Abstract
Since its publication, Aharonov and Vaidman’s threebox paradox has undergone three major advances: i). A noncounterfactual scheme by the same authors in 2003 with strong rather than weak measurements for verifying the particle’s subtle presence in two boxes. ii) A realization of the latter by Okamoto and Takeuchi in 2016. iii) A dynamic version by Aharonov et al. in 2017, with disappearance and reappearance of the particle. We now combine these advances together. Using photonic quantum routers the particle acts like a quantum “shutter.” It is initially split between Boxes A, B and C, the latter located far away from the former two. The shutter particle’s whereabouts can then be followed by a probe photon, split in both space and time and reflected by the shutter in its varying locations. Measuring the former is expected to reveal the following timeevolution: The shutter particle was, with certainty, in boxes A+C at t_{1}, then only in C at t_{2}, and finally in B+C at t_{3}. Another branch of the split probe photon can show that boxes A+B were empty at t_{2}. A Belllike theorem applied to this experiment challenges any alternative interpretation that avoids disappearancereappearance in favor of local hidden variables.
Introduction
How literally should quantum formalism be taken? Ever since its advent, it has predicted phenomena which, by classical physics, are inconceivable. In response, several interpretations proposed that this is not the case, but it is rather some local hidden variables that operate underneath. Other advances, however, in the form of new mathematical proofs and experiments, proved that the bare formalism, no matter how odd, provides an accurate description.
Such is the familiar state revealed by the doubleslit experiment, featuring in every introductory text on QM, to which Feynman^{1} has referred as presenting the theory’s core mystery. Consider its refined version. When a single particle traverses a MachZehnder Interferometer (MZI), its state is given by the simple expression
which, in everyday language, reads: The particle traverses in some sense both paths 1 and 2, assuming a definite path only under position measurement taken during its passage.
Upon position measurement, this inevitably invokes the notorious measurement problem and the contentious “collapse.” Understandably, then, attempts were made to show that this superposition reflects only our subjective ignorance, while the particle itself goes on one definite side. Eventually, however, singleparticle interference experiments indicated that both paths are simultaneously traversed. So much so, that even when one path is measured and yields no detection, InteractionFree Measurement (IFM) ensues, destroying the interference effect just like a positive click. Quantum superposition, then, is the case just as (1) indicates.
Such was also the case with the EPR state. Its familiar twospins version obeys the relation
which, in everyday language, reads: Each particle’s spin (like the particle’s position in Eq. 1) is both ↓ and ↑ along every direction, yet maximally correlated with the other, becoming definite in a certain direction for both particles only under measurement performed on either one of them.
This state entails the (in)famous issue of quantum nonlocality. And here too, attempts to restore locality were proposed, again to be ruled out eventually, this time by Bell’s^{2} theorem and the subsequent tests. Further attempts were still made to find loopholes in these advances, all countered so far.
These advances have come full circle when Bell’s theorem was extended by Hardy^{3,4} also for the above, older case of the doubleslit/MZI with a single particle. One extension is Hardy’s paradox^{3}, where a particle and its antiparticle interact but do not annihilate. Their interference patterns indicate that nonlocal correlations have existed, all along, within each single wavefunction^{5}. In another setting^{4}, a single photon is entangled with two atoms placed along the two MZI’s paths. These atoms thereby become an EPRBell pair, proving, again, that their nonlocal correlation were inherited from the earlier nonlocal correlation between the two halves of the single particle’s wavefunction.
In summary, the two most basic characteristics of QM merit being taken literally: i) A single particle propagates like a wave, somehow traversing all possible paths with its expanding front; and ii) Within a single wavefunction, a local interaction with a measuring device instantly affects the entire wavefunction, regardless of its spread in spacetime, while preserving relativistic causality.
The main contribution of this paper is a set of feasible experimental proposals for testing the nonlocality of a single particle, and specifically, its ability to disappear and reappear, with certainty, across various locations at certain instances. This unique dynamics is indicated by the particle’s reflecting a probe photon from all these spacetime points, with the aid of photonic quantum routers. The extreme nonlocality in this setup is proved with a Belllike setting.
The paper is organized as follows. In “Time to Accept Even Greater Oddities” we present a surprising prediction regarding the interaction of a probe photon with a pre and postselected shutter photon. In the next two sections we present the theoretical framework which rigorously derives this prediction from the standard quantum formalism. Then in “The StrongMeasurement Version of the Three Boxes Paradox and its Realization” we present an actual experiment, already performed, testing an earlier prediction of that framework, thereby setting the stage for the following. We proceed on to the newer predictions in “The DisappearingReappearing Particle” and in “Bell’s Inequality Ruling Out the ‘All Along’ Alternative”. Then come the main sections “The Proposed Realization of the DisappearingReappearing Particle Experiment” and “Experimental Setup and further Practical Considerations,” where we present a feasible experimental setup for testing our predictions, with a few variants. We mention a stronger variant in “A Stricter Version: Measuring Both Appearances and Disappearances with the Same Probe Photon” and finally return to the question of nonlocality in “An EPRBellType Validation.” We conclude with some general implications in “Discussion”.
Methods and Results
We submit that the same credibility assigned to the above states (1), (2) goes, for the same reasons, to even more intriguing phenomena recently derived from quantum theory. These can be summarized in equations (4), (6) below, already implicit in (1), (2) and presented in what follows with their own BellHardy type proofs.
Time to Accept Even Greater Oddities
The first intriguing prediction is Aharonov and Vaidman’s^{6,7}, which has recently won empirical support by Okamoto and Takeuchi^{8}. They considered the interaction (via a quantum router) between an incident (In) probe photon (pp) and another pre and postselected photon acting as a shutter (sp):
where A, B, C denote three locations in space, i.e. three boxes, and \({\rm{R}}i\rangle /{\rm{T}}i\rangle \) stand for reflection/transmission from/through box i.
The probe photon was prepared in a superposition \({{\rm{In}}\rangle }_{pp}={\alpha }_{1}{A\rangle }_{pp}+{\alpha }_{2}{B\rangle }_{pp},\) where α_{1} and α_{2} are two arbitrary complex coefficients satisfying \({{\alpha }_{1}}^{2}+{{\alpha }_{2}}^{2}=1\), while the shutter photon was pre and postselected in \({\psi \rangle }_{sp}=\frac{1}{\sqrt{3}}({A\rangle }_{sp}+{B\rangle }_{sp}+{C\rangle }_{sp})\) and \({\phi \rangle }_{sp}=\frac{1}{\sqrt{3}}({A\rangle }_{sp}+{B\rangle }_{sp}{C\rangle }_{sp})\) as in the original 3box paradox^{6} (not to be confused with the “disappearing and reappearing” paradox described below).
According to Eq. 3 and the preparation above, before postselection the probe and shutter photon become entangled as follows:
However, as shown in^{6,7}, the combination of pre and postselection implies that, if we look for the shutter photon in Box A/B, it will be found with certainty in A/B, respectively. While this holds for the time between pre and postselection, apparently making it inaccessible for verification, it becomes accessible by its entanglement with the probe photon (Eq. 4). Indeed, upon postselection of the shutter, the probe state while retaining its initial coherent superposition, becomes
which, in everyday language, reads: a single probe photon is completely reflected by a single shutter photon retrodicted to have simultaneously resided in two boxes A and B.
Here, quantum superposition (1) is revealed not only by the particle’s passive passage along two paths, but by its active interaction with another particle in both locations (regarding the shutter’s unique mode of presence in Box C, see discussion below). This interaction resembles Quantum Oblivion^{9}, where only one party seems to be affected by the interaction.
In what follows we discuss a fourth equation of this kind, revealing, in addition to the former three (1, 2, and 5), a unique evolution in time. A particle, again operating as a shutter and superposed over three boxes, seems to abruptly disappear and reappear between them (see Figs 1 and 2 for a detailed experimental scheme, which will be explained in detail in the following sections). This timeevolution is revealed by the probe photon being prepared with the normalized spatiotemporal superposition
which means that it has a chance \({{\alpha }_{1}}^{2}\)to be in Box A at time t_{1}, \({{\alpha }_{3}}^{2}\) to be in Box C at time t_{1} and so on. Thanks to their interaction (Eq. 3), we predict that this probe photon would be reflected by the pre and postselected shutter from all these varying positions in space and time. The probe photon can thus undergo interference of all its five spatiotemporal reflected trajectories
which, in everyday language, reads: The shutter photon seems to reside in boxes A and C first, then only in C, and finally in B and C, reflecting the probe photon from all these varying locations.
Here too, one would suggest alternative accounts for the position of the shutter photon based on local hidden variables instead of this extravagant disappearancereappearance scenario. But here again, the above BellHardy proof ^{2,4} for singleparticle nonlocality rules out this alternative, thereby warranting an experimental realization for the explicitly nonlocal account straightforwardly given by Eq. 7. An even more straightforward proof is given in the following sections.
The Two StateVector Formalism
For explaining the curious predictions of (5) and (7) we begin with a brief introduction to the specific formalism that has produced them. First, the TwoStateVector Formalism (TSVF)^{10,11,12,13} is fully consistent with the standard quantum formalism: All the former’s predictions could equally be derived by the latter. Why, then, has that never happened? The main reason is the perfect timesymmetry underlying TSVF, so alien to quantummechanical intuition when measurement is involved. Once, however, TSVF does derive a prediction, it always turns out to be in full accord with quantum theory.
The TSVF predictions usually refer to the evolution of a particle between two consecutive quantum measurements. If, for instance, the particle is prepared at \(t={t}_{i}\) with \(\psi \rangle \) and later at \(t={t}_{f}\) found at a nonorthogonal state \(\phi \rangle \), then for all \({t}_{i} < t < {t}_{f}\) it can be described by the twostate \({}_{{t}_{f}}\langle \phi \,{\psi \rangle }_{{t}_{i}}\).
During this interval, and under special combinations of pre and postselected states, the particle is also expected to possess some intriguing properties, such as unusually small/large or even complex physical variables^{14,15}, or a spin separated from its mass^{16}, etc.
There is, however, an inherent obstacle for the validation of these predictions: The intermediate properties cannot, in general, be validated by an ordinary, projective measurement, because then it would not be a betweenmeasurements state! This was the motivation for inventing weak measurements^{17,18}: Let the coupling between particle and measuring apparatus be very weak, thereby highly inflicted by noise, but take many such measurements on an ensemble of particles (preferably in the same state) and average them out. Then, in compliance with basic statistical laws, the (constant) signal will strongly prevail over the (varying) noise, revealing the predicted phenomena without paying with the usual “collapse.”
This technique, however, has invoked some skepticism, mainly related to its statistical character which allows more conservative explanations. This challenge merits consideration. If “The weight of evidence for an extraordinary claim must be proportioned to its strangeness”^{19}, it is strong projective measurement that should validate TSVF predictions.
Advances: TSVF with Strong Measurements, Delayed Measurements, and the BellHardy Nonlocality Proof
These indeed were the advances made over the years, mostly recently. Experimental settings involving strong measurements were offered to test extraordinary TSVF predictions like the “Cheshire cat”^{16}, the “quantum pigeonhole”^{20,21} and the “disappearing and reappearing particle”^{22,23}. Also, as noted above, a protocol involving strong measurements proposed by Aharonov and Vaidman^{7} was experimentally realized by Okamoto and Takeuchi^{8}, to be elaborated in greater detail below.
One ingenious advance was made by an earlier work apparently not employing TSVF, yet doing just that. Hardy^{4} has analyzed a single photon traversing an MZI and subtly interacting with two superposed atoms placed on its two paths. The photon is then selected for the cases where its interference is disturbed. Only then the two atoms, which interacted with the photon earlier, become EPR entangled. This setting belongs to the TSVF family for the simple reason that it addresses the photon’s state which prevailed between two measurements, with the advance that the intermediate state is now subjected not to a weak but rather to a delayed strong measurement: The two atoms remain superposed and entangled till after the photon’s postselection, keeping the photon’s earlier state “alive” even much later.
These advances are employed in what follows. Between the particle’s pre and postselections, it is entangled with a photon, which later reveals the odd dynamics predicted for the intermediate timeinterval.
The StrongMeasurement Version of the Three Boxes Paradox and its Realization
The three boxes paradox^{6} is based on a counterfactual: “Had one measured the particle in only Box A, it would be there, and similarly for B” This is an extravagant type of nonlocality, where the very act of position measurement in a certain location seems to force the particle to “collapse” just there. Yet it is a retrodiction, holding only for a past state after the postselection, which is by definition no longer accessible. This seems to rob the paradox much of its acuity.
This obstacle was overcome by Aharonov and Vaidman^{7} in the general N boxes case, with a scheme analogous to Hardy’s delayed measurement^{4}. This time, then, it is a strong projective measurement of a particle operating as a “shutter.” The experiment has two stages: (i) The shutter particle, after the preselection, superposed over the three boxes, is coupled with a superposed probe particle. (ii) Then, after the former’s postselection, the probe particle is subjected to a measurement which reveals the shutter particle’s intermediate location. The retrodiction thus turns into a standard prediction: In all cases where the shutter’s postselection succeeds, the probe particle is reflected from all the N1 boxes, demonstrating the shutter’s simultaneous existence in all of them.
Okamoto and Takeuchi^{8} have recently tested this prediction of the TSVF for N = 3 in a quantum optics setup employing a novel photonic quantum router^{24} which enables the shutter and the probe photons to interact. Their results show that the shutter did not randomly occupy one of the two boxes, i.e., it did not “collapse.” Rather (within experimental accuracy limitations), it has reflected the probe photon from both boxes. The latter was therefore measured (in a manner akin to interference) on both its possible return paths. Notice the novelty: Quantum superposition, so far demonstrated only by the particle’s passive passage through multiple paths, is now made apparent with the particle’s active operation on another particle, at all these locations.
The DisappearingReappearing Particle
We now come to our experiment. Whereas the three box paradox^{6} presents an intriguing state – the particle reflecting the probe photon from two locations – the disappearingreappearing particle Gedankenexperiment^{22} gives an even more intriguing evolution: The particle is bound to be found in mutuallyexclusive boxes at different times. And here again, the prediction is for strong (projective) measurements.
The preselection (in fact a preparation) is.
and the postselection.
We further introduce a time evolution \(H=\varepsilon {\sigma }_{x}\) during \(0\le t\le {t}_{f},\) allowing the particle to move between Boxes A and B (according to this notation, σ_{ x } is the Paulix matrix, \(A\rangle \equiv \uparrow \rangle \) and \(B\rangle \equiv \downarrow \rangle \)). Specifically, after a period of time denoted by t_{2}, a particle starting in A would be evenly superposed in A and B. After time t_{3} this particle would completely move to Box B and after time t_{f} it would return to A with a negative phase, but then we will project its state on Eq. 9. To simulate this evolution in our proposed photonic experiment, we would later use two beamsplitters.
Surprisingly, we have three different predictions (see also^{22,23}) for the three instants during this interval, depending on the moment \({t}_{1}\approx 0/{t}_{2}/{t}_{3}\) in which we decide to open the boxes and look for the particle:
where \({{\rm{\Pi }}}_{j}\equiv j\rangle \langle j\) is the projection operator (in the Heisenberg picture) onto box \(j=A,B,C\). These predictions are based on the corresponding weak values which coincide with the eigenvalues of the corresponding operators, and hence also imply the counterfactual results of a projective measurement, had it been performed^{6}.
In other words, a particle obliged by the formalism to reside in Box A at time t_{ 1 }, is also obliged, with the same certainty, to “disappear” from it at time t_{2}, even though there is no tunneling from Boxes A and B to Box C. We note that if the values in Eqs 9–11 are understood as weak values, they can be all weakly measured at the same experimental run^{22}. Here, however, we present a strong simultaneous validation of these predictions, as follows.
It is timesymmetry that obliges this prediction. Followed from past to future, this formulation gives a reasonable account: Once you have prepared the preselected state (8), then, if you get either (10), (11) or (12), your probability to get the postselection (9) goes up from 11% to 33%.
But the same evolution followed conversely (just as if pre and postselection were interchanged), is odd: If you did not perform any intermediate measurement, just obtained the pre and postselections (8) and (9), then all three retrodictions (10,11,12) hold with 100% certainty.
The TSVF explains this odd sequence of mutuallyinconsistent locations as follows. In Eqs 10–12 we focused on properties that can be found with certainty when performing a strong projective measurement. However, the weak values \({\langle {{\rm{\Pi }}}_{B}({t}_{1})\rangle }_{w}={\langle {{\rm{\Pi }}}_{A}({t}_{3})\rangle }_{w}=\,1\), which can be validated through weak measurements, show that, in a sense, the total number of particles within Boxes A and B has been 0 all the time. Interestingly, at a time different from t_{ 2 } this 0 is achieved through a sum of two weak values–one positive and the other negative. Whereas a positive weak value indicates an ordinary effective interaction (although sometimes weaker or stronger than the one expected, due to weak values deviating from the spectrum of the measured operator), a negative weak value suggests an effective interaction with a minus sign. The particle therefore effectively behaves as if all its properties (such as mass and momentum) are negative, thereby termed Negaparticle. Notice that although the logic is that of weak values, in the present case they coincide with the eigenvalues of the projection operators, hence applicable also in the strong sense^{6} which we examine here.
Moreover, at t_{2} due to the tunneling between A and B described above, the particle’s positive and negative weak values fully cancel each other, hence we expect its “disappearance.” Then at t_{3}, when they part again, the particle “reappears.” This happens continuously in time and in a selfconsistent manner from the perspective of weak values.
Bell’s Inequality Ruling Out the “All Along” Alternative
The above claims may seem unusual, even in comparison with other quantum effects. It would therefore be natural to consider a more moderate, even trivial alternative: The pre and postselections give mere subensembles. This would be what Griffiths denotes “a family of histories”^{25}, comprised of three distinct groups, each of which has its own properties:

i)
Particles that went from the beamsplitter (BS) only to Box B at t_{1} and then tunneled to A at t_{3}, therefore not yielding the postselection (8). This family is ruled out by the postselection.

ii)
Particles that went to A at t_{1} and then tunneled to B later, at t_{3}, yielding the postselection (8) after being measured at either t_{1} or t_{3}.

iii)
Particles that went to C at t_{1} and yield the postselection (9) after being measured at any intermediate time.
Perhaps, then, no abrupt position changes occur because each individual particle has only one history. It may have travelled from A to B if found in either, but if found at any time in C, then it has been there all along. Similarly, if it is not found in A and B at t_{2} it has been absent from them all along.
The flaw of this alternative is straightforward. Being “all along” somewhere is, by definition, a local hidden variable. Here the abovementioned BellHardy proof ^{2,4} holds: Our single particle’s wavefunction, split into three, can exhibit nonlocal features, similar to those of an EPR pair or more specifically the W tripartite state^{26} (although weaker in strength), this time apparently giving rise to a nonlocal disappearance and reappearance. For, have we inserted three atoms into the three boxes in Eq. 8, as Hardy did with two atoms in^{4}, the single superposed photon would have entangled them in a W state (up to a relative phase which can be cancelled out) for which specific Belllike inequalities are violated^{27} implying the failure of local realism.
A similar shortcoming of the Consistent Histories formalism gave rise to the Entangled Histories formalism^{28} and was recently described in^{29,30}.
Below, following the proposal for a laboratory realization of this experiment, we present a more straightforward Belltype proof specifically adapted to the present shutterprobe photons pair.
The Proposed Realization of the DisappearingReappearing Particle Experiment
Based on the OkamotoTakeuchi^{8} realization of Aharonov and Vaidman’s^{7} protocol, our proposal for a realization of the disappearingreappearing particle naturally follows. It is based on the probe photon’s being superposed in time, such as the time of emission/arrival, enabling temporal interference^{31,32}.
Preparing temporal superposition is easy. Split a single photon into three equalintensity beams (see e.g.^{33}) and delay the 2^{nd} and 3^{rd} beams. The three beams therefore pass through the 3boxes system at t_{1}, t_{2} and t_{3}, respectively.
At each instant, direct the corresponding 1/3 beam to one or more boxes in accordance with the TSVF prediction about the shutterparticle’s position at that instant (Eqs 10–12), as follows:

i.
At t_{1}, the first 1/3 beam is split again, this time spatially, by a simple BS into 2, and goes to Boxes A + C.

ii.
At t_{2}, the second 1/3 goes only to C.

iii.
At t_{3}, the last 1/3, again split by simple BS, goes to B+C.
Then, after returning from all the boxes at all times, the probe is delayed and reunited in a manner precisely reverse to the above splits: first spatially and then temporally. Once all splits are completely undone, we measure the photon to see whether its initial quantum state is restored by interference.
Notice the required carefulness: To restore the photon’s initial coherent superposition, a shutter must exist with certainty at all these boxes: (A+C)(t_{1}), C(t_{2}), (B+C)(t_{3}). Failure of the shutter to be present at any box at the right instance would ruin the coherent superposition. Furthermore, there can be no other certain shutter positions during these times, i.e. this set of 5 slits closed by 1 shutter is maximal.
In addition to this direct test of the shutterparticle’s presence, there is a complementary test, measuring its varying absence: Send a photon to A and B, the boxes predicted to be empty at t_{2}. This, conveniently, does not involve complicated momentum exchange between shutter and photon. Here too, only the specific combination of transmitted beams through (A+B)(t_{2}), restores the photon’s initial state with certainty.
Experimental Setup and further Practical Considerations
For the experimental implementation of this theoretical prediction, the realization of a quantum shutter is the most important task. For this purpose, we propose using a photonic quantum router (PQR) as shown in Fig. 2a^{8}. The PQR consists of a twomode nonlinear sign shift (NS) gate embedded in a MachZehnder interferometer^{8}. When there is no control photon input to mode c_{ in }, the photon input to a_{ in } is routed to mode a_{ out }. In contrast, the photon is routed to mode b_{ out } when a control photon is input to mode c_{ in }.
Figure 2b shows a photonic scheme using PQRs for our experiment. The input mode b_{ in } of PQR in Fig. 2a is omitted in Fig. 2b for visibility. In this scheme, the quantum shutter is represented by a shutter photon (SP), and the interaction between the quantum shutter and the probe photon (PP) is realized by PQRs. By passing through the beamsplitters and the phase shifter of π/2, the SP is prepared in the preselected state \(({A\rangle }_{sp}+i{B\rangle }_{sp}+{C\rangle }_{sp})/\sqrt{3}\) which corresponds to Eq. (8). Here, the continuous evolution in time as described by^{4} is realized discretely; BS1 represents the tunneling between the boxes A and B between times t_{1} and t_{2}. Similarly, BS2 is for the tunneling between times t_{2} and t_{3}. Finally, SP is postselected by the state \(({A\rangle }_{sp}i{B\rangle }_{sp}+{C\rangle }_{sp})/\sqrt{3}\) by passing through the beamsplitters and the phase shifters \((\pi ,3\pi /2)\). Note that the postselected state is slightly different from that in Eq. (9) because the time evolution from t_{3} to t_{f} can be embedded within the postselected state, without affecting the result.
The PP is divided into 5 spatiotemporal beams, guided to five PQRs as follows. At t_{1}, while passing through PQRs, the first two PP beam interact with the SP in the modes (boxes) A and C. Similarly, PQRs provide the interactions between the PP and the SP in spatiotemporal modes C(t_{2}), B(t_{3}) and C(t_{3}). As a result of the interaction at PQR, the PP is routed to the right output mode (b_{ out } in Fig. 2a) when the SP is in the same PQR. In contrast, when the SP is not input to PQR, the PP will be routed to the left mode (a_{ out } in Fig. 2a) and discarded (represented by “x” in the figures). After the interactions with the SP, the 5 spatiotemporal paths of the PP are remerged by the beamsplitters. Since the pre and postselected SP appears in all the PQRs, the PP (due to interference) will be perfectly reflected with maintaining the coherence at each PQR^{8}. As a result, the PP will be found at SPD2 with a probability of 100% when SP is detected at SPD1. Note that it is possible to reduce the number of the PQRs from 5 to 3 by recycling the PQR at mode C with an active optical path control.
To demonstrate disappearance and reappearance even simpler probeshutter interactions suffice. Figure 3 shows such a schemes for simplified, more feasible experiments. Figure 3a shows a scheme for an experiment where the SP is in \({A\rangle }_{sp}\) at time t_{1}, \({C\rangle }_{sp}\) at time t_{2} and \({B\rangle }_{sp}\) at time t_{3} can be confirmed. This scheme consists of three PQRs to check the reflections of PP from the spacetime modes A(t_{1}), C(t_{2}) and B(t_{3}). The total reflection brought about by the SP’s disappearance and reappearance gives the unity detection probability of the PP at SPD2 when the SP is detected at SPD1. Figure 3b further shows the simplest test for disappearance of the SP from box A, which has two PQRs to check the reflections from the two spacetime modes A(t_{1}), and C(t_{2}). The unity detection probability at SPD2 proves the SP’s absence from the mode A at t_{1}.
For NS gates, one can use heralding NS gates (singlemodeinput) employing linear optics and single photons^{34,35}, which has been successfully demonstrated experimentally^{36}. With the help of ancillary photons, the operation of these NS gates is 100% successful when a heralding signal is output. The twomodeinput NS gate required in Fig. 2a is easily constructed using two singlemodeinput NS gates^{34,35,36,37} embedded in an interferometer. Therefore, in order to realize one PQR (Fig. 2a), two ancillary photons are required. Thus, the total number of 12 photons is required for implementing the scheme shown in Fig. 2b. The experiment using 12 photons seems technically difficult with current technology, but the recent rapid progresses on single photon sources^{37,38} may make it possible in the near future. The feasibility of experimental implementation of simplified versions shown in Fig. 3a,b is much higher, because the total number of required photons for these schemes is 8 and 6, respectively.
Alternatively, one could also use postselectionbased NS gates^{39,40,41}, which are only successful when the number of incident photons from two input ports are kept at either of the two output ports. The advantage is that these gates do not require ancillary photons. However, one needs to know the number of photons at each of the outputs at every PQRs using nondestructive photon number measurement, which is technically very difficult. Otherwise, one could try to estimate the successful events using the photon detection at the final output ports and unconnected outputs of PQRs (shown by “x” in the figures) as done in a similar experimental demonstration^{8}.
A Stricter Version: Measuring Both Appearances and Disappearances with the Same Probe Photon
The complex splitting of the probe photon in time and space, as well as the insertion of multiple PQRs, are feasible yet challenging, which is why a less complex laboratory version has been described above. Yet ignoring feasibility for a moment, an even more complex splitting is worth considering, where both the ideal proposal presented in Sec. “The Proposed Realization of the DisappearingReappearing Particle Experiment” and the control measurement at the end of that section are attained together.
To reveal both disappearance and reappearance it one setting, simply split the photon into six beams, such that even at t_{2} it is spatially split into two. Now send these two rays at t_{2} to Boxes A and B. The two PQRs in boxes A and B are now both switched in order to transmit the photon if no shutter photon is present. Here, the test is stricter: Any deviation from the sequence of transmissions, at t_{1} and t_{3} in the presence of a shutter and at t_{2} in its absence, would fail to preserve the coherence of the probe photon (Fig. 4).
An EPRBellType Validation
Earlier, discussing the original disappearingreappearing particle case with no measurement, we have employed the BellHardy proof to rule out alternative interpretations to the predicted dynamics of the particle within the boxes. Now that two particles are involved, namely the shutter and probe photon, a more straightforward proof for the particles’ nonlocal dynamics is at hand. It is based on the entangled state in Eq. 4 and its more complex spatiotemporal analogue preceding the outcome in Eq. 7. Okamoto and Takeuchi have already emphasized in^{8} that the shutter and probe photons become, after interacting, an entangled pair. In our case, the state is not maximally entangled, containing a few terms that somewhat weaken the nonlocal correlations, but nevertheless we can employ the general theorem stating that “All entangled quantum states are nonlocal”^{42}. This assures us that there does not exist any local hidden variables model which can account for the predicted outcomes.
On a more intuitive level, the probeshutter entangled state prior to postselection is amenable to Belllike tests. For this propose we may allow each particle to undergo either one of two different measurements, at the experimenters’ choices. Following is a highly idealized scheme for this purpose: Let Alice and Bob collect the shutter and probe photons, respectively, prior to their unification. That is, Alice takes the three boxes A, B, and C where the shutter resides in superposition. Bob collects the reflected probe photon in five separate cavities, according to its reflection from the appropriate boxes at the appropriate times \(i:A({t}_{1}),ii:C({t}_{1}),iii:C({t}_{2}),iv:B({t}_{3}),v:C({t}_{3})\). Each of them has the choice to either

Open each of the boxes/cavities for a projective position measurement and find where the shutter/probe photon is;
or

Reunite all of them and measure the shutter/probe photon for a noncommuting variable which is a coherent superposition of its possible positions (i.e. projecting on the complete superposed state with no relative phases). This measurement is akin to interference, measuring the photon’s momentum.
The entangled state of the probe and shutter photons prior to postselection implies, according to^{42}, that each experimenter’s choice of measurement nonlocally affects the outcome obtained by the other, in a way which cannot be reproduced using local hidden variables. But now, interestingly, the outcome is not a state but rather the particle’s entire evolution.
Thus, our experiment entails (to various degrees depending on the α_{ j } coefficients and the choices of measurements performed by the parties) nonlocal correlations between the probe and shutter photons. As such, the shutter’s varying positions cannot be accounted for by any local hidden variables model.
Discussion
To summarize, we can better comprehend the bearing of the suggested experiment on the foundations of QM by considering again the simple state in Eq. 1, slightly adapted to the present context by splitting the wavefunction into three:
Everyone has by now become accustomed to regarding this state as a fundamental quantummechanical state which allows a particle to equally reside in three places. Now, however, we have gained further insight into the nature of this superposition: Under a certain postselection, the system’s description is much more peculiar, namely the one pointed out by Aharonov et al.^{6,7}. The particle would, in retrospect, appear in whatever location it is looked for – A or B. Apparently, the restriction “would, in retrospect” turns this prediction into a mere counterfactual, but this is not the case anymore. Thanks to Okamoto and Takeuchi’s^{8} experiment, we know that a delayed measurement can straightforwardly access this peculiar past state: Just let another probe particle, also split into two, interact with the shutter particle, but do not measure any particle yet. Then make the postselection for the shutter. Finally, measure the probeparticle. The TSVF prediction, as experiment^{8} has shown, can be now verified by the latter’s coherence.
In this paper, then, an experiment is proposed to access an even more peculiar past state, in fact a past evolution: Under the appropriate pre and postselections, a delayed measurement should reveal the superposed particle’s disappearance and reappearance between distant boxes. Here too, a probe photon appropriately split to interact with the shutter at the appropriate times and locations can verify this unique dynamics.
Note, in passing, that this is also an example of the opportunities provided by pre and postselected quantum routers for photonic quantum computers. Further TSVF predictions with possible applications merit research along this line.
This work also sheds light on the controversial question regarding the past of a quantum particle^{43,44}. It has been suggested in^{43}, and later tested in^{44}, that a particle has been located wherever it has left a weak trace, i.e. a nonzero weak value. In this paper we have demonstrated a stronger notion of a past – a particle has been located wherever it has acted as a shutter. As the terms suggest, a strong presence implies a weak one, but not vice versa. Unlike the former criterion suggested by Vaidman, this criterion can be tested with standard projective measurements and moreover, the single particle’s past can, in theory, be verified with certainty.
Whenever the weak values do not coincide with the eigenvalues of a dichotomic operator, these predictions can be verified only with the aid of weak measurements. Nevertheless, this weak reality outlines a richer story underlying the predictions with strong (projective) measurements.
Let us then rephrase the question posed in the beginning: How literally should the TSVF retrodictions be taken? Our answer is based on two facts: i) TSVF is fully consistent with quantum theory. ii) Its counterfactual retrodictions have recently become ordinary predictions thanks to the introduction of delayed measurements. From these advances, a novel aspect of quantum uncertainty emerges. Perhaps “superposition” is actually a collection of many ontic states (or better, twotime ontic states^{45}), which postselection helps to isolate and measure under greater resolution. Some of these superpositions are even more unique, involving, e.g., “collapse” of the particle’s position into whatever box opened to find it^{6}, or possessing unusual momenta^{46}. These phenomena can never be observed in real time, thereby avoiding violations of causality and other basic principles of physics. Yet the proof for their existence is as rigorous as the known proofs for quantum superposition and nonlocality – all of which are posthoc. Very likely, other interesting forms of superposition are awaiting derivation and experiment, to advance our understanding of quantum reality.
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Acknowledgements
It is a pleasure to thank Yakir Aharonov for many illuminating discussions. E.C. was supported by the Canada Research Chairs (CRC) Program. R.O. was supported by PRESTO, JST (No. JPMJPR15P4) and JSPSKAKENHI (No. JP17H02936). S.T. was supported by JSPSKAKENHI (No. JP26220712) and CREST, JST (No. JPMJCR1674).
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A.C.E. and E.C. analyzed the theoretical aspects of the work. R.O. and S.T. analyzed the experimental aspects of the work. The writing and figures preparation were correspondingly shared between A.C.E., E.C., R.O. and S.T. who also discussed together the synergy between theory and experiment.
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Elitzur, A.C., Cohen, E., Okamoto, R. et al. Nonlocal Position Changes of a Photon Revealed by Quantum Routers. Sci Rep 8, 7730 (2018). https://doi.org/10.1038/s4159801826018y
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