectone collapsed eigenstate, then it does not mean that the particle was already in that eigenstateprior to the measurement (e.g., it does not mean that the particle ”went through one slit,””took one arm,” or was already in a polarization or spin eigenstate).The separation fallacy will be first illustrated in a non-technical manner for the first four exper-iments. Then the lessons will be applied in a slightly more technical discussion of quantum eraserexperiments–where, due to the separation fallacy, incorrect inferences about retrocausality have beenrampant. In the well-known setup for the double-slit experiment, if a detector D is placed a small finitedistance after slit 1 so a particle ”going through the other slit” cannot reach the detector, then ahit at the detector is usually interpreted as ”the particle went through slit 1.”Figure 1aBut this is incorrect. The particle is in a superposition state, which we might represent schemat-ically as | Slit (cid:105) + | Slit (cid:105) , that evolves until it hits the detector which projects (or collapses) the2uperposition to one of (the evolved versions of) the slit-eigenstates. The particle’s state was notcollapsed earlier so it was not previously in the | Slit (cid:105) eigenstate, i.e., it did not ”go through slit 1.”Thus what is called ”detecting which slit the particle went through” is a misinterpretation. It isonly placing a detector in such a position so that when the superposition projects to an eigenstate,only one of the eigenstates can register in that detector. It is about detector placement ; it is notabout which-slit.By erroneously talking about the detector ”showing the particle went through slit 1,” we imply atype of retro-causality. If the detector is suddenly removed after the particle has passed the slits, thenthe superposition state continues to evolve and shows interference on the far wall (not shown)—inwhich case people say ”the particle went through both slits.” Thus the ”bad talk” makes it seemthat by removing or inserting the detector after the particle is beyond the slits, one can retro-causethe particle to go through both slits or one slit only.This sudden removal or insertion of detectors that can only detect one of the slit-eigenstates isa version of Wheeler’s delayed choice thought-experiment [10].Figure 1b. Wheeler’s Delayed Choice 2 Slit ExperimentIn Wheeler’s version of the experiment, there are two detectors which are positioned behindthe removable screen so they can only detect one of the projected (evolved) slit eigenstates whenthe screen is removed. The choice to remove the screen or not is delayed until after a photon hastraversed the two slits.”In the one case [screen in place] the quantum will ... contribute to the record of atwo-slit interference fringe. In the other case [screen removed] one of the two counters willgo off and signal in which beam–and therefore from which slit–the photon has arrived.”[10, p. 13]The separation fallacy is involved when Wheeler infers from the fact that one of the specially-placed detectors went off that the photon had come from one of the slits–as if there had been aprojection to one of the slit eigenstates at the slits rather than later at the detectors.Such descriptions using the separation fallacy are unfortunately common and have generated aspate of speculations about retrocausality. Consider a Mach-Zehnder-style interferometer with only one beam-splitter (e.g., half-silvered mirror)at the photon source which creates the photon superposition: | T (cid:105) + | R (cid:105) (which stand for ”Transmit”to the upper arm or ”Reflect” into the lower arm at the first beam-splitter).3igure 2: One beam-splitterWhen detector D registers a hit, it is said that ”the photon was reflected and thus took thelower arm” of the interferometer and similarly for D and passing through into the upper arm. Thisis the interferometer analogue of putting two up-close detectors after the two slits in the two-slitexperiment.And this standard description is incorrect for the same reasons. The photon stays in the su-perposition state until the detectors force a projection to one of the (evolved) eigenstates. If theprojection is to the evolved | R (cid:105) eigenstate then only D will get a hit, and similarly for D andthe evolved version of | T (cid:105) . The point is that the placement of the detectors (like in the double-slitexperiment) only captures one or the other of the projected eigenstates.Now insert a second beam-splitter as in the following diagram.Figure 3: Two beam-splittersIt is said that the second beam-splitter ”erases” the ”which-way information” so that a hit ateither detector could have come from either arm, and thus an interference pattern emerges.But this is also incorrect. The superposition state | T (cid:105) + | R (cid:105) (which contains no which-wayinformation) is further transformed at the second beam-splitter to the superposition | T , T (cid:105) + | T , R (cid:105) + | R , T (cid:105) + | R , R (cid:105) that can be regrouped according to what can register at each detector:4 | T , R (cid:105) + | R , T (cid:105) ] D + [ | T , T (cid:105) + | R , R (cid:105) ] D .The so-called ”which-way information” was not there to be ”erased” since the particle did nottake one way or the other in the first place. The second beam-splitter only allows the superpositionstate [ | T , R (cid:105) + | R , T (cid:105) ] D to be registered at D or the superposition state [ | T , T (cid:105) + | R , R (cid:105) ] D to be registered at D . By using a phase shifter φ , an interference pattern can be recorded at eachdetector since each one is now detecting a superposition that will involve interference.By inserting or removing the second beam-splitter after the particle has traversed the first beam-splitter (as in [10]), the separation fallacy makes it seem that we can retro-cause the particle to gothrough both arms or only one arm.The point is not the second beam-splitter but the detectors being able to register the collapseto either eigenstate and thus the interference between them. Instead of inserting the second beam-splitter, we could rig up more mirrors, a lens, and a single detector so that when the single detectorcauses the collapse, then it is will register either arm-eigenstate.Figure 4: Detector placed to register all hitsThis might also be (mis)interpreted as ”erasing” the ”which-way information” but in fact thephoton did not go through just one arm so there was no such information to be erased. The point isthe positioning of the detector so that it detects the evolved superposition | T (cid:105) + | R (cid:105) that will showinterference. Any setup that would allow a detector to register both collapsed eigenstates (and thus toregister the interference effects of the evolving superposition) would ipso facto be a setup that couldbe (mis)interpreted as ”erasing” the ”which-way information.” That is why the separation fallacyis so persistent in the interpretation of which-way interferometer and other quantum separationexperiments. Another common textbook example of the separation fallacy is the treatment of polarization ana-lyzers such as calcite crystals that are said to create two orthogonally polarized beams in the upperand lower channels, say | v (cid:105) and | h (cid:105) from an arbitrary incident beam.5igure 5: vh -analyzerThe output from the analyzer P is routinely described as a ”vertically polarized” beam and”horizontally polarized” beam as if the analyzer was itself a measurement that collapsed or projectedthe incident beam to either of those polarization eigenstates. This seems to follow because if onepositions a detector in the upper beam then only vertically polarized photons are observed andsimilarly for the lower beam and horizontally polarized photons. A blocking mask in one of thebeams has the same effect as a detector to project the photons to eigenstates. If a blocking mask ininserted in the lower beam, then only vertically polarized photons will be found in the upper beam,and vice-versa.But here again, the story is about detector (or blocking mask) placement (”placement” is moreprecise than ”contextuality”); it is not about the analyzer supposedly projecting a photon intoone or the other of the eigenstates. The analyzer puts the incident photons into a superpositionstate, an entangled superposition state that associates polarization and the spatial channel. If adetector is placed in, say, the upper channel, then that is the measurement that collapses the evolvedsuperposition state. If the collapse is to the vertical polarization eigenstate then it will register onlyin the upper detector and similarly for a collapse to the horizontal polarization eigenstate for anydetector placed in the lower channel. Thus it is misleadingly said that the upper beam was alreadyvertically polarized and the lower beam was already horizontally polarized as if the analyzer hadalready done the projection to one of the eigenstates.If the analyzer had in fact induced a collapse to the eigenstates, then any prior polarization ofthe incident beam would be lost. Hence assume that the incident beam was prepared in a specificpolarization of, say, | ◦ (cid:105) half-way between the states of vertical and horizontal polarization. Thenfollow the vh -analyzer P with its inverse P − to form an analyzer loop [3].Figure 6: vh -analyzer loopThe characteristic feature of an analyzer loop is that it outputs the same polarization, in thiscase | ◦ (cid:105) , as the incident beam. This would be impossible if the P analyzer had in fact renderedall the photons into a vertical or horizontal eigenstate thereby destroying the information about thepolarization of the incident beam. But since no collapsing measurement was in fact made in P orits inverse, the original beam can be the output of an analyzer loop.Very few textbooks realize there is even a problem with presenting a polarization analyzer suchas a calcite crystal as creating two beams with orthogonal eigenstate polarizations—rather thancreating a tagged superposition state so that appropriately positioned detectors can detect only oneeigenstate when the detectors cause the projections to eigenstates.One (partial) exception is Dicke and Wittke’s text [1]. At first they present polarization analyzersas if they measured polarization and thus ”destroyed completely any information that we had aboutthe polarization” [1, p. 118] of the incident beam. But then they note a problem:6The equipment [polarization analyzers] has been described in terms of devices whichmeasure the polarization of a photon. Strictly speaking, this is not quite accurate.” [1,p. 118]They then go on to consider the inverse analyzer P − which combined with P will form ananalyzer loop that just transmits the incident photon unchanged.They have some trouble squaring this with their prior statement about the P analyzer destroyingthe polarization of the incident beam but they, unlike most texts, struggle with getting it right.”Stating it another way, although [when considered by itself] the polarization P com-pletely destroyed the previous polarization Q [of the incident beam], making it impossibleto predict the result of the outcome of a subsequent measurement of Q , in [the analyzerloop] the disturbance of the polarization which was effected by the box P is seen to berevocable: if the box P is combined with another box of the right type, the combinationcan be such as to leave the polarization Q unaffected.” [1, p. 119]They then go on to correctly note that the polarization analyzer P did not in fact project theincident photons into polarization eigenstates.”However, it should be noted that in this particular case [sic!], the first box P in [thefirst half of the analyzer loop] did not really measure the polarization of the photon: nodetermination was made of the channel ... which the photon followed in leaving the box P .” [1, p. 119]There is some classical imagery (like Schr¨odinger’s cat running around one side or the otherside of a tree) that is sometimes used to illustrate quantum separation experiments when in fact itonly illustrates how classical imagery can be misleading. Suppose an interstate highway separatesat a city into both northern and southern bypass routes–like the two channels in a polarizationanalyzer loop. One can observe the bypass routes while a car is in transit and find that it is in onebypass route or another. But after the car transits whichever bypass it took without being observedand rejoins the undivided interstate, then it is said that the which-way information is erased so anobservation cannot elicit that information.This is not a correct description of the corresponding quantum separation experiment sincethe classical imagery does not contemplate superposition states. The particle-as-car is in a taggedsuperposition of the two routes until an observation (e.g., a detector or ”road block”) collapses thesuperposition to one eigenstate or the other. Correct descriptions of quantum separation experimentsrequire taking superposition seriously–so classical imagery should only be used cum grano salis .This analysis might be rendered in a more technical but highly schematic way. The photonsin the incident beam have a particular polarization | ψ (cid:105) such as | ◦ (cid:105) in the above example. Thispolarization state can be represented or resolved in terms of the vh -basis as: | ψ (cid:105) = (cid:104) v | ψ (cid:105) | v (cid:105) + (cid:104) h | ψ (cid:105) | h (cid:105) .The effect of the vh -analyzer P might be represented as tagging the vertical and horizontalpolarization states with the upper and lower (or straight) channels so the vh -analyzer puts anincident photon into the superposition state: (cid:104) v | ψ (cid:105) | v (cid:105) U + (cid:104) h | ψ (cid:105) | h (cid:105) L , not into an eigenstate of | v (cid:105) in the upper channel or an eigenstate | h (cid:105) in the lower channel.If a blocker or detector were inserted in either channel, then this superposition state wouldproject to one of the eigenstates, and then (as indicated by the tags) only vertically polarized photonswould be found in the upper channel and horizontally polarized photons in the lower channel.7he separation fallacy is to describe the vh -analyzer as if the analyzer’s effect by itself was toproject an incident photon either into | v (cid:105) in the upper channel or | h (cid:105) in the lower channel–insteadof only creating the above tagged superposition state. The mistake of describing the unmeasuredpolarization analyzer as creating two beams of eigenstate polarized photons is analogous to themistake of describing a particle as going through one slit or the other in the unmeasured-at-slitsdouble-slit experiment–and similarly for the other separation experiments.It is fallacious to reason that ”we know the photons are in one polarization state in one channeland in the orthogonal polarization state in the other channel because that is what we find when wemeasure the channels,” just as it is fallacious to reason ”the particle has to go through one slit oranother (or one arm or another in the interferometer experiment) because that is what we find whenwe measure it.” This purely operational (or ”Copenhagen”) description does not take superpositionseriously since a superposition state is not ”what we find when we measure.”In the analyzer loop , no measurement (detector or blocker) is made after the vh -analyzer. It isfollowed by the inverse vh -analyzer P − which has the inverse effect of removing the U and L tagsfrom the superposition state (cid:104) v | ψ (cid:105) | v (cid:105) U + (cid:104) h | ψ (cid:105) | h (cid:105) L so that a photon exits the loop in the untaggedsuperposition state: (cid:104) v | ψ (cid:105) | v (cid:105) + (cid:104) h | ψ (cid:105) | h (cid:105) = | ψ (cid:105) .The inverse vh -analyzer does not ”erase” the which-polarization information since there was nomeasurement–to reduce the superposition state to eigenstate polarizations in the channels of theanalyzer loop–in the first place. The inverse vh -analyzer does erase the which-channel tags so theoriginal state (cid:104) v | ψ (cid:105) | v (cid:105) + (cid:104) h | ψ (cid:105) | h (cid:105) = | ψ (cid:105) is restored (which could be viewed as a type of interferenceeffect, e.g., [3, Sections 7-4, 7-5]). We have seen the separation fallacy in the standard treatments of the double-slit experiment, which-way interferometer experiments, and in polarization analyzers. In spite of the differences betweenthose separation experiments, there was that common (mis)interpretative theme. Since the ”logic”of the polarization analyzers is followed in the Stern-Gerlach experiment (with spin playing the roleof polarization), it is not surprising that the same fallacy occurs there.Figure 7: Stern-Gerlach ApparatusAnd again, the fallacy is revealed by considering the Stern-Gerlach analogue of an analyzer loop.One of the very few texts to consider such a Stern-Gerlach analyzer loop is
The Feynman Lectureson Physics: Quantum Mechanics (Vol. III) where it is called a ”modified Stern-Gerlach apparatus”[2, p. 5-2]. 8igure 8: Stern-Gerlach LoopOrdinarily texts represent the Stern-Gerlach apparatus as separating particles into spin eigen-states denoted by, say, + S, S, − S . But as in our other examples, the apparatus does not projectthe particles to eigenstates. Instead it creates a superposition state so that with a detector in acertain position, then as the detector causes the collapse to a spin eigenstate, the detector will onlysee particles of one spin state. Alternatively if the collapse is caused by placing blocking masks overtwo of the beams, then the particles in the third beam will all be those that have collapsed to thesame eigenstate. It is the detectors or blockers that cause the collapse or projection to eigenstates,not the prior separation apparatus.We previously saw how a polarization analyzer, contrary to the statement in many texts, doesnot lose the polarization information of the incident beam when it ”separates” the beam (into apositionally-tagged superposition state). In the context of the Stern-Gerlach apparatus, Feynmansimilarly remarks:”Some people would say that in the filtering by T we have ’lost the information’ aboutthe previous state (+ S ) because we have ’disturbed’ the atoms when we separated theminto three beams in the apparatus T. But that is not true. The past information is notlost by the separation into three beams, but by the blocking masks that are put in. . . .”[2, p. 5-9 (italics in original)] We have seen the same fallacy of interpretation in two-slit experiments, which-way interferometerexperiments, polarization analyzers, and Stern-Gerlach experiments. The common element in all thecases is that there is some ’separation’ apparatus that puts a particle into a certain superposition ofeigenstates in such a manner that when an appropriately positioned detector induces a collapse to aneigenstate, then the detector will only register one of the eigenstates. The separation fallacy is thatthis is misinterpreted as showing that the particle was already in that eigenstate in that position asa result of the previous ’separation.’ The quantum erasers are elaborated versions of these simplerexperiments, and a similar separation fallacy arises in that context.
A simple quantum eraser can be devised using a single beam of photons as in [5]. We start with thestandard two-slit setup. 9igure 9: Two-slit setupAfter the two slits, a photon could be schematically represented as being in a superpositionstate | s (cid:105) + | s (cid:105) (where s s v, h polarizersAfter the two slits, a photon is in a state that entangles the spatial slit states and the polar-ization states which might be represented as: | s (cid:105) ⊗ | h (cid:105) + | s (cid:105) ⊗ | v (cid:105) (for a discussion of this type ofentanglement, see [7]). But as this superposition evolves, it cannot be separated into a superpositionof the slit-states as before, so the interference disappears.Then a +45 o polarizer is inserted between the two-slit screen and the wall. This transforms theevolving state to: | s (cid:105) ⊗ | o (cid:105) + | s (cid:105) ⊗ | o (cid:105) = [ | s (cid:105) + | s (cid:105) ] ⊗ | o (cid:105) so that the | s (cid:105) + | s (cid:105) term will show interference in a ”fringe” pattern when the 45 o polarizedphotons hit the wall. If we had inserted a − o polarizer, then again interference in an ”antifringe”10attern would appear as the − o polarized photons hit the wall. The sum of the fringe and antifringepatterns gives the no-interference pattern of the previous figure.Figure 11: +45 o polarizer and fringe patternA common description of this type of quantum eraser experiment is that the insertion of the h, v polarizers ”marks” the photons with ”which-slit information” (Figure 10) that destroys theinterference–even if the horizontal or vertical polarization is not measured at the wall. If the hor-izontal or vertical polarization was measured at the wall, then the evolved superposition state | s (cid:105) ⊗ | h (cid:105) + | s (cid:105) ⊗ | v (cid:105) would collapse to the evolved version of | s (cid:105) (if h was found) or | s (cid:105) (if v was found). This is said to reveal the so-called ”which-slit information” that the photon wentthrough slit 1 or slit 2, i.e., that at the slits, the photon was already in the state | s (cid:105) or | s (cid:105) insteadof being in the entangled superposition state. By incorrectly inferring that the photon was in onestate or the other at the slits–while it would have to ”go through both slits” to yield the interferencepattern obtained by inserting the 45 o polarizer–we seem to be able to retrocause the particle togo through one slit or both slits by withdrawing or inserting the 45 o polarizer after a photon hastraversed the two slits.It is precisely the separation fallacy that leads to this inference of retrocausality. In the situationof Figure 10, the photon superposition state | s (cid:105) ⊗ | h (cid:105) + | s (cid:105) ⊗ | v (cid:105) evolves until it hits the wall. Theslit states are indeed marked, tagged, labelled, or entangled with polarization states but this isincorrectly called ”which-way information” as if it could ”reveal” that the photon was in the state | s (cid:105) or | s (cid:105) at the slits, i.e., that it went through slit 1 or slit 2.Also it might be noted that the insertion of a +45 o or − o polarizer does not ”restore” theoriginal interference pattern of Figure 9 but picks out the fringe or antifringe interference patternsout of the Figure 10 ”mush” of hits. We now turn to one of the more elaborate quantum eraser experiments [9].11igure 12: Setup with two slitsA photon hits a down-converter which emits a ”signal” p -photon entangled with an ”idler” s -photon with a superposition of orthogonal | x (cid:105) and | y (cid:105) polarizations so the overall state is: | Ψ (cid:105) = √ (cid:104) | x (cid:105) s ⊗ | y (cid:105) p + | y (cid:105) s ⊗ | x (cid:105) p (cid:105) .The lower s -photon hits a double-slit screen, and will show an interference pattern on the D s detector as the detector is moved along the x -axis.Next two quarter-wave plates are inserted before the two-slit screen with the fast axis of the oneover slit 1 oriented at | +45 ◦ (cid:105) to the x-axis and the one over the slit 2 with its fast axis oriented at |− ◦ (cid:105) to the x -axis. Figure 13: Setup with λ/ s s Ψ (cid:105) = (cid:104)(cid:16) | L (cid:105) s ⊗ | y (cid:105) p + i | R (cid:105) s ⊗ | x (cid:105) p (cid:17) + (cid:16) i | R (cid:105) s ⊗ | y (cid:105) p − i | L (cid:105) s ⊗ | x (cid:105) p (cid:17)(cid:105) .Then by measuring the linear polarization of the p -photon at D p and the circular polarizationat D s , ”which-slit information” is said to be obtained and no interference pattern recorded at D s .For instance measuring | x (cid:105) at D p and | L (cid:105) at D s imply s
2, i.e., slit 2. But as previously explained,this does not mean that the s -photon went through slit 2. It means we have positioned the twodetectors in polarization space , say to measure | x (cid:105) polarization at D p and | L (cid:105) polarization at D s , soonly when the superposition state collapses to | x (cid:105) for the p -photon and | L (cid:105) for the s -photon do weget a hit at both detectors.This is the analogue of the one-beam-splitter interferometer where the positioning of the de-tectors would only record one collapsed state which did not imply the system was all along in thatparticular arm-eigenstate. The phrase ”which-slit” or ”which-arm information” is a misnomer inthat it implies the system was already in a slit- or way-eigenstate and the so-called measurementonly revealed the information. Instead, it is only at the measurement that there is a collapse orprojection to an evolved slit-eigenstate (not at the previous separation due to the two slits).Walborn et al. indulge in the separation fallacy when they discuss what the so-called ”which-pathinformation” reveals.Let us consider the first possibility [detecting p before s ]. If photon p is detected withpolarization x (say), then we know that photon s has polarization y before hitting the λ/ | Ψ (cid:105) ], it is clear thatdetection of photon s (after the double slit) with polarization R is compatible only withthe passage of s through slit 1 and polarization L is compatible only with the passageof s through slit 2. This can be verified experimentally. In the usual quantum mechanicslanguage, detection of photon p before photon s has prepared photon s in a certain state.[9, p. 4]Firstly, the measurement that p has polarization x after the s photon has traversed the λ/ s photon to already have ”polarization y before hitting the λ/ p is measured with polarization x , then the two particlesystem is in the superposition state: i | R (cid:105) s ⊗ | x (cid:105) p − i | L (cid:105) s ⊗ | x (cid:105) p = [ i | R (cid:105) s − i | L (cid:105) s ] ⊗ | x (cid:105) p which means that the s photon is still in the slit-superposition state: i | R (cid:105) s − i | L (cid:105) s . Then onlywith the measurement of the circular polarization states L or R at D s do we have the collapse to(the evolved version of) one of the slit eigenstates s s s through slit 1” or ”slit 2”, i.e., s s
2, instead ofthe photon s being in the tagged superposition state | Ψ (cid:105) after traversing the slits.Let us take a new polarization space basis of | + (cid:105) = +45 ◦ to the x -axis and |−(cid:105) = − ◦ to the x -axis. Then the overall state can be rewritten in terms of this basis as (see original paper for thedetails): | Ψ (cid:105) = (cid:104) ( | + (cid:105) s − i | + (cid:105) s ) ⊗ | + (cid:105) p + i ( |−(cid:105) s + i |−(cid:105) s ) ⊗ |−(cid:105) p (cid:105) .13igure 14: Setup with D p polarizerThen a | + (cid:105) polarizer or a |−(cid:105) polarizer is inserted in front of D p to select | + (cid:105) p or |−(cid:105) p respectively.In the first case, this reduces the overall state | Ψ (cid:105) to | + (cid:105) s − i | + (cid:105) s which exhibits an interferencepattern, and similarly for the |−(cid:105) p selection. This is misleadingly said to ”erase” the so-called ”which-slot information” so that the interference pattern is restored.The first thing to notice is that two complementary interferences patterns, called ”fringes” and”antifringes,” are being selected. Their sum is the no-interference pattern obtained before insertingthe polarizer. The polarizer simply selects one of the interference patterns out of the mush of theirmerged non-interference pattern. Thus instead of ”erasing which-slit information,” it selects one oftwo interference patterns out of the both-patterns mush.Even though the polarizer may be inserted after the s -photon has traversed the two slits, thereis no retrocausation of the photon going though both slits or only one slit as previously explained.One might also notice that the entangled p -photon plays little real role in this setup (as opposedto the ”delayed erasure” setup considered next). Instead of inserting the | + (cid:105) or |−(cid:105) polarizer infront of D p , insert it in front of D s and it would have the same effect of selecting | + (cid:105) s − i | + (cid:105) s or |−(cid:105) s + i |−(cid:105) s each of which exhibits interference. Then it is very close to the one-photon eraserexperiment of the last section. If the upper arm is extended so the D p detector is triggered last (”delayed erasure”), the same resultsare obtained. The entangled state is then collapsed at D s . A coincidence counter (not pictured) isused to correlate the hits at D s with the hits at D p for each fixed polarizer setting, and the sameinterference pattern is obtained. 14igure 15: Setup for ”delayed erasure”The interesting point is that the D p detections could be years after the D s hits in this delayederasure setup. If the D p polarizer is set at | + (cid:105) p , then out of the mush of hits at D s obtained yearsbefore, the coincidence counter will pick out the ones from | + (cid:105) s − i | + (cid:105) s which will show interference.Again, the years-later D p detections do not retrocause anything at D s , e.g., do not ”erase which-way information” years after the D s hits are recorded (in spite of the ”delayed erasure” talk). Theyonly pick (via the coincidence counter) one or the other interference pattern out of the years-earliermush of hits at D s .”We must conclude, therefore, that the loss of distinguishability is but a side effect,and that the essential feature of quantum erasure is the post-selection of subensembleswith maximal fringe visibility.” [8, p. 79]The same sort of analysis could be made of the delayed choice quantum eraser experimentdescribed in the paper by Kim et al. & Scully [6]. Brian Greene [4, pp. 194-199] gives a goodinformal analysis of the Kim et al. & Scully experiment which avoids the separation fallacy and thusavoids any implication of retrocausality. References [1] Dicke, Robert H. and James P. Wittke 1960.
Introduction to Quantum Mechanics . Reading MA:Addison-Wesley.[2] Feynman, Richard P., Robert B. Leighton and Matthew Sands 1965.
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Physical ReviewA . 45 (11 (1 June)): 7729-7736.[8] Kwiat, P. G. , P. D. D. Schwindt and B.-G. Englert 1999. What Does a Quantum Eraser ReallyErase? In
Mysteries, Puzzles, and Paradoxes in Quantum Mechanics . Rodolfo Bonifacio ed.,Woodbury NY: American Institute of Physics: 69-80.[9] Walborn, S. P., M. O. Terra Cunha, S. Padua and C. H. Monken 2002. Double-slit quantumeraser.
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