aa r X i v : . [ qu a n t - ph ] D ec On unitary evolution and collapsein Quantum Mechanics
Francesco Giacosa
Institute for Theoretical Physics,J. W. Goethe University, Max-von-Laue-Str. 1, 60438,Frankfurt am Main, Germany
August 2, 2018
Abstract
In the framework of an interference setup in which only two outcomesare possible (such as in the case of a Mach-Zehnder interferometer), wediscuss in a simple and pedagogical way the difference between a standard,unitary quantum mechanical evolution and the existence of a real collapseof the wave function. This is a central and not-yet resolved questionof Quantum Mechanics and indeed of Quantum Field Theory as well.Moreover, we also present the Elitzur-Vaidman bomb, the delayed choiceexperiment, and the effect of decoherence. In the end, we propose twosimple experiments to visualize decoherence and to test the role of anentangled particle.
Quantum Mechanics (QM) is a well-established theoretical construct, whichpassed countless and ingenious experimental tests [1]. Still, it is renowned thatQM has some puzzling features [2, 3, 4, 5, 6]: are macroscopic distinguishablesuperpositions (Schr¨odinger-cat states) possible or there is a limit of validityof QM? Do measurements imply a non-unitary (collapse-like) time evolutionor are they also part of a unitary evolution? In the latter case, should wesimply accept that the wave function splits in many branches (i.e., parallelworlds), which decohere very fast and are thus independent from each other? Itis important to stress that these issues are not only central in nonrelativistic QMbut apply also in relativistic Quantum Field Theory. Namely, the generalizationto quantized fields does not modify the role of measurements.In this work we discuss in a introductory way some of the questions men-tioned above. We study the quantum interference in an idealized two-slit ex-periment and we analyze the effect that a detector measuring “which path hasbeen taken” has on the system. In particular, we shall concentrate on the1ollapse of the wave function, such as the one advocated by collapse models[5, 6, 7, 8, 9, 10, 11, 12] and show which are the implications of it.Variants of our setup also lead us to the presentation of the famous Elitzur-Vaidman bomb [13] and to delayed choice experiments [14, 15]. Thus, we candescribe in a unified framework and with simple mathematical steps (typical ofa QM course) concepts related to modern issues and experiments of QM.Besides the pedagogical purposes of this work, we also aim to propose twoexperiments (i) to see decoherence at work in an interference setup with onlytwo possible outcomes and (ii) to test the dependence of the interference on anidler entangled particle.
We consider an interference setup as the one depicted in Figs. 1 and 2. Aparticle P flies toward a barrier which contains two ‘slits’ and then flies furtherto a screen S . Usually in such a situation there is a superposition of waveswhich generates on the screen S many maxima and minima. We would like toavoid this unnecessary complication here but still use the language of a double-slit experiment in which a sum over paths is present. To this end, we assumethat the particle can hit the screen in two points only, denoted as A and B ,see the discussion below. All the issues of QM can be studied in this simplifiedframework. We assume also to ‘sit on’ the screen S : when the particle hits A or B we ‘see’ it.First, we consider the case in which only the left slit is open (Fig. 1, leftside). In order to achieve our goal, the slit is actually not a simple hole in thebarrier (out of which a spherical wave would emerge) but a more complicatedfilter which projects the particle either to a straight trajectory ending in A orto a straight trajectory ending in B, see Fig. 1. In the language of QM, thissituation amounts to a wave function | L i associated to the particle which hasgone through the left slit, which is assumed to be: | L i = 1 √ | A i − | B i ) . (1)Then, by simply using the Born rule (i.e., by squaring the coefficient multiplying | A i or | B i ), we predict that the particle ends up either in the endpoint A withprobability 50% or in the endpoint B with probability 50%. This is indeedwhat we measure by repeating the experiment many times. As we see, theprobability is -for us observer on the screen S - a fundamental ingredient of QM,which however enters only in the very last step, i.e. when the measurementcomes into the game. The state | L i is an equal (antisymmetric) superpositionof | A i and | B i , but in a single experiment we do not find a pale spot on A anda pale spot on B : we always find the particle either fully in A or in B. It isonly after many repetitions of the experiment that we realize that the outcome A and the outcome B are equally probable.2 |L>= |A>−|B> |R>= |A>+|B> A 50% B 50%
Particle P
Left slit (open)
Right slitA 50% B 50%
Particle P
Left slit (open) (closed)
Right slit S = screen (closed)
Figure 1: Hypothetical experiment with only two possible outcomes ( A and B ).Left: only the left slit is open. Right: only the right slit is open. Note, each slitis not a simple hole but acts as a filter which projects the particle either to atrajectory with endpoint A or to a trajectory with endpoint B .If only the right slit is open (Fig. 1, right side), we have a similar situationin which only two trajectories ending in A and in B are present. The wavefunction of the particle after having gone through the right slit is denoted by | R i and is described by the orthogonal combination to | L i : | R i = 1 √ | A i + | B i ) . (2)Then, also in this case one finds the particle in 50% of cases in A and 50% in B. We now turn to the case in which both slits are open, see Fig. 2. The wavefunction of the particle is assumed to be the sum of the contributions of the twoslits: | Ψ i = 1 √ | L i + | R i ) , (3)i.e. the contributions of both slits add coherently. A simple calculation showsthat | Ψ i = | A i , (4)which means that the particle P always hits the screen in A and never in B. Namely, in A we have a constructive interference, while in B we have a destructive interference. (Notice that the points A and B are not equidistantfrom the two slits. However, we take the two slits as being close to each otherand the points A and B as being far from each other: the difference between thesegments LA and RA (and so between LB and RB ) is assumed to be negligible3 article P Left slit (open)
Right slit (open)
S A 100% B 0% S ψ Figure 2: Same setup of Fig. 1, but now both slits are open: interference takesplace and all particles hit the screen in A .4uch that the two contributions of the wave packet of the particle P from theleft and right slit arrive almost simultaneously and the depicted interferenceeffect takes place).In conclusion, we have chosen the language of a two-slit experiment becauseit is the most intuitive. The price to pay is a slit acting as a filter and not asa simple hole. However, one can easily build analogous setups as the one heredescribed by using photon polarizations, electron spins or equivalent quantumobjects, or by using a Mach-Zehnder interferometer, see details in Sec. 2.3.3. As a next step we put a detector D right after the two slits (both open). D measures through which hole the particle has passed, without destroying it (seeFig. 3). We analyze the situation in two ways: first, by assuming the collapse ofthe wave function as induced by D and, second, by studying the entanglementof the particle with the detector. Note, we still assume that we sit on (or watch)the screen S only, but we are not directly connected to the detector D. Collapse : In this case we assume that the detector D generates a collapseof the wave function. Suddenly after the interaction with D, the state of theparticle P collapses into | L i with a probability of 50% or into | R i with a prob-ability of 50%. Then, the state is described by either | L i or | R i , but not anylonger by the superposition of them. As a consequence, we have in half of thecases a situation analogous to having only the left slit open and in the otherhalf to having only the right slit open.What we will then see on the screen S ? The probability to find the particlein A is given by P [ A ] = P [ L, A ] + P [ R, A ] = 12 ·
12 + 12 ·
12 = 12 (5)where P [ L, A ] = 1 / D has measured theparticle going through the left slit and then the particle has hit the screen in A. Similarly, P [ R, A ] = 1 / D has measured theparticle going through the right slit before the latter hits A. A similar descriptionholds for P [ B ] = 1 / P [ B ] = P [ L, B ] + P [ R, B ] = 12 ·
12 + 12 ·
12 = 12 . (6)The collapse is obviously part of the standard interpretation of QM, in whicha detector is treated as a classical object which induces the collapse of the quan-tum state. As a result, there is no interference on the screen S . As renowned,the standard interpretation does not put any border between what is a classi-cal system and what is a quantum system. Nevertheless, one can interpret thecollapse postulate as an effective description of a physical process. Namely, intheories with the collapse of the wave function, the collapse is a real physicalphenomenon which takes place when one has a macroscopic displacement of the5 > Ψ | > Ψ = |L> with prob. 50% | > Ψ with prob. 50% = |R>| > Ψ = 2|L>|D > + |R>|D > RL = |D > = |D > R = |D > L D With collapse:Without collapse: D RL 0
States of the detector :
RL 0 RL 0
A 50% B 50%
Particle P
Left slit (open)
Right slit (open)
Figure 3: A detector which measures which slit the particle has gone through isplaced just after the slits. The wave functions for the collapse and no-collapsescenarios are depicted.position wave function of the detector (or, more generally, of the environment).In this framework, somewhere in between the quantum world and the classi-cal macroscopic world, a new physical process takes place which realizes thecollapse: this could be, for instance, the stochastic hit in the Ghirardi-Rimini-Weber model [5, 7, 8] or the instability due to gravitation in the Penrose-Diosiapproach [6, 10, 11]. Neglecting details, the main point is that such collapsetheories realize physically the collapse which is postulated in the standard in-terpretation and liberates it from inconsistencies. Still, it is an open and wellposed physical question if (at least one of) such collapse theories are (is) correct.
No-collapse:
In this case we do not assume that the detector D generatesa collapse of the wave function, but we enlarge the whole wave function of thesystem by including also the wave function of the detector. We assume that,prior to measurement, the detector is in the state | D i (we can, for definiteness,think of a old-fashion indicator which points to 0, see Fig. 3). Then, when bothslits are open, the state of the whole system just after having passed throughthem but not yet in contact with the detector D , is given by | Ψ i = 1 √ | L i + | R i ) | D i . (7)Then, the particle-detector interaction induces a (we assume very fast) time6volution which generates the following state: | Ψ i = 1 √ | L i | D L i + | R i | D R i ) , (8)where | D L i ( | D R i ) describes the pointer of the detector pointing to the left(right). Thus, no collapse is here taken into account, because the whole wavefunction still includes a superposition of | L i and | R i , which, however, are nowentangled with the detector states | D L i and | D R i , respectively.An important point is that the overlap of | D L i and | D R i is small: h D L | D R i ≃ N atoms, where N is of the order of the Avogadro constant. Theatom α of the pointer is in a superposition of the type ( ψ αL ( ~x ) + ψ αR ( ~x )) / √ ψ αL ( ~x ) ( ψ αR ( ~x )) is the wave function of the atom when the pointer pointsto the left (right). We have: h D L | D R i = N Y α =1 Z d x ( ψ αL ( ~x )) ∗ ψ αR ( ~x ) . (10)The quantity R d x ( ψ αL ( ~x )) ∗ ψ αR ( ~x ) = λ α is such that | λ α | <
1. For a largedisplacement, λ α is itself a very small number (small overlap), but the crucialpoint is to observe that h D L | D R i is the product of many numbers with modulussmaller then 1. Assuming that λ α = λ for each α (each atom gets a similardisplacement: this assumption is crude but surely sufficient for an estimate), weget h D L | D R i ≃ λ N , (11)which is extremely small for large N. Even if we take λ = 0 .
99 (which is indeedquite large and actually overestimates the overlap of the wave functions of anatom belonging to macroscopic distinguishable configuration), we obtain h D L | D R i ≃ . N A ∼ − (12)which is tremendously small.After having clarified the de facto orthogonality of | D L i and | D R i , we rewritethe full wave function of the system | S i as | Ψ i = 12 [ | A i ( | D R i + | D L i ) + | B i ( | D R i − | D L i )] . (13)Then, the probability to find the particle P in A is obtained (now by using theBorn rule, because we are observing the screen S ): P [ A ] = P [ L, A ] + P [ R, A ] = 12 ·
12 + 12 ·
12 = 12 (14)7here P [ L, A ] = 1 / | A i | D L i and P [ R, A ] = 1 / | A i | D R i . A similarsituation holds for P [ B ] = 1 / . Thus, also in this case the presence of D causesthe disappearance of interference.The same result is obtained if we use the formalism of the statistical operator,which is defined by ˆ ρ = | Ψ i h Ψ | (see, for instance, Refs. [1, 5]). Upon tracingover the detector states (environment states) the reduced statistical operatorreads (we use here h D L | D R i = 0):ˆ ρ red = h D L | ˆ ρ | D L i + h D R | ˆ ρ | D R i = (cid:0) | A i | B i (cid:1) (cid:18) (cid:19) (cid:18) h A |h B | (cid:19) , (15)where the diagonal elements represent p [ A ] = p [ B ] = 1 / | D L i and | D R i . Sum up:
We find that, for us sitting on the screen S , the very same outcome ,i.e. the absence of interference, is obtained by applying the collapse postulate asan intermediate step due to the detector D or by considering the whole quantumstate -including the detector D - and by applying the Born rule only in the veryend. This equivalence holds as long as the (anyhow very small) overlap of thedetector states of Eq. (12) is neglected (see also the related discussion in Sec.3). The question is then: do we need the collapse? The second calculation(no-collapse) seems to answer us: ‘no, we don’t’. In this respect, one has asuperposition of macroscopic distinct states, which coexist and are nothing elsebut the branches of the Everett’s or many worlds interpretation (MWI) of QM[16]. Thus, assuming that no collapse takes place brings us quite naturally tothe MWI [3, 17, 18, 19, 20].However, care is needed: in fact, the ‘no collapse’ assumption is a generalstatement and means also that there is no collapse when the particle P hitsthe screen S (where our own wave function is part of the game). Let us clarifybetter this point by going back to the very first case we have studied, in whichonly the left slit was open and no detector D was present (Fig. 1, left part).The wave function of the particle just before hitting the screen is given by | L i = ( | A i − | B i ) / √
2. But then, after the hit and assuming no collapse, thewhole wave function -including us, who are the observers - reads: | Ψ i = 1 √ | A i | Screen recording A and we observing A i− √ | B i | Screen recording B and we observing B i . (16)The question is why the coefficient in front of the vector | A i | Screen recording A and we observing A i subjective probability of observing A for the observer (us)sitting on the screen. In other words, how does the MWI explain the proba-bilities according to the Born rule? The Born rule seems to be an additionalpostulate, which has to be put ad hoc into it. This situation is however not sat-isfactory, because the main idea of the MWI is to eliminate the collapse from thedescription of the QM and consequently to derive the standard Born probabil-ities. Although there are attempts to show that there is no need of postulatingthe Born rule in this context [21] (see also Ref. [22]), no agreement has beenreached up to now [5, 23, 24, 25]. This is indeed an argumentation in favour ofthe possibility that a collapse really takes place. Surely, ‘real collapse’ scenariosdeserve to be studied theoretically and experimentally [5, 6, 7].Note, up to now we did not mention the decoherence, see e.g. Refs. [2, 26,27, 28, 29] and refs. therein. This is possible because we have put a detectorthat makes a measurement by evolving from the state | D i into two (almost)orthogonal states | D L i and | D R i , but actually one can interpret this fast changeof the detector state as the result of a decoherence phenomenon. This is how-ever a rather peculiar decoherence, because we have prepared the detector ina particular (low entropic) | D i state, which is ‘ready to’ evolve into | D L i and | D R i as soon as it interacts with the particle P . In Sec. 3 we will describe whatchanges when the environment, instead of the detector, is taken into account. A simple change of the setup allows us to present the famous Elitzur-Vaidmanbomb, first described in Ref. [13] and then experimentally verified in Ref. [30].We substitute the detector with a ‘bomb’, which can be activated by the particle P . We place the bomb only in front of the left slit, see Fig. 4. This means that,if only the left slit is open, the bomb explodes soon after the particle has gonethrough the slit. If, instead, only the right slit is open, it doesn’t explode.For definiteness and simplicity we assume that the particle is not destroyed norabsorbed by the bomb.Just as previously, we can interpret the experiment applying either the col-lapse or by studying the whole wave function. In the collapse approach, thebomb simply makes a measurement. When both slits are open the wave func-tion, before the interaction with the bomb, is given by | Ψ i = ( | L i + | R i ) / √ . Notice that in the second case the bomb is doing a null measurement. Thevery fact that the bomb does not explode means that the particle went to theright slit (we assume 100% efficiency in our ideal experiment). When the bombexplodes there is a collapse into | Ψ i = | L i , when it doesn’t into | Ψ i = | R i . Then, we have a situation which is very similar to the case of the detector D which we have studied previously: no interference on the screen S is observed,but we observe the particle in the endpoint A and B with probability 1 / > Ψ B A 50% B 50%
Particle P
Left slit (open)
Right slit (open)
Figure 4: Variant of the Elitzur-Vaidman experiment: a bomb is placed justafter the left slit.tion is given by (after interaction with the bomb) | Ψ i = 1 √ | L i | B E i + | R i | B i )= 12 [ | A i ( | B i + | B E i ) + | B i ( | B i − | B E i )] (17)where | B i is the state describing the unexploded bomb and | B E i the explodedone. Obviously, as in Eq. (12), we have h B E | B i ≃ . Again and just as beforeno interference is seen on S but the two outcomes A and B are equiprobable.Clearly, no difference between assuming the collapse or not is found, but theinteresting fact is that the non-explosion of the bomb is enough to destroyinterference.If, instead of the bomb we put a fake bomb (referred to as the dud bomb,which has the very same aspect of the real functioning bomb but does notinteract at all with the particle P ), the wave function of the system is given by | Ψ i = 1 √ | L i + | R i ) (cid:12)(cid:12) B dud (cid:11) = | A i (cid:12)(cid:12) B dud (cid:11) (18)where (cid:12)(cid:12) B dud (cid:11) describes the wave function of the dud bomb. In this case, thereis interference and the particle P always ends up in A .10hen, the amusing part comes: if we do not know if the bomb is a dud ornot, we can –in some but not all cases– find out by placing it in front of theleft slit. If there is no explosion and the particle ends up in B, we deduce forsure that the bomb is real. Namely, this outcome is not possible for a dud,see Eq. (18). Note, we have deduced that the bomb is ‘good’ without makingit explode (that would be easy: just send the particle P toward the bomb, ifit goes ‘boom’ it was real). This situation occurs in 25% of cases in which afunctioning bomb is placed behind the slit, see Eq. (17): we can immediately‘save’ 25% of the good bombs. Conversely, in 50% of cases the good bombsimply explodes and we lose it (then, the particle P goes to either A (25%) orto B (25%)). In the remaining 25% the good bomb does not explode, but theparticle P hits A. Then, we simply do not know if the bomb is good or fake:this situation is compatible with both hypotheses. We can, however, repeat theexperiment: in the end, we will be able to save 1 / / · / ... = 1 / Another interesting configuration is obtained by assuming that a second entan-gled particle, denoted as I (for idler), is emitted when P goes through the slit(s).The system is built in the following way: if the particle P goes through the leftslit, the particle I is described by the state | I L i . Similarly, when the particle P goes to the right slit, the particle I is described by the state | I R i . We assumethat the two idler states are orthogonal: h I L | I R i = 0 . This situation resemblesclosely that of delayed choice experiments [14, 15].When both slits are open the whole wave function of the system is given by: | Ψ i = 1 √ | L i | I L i + | R i | I R i )= 12 [ | A i ( | I R i + | I L i ) + | B i ( | I R i − | I L i )] . (19)The particle I is entangled with P , but being the latter a microscopic object,we surely cannot apply the collapse hypothesis because the particle I is not ameasuring apparatus.Do we have interference on the screen S in this case? The answer is clear:no. The states | A i | I L i , | A i | I R i , | B i | I L i , | B i | I R i represent a basis of thissystem, thus the probability to obtain | A i (that is, the probability of P hitting S in A ) is 1 / / / . So for B. The presence of the entangled idler statedestroys the interference on S. It is sometime stated that this result is a consequence of the fact that thestate of the idler particle I carries the information of which way P has followed.For this reason, the interference has disappeared (this is a modern reformula-tion of the complementarity principle). However, such expressions, althoughappealing, are often too vague and need to be taken with care.As a next step we study what happens if we perform a measurement on theparticle I . We study separately two distinct types of measurements.11 easuring I in the | I L i - | I R i basis. First, we perform a measurement whichtells us if the state of the idler particle is | I L i or | I R i . For simplicity, we apply thecollapse hypothesis (as usual, the results would not change by keeping track ofthe whole unitary quantum evolution). But first, we have to clarify the followingissue: when do we perform the measurement on I ? We have two possibilities: • If we measure the state of I before the particle S hits the screen, the wavefunction reduces to | L i | I L i or to | R i | I R i with 50% probability, respec-tively. Then, the screen S performs a second measurement: we find -asusual- 50% of times A (25% | A i | I L i and 25% | A i | I R i ) and 50% of times B (25% | B i | I L i and 25% | B i | I R i ). • If, instead, the particle P arrives first on the screen S , the quantum statecollapses into | A i ( | I R i + | I L i ) / √ A has clicked), orinto | B i ( | I R i − | I L i ) / √ B has clicked). Thesubsequent measurement of the I particle will then give | I L i or | I R i (50%each).In conclusion, we realize that it is absolutely not relevant which experimentis done before the other. In particular, for us sitting on the screen S , it doesnot matter at all when and if the measurement of the idler state is performed.We simply see no interference. Measuring I in the ( | I R i + | I L i ) / √ - ( | I R i − | I L i ) / √ basis. Being theparticle P entangled with another particle and not with a macroscopic state, wecan also decide to perform a different kind of measurement on I. For instance,we can put a detector measuring I by projecting onto the basis ( | I R i + | I L i ) / √ | I R i − | I L i / √
2. If we do this measurement before the particle P has hitthe screen S, we have the following outcome as a consequence of the collapseinduced by the I -detector: | Ψ i = | A i ( | I R i + | I L i ) / √ | Ψ i = | B i ( | I R i − | I L i ) / √ P will surely hit S in A, in the latter in B. One sometimes interpret the experiment in the following way: the detectormeasuring the state of I as being either ( | I R i + | I L i ) / √ | I R i − | I L i / √ | I R i + | I L i ) / √ P in the position A , just as thecase with two open slits (Fig. 2). In the other case, when the detector measures( | I R i − | I L i ) / √
2, we also have a kind of interference in which the final position B is the only outcome. In the language of Ref. [14], one speaks of ‘fringes’ inthe former case, and of ‘anti-fringes’ in the latter.However, care is needed: for us sitting on S , if we do not know whichmeasurement is performed on I, we simply see that no interference occurs (50%- A and 50%- B ). But, if we could then speak with a colleague working with the12 -detector, we would realize that, each time we have measured A he has foundthe state ( | I R i + | I L i ) / √ , while each time we have measured B he has found( | I R i − | I L i / √ . Thus, we have a correlation of our results (measurement ofthe screen S ) with those of the I -detector. This is actually no surprise if welook at the quantum state of Eq. (19). This statement is indeed more precisethan the statement of having interference because we have erased the which-wayinformation. Namely, we do not have interference.Indeed, we can perform the measurement of I even after (in principle muchtime after) the screen S has measured P in either A or B. Here the name ‘delayedchoice’ comes from: we choose if we retain the which-way information or not.Still, the result is the same because there is no influence on the time-orderingof the measurements. If the measurement of the screen S occurs first, we havea collapse onto the very same Eqs. (20)-(21). Then, a measurement of the idlerparticle I would simply find either ( | I R i + | I L i ) / √ A ) and( | I R i − | I L i ) / √ B ). For sure, there is no change of the pastby a measurement of the idler state, but simply a correlation of states. Still,such a very interesting setup visualizes many of the peculiarities of QM and canbe used for quantum cryptography. In a two-slit experiment all the peculiarities of QM are evident due to the factthat the particle P follows (at least) two paths at the same time. This isextremely fascinating as well as counterintuitive for our imagination based ona childhood with rolling ‘classical’ marbles. However, as already mentioned inSec. 1.1, a simple implementation of the two-slit experiment does not produceonly two possible outcomes, but gives rise to a superposition of waves with manymaxima and minima. In the following we present two possible realizations ofour Gedankenexperiment which do not make use of slits.An interference experiment in which only two outcomes are possible can berealized by using particles with spin 1 / | h i and | v i respectively). Inour analogy, the state | h i corresponds to the state of our particle P coming outfrom the left slit, | h i ≡ | L i , and similarly | v i from the right slit, | v i ≡ | R i . Then, we place a detector which acts as the screen S by making a measurementin the basis | A i = ( | v i + | h i ) / √ | B i = ( | v i − | h i ) / √
2. In addition, we canplace a second detector which plays the role of the detector D by measuring thepolarization in the | h i - | v i basis. Indeed, in this case we do not need to send thephotons along two different paths, because the polarization d.o.f. is enough forour purposes.Another possible realization of our setup is the Mach-Zehnder interferometer13 DA 50% PDB 50%PDA 50% PDB 50% M M γ M M γ PDPD B 0%A 100% (c)(a) (b)
M M γ BSBSBSBS
Figure 5: The Mach-Zehnder interferometer. M stands for mirror, BS for beamsplitter, and PD for photon detector. The case (a) is analogous to having onlythe left slit open (Fig. 1, left side), (b) to only the right slit open (Fig. 1, rightside), (c) to both slits open (Fig. 2).[31], see Fig. 5, which makes use of beam splitters. When a photon is sentto the path of Fig. 5.a (denoted as path-1), both photon counters A and B can detect the photon with a probability of 50% . For our analogy, we have | path -1 i ≡ | L i . Similarly, when the photon is sent to the path of Fig. 5.b(path-2), we hear a click in A or in B with 50% probability. For the analogy: | path -2 i ≡ | R i . When a beam splitter is put in the beginning of the setup, afterthe photon passes through, we get a superposition ( | path -1 i + | path -2 i ) / √ D , the bomb, entangled particle(s) aswell as the environment can be easily carried out.In the end, notice that Mach-Zehnder interferometers can be constructed byusing neutrons instead of photons. The so-called neutron interferometers (seethe recent review paper [32] and refs. therein) can be very well controlled andallow to experimentally study quantum systems to a great level of accuracy. In this section we show that there is a difference between the collapse and no-collapse scenarios. To this end, instead of having a detector, a bomb, or an idlerentangled state, we assume that the space between the slits and the screen isnot the vacuum. Then, we study the time evolution of the environment whichinteracts with the particle P . This interaction is assumed to be soft enough notto absorb or kick away the particle in such a way that the final outcomes on thescreen S are still the endpoints A or B .Before the particle P goes through the slit(s), the environment is describedby the state | E i . First, we study the case in which only the left slit is open.Denoting as t = 0 the time at which P passes through the left slit, the wave14unction of the environment evolves as function of time t as | Ψ( t ) i = | L i | E L ( t ) i , (22)where by construction | E L (0) i = | E i . Similarly, if only the right slit is open, atthe time t the system is described by | Ψ( t ) i = | R i | E R ( t ) i with | E R (0) i = | E i .We now turn to the case in which both slits are open. It is important to stressthat, by assuming a weak interaction of the particle P with the environment,we surely do not have -at first- a collapse of the wave function, but an evolutionof the whole quantum state given by: | Ψ( t ) i = 1 √ | L i | E L ( t ) i + | R i | E R ( t ) i )= 12 [ | A i ( | E R ( t ) i + | E L ( t ) i ) + | B i ( | E R ( t ) i − | E L ( t ) i )] . (23)This is indeed very similar to the detector case, but there is a crucial aspectthat we now take into consideration. The state | E L ( t ) i and | E R ( t ) i coincide at t = 0 and then smoothly depart from each other. At the time t we assume tohave c ( t ) = h E L ( t ) | E R ( t ) i = e − λt . (24)(where c ( t ) is taken to be real for simplicity). This is nothing else than a gradualdecoherence process. The states of the environment entangled with | L i and | R i overlap less and less by the time passing. The constant λ describes the speedof the decoherence and depends on the number of particles involved and theintensity of the interaction. Note, strictly speaking, this non-orthogonality isalso present in the case of the detector (if no collapse is assumed), but theoverlap is amazingly small, see the estimate in Eq. (12). (In the case of thedetector D of Sec. 2.2, λ is very large and consequently λ − is a very shorttime scale, shorter than any other time scale in the setup of Fig. 3. For thatreason we assumed that the detector state evolved for all practical purposesinstantaneously from the ready-state (pointer up) to pointing either to the leftor to the right.)Now we ask the following question: what is the probability that the particle P hits the screen in A ? We assume that the particle P hits the screen at thetime τ. At this instant, the state is given by | Ψ( τ ) i with h E L ( τ ) | E R ( τ ) i = c ( τ ).We now present the mathematical steps leading to p [ A, τ ], which, althoughstill simple, are a bit more difficult than the previous ones. The reader who isonly interested in the result can go directly to Eq. (29).At the time τ we express the state | E L ( τ ) i as | E L ( τ ) i = c ( τ ) | E R ( τ ) i + X α b α ( τ ) (cid:12)(cid:12) E αR, ⊥ ( τ ) (cid:11) (25)where the summation over α includes all states of the environment which are or-thogonal to | E R ( τ ) i : D E αR, ⊥ ( τ ) | E R ( τ ) E = 0. This expression is possible becausethe set { E R ( τ ) , E αR, ⊥ ( τ ) } represents a orthonormal basis for the environment15tate. Its explicit expression is be extremely complicated, but we do not needto specify it. The normalization of the state | E L ( τ ) i implies that | c ( τ ) | + X α | b α ( τ ) | = 1 . (26)Then, the state of the system at the instant τ is given by the superposition | Ψ( τ ) i = 12 [1 + c ( τ )] | A i | E R ( τ ) i + 12 | A i X α b α ( τ ) (cid:12)(cid:12) E αR, ⊥ ( τ ) (cid:11) + 12 [1 − c ( τ )] | B i | E R ( τ ) i + 12 | B i X α b α ( τ ) (cid:12)(cid:12) E αR, ⊥ ( τ ) (cid:11) . (27)At the time τ the probability of the particle P hitting A is given by p [ A, τ ] = 14 | c ( τ ) | + 14 X α | b α ( τ ) | = 14 | c ( τ ) | + 14 (cid:16) − | c ( τ ) | (cid:17) , (28)where in the last step we have used Eq. (26). A simple calculation leads to p [ A, τ ] = 12 + 12 c ( τ ) = 12 + 12 e − λτ . (29)A similar calculation leads to the probability of the particle P hitting S in B as p [ B, τ ] = 12 − c ( τ ) = 12 − e − λτ . (30)We see that ‘a bit’ of interference is left (no matter how large the time interval τ is): p [ A, τ ] − p [ B, τ ] = e − λτ , (31)showing that there is always an (eventually very slightly) enhanced probabilityto see the particle in A rather than in B. Notice that the very same result is found by using the reduced statisticaloperator:ˆ ρ red ( τ ) = h E R ( τ ) | ˆ ρ ( τ ) | E R ( τ ) i + X α (cid:10) E αR, ⊥ ( τ ) | ˆ ρ ( τ ) | E αR, ⊥ ( τ ) (cid:11) = (cid:0) | A i | B i (cid:1) (cid:18) p [ A, τ ] c ( τ ) c ( τ ) p [ B, τ ] (cid:19) (cid:18) h A |h B | (cid:19) (32)where ˆ ρ ( τ ) = | Ψ( τ ) i h Ψ( τ ) | . The diagonal elements are the usual Born probabil-ities, while the non-diagonal elements quantify the overlap of the two branchesand become very small for increasing time. (A related subject to the quantumevolution described here is that of the weak measurement, in which the ‘mea-surement’ is performed by a weak interaction and thus a unitary evolution of thewhole system is taken into account, see the recent review [33] and refs. therein.)16ll these considerations do not require any collapse of the wave function dueto the environment (see also Ref. [34]). Indeed, if we replace the environmentwith the detector D of Sec. 2 (which was nothing else than a particular en-vironment), the whole discussion is still valid (but see the comments on timescale after Eq. (24)). The only point when the Born rule enters is when we seethe particle being either in A or in B, but -as we commented previously- in thisno-collapse MWI scenario, we do not know why the Born rule applies [23, 24].In this sense, decoherence alone is not a solution of the measurement problem[35]. The wave function is still a superposition of different and distinguishablemacroscopic states. Still, because of decoherence, these states (branches) be-come almost orthogonal, thus decoherence is an important element of the MWIalthough it does not explain the emergence of probabilities.What do theories with the collapse of the wave function predict? As longas few particles of the environment are involved (i.e., at small times), for surewe do not have any collapse and the entanglement in Eq. (23) is the correctdescription of the system. Namely, we know that interference effects occurfor systems which contains about 1000 (and even more) particles [36]. But,if we wait long enough we can reach a critical number of particles at whichthe collapse takes place. Thus, simplifying the discussion as much as possible,according to collapse models there should be a critical time-interval τ ∗ at whichthe probability p [ A, τ ] suddenly jumps to 1 / p [ A, τ ] = + e − λτ for τ < τ ∗ ; for τ ≥ τ ∗ . (33)Indeed, such a sudden jump is an oversimplification, but is enough for ourpurposes: it shows that a new phenomenon, the collapse, takes place. In Fig. 6we show schematically the difference between the ‘no-collapse’ and the ‘collapse’cases. Obviously, if τ ∗ is very large, it becomes experimentally very difficult todistinguish the two curves, but the qualitative difference between them is clear.In Ref. [38] the gradual appearance of decoherence due to interaction ofelectrons with image charges has been experimentally observed. This is analo-gous to our Eq. (29). (For other decoherence experiment see Ref. [29] and refs.therein.) Indeed, it would be very interesting to study decoherence in a setupwith only two outcomes, for instance with the help of a Mach-Zehnder inter-ferometer or by using neutron interferometers. Namely, even if the distinctionbetween collapse/non-collapse is not yet reachable [7], a clear demonstration ofdecoherence and the experimental verification of Eq. (29) would be useful on itsown.As a last step, we show that the behavior p [ A, t ] = 1 / ∀ t > τ ∗ is a pe-culiarity of the collapse approach which is impossible if only a unitary evo-lution is taken into account. The proof makes use of the Hamiltonian H of the whole system (particle+slits+environment), for which we assume that h R | H | L i = h L | H | R i = 0, i.e. the full Hamiltonian does not mix the states | L i and | R i . (This is indeed a quite general assumption for the type of problems17 .250.500.751.00 0 2 4 6 8 10 τ∗ τ [a.u.] p[A, ] τ Figure 6: The quantity p [ A, τ ] is plotted as function of τ. The dashed linerepresents the prediction of the unitary evolution of Eq. (23). The solid linerepresents the prediction of the collapse hypothesis of Eq. (33): if the detectionof the screen takes place for τ larger than the critical value τ ∗ , the state hascollapsed to either A or B, therefore p [ A, τ > τ ∗ ] = 1 /
2. Note, we use arbitraryunits. The choice of τ ∗ is also arbitrary and serves to visualize the effect (it isexpected to be much larger in reality).18hat we study: once the particle has gone through the left slit, its wave functionis | L i and stays such (and viceversa for | R i ) . Similarly, in the example of a(photon or neutron) Mach-Zehnder interferometer, after the first beam-splitterthe path is either the lower or the upper and the whole Hamiltonian does notmix them.) It then follows that: | Ψ( t ) i = e − iHt √ | L i | E i + | R i | E i )= 1 √ (cid:0) | L i e − iH L t | E i + | R i e − iH R t | E i (cid:1) (34)where we have expressed | E L ( t ) i = e − iH L t | E i and | E R ( t ) i = e − iH R t | E i byintroducing the Hamiltonians H L = h L | H | L i and H R = h R | H | R i which act inthe subspace of the environment. (These expressions hold because H n | L i | E i = | L i H nL | E i for each n ). The overlap c ( t ) defined in Eq. (24) can be formallyexpressed as c ( t ) = h E L ( t ) | E R ( t ) i = D E (cid:12)(cid:12)(cid:12) e − i ( H R − H L ) t (cid:12)(cid:12)(cid:12) E E . (35)Being H L and H R Hermitian, also H R − H L is such. For a finite number ofdegrees of freedom of the system, the quantity c ( t ) shows a (almost) periodicbehavior and returns (very close) to the initial value 1 in the so-called Poincar´eduration time (which can be very large for large systems). It is then excludedthat c ( t ) vanishes for t > τ ∗ . (At most, it can vanish for certain discrete times,see Sec. 4, but not continuously). Even in the limit of an infinite numberof states, the quantity c ( t ) does not vanish but approaches smoothly zero for t → ∞ . As a last example, we design an ideal setup in which the environment is repre-sented again by a single particle, the idler state (see Sec. 2.3.2). However, weassume now that a time-evolution of the idler state takes place: | Ψ( t ) i = 1 √ | L i | E L ( t ) i + | R i | E R ( t ) i ) , (36)with the ‘environment’ states now expressed in terms of the orthonormal idler-basis {| I i , | I i} . | E L ( t ) i = | I i , (37) | E R ( t ) i = cos( ωt ) | I i + sin( ωt ) | I i . (38)Thus, while | E L ( t ) i = | I i is a constant over time, we assume that | E R ( t ) i rotates in the space spanned by | I i and | I i . Then, we can rewrite | Ψ( t ) i as | Ψ( t ) i = 12 | A i [(1 + cos( ωt )) | I i + sin( ωt ) | I i ]+ 12 | B i [( − ωt )) | I i + sin( ωt ) | I i ] . (39)19 .250.500.751.00 0 2 4 6 8 10 τ [a.u.] p[A, ] τ Figure 7: Quantity p [ A, τ ] as function of τ in the case of entanglement with anidler state according to Eq. (39).The probability p [ A, τ ] is given by p [ A, τ ] = 12 + 12 cos( ωτ ) (40)where τ is the time at which the particle P hits the screen.In conclusion, in a real implementation of this simple idea, it would beinteresting to see the appearance and the disappearance of interference (withboth fringes and antifringes) as function of the time of flight τ , see Fig. 7. Itshould be however stressed that the full interaction Hamiltonian does not acton the idler state alone. Indeed, the corresponding Hamiltonian has the form H = α ( | R i | I i h R | h I | + h.c.) . (41)This is indeed a quite peculiar type of interaction because the idler state rotatesonly if the particle P is in the state | R i (in the language of Sec. 4, it means: H L = 0 , H R = α ( | I i h I | + h.c.) . ). This implies that the spatial trajectory ofboth states | I i and | I i must be the same, otherwise the overlap h E L ( t ) | E R ( t ) i would be an extremely small number and the effect that we have describedwould not take place. We have presented an ideal interference experiment in which we have comparedthe unitary evolution and the existence of a collapse of the wave function. Wehave analyzed the case in which a detector measures the which-way informa-tion and we have shown that the collapse postulate as well as the no-collapse20nitary evolution lead to the same outcome: the disappearance of interferenceon the screen. In the unitary (no-collapse) evolution, this is true only if thestates of the detector are orthogonal. This is surely a very good, but not exact,approximation. It was then possible to describe within the very same Gedanken-experiment two astonishing quantum phenomena: the Elitzur-Vaidman bomband the delayed-choice experiment.We have then turned to a description of the entanglement with the environ-ment. The phenomenon of decoherence ensures that the interference smoothlydisappears. However, as long as the quantum evolution is unitary, it never disap-pears completely. Conversely, the real collapse of the wave function introducesa new kind of dynamics which is not part of the linear Schr¨odinger equation.While the details differ according to which model is chosen [7], the main featuresare similar: a quantum state in which one has a delocalized object (superposi-tion of ‘here’ and ‘there’) is not a stable configuration, but is metastable anddecays to a definite position (either ‘here’ or ‘there’). In conclusion, the collapseand the no-collapse views are intrinsically different, as Fig. 5 shows. At a fun-damental level, the unitary (no-collapse) evolution leads quite naturally to themany worlds interpretation in which also detectors and observers are includedin a superposition (for a different view see the Bohm interpretation [39]).Even if the distinction between the collapse and the no-collapse alternativesis probably still too difficult to be detected at the moment, the demonstrationof decoherence in an experiment with two final states would be an interestingoutcome on its own (see the dashed curve in Fig. 6) . Also a situation in whichan entangled particle is emitted in such a way that an ‘oscillating interference’takes place (see Fig. 7) might be an interesting possibility.A further promising line of research to test the existence of the collapse ofthe wave function is the theoretical and experimental study of unstable quan-tum systems. The non-exponential behavior of the survival probability for shorttimes renders the so-called Zeno and Anti-Zeno effects possible [40, 41, 42, 43]:these are modifications of the survival probability due to the effect of the mea-surement, which have been experimentally observed [44]. The measurement ofan unstable system (for instance, the detection of the decay products) can bemodelled as a series of ideal measurements in which the collapse of the wavefunction occurs, but can also be modelled through a unitary evolution in whichthe wave function of the detector is taken into account and no collapse takesplace [45, 46, 47]. Then, if differences between these types of measurementappear, one can test how a detector is performing a certain measurement [48].Quite remarkably, such effects are not restricted to nonrelativistic QM, but holdpractically unchanged also in the context of relativistic quantum field theory [49]and are therefore applicable in the realm of elementary particles.In conclusions, Quantum Mechanics still awaits for better understanding inthe future. It is surely of primary importance to test the validity of (unitary)standard QM for larger and heavier bodies. In this way the new collapse dy-namics, if existent, may be discovered.
Acknowledgment:
These reflections arise from a series of seminars on21Interpretation and New developments of QM’ and lectures ‘Decays in QM andQFT’, which took place in Frankfurt over the last 4 years. The author thanksFrancesca Sauli, Stefano Lottini, Giuseppe Pagliara, and Giorgio Torrieri foruseful discussions. Stefano Lottini is also aknwoledged for a careful reading ofthe manuscript and for help in the preparation of the figures.
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