TThe Delayed-Choice Quantum Eraser Leaves No Choice
Tabish Qureshi ∗ Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi, India.
A realizable delayed-choice quantum eraser, using a modified Mach-Zehnder (MZ) interferometerand polarization entangled photons, is theoretically analyzed here. The signal photon goes through amodified MZ interferometer, and the polarization of the idler photon provides path information for thesignal photon. The setup is very similar to the delayed-choice quantum eraser experimentally studiedby the Vienna group. In the class of quantum erasers with discrete output states, it is easy to see thatthe delayed mode leaves no choice for the experimenter. The which-way information is always erased,and every detected signal photon fixes the polarization state of the idler, and thus gives information onprecisely how the signal photon traversed the two paths. The analysis shows that the Vienna delayed-choice quantum eraser is the first experimental demonstration of the fact that the delayed mode leaves nochoice for the experimenter, and the which-way information is always erased. Additionally it is shown thatthis argument holds even in a conventional two-slit quantum eraser. Every photon registered anywhere onthe screen, fixes the state of the two-state which-way detector in a unique mutually unbiased basis. In thedelayed-choice quantum eraser experiments, the role of mutually unbiased basis sets for the which-waydetector, has been overlooked till now.
I. INTRODUCTION
The concept of wave-particle duality started out as a de-bate on the corpuscular nature versus wave nature of light.With the advent of quantum mechanics, a new languageemerged which described this concept, namely the Bohr’sprinciple of complementarity [1]. According to Bohr, thetwo natures of quantum objects, which we shall refer to asquantons here, the wave and particle natures complementeach other. However, the two natures are also mutuallyexclusive so that an experiment which brings out one na-ture, necessarily hides the other. The two-slit interferenceexperiment, where one additionally tries to probe which ofthe two slits the quanton went through, became a test-bedfor the concept of wave-particle duality right from the timeof its inception [2]. It soon became clear that if one triesto get the which-path or which-way information about thequanton, the interference is destroyed. Jaynes [3] came upwith an interesting idea that there can be ways in which theacquired which-way information can be erased such thatthe destroyed interference can be brought back, in perfectharmony with the concept of wave-particle duality. Themodern formulation of ”quantum eraser” was proposed byScully and Dr¨uhl [4]. What made their proposal more excit-ing was their suggestion that one may choose to erase thewhich-way information much after the quanton had regis-tered on the screen, and the interference could still be re-covered. This initiated a lively debate on the subject whichcontinues till date [5–14]. The concept of quantum eraserimplied that the experimenter could choose to retain thewhich-way information or erase it, and as a consequence,can force the quanton to behave either as a particle or awave. The “delayed-choice” quantum eraser went a leapfurther by suggesting that the quanton which traveled thetwo paths and hit the screen, may be forced to behave like ∗ [email protected] PBS2BS2CrystalMirror D D Polarizationrotater
MirrorPBS1Laser SignalIdler Q Q FIG. 1. A schematic diagram of a quantum eraser setup us-ing entangled photons, and a modified Mach-Zehnder inter-ferometer. The two paths of the signal photon end up gettingentangled with the polarization states of the idler photon. a particle or a wave by a choice made the experimenter af-ter it has already hit the screen. This kind of thinking ledto a talk of ”retrocausality” in the delayed-choice quantumeraser experiment, which is still being hotly debated [9–13].With the advances in experimental techniques the quan-tum eraser, with or without delayed-choice, was realized invarious ways [15–26], and several other proposals were made[27–30]. It has also been demonstrated that the idea ofquantum eraser should also work for three-path interference[31]. However, no experimental progress has been in thatdirection yet. It may be pertinent to mention a new class ofdelayed-choice experiments with a quantum twist, that were a r X i v : . [ qu a n t - ph ] O c t recently studied [32–37]. The idea in those experiment wasexplore the possibility of a quantum superposition of waveand particle behavior.The current debate mainly revolves around the interpreta-tion of delayed-choice quantum eraser. A widely held view,due to Englert, Scully and Walther [5], is that the choice toretain or erase the information regarding which of the twopaths the quanton followed, always rests with the experi-menter. While this view is quite acceptable for the normalquantum eraser, it is hard to digest for many people whenapplied to the quantum eraser experiment carried out in thedelayed mode. According to this view, even in the delayedmode, the experimenter chooses whether the quanton dis-plays wave nature or particle nature. However, the authorsof this view do not comment on how one should interpretthe ”actual behavior” of the quanton in such experiments.Questions like if the quanton shows particle nature in thedelayed mode, does it actually follow only one of the twopaths, are left unanswered. Here we re-investigate the de-layed choice quantum eraser, by proposing a realizable ex-periment using entangled photons, and try to find answersto the questions which are under debate. II. QUANTUM ERASER WITH A MACH-ZEHNDERSETUP
Let us consider an experimental setup as shown in Fig.1, where there is a spontaneous parametric down conver-sion (SPDC) source producing pairs of photons which areentangled in polarization such that the state is given by | Ψ (cid:105) = √ ( | V s (cid:105)| H i (cid:105) + | H s (cid:105)| V i (cid:105) ) | ψ (cid:105)| φ (cid:105) (1)where H and V denotes the horizontal and vertical polar-ization states, the labels s, i denote the signal and idlerphotons, respectively. The spatial states for the signaland idler photons are denoted by | ψ (cid:105)| φ (cid:105) , respectively.A Mach-Zehnder interferometer can be easily analyzed us-ing quantum mechanics [38, 39]. After the signal photonpasses through the polarizing beam-splitter PBS1, the statechanges to | Ψ (cid:105) = U P BS | ψ (cid:105) = √ ( | V s (cid:105)| H i (cid:105)| ψ (cid:105) + | H s (cid:105)| V i (cid:105)| ψ (cid:105) ) | φ (cid:105) , (2)where the | ψ (cid:105) , | ψ (cid:105) represent the states of the signal pho-ton in the upper and lower path of the Mach-Zehnder in-terferometer, respectively. One would notice that the twopaths of the signal photon are now entangled with the po-larization states of the two photons. The signal photon inthe upper path (1) passes through half-wave plate which ro-tates the polarization by 90 degrees, flipping the | V s (cid:105) stateto | H s (cid:105) state, so that the state now reads | Ψ (cid:105) = √ ( | H i (cid:105)| ψ (cid:105) + | V i (cid:105)| ψ (cid:105) ) | H s (cid:105)| φ (cid:105) . (3)This process now entangled the two paths of the signalphoton with the polarization states of the idler photon. Interestingly, the polarization of the signal photon is nowdisentangled from that of the idler. Separating out thehorizontal and vertical components of the idler photon canyield information on which of the two paths the signal pho-ton followed, simply because (cid:104) V i | Ψ (cid:105) = | ψ (cid:105)| V s (cid:105)| φ (cid:105) , and (cid:104) H i | ψ (cid:105) = | ψ (cid:105)| V s (cid:105)| φ (cid:105) .On the other hand, (3) can also be rewritten as | Ψ (cid:105) = {| R i (cid:105) ( | ψ (cid:105) − i | ψ (cid:105) )+ | L i (cid:105) ( | ψ (cid:105) + i | ψ (cid:105) ) }| V s (cid:105)| φ (cid:105) , (4)where | R i (cid:105) = √ ( | H i (cid:105) + i | V i (cid:105) ) , | L i (cid:105) = √ ( | H i (cid:105)− i | V i (cid:105) ) rep-resent the left and right circular polarization states, respec-tively. If one measured the circular component of polariza-tion of the idler photon, obtaining the state | R i (cid:105) would tellone that the state of the signal photon is √ ( | ψ (cid:105) − | ψ (cid:105) ) ,and obtaining the state | L i (cid:105) would tell one that it is √ ( | ψ (cid:105) + | ψ (cid:105) ) . With this, our arrangement for obtain-ing path information is fully in place. Obtaining the state | R i (cid:105) of the idler photon tells us that the signal photon fol-lowed both paths, exactly as it would, if there were no path-detecting mechanism in place , except for a phase differenceof − π/ between the two paths. Obtaining the state | L i (cid:105) of the idler tells us again that the signal photon followedboth paths, but now with a phase difference of π/ betweenthe two paths. Quantum mechanics then implies that ifone measures the polarization of the idler in the horizontal-vertical (linear polarization) basis, while the signal photon isstill traveling, one can force it to follow one of the two MZpaths. On the other hand, by measuring the polarizationof the idler in the circular basis, one can force the signalphoton to follow both the paths. The choice lies with theexperimenter.The effect of the second beam-splitter, on the two com-ponents | ψ (cid:105) , | ψ (cid:105) , is the following [38] U BS | ψ (cid:105) = √ ( | D (cid:105) + i | D (cid:105) ) U BS | ψ (cid:105) = √ ( i | D (cid:105) + | D (cid:105) ) , (5)where U BS represents the unitary evolution due to themirrors and the second beam-splitter BS2, and | D (cid:105) , | D (cid:105) are the states at the detectors D , D , respectively. If thestate of the signal photon is | ψ (cid:105) or | ψ (cid:105) , in both the sit-uations it is equally likely to hit D or D . This implies aloss of interference, resulting from extraction of which-pathinformation by the idler. So, obtaining the horizontal orvertical state of the idler, destroys the interference of thesignal photon, but yields its precise path information.However, if one measured the circular polarization of theidler, and obtained the state | R i (cid:105) , it would tell us that thestate of the signal photon would be √ ( | ψ (cid:105) − i | ψ (cid:105) ) . Thesecond beam-splitter BS2 would take this state only to D : U BS √ ( | ψ (cid:105) − i | ψ (cid:105) ) = | D (cid:105) . This implies interferencewith the bright fringe at D and the dark one at D . Onthe other hand, obtaining the state | L i (cid:105) would tell us thatthe state of the signal photon would be √ ( | ψ (cid:105) + i | ψ (cid:105) ) .The beam-splitter BS2 would take this state only to D : U BS √ ( | ψ (cid:105) + i | ψ (cid:105) ) = i | D (cid:105) . This also implies inter-ference, but with the bright fringe at D and the dark oneat D . Both these situations describe the phenomenon ofquantum erasure, where the lost interference comes backif the which-path information is erased. However, the twointerferences are mirror images of each other, and takentogether, they cancel each other out.Let us now look at the delayed mode where no measure-ment is made on the idler photon, the path of the idler beingmuch longer, and the signal photon reached the detectors.For example, in one experiment performed by the Viennagroup [16], the idler photon travels a distance of 144 kilo-meters before it reaches the analyzing detectors, whereasthe MZ paths are of the order of 2 meters. In our setup,the final state of the two photons, just before the signalphoton hits the detectors, is given by | Ψ (cid:105) = U BS | Ψ (cid:105) = √ U BS ( | H i (cid:105)| ψ (cid:105) + | V i (cid:105)| ψ (cid:105) ) | H s (cid:105)| φ (cid:105) = [ | H i (cid:105) ( | D (cid:105) + i | D (cid:105) ) + | V i (cid:105) ( i | D (cid:105) + | D (cid:105) )] | H s (cid:105)| φ (cid:105) . (6)This state indicates that D and D are equally likely toregister the signal photon, as |(cid:104) D | Ψ (cid:105)| = |(cid:104) D | Ψ (cid:105)| =1 / , which in turn implies no interference.An interesting scenario emerges if one rewrites the state(6) as | Ψ (cid:105) = √ ( | D (cid:105)| R i (cid:105) + i | D (cid:105)| L i (cid:105) ) | V s (cid:105)| φ (cid:105) . (7)This state indicates that if the signal photon registers at D , it fixes the polarization state of the idler to the rightcircular state | R i (cid:105) , and if it registers at D , it fixes polariza-tion state of the idler to the left circular state | L i (cid:105) . But thestates | R i (cid:105) , | L i (cid:105) correspond to the erased which-path infor-mation, and tell us that the signal photon followed both thepaths, and not one of the two. In the delayed mode, theexperimenter no longer has the choice to seek either which-path information or quantum eraser. This runs counter tothe widely accepted notion that the choice of which-pathinformation or quantum eraser, lies with the experimenterin the delayed mode [5]. Not only does the registered sig-nal photon tell us the that the which-path information iserased, it tells us precisely how the signal photon traversedthe two paths, and the phase difference between the twopaths, by virtue of (4). This correlation, of course, can alsobe used to recover the lost interference, constituting theusual quantum eraser.More interesting is the fact the correlation between theleft-right circularly polarized states of the idler, and the de-tectors D and D of the modified MZ setup has alreadybeen experimentally observed in the Vienna delayed choicequantum eraser experiment [16]. However, its implicationwas not recognized for want of an analysis similar to the onepresented here. The equivalence between the setup studiedhere and the one implemented by the Vienna group can beeasily seen. They used, what they call, a ’hybrid entangler’to achieve entanglement between the two paths of one pho-ton and the polarization states of a causally disconnectedphoton. Although the authors go an extra step by varying C oun t s Position (arbitary units)
FIG. 2. Simulated results of the quantum eraser setup usingentangled photons, and a modified Mach-Zehnder interferom-eter (see FIG. 1). Points represent coincident counts of D (red squares) and D (blue circles) with one particular stateof the idler, as a function of the position of BS1. Solid linesare the corresponding theoretical curves. Compare with fig-ure 3D of Ref. [16]. The peak will be shifted if the coincidenceis done with a state of a different mutually unbiased basis ofthe polarization of the idler photon. the position of PBS1, and observing the counts of each de-tector (coincident with remote photon), there is a centralposition of PBS1 for which one detector gives maximumcounts, and the other one gives almost zero (see figure 3Dof Ref. [16]). FIG. 2 shows simulated data of our suggestedexperiment, and is closely similar to the Vienna experimentresults. The maximum and minimum counts at the cen-tral position are the equivalent of the bright and dark fringeof the traditional two-slit experiment. At this position ofPBS1, registering of a photon at a particular detector, fixesthe polarization state of the other photon which is 144 kmaway. In the experiment, this emerges as the prefect corre-lation between the two observations.The prevalent belief [5] says that even in the delayedmode, observing the idler in the horizontal-vertical basis,gives one the path-information about the signal photon.The preceding analysis shows that this is incorrect. Forexample, if the signal photon registers at D , (7) tellsus that the polarization state of the idler is | R i (cid:105) . Since | R i (cid:105) = √ ( | H i (cid:105) + i | V i (cid:105) ) , if one insists on measuring thepolarization in the horizontal-vertical basis, one will get thetwo results with equal probability, and hence no interfer-ence to speak of. However, in this case, getting a | H i (cid:105) or | V i (cid:105) does not give one any path information. This is sim-ply because there is a correlation between | D (cid:105) , | D (cid:105) and | R i (cid:105) , | L i (cid:105) (by virtue of (7)), and getting a (say) | D (cid:105) de-stroys the possibility of using (3) to infer path information[14]. So, the loss of interference is not because of obtainingany path information by looking at | H i (cid:105) and | V i (cid:105) states.The interference is lost anyway, unless one correlates withthe | R i (cid:105) , | L i (cid:105) states. III. MOVABLE BEAM-SPLITTER
An objection can be raised that the preceding analysisholds only for certain fixed locations of BS1 or BS2, as onlyfor those locations one of the detectors D , D will showzero count (destructive interference). In the following wewill show that that is not the case, and this argument can bemade quite general. Let us suppose that the beam-splitterBS1 is movable, so that its position leads to a phase factorof e πix/λ for the upper path, where λ is the wavelengthof the light used in the experiment. For x = 0 the twopath lengths are the same, and all the preceding argumentsgo through. For an arbitrary x , the final state of the twophotons, just before the signal photon hits the detectors,instead of (6), is now given by | Ψ (cid:48) (cid:105) = √ U BS ( | H i (cid:105)| ψ (cid:105) e πixλ + | V i (cid:105)| ψ (cid:105) ) | H s (cid:105)| φ (cid:105) = [ | H i (cid:105) e πixλ ( | D (cid:105) + i | D (cid:105) )+ | V i (cid:105) ( i | D (cid:105) + | D (cid:105) )] | H s (cid:105)| φ (cid:105) . (8)In terms of the states | R i (cid:105) , | L i (cid:105) , the above can be writtenas | Ψ (cid:48) (cid:105) = | D (cid:105) (cid:16) ( e πixλ + 1) | R i (cid:105) + ( e πixλ − | L i (cid:105) (cid:17) + i | D (cid:105) (cid:16) ( e πixλ − | R i (cid:105) + ( e πixλ + 1) | L i (cid:105) (cid:17) . (9)Now there is no correlation between the states | R i (cid:105) , | L i (cid:105) and the detector states | D (cid:105) , | D (cid:105) . So a photon registeredat the detectors cannot not tell us if the state of the idlerwill be | R i (cid:105) or | L i (cid:105) .However, one should realize that there is nothing sacredabout the basis | R i (cid:105) , | L i (cid:105) chosen for the idler photon. Giventhe polarization states | H i (cid:105) , | V i (cid:105) , there exist an infinite num-ber of mutually unbiased basis states which can be used forthe purpose. The basis defined by | R i (cid:105) , | L i (cid:105) happens tobe just one such basis which is unbiased with respect to | H i (cid:105) , | V i (cid:105) . One might as well choose the following basisstates for the polarization of the idler photon | P i (cid:105) = 1 √ e iθ | H i (cid:105) + i | V i (cid:105) ) , | Q i (cid:105) = 1 √ e iθ | H i (cid:105) − i | V i (cid:105) ) , (10)where θ is an arbitrary phase factor. The state of the twophotons, just before the signal photon enters BS2 | Ψ (cid:48) (cid:105) = √ ( e πixλ | H i (cid:105)| ψ (cid:105) + | V i (cid:105)| ψ (cid:105) ) | H s (cid:105)| φ (cid:105) , (11)can be written in terms of this new basis as | Ψ (cid:48) (cid:105) = { e i ( πxλ − θ ) | P i (cid:105) ( | ψ (cid:105) − i | ψ (cid:105) )+ e i ( πxλ − θ ) | Q i (cid:105) ( | ψ (cid:105) + i | ψ (cid:105) ) }| V s (cid:105)| φ (cid:105) . (12)This shows that the states of any basis unbiased with re-spect to | H i (cid:105) , | V i (cid:105) get correlated to √ ( | ψ (cid:105) + i | ψ (cid:105) ) and √ ( | ψ (cid:105) − i | ψ (cid:105) ) , and thus are indicators of the photon fol-lowing both paths. Now, if one chooses the basis such that θ = πxλ , the state of the two photons, just before the signalphoton hits the detectors, is given by | Ψ (cid:48) (cid:105) = √ ( | D (cid:105)| P i (cid:105) + i | D (cid:105)| Q i (cid:105) ) | V s (cid:105)| φ (cid:105) . (13)This means that every signal photon registered at the detec-tors, fixes the polarization state of the idler in this particularbasis. So, for every position of the beam-splitter BS1, thereexists a basis (10) for the idler, the states of which are per-fectly correlated with two detectors of the signal photon.While it is true that (9) also indicates that every signalphoton registered at (say) D , fixes the state of the idler,but there is no way to know, a priori, how that state wouldbe read. In the present case, looking at the position of thebeam-splitter BS1, one can choose the basis (10) in whichto measure the polarization of the idler so that the resultsof the two are perfectly correlated. Choosing the basis mayamount to choosing the angle by which the polarization ofthe idler has to be rotated, or something equally straight-forward. Each detected signal photon tells one whether thestate of the idler is | P i (cid:105) or | Q i (cid:105) , and consequently also tellsthat the signal photon followed both paths, and not oneof the two. Since this experiment can be performed, andthe correlation measured, at least in principle, it tells usthat in the delayed mode of the quantum eraser, which-wayinformation is always erased. IV. THE TWO-SLIT WHICH-WAY EXPERIMENT
Various other delayed choice quantum eraser experimentshave been performed using conventional double-slit inter-ference. One might wonder if the arguments presented inthe preceding discussion hold for the two-slit delayed choicequantum eraser experiments too. This is the question weaddress in the following analysis. Consider a two-slit in-terference experiment with a two-state which-way detector,as shown in FIG. 3. Without specifying the nature of thewhich-way detector, we assume that its effect is to entanglethe two photon paths with the two states of the which-waydetector, such that the combined state, when the photonreaches the screen, is given by Ψ( x ) = √ ( ψ ( x ) | d (cid:105) + ψ ( x ) | d (cid:105) ) , (14)where | d (cid:105) , | d (cid:105) are orthonormal states of the which-waydetector. One can now define a mutually unbiased basisby the states | d θ ± (cid:105) = √ ( e iθ | d (cid:105) ± | d (cid:105) ) . The state of thephoton and which-way detector may be rewritten in the newbasis Ψ( x ) = (cid:16) [ e − iθ ψ ( x ) + ψ ( x )] | d θ + (cid:105) +[ e − iθ ψ ( x ) − ψ ( x )] | d θ − (cid:105) (cid:17) . (15)The two states of the photon, corresponding to thewhich-way detector states | d θ ± (cid:105) are given by ψ ± ( x ) = yD Which-waydetector Double slitScreen1 2Source
FIG. 3. A schematic diagram of a two-slit interference experi-ment with which-way detection. The two paths of the photonget entangled with the two states of the which-way detector. √ [ e − iθ ψ ( x ) ± ψ ( x )] , respectively. One can do a rig-orous wave-packet analysis of the dynamics of the photon,and find the two states to have the following typical form[2] ψ ± ( x ) = A ( x )[1 ± cos (cid:0) πxdλD − θ (cid:1) ] , (16)where d is the separation between the two slits, D is thedistance between the slits and the screen, and A ( x ) is anenvelope function. For θ = 0 , ψ + ( x ) represents an inter-ference pattern with a central peak at x = 0 . On the otherhand, ψ − ( x ) represents a similar, but shifted interferencepattern, with a minimum at x = 0 (see FIG. 4). One cansee that if a photon is detected at x = 0 , it can only belongto ψ + ( x ) , because | ψ − ( x = 0) | = 0 . Then, from (15) oneinfers that the state of the which-way detector is | d (cid:105) , andnot | d − (cid:105) . One can then conclude that the photon traveledboth the paths, and not one of the two. The same argu-ment can be made for all values of x where ψ + ( x ) has apeak.But what about the photons which land at positionswhere ψ + ( x ) does not have a peak? In that case one canchoose a different basis for the which-way detector statessuch that θ = πxdλD . Remember that the interference pat-terns are obtained only in coincidence with the which-waydetector states, and in coincidence with | d θ ± (cid:105) , the interfer-ence patterns will be shifted (see FIG. 4). They will beshifted in such a way that | ψ − ( x ) | = 0 for that particular x . One can then logically conclude that the state of thewhich-way detector is | d θ + (cid:105) , and not | d θ − (cid:105) , and the which-way information is erased. This again indicates that thephoton followed both the paths. This correlation can be ex- | ψ + ( x ) | ( a r b i t r a r y un i t s ) x (dimensionless) FIG. 4. Typical recovered two-slit interference because ofquantum erasure. Recovered interference | ψ + ( x ) | for θ = 0(solid blue line), | ψ − ( x ) | for θ = 0 (dotted green line), and | ψ + ( x ) | for θ (cid:54) = 0 (dot-dashed red line). perimentally seen, as looking at the values of x at which thephoton is registered, one can choose a mutually unbiasedbasis of the which-way detector states | d θ ± (cid:105) , specified by θ = πxdλD . Thus, for every photon registered on the screen,one can choose a mutually unbiased basis of the which-waydetector states, whose measurement result is predicted bythe registered photon. So, even in a two-slit delayed-choicequantum eraser, every registered photon fixes the state ofthe which-way detector in a knowable basis, and thus alwayserases the which-way information.Some comments on how one can choose a different mu-tually unbiased basis for the which-way detector in delayed-choice quantum eraser experiments. In the experiment ofKim et.al. [20], the idler photon, after traversing two paths,is recombined using a beam-splitter. Changing the relativelengths of the two paths would amount to choosing a differ-ent mutually unbiased basis for the idler. The recombiningbeam-splitter may be moved in synchrony with the movabledetector for the signal photon [20]. In the experiment ofScarcelli et.al. [24], quantum erasing is achieved by lettingthe idler pass through a fixed narrow slit, and observing thesignal photon in coincidence with it. Here, changing thebasis of the path-detecting photon (idler) can be achievedby changing the position of the narrow slit through whichthe idler passes. V. CONCLUSIONS
The correlation in (7) emerges in a straightforward fash-ion in a MZ like setup where there are only two discreteoutput states. As the Vienna experiment [16] is the firstexperiment to implement a delayed-choice quantum eraserin a MZ setup, it is also the first one to experimentallydemonstrate that in the delayed mode, which-way informa-tion is always erased, and the photon always follows boththe paths. In the traditional two-slit experiment, this ef-fect is hidden because there is a continuous set of positionson the screen where the photon can register. However, wehave shown that even in the two-slit delayed choice quan-tum eraser, for every photon detected on the screen, thereexists a basis in which the state of the which-way detectorgets fixed by the act of photon hitting the screen. This ba-sis can be known from the position of the photon, and the corresponding measurement can be made on the which-waydetector to test the correlation. In the light of this analysis,and the results of the Vienna experiment, the long held no-tion that in the delayed mode, the experimenter has a choicebetween reading the which-way information or erasing it,should be given up. In the delayed mode, the which-wayinformation is always erased. This takes the mystery out ofthe delayed-choice quantum eraser, and renders irrelevantany talk of retrocausality. [1] N. Bohr, “The quantum postulate and the recent devel-opment of atomic theory,”
Nature (London) , 580-591(1928).[2] T. Qureshi, R. Vathsan, “Einstein’s recoiling slit experi-ment, complementarity and uncertainty,”
Quanta , 58-65(2013).[3] E. Jaynes, in Foundations of Radiation Theory and Quan-tum Electrodynamics , ed. A.O. Barut (Plenum, New York1980), pp. 37.[4] M. O. Scully and K. Dr¨uhl,
Phys. Rev. A , 2208 (1982).[5] B.-G. Englert, M. O. Scully, H. Walther, “Quantum era-sure in double-slit interferometers with which-way detec-tors,” Am. J. Phys. , 325 (1999).[6] U. Mohrhoff, “Objectivity, retrocausation, and the exper-iment of Englert, Scully, and Walther,” Am. J. Phys. ,330 (1999).[7] Y. Aharonov, M.S. Zubairy, “Time and the quantum:erasing the past and impacting the future,” Science ,307(5711):875–879 (2005).[8] B.J. Hiley, R.E. Callaghan, “What is erased in the quan-tum erasure?,”
Foundations of Physics , 36(12):1869-1883(2006).[9] D. Ellerman, “Why delayed choice experiments do Not im-ply retrocausality,”
Quantum Stud.: Math. Found. , 183(2015).[10] J. Fankhauser, “Taming the delayed choice quantumeraser,” Quanta , 44 (2019).[11] R.E. Kastner, “The ‘delayed choice quantum eraser’ neithererases nor delays,” Found. Phys. , 717 (2019).[12] R.E. Kastner, The Transactional Interpretation of QuantumMechanics: The Reality of Possibility (Cambridge Univer-sity Press, Cambridge, 2012).[13] R.E. Kastner,
Adventures in Quantumland: Exploring OurUnseen Reality (World Scientific, Singapore, 2019).[14] T. Qureshi, “Demystifying the delayed-choice quantumeraser,”
Eur. J. Phys. , 055403 (2020).[15] X. Ma, J. Kofler, A. Zeilinger, “Delayed-choice gedankenexperiments and their realizations,” Rev. Mod. Phys. ,015005 (2016).[16] X. Ma, J. Kofler, A. Qarry, N. Tetik, T. Scheidl, R. Ursin,S. Ramelow, T. Herbst, L. Ratschbacher, A. Fedrizzi, T.Jennewein, A. Zeilinger, “Quantum erasure with causallydisconnected choice,” Proc. Natl. Acad. Sci. U.S.A. ,1221 (2013).[17] A. G. Zajonc, L. Wang, X.Y Zou, L. Mandel, “Quantumeraser,”
Nature , 507 (1991).[18] P. G. Kwiat, A. Steinberg, R. Chiao, “Observation of aquantum eraser: A revival of coherence in a two-photoninterference experiment,”
Phys. Rev. A , 7729 (1992). [19] T. J. Herzog, P. G. Kwiat, H. Weinfurter, A. Zeilinger,“Complementarity and the quantum eraser,” Phys. Rev.Lett. , 3034 (1995).[20] Y.-H. Kim, Rong Yu, Sergei P. Kulik, Y. Shih and M. O.Scully, “Delayed ’choice’ quantum eraser,” Phys. Rev. Lett. , 1 (2000).[21] S. P. Walborn, M. O. Terra Cunha, S. P´adua, C. H. Monken,“Double-slit quantum eraser,” Phys. Rev. A , 033818(2002).[22] H. Kim, J. Ko, and T. Kim, “Quantum-eraser experimentwith frequency-entangled photon pairs,” Phys. Rev. A ,054102 (2003).[23] U. L. Andersen, O. Gl¨ockl, S. Lorenz, G. Leuchs, R. Filip,“Experimental demonstration of continuous variable quan-tum erasing,” Phys. Rev. Lett. , 100403 (2004).[24] G. Scarcelli, Y. Zhou, Y. Shih, “Random delayed-choicequantum eraser via two-photon imaging,” Eur. Phys. J. D , 167 (2007).[25] L. Neves, G. Lima, J. Aguirre, F.A. Torres-Ruiz, C. Saave-dra, A. Delgado, “Control of quantum interference in thequantum eraser,” New J. Phys. , 073035 (2009).[26] M.B. Schneidera, I.A. LaPuma, “A simple experiment fordiscussion of quantum interference and which-way measure-ment,” Am. J. Phys. , 266 (2002).[27] A. Bramon, G. Garbarino, B.C. Hiesmayr, “Quantum mark-ing and quantum erasure for neutral kaons,” Phys. Rev.Lett. , 020405 (2004).[28] T. Qureshi, Z. Rahman, “Quantum eraser using a modifiedStern-Gerlach setup,” Prog. Theor. Phys. , 71 (2012).[29] R.D. Barney, J-F. S. Van Huele “Quantum coherence recov-ery through Stern–Gerlach erasure,”
Phys. Scr. , 105105(2019).[30] M. Chianello, M. Tumminello, A. Vaglica, and G. Vetri,“Quantum erasure within the optical Stern-Gerlach model,” Phys. Rev. A , 053403 (2004).[31] N.A. Shah, T. Qureshi, “Quantum eraser for three-slit in-terference,” Pramana J. Phys. , 80 (2017).[32] R. Ionicioiu, D.R. Terno, “Proposal for a quantum delayed-choice experiment,” Phys. Rev. Lett. , 230406 (2011).[33] R. Auccaise, R.M. Serra, J.G. Filgueiras, R.S. Sarthour, I.S.Oliveira, L.C. C´eleri, “Experimental analysis of the quan-tum complementarity principle,”
Phys. Rev. A , 032121(2012).[34] A. Peruzzo, P. Shadbolt, N. Brunner, S. Popescu, J.L.O’Brien, “A quantum delayed-choice experiment,” Science , 634 (2012).[35] F. Kaiser, T. Coudreau, P. Milman, D.B. Ostrowsky,S. Tanzilli, “Entanglement-enabled delayed-choice experi-ment,”
Science , 637 (2012). [36] T. Qureshi, “Quantum twist to complementarity: A dualityrelation,”
Prog. Theor. Exp. Phys. , 041A01 (2013).[37] J-S. Tang, Y-L. Li, C-F. Li, G-C. Guo, “Revisiting Bohr’sprinciple of complementarity with a quantum device,”
Phys.Rev. A , 014103 (2013). [38] V. Scarani, A. Suarez, “Introducing quantum mechan-ics: One-particle interferences” Am. J. Phys. , 718-721(1998).[39] C. Ferrari, B. Braunecker, “Entanglement, which-way mea-surements, and a quantum erasure,” Am. J. Phys.78