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ISRN Mathematical Physics
Volume 2012 (2012), Article ID 327278, 37 pages
Quantum Histories and Quantum Complementarity
Department of Psychiatry and Neurobiology, Graduate School of Medical Science, Kanazawa University, 13-1 Takaramachi, Kanazawa, Ishikawa 920-8641, Japan
Received 6 August 2011; Accepted 27 September 2011
Academic Editor: F. Sugino
Copyright © 2012 Danko Georgiev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We discuss the origin of quantum complementarity using Feynman's sum-over-histories approach to Quantum Mechanics (QM). We analyze two experimental setups introduced by W. G. Unruh and S. Afshar and show how one should consistently calculate the existence of which-way information using interfering quantum histories. Then we discuss the possible problems associated with the notion of which way information and provide an explanation why Englert-Greenberger duality relation cannot be violated experimentally if the standard QM postulates are accepted.
In the early days of quantum mechanics (QM) it was thought that quantum complementarity results from Heisenberg uncertainty principle; however later works have shown that the quantum complementarity could be more fundamental than previously thought and might be enforced via entanglements between the evolving quantum particle and measuring devices [1, 2]. For interferometric (or double slit) setups the quantum complementarity has been formulated in a mathematical form known as Englert-Greenberger duality relation: where stands for visibility of interference fringes and stands for distinguishability of photon paths. If we denote by and the two photon beams passing through each interferometer arm (or each slit) and if we assume that and are pure states (cf. [3, 4]) then for and we will have
Simple arithmetic substitution of (1.2) and (1.3) in (1.1) shows that indeed there is an equality: . Inequality will be achieved if and are mixed states; however in such a case the expressions for and should take into account the degree of coherence within each beam and thus differ from (1.2) and (1.3) (cf. [3, 4]). In the subsequent discussion we will consider setups in which and are pure states therefore for our purpose we could safely use (1.2) and (1.3) without being mathematically inconsistent.
In 2004, Afshar and Cramer announced that Afshar had found an experimental way of showing that the principle of quantum complementarity is wrong [5, 6]. Particularly their claim was that the Englert-Greenberger duality relation is violated in a double-slit lens setup where one may both (1) verify the existent quantum interference pattern by placing wire grid at the dark fringes at the focal plane of the lens and subsequently (2) obtain the which-way information by detecting the photon at the image plane of the lens and finding out through which slit the photon passed. Furthermore, Cramer considered the results of Afshar experiment as an evidence against Copenhagen Interpretation of QM and Bohr’s antirealism (cf. Appendix). Later in 2004, Unruh proposed and analyzed an alternative interferometric experiment, which is equivalent to Afshar’s setup but simpler for comprehension. Regretfully, Unruh defended the principle of complementarity (as well as Bohr’s antirealism) via inconsistent mathematical reasoning, which lead to a heated debate both in Internet and academic journals.
Before we discuss the different viewpoints expressed in the complementarity debate, we would like to point out that under standard QM postulates are understood the unitary evolution of the wavefunction of a given quantum system described by the Schrödinger equation: and the Born rule: according to which the probability density function for observing a quantum system at position and at time is obtained from the square of its wavefunction (cf. ).
All other auxiliary assumptions that make it possible to attribute some physical meaning to the performed calculations should be considered interpretation dependent and could potentially lead to inconsistencies (paradoxes). Two problematic auxiliary assumptions present in the Copenhagen Interpretation of QM and Bohr’s antirealism, which should not be classified as standard QM viewpoint, are the following. (A1) Results or conclusions derived from investigation of two alternative single-slit setups could be extrapolated to a coherent version of the setup in which both slits are open [8–13]. (A2) Introduction of an obstacle (absorber) at a place where the wavefunction of given quantum system is zero () could fundamentally change the results or the conclusions derived from the experimental setup [11–14].
Some of the most notable positions expressed in the complementarity debate are as follows. (P1) Quantum complementarity is violated; there are both which-way information in Afshar’s setup and detectable interference fringes, hence [6, 8–10, 15–17]. Afshar’s setup is fundamentally different from Unruh’s setup. (P2) Quantum complementarity is not violated in Afshar’s setup; it is true that ; however Englert-Greenberger duality relation has not been derived for Afshar’s setup [18, 19]. (P3) Quantum complementarity is not violated in Afshar’s setup, , without wire grid on the photon path there is which-way information in Afshar’s setup; however putting the wire grid at the dark interference fringes (partially) erases the which-way information [11, 12, 14]. (P4) Quantum complementarity is not violated in Afshar’s setup, , without wire grid on the photon path there is which-way information in Afshar’s setup, putting the wire grid simply does not detect interference pattern; a photon detected at the image plane of the lens reveals the which-way information and a photon detected at the focal plane of the lens reveals the interference pattern; however since a photon cannot be detected twice there is no paradox to be solved . (P5) Quantum complementarity is not violated in Afshar’s setup, , without wire grid on the photon path there is which-way information in Afshar’s setup and also there is destructive interference at the dark fringes provided that the interference is only calculated but not measured; however putting the wire grid measures the interference and therefore (partially) erases the which-way information. Afshar’s setup is equivalent to Unruh’s setup . (P6) Quantum complementarity is not violated in Afshar’s setup, , without wire grid on the photon path there is no which-way information in Afshar’s setup, putting the wire grid at the dark interference fringes is irrelevant for the existence of which-way information. Afshar’s setup is completely different from Unruh’s setup . (P7) Quantum complementarity is not violated, , without wire grid on the photon path there is no which-way information in Afshar’s setup, and putting the wire grid at the dark interference fringes is irrelevant for the existence of which-way information [22, 23]. (P8) Quantum complementarity is not violated, , without wire grid on the photon path there is no which-way information in Afshar’s setup, and putting the wire grid at the dark interference fringes is irrelevant for the existence of which-way information. Afshar’s setup is equivalent to Unruh’s setup. The pseudo-paradox requires at least 8 alternative quantum histories and troubles result from inconsistent calculation of interferences among these 8 quantum histories [24–26].
As it can be seen the various positions might agree on some statements, but disagree in the details. In this work, we will try to address each of the positions (P1)–(P8) expressed in the complementarity debate. We will provide complete description of Unruh’s and Afshar’s setups using Feynman’s sum-over-histories approach, and we will show that the quantum complementarity results from the requirement for mathematical consistency of the theory. In Section 2 we will briefly introduce the reader to three important concepts, these of quantum history, history Hilbert space, and branch vectors in standard Hilbert space, in order to be able to tackle the problem of mathematical consistency in the context of QM experiments. In Section 3 we will describe Unruh’s setup and will provide a geometrically intuitive description of the mutually exclusive possible ways in which the 8 quantum histories can interfere among each other. In Section 4 we will provide complete mathematical description of Unruh’s setup using Feynman’s sum over histories. In Section 5 we will introduce Afshar’s setup and will prove the equivalence between Afshar’s setup and Unruh’s setup. In Section 6 we will solve numerically the Fresnel integrals for Afshar’s setup and will show that the interference pattern in coherent setup is present regardless of the presence or absence of wire grid on photon paths. In Section 7 we will compute the visibility and distinguishability for Unruh’s and Afshar’s setups in two mutually exclusive cases: coherent setup versus incoherent setup. Then we will compare the proposed consistent mathematical treatment of those two setups with the inconsistent antirealist solution of the problem advocated by Unruh . In Section 8 we will show that because the philosophical troubles with the notion of which-way information result from the Schrödinger equation, the which-way problem can be formulated even in Bohmian QM where the particle hidden trajectories through the slits can be easily inferred from the so-called Guiding equation (in other words the Guiding equation in Bohmian QM cannot resolve quantum complementarity issues). At the end we will briefly discuss what has been gained from the analysis of Afshar experiment using Feynman’s sum-over-histories approach and will explain why Englert-Greenberger duality relation cannot be violated experimentally if the standard QM postulates are accepted.
2. Quantum Histories
Before we discuss Unruh’s and Afshar’s setups we will introduce several basic concepts, which will be used throughout our exposition. A central concept in the definition of which-way information is the sequence of events or quantum history. The concept of a quantum history originates from Feynman’s path integral (sum-over-histories) formulation of QM [27, 28]. Modern interpretation of QM based on Feynman’s work and utilizing the notion of decoherent quantum histories was further developed by Griffiths [29–32], Hartle [33, 34], Isham , Omnès , and others. In order to avoid confusion we would like to point out that the current work is based upon Feynman’s sum-over-histories approach and does not require decoherent quantum histories.
In order to be able to meaningfully express in mathematical form the possibility for a quantum particle to pass through one slit, then to pass through region with interference fringes and finally to be registered by a detector, we need something more than just writing ordinary state vector in the standard Hilbert space of the usual quantum formulation (cf. ). For example, let us discuss the tossing of a quantum coin which can end up either tails or heads . The probability to obtain each observable outcome (tails or heads) is given by the Born rule , where . Still, however we do not have a vocabulary to talk about a sequence of outcomes obtained at different times . We might want to discuss two different sequences of coin tossing, in which we have obtained, for example, (1) tails, tails, tails, heads, versus (2) tails, heads, tails, heads. Looking only at the last outcome (i.e., the state vector at a given time) we do not have sufficient information to distinguish between sequence (1) and (2). In addition, there is no single observable within the standard Hilbert space , which corresponds to any sequence of coin tosses.
2.1. History Hilbert Space and History Projection Operators
Let us take the closed quantum system to be described by a quantum state in . Each quantum history could be described by giving a sequence of alternatives at a series of times . Alternatives at a moment of time are presented by an exhaustive set of orthogonal projection operators (or simply projectors) , . Each projector satisfies , where denotes the conjugate transpose of . Any nonzero ket generates a one-dimensional subspace , often called a ray or a pure state, consisting of all scalar multiples of , that is to say, the collection of kets of the form , where is any complex number. The projector onto is the dyad:
In the following discussion we will assume that each ket is normalized (which means ), therefore . The symbol for the projector projecting onto the ray generated by is not part of standard Dirac notation, but it is very convenient, and will be used throughout this exposition (cf. [32, page 29]).
Because the statement that a certain quantum history is realized is itself a proposition, it follows that the set of all such quantum histories should possess a lattice structure analogous to the lattice of single-time propositions in standard quantum logic . In particular, a quantum history proposition could be represented by a history projection operator in a new type of Hilbert space called the history Hilbert space defined as a tensor product: where is a copy of the Hilbert space used to describe the system at time and is a variant of the tensor product symbol . We can equally well write but it is helpful to have a distinctive notation (i.e., ) for a tensor product when the factors in it refer to different times , and reserve the symbol for a tensor product of spaces at a single time (cf. [32, page 96]).
If the initial state at of the studied quantum system is in , the history of alternatives in is represented by the corresponding chain of projections called a history projection operator:
Note that the superscripts in (2.3) are labels, not powers. This usage need not cause any confusion, since the square of a projector is the projector itself, and thus there is never any need to raise it to some power.
Although we have abbreviated the whole chain by a single index in the left-hand side of the equation (cf. ), in the following exposition it will be impractical to use Greek letters for denoting different history projection operators , , , and so forth, because we have to memorize what stand for. Therefore it would be much easier if we have each history projection operator written in full notation instead of simply :
In order to complete the sample space we could add one more history: to which could be assigned zero probability because we required that the initial state of the studied quantum system is .
2.2. Chain Operators on the Standard Hilbert Space
The time development of a quantum system in the histories perspective is given by the time-dependent Schrödinger equation (1.4), which is used as a tool to calculate the probabilities of different histories. If we integrate the Schrödinger equation from time to starting from an arbitrary initial state , due to the fact the equation is linear, the dependence of the state at time upon the initial state can be written in the form
With the use of time development operators, for the history in (2.3) can be defined a chain operator (also known as class operator) that operates on the standard Hilbert space : and its adjoint is given by the expression
Notice that the adjoint is formed by replacing each in (2.3) separating from by . Therefore there is a one-to-one mapping between each that is an operator on the history Hilbert space and each that is an operator on the standard Hilbert space [32, page 120]. In the subsequent discussion chain operators could also be written in full notation instead of simply .
It is more natural to use when we refer to each quantum history, whereas it is more effective to use when we calculate the quantum amplitudes associated with each quantum history. However, the main purpose of using both history projection operators and chain operators is to stress on the fact that our arguments could be represented both in and , from which follows that the validity of our results does not depend on the introduction of history Hilbert space and history projection operators .
2.3. Branch Vectors in the Standard Hilbert Space
Due to the one-to-one mapping between and , to each quantum history from the history Hilbert space can be assigned a branch vector in the standard Hilbert space . Insofar the history projection operators in and the branch vectors in represent the same underlying quantum reality; it is not surprising that one can construct one-to-one correspondence between them. Such one-to-one correspondence between and implies that the validity of our results should not depend on the choice of QM interpretation, because branch vectors are defined in and thus present in all QM interpretations.
2.4. Feynman’s Sum-over-Histories versus Decoherent Histories
In the usual consistent histories approach advocated by Griffiths  it is required that one works within a consistent family of quantum histories (also known as decoherent histories). The consistency (decoherence) condition for a set of quantum histories requires that the chain operators and in associated with any two different quantum histories and in the sample space be mutually orthogonal in terms of the operator inner product:
Therefore one is limited to work only with sample spaces of quantum histories for which the consistency (decoherence) condition is fulfilled. According to Griffiths discussing a family of quantum histories which are not decoherent is meaningless [32, page 123].
Noteworthy, the decoherence condition is absent in Feynman’s sum-over-histories approach where the quantum probability amplitude for an event is given by adding together the contributions of all the histories in configuration space leading to the event in question. The contribution of each history to the amplitude is proportional to , where is reduced Planck constant and is the action of that history. The summation over all histories builds the concept of superposition, and thus the possibility of quantum interference, directly into the formulation of the theory. Because thorough elucidation of the origin of Afshar’s pseudo-paradox requires discussion of quantum interferences among a family of 8 non-decoherent quantum histories that could interfere in two mutually exclusive ways, in the following exposition we will adopt Feynman’s original approach. In this latter case the requirement for mathematical consistency should not be confused with the decoherence condition given by (2.10).
2.5. Coarse Graining
The coarse graining in the sum-over-histories approach consists in partitioning the fine-grained particle paths , into an exhaustive set of exclusive classes corresponding to regions of the configuration space , at a sequence of times , . Each quantum history defined in the above manner is coarse-grained because alternatives are not specified at every time but only at some times (cf. ).
In order to clarify the coarse-graining procedure we will provide as an example the construction of a coarse-grained position basis in one dimension. Let us first consider one-dimensional fine-grained wavefunction that is acceptable in QM, namely, single valued, continuous, nowhere infinite and with piecewise continuous first derivative. Such a function can be normalized so that . Since the position is a continuous variable, we should write uncountable number of position kets in rigged Hilbert space . For each position ket we have
Further, it can be seen that
In order to normalize the position kets, we might plug in arbitrary position ket instead of which implies . Thus if it follows that , while for one obtains the -infinity, which is removable after integration [40, pages 149–151]. The complex number is the amplitude to find a particle in a state at position .
With these preliminary notes in mind, for a fine-grained wavefunction confined within the interval : we could introduce a coarse-grained basis , and finite-dimensional coarse-grained Hilbert space such that, for arbitrary position ket in , where is the size of the “grain” in the coarse graining, and the Heaviside theta function is defined as
Thus (2.15) is just a neat way to summarize the following cases:
The quantum amplitudes for each vector in the coarse-grained Hilbert space are computed from the fine-grained wavefunction :
The coarse-grained wavefunction in can be expressed in the coarse-grained position basis as a sum of finite number of elements:
The possible countably infinite number of coarse-grained vectors for or are discarded from the coarse-grained description due to the fact that they contribute to the sum in with amplitudes . This is our rationale for describing the subsequent interferometer setup with a finite coarse-grained position basis. Simply, we do not need to account for the huge number of coarse-grained positions filling the space outside the interferometer arms or filling the space throughout the Universe, because we assume an ideal case where the probability for the photon to be detected outside of the interferometer is zero, or in other words we assume a standard particle in a box description with an infinite potential at the walls of the box (interferometer). The above one-dimensional description can be easily generalized to 3 dimensions.
Because for the construction of the coarse-grained Hilbert space one starts from a fine-grained rigged Hilbert space , the final result is to produce a simplified description of the same physical reality. The construction of the coarse-grained function leads to a reduction in the number of dimensions of the Hilbert space and the coarse-grained wavefunction contains less information compared to the fine-grained wavefunction because the mapping is many to one. Indeed distinct fine-grained wavefunctions , may produce the same coarse-grained wavefunction if they satisfy the equation:
The benefits of the coarse-grained description are that it highlights the important features of the underlying physics and removes the inessential details that may distract or even frustrate the reader.
3. Unruh’s Setup
Unruh proposed a double Mach-Zehnder interferometric setup as a formal (and much easier to discuss) one-to-one model of Afshar’s setup . In Unruh’s setup we arrange 3 half-silvered mirrors (beam splitters) and 4 fully silvered mirrors in such a fashion so that we obtain two Mach-Zehnder interferometers in a sequence (see Figures 1–6). Mathematically the action of the fully silvered mirrors is to reflect an incoming photon at angle with a phase delay of . With the use of Euler’s formula it can be shown that in the complex plane the action of the fully silvered mirror amounts to multiplication by of the incoming quantum amplitude. The action of the half-silvered mirror is to coherently reflect of the incoming photon amplitude with phase delay of and at the same time transmit without phase delay another of the photon amplitude (such a treatment is standard, cf. [24, 25], and acknowledged to be correct by Unruh). The coarse graining of the setup that is proposed here is one interferometer arm and takes into account the events at each fully silvered mirror, whereas Unruh used a coarser coarse graining where he considered just interferometer segments between beam splitters and did not distinguish between states before or after the mirror reflections . The photons that exit from the interferometer arms are assumed to be “detected” by detectors or , which just trap the photon within a QED cavity. This assumption is introduced so that we can describe the process of photon detection with a projector onto path 5 or path 6 (see Figures 1–6 for labeling of the interferometer paths).
In his original exposition Unruh investigates two cases in which there are obstacles (absorbers) either on path 1 or path 2 of the interferometer. Unruh’s intention is to produce two incoherent setups, which are taken together in coherent superposition to reproduce the situation in which both paths are free from obstacles. It can be shown however that Unruh’s intention is to be correctly mathematically modeled by either removing the first beam splitter (Figure 1) or replacing the first beam splitter with fully silvered mirror (Figure 2) . The advantage of the latter proposal is that one can directly make quantum coherent superposition of the two single-path setups, while in Unruh’s original proposal (putting obstacles on photon paths) one has to artificially “eliminate” from the superposition the excited electron states in the obstacles (which result from photon absorption by the obstacles).
In the first setup (Figure 1) we remove beam splitter 1 (BS1) and see that the photon exits at path 6 and is thus detected by detector 1 (). This is because the beam passing via path 1 produces two branches at paths 3 and 4, which destructively interfere at path 5 and constructively interfere at path 6. We will see later that it is the existence of destructive interference at arm 5 that is important for arising of Afshar’s pseudo-paradox.
In the second setup (Figure 2) we replace beam splitter 1 (BS1) with fully silvered mirror which reflects the photon and see that the photon exits at path 5 and is thus detected by detector 2 (). This is because the beam passing via path 2 also produces two branches at paths 3 and 4, which destructively interfere at path 6 and constructively interfere at path 5. Again, we note that the existence of destructive interference at arm 6 is essential for arising of Afshar’s pseudo-paradox.
If we now open both arms (Figure 3) there will be branch vectors (corresponding to 8 alternative quantum histories) which can destructively interfere in two exclusive ways. At each detector there are exactly 3 branch vectors, which are with the same phase and a single branch vector which is out of phase by radians. The question is how we decide which branch vectors cancel out and which branch vectors do not? The answer lies within the claim for presence or absence of destructive interference at arm 4 of the interferometer. Moreover, whether we put obstacles on arm 4 is irrelevant from mathematical viewpoint as we will see in Section 3.2. One can now understand why Afshar’s pseudo-paradox occurs—if the branch vectors at arm 4 are assumed to destructively interfere (see Figure 4), then the destructive interferences at path 5 and path 6 (shown in Figures 1 and 2) cannot be inferred any more.
If there are no labels on the photon paths 1 and 2 (e.g., there are no different polarization filters) straightforward calculations yield destructive interference at path 4 and no which-way information (Figure 4). As explained in [24, 25] one can simply think that the recombined beam is now located at path 3 and the which-way information is erased similarly to a single Mach-Zehnder setup. Then beam splitting of this beam located at path 3 cannot produce any which-way information (Figure 4), and this will be verified by the sum-over-histories calculation in Section 4.
Notice that if there are, say, two different polarization filters on paths 1 and 2 (e.g., right circular and left circular ), due to entanglement of the photon paths 1 and 2 with the photon polarizations there will be no quantum interference at path 4 and there will be which-way information at paths 5 and 6 (Figure 5). In such a case the standard Hilbert space describing the photon state cannot have a single dimension for passage through path 1, , and a single dimension for passage through path 2, , instead one needs a tensor product Hilbert space, with doubled number of dimensions in order to account for the photon polarizations, which are entangled with the photon paths. If the two different polarizations are right circular and left circular , then the newly produced tensor product Hilbert space will have doubled number of dimensions: , , , ,, , . In this higher-dimensional space there is no overlap and no destructive interference at path 4. This is the only mechanism possible by which one could avoid destructive interference and preserve the which-way information, namely, to increase the dimensions of the Hilbert space (via entanglements) in such a manner that there are at least two orthogonal possibilities for passage along , , , , , and .
3.1. Rebuttal of Auxiliary Assumption (A1)
A central feature of both Unruh’s argument  and Afshar’s argument [8–10] is the auxiliary assumption (A1) formulated in Section 1, that is, both authors rely on preliminary analysis of single-slit setups with either slit 1 (path 1) open or slit 2 (path 2) open. Only after they do this preliminary analysis with either one of the two slits open, but not both, Unruh and Afshar extrapolate to setup with both slits open. However it was noted in [24, 25] that the thought experiments with only one slit open implicitly assume occurrence at different thought times. First, at time one discusses what will happen if only slit 1 is open. Then one goes on to discuss another time in which one has only slit 2 open. The central point, which we would like to stress upon, is that these thought times and effectively play the role of polarization filters positioned at the slits, or in other words one mathematically produces a tensor product Hilbert space with doubled number of dimensions , , , ,, , , where the different times and are entangled with the photon paths. As it is accepted in physics, time in the current exposition stands for the reading of a clock, that is, one can easily imagine an entangled clock located near the interferometric setup, which reads two different times for each single-slit case.
In both ways (either inserting different polarization filters or thinking of two different times with only one slit open at a time) one has a mixed setup and mixed photon density matrix. That is why there is no destructive interference at path 4 (Figure 5) and photons from path 1 destructively interfere at path 5, and end at path 6 (), and photons from path 2 destructively interfere at path 6, and end at path 5 ().
However, the logic that holds for mixed setups is not directly applicable to quantum coherent setups in which both paths are open and in which nothing forbids the destructive interference from happening. Mixed setups (incoherently superposed setups) are not equivalent to quantum coherent setups, and these two types of setups can be distinguished by the presence/absence of interference between the two paths or slits.
Surprisingly, for a coherent setup with both paths open Unruh claims that despite the fact one knows that there is interference at path 4, still photons coming at path 5 and 6 do have which-way information from the corresponding slits . According to Unruh, only if there is obstacle on path 4 to measure the interference the which-way information will be lost. Thus Unruh insists that only measurement of the quantum interference destroys the which-way information at paths 5 and 6, while inferring (calculating) that there is interference can be peacefully done along with the claim for existent which-way information. In Unruh’s antirealistic philosophy inferring something is not measuring something (and this presumably is the essence of quantum weirdness). According to Unruh one can know that there is destructive interference along with the which-way information claim, but in this case one cannot measure the interference. The ground for the which-way information claim is the incoherent mixture of the single-path setups described in Figures 1 and 2. From existent which-way information (“which-way correlation” in Unruh’s words) in the single-path setups Unruh infers (extrapolates) that this should hold in coherent setup with both paths open due to the linearity of QM. Some standard textbooks claim that due to linearity of QM the waves superimpose “without permanently destroying or disrupting either wave” , yet such statement seems to ignore the fact that destructive interference means exactly permanent destroying or disrupting of some waves (branch vectors). The linearity of QM cannot justify the usage of the auxiliary assumption (A1), because incoherent quantum superpositions are not equivalent to coherent superpositions. What we proved in [24–26] based on Feynman’s sum-over-histories approach is that quantum complementarity is enforced at the level of addition or subtraction of individual branch vectors in QM and importantly that it matters, which branch vectors one adds and which branch vectors one cancels from the calculation (complete outline of the calculation will be provided in Section 4). Thus, quantum complementarity results from the requirement for mathematical consistency of branch vector addition and subtraction in cases where there are more than two exclusive ways to do the addition/subtraction of branch vectors. The linearity of QM cannot be used to infer conclusions from single-path experiments (incoherent superpositions) to coherent double-path experiments, especially if one has to decide existence/nonexistence of which-way information. Furthermore, our arguments do not violate the superposition principle because the existence of more than two exclusive ways to do the addition/subtraction of branch vectors does not change the value of the quantum amplitudes resulting from the summation. Instead, Afshar’s pseudo-paradox is born exactly because the superposition principle cannot differentiate between the two exclusive ways to do the addition/subtraction of branch vectors.
Here, we defend the viewpoint that knowledge of interference pattern derived from calculation is sufficient to erase the which-way information. In other words, the issue of mathematical consistency is not subject to experimental measurement. If the calculation shows that there is a destructive interference, then we do not need to measure it in order to prove it. Calculated interference is inconsistent with the which-way information claim. Even if we only calculate but do not perform measurement of the interference it will be inconsistent to claim that there is which-way information because the interference is not measured experimentally. Our opinion is thus contrasting Unruh’s claim that we can calculate the interference and still consistently claim that there is which-way information because the interference is not measured experimentally .
If one infers the existent interference at path 4 and still claims that there is which-way information (as Unruh did in ) then the only way is to have beam splitter BS3, which is able to violate QM principles and separate path 1 from path 2, which at path 3 are indistinguishable [24, 25]. In other words BS3 needs to be able to “distinguish” the past of the branch vectors which at the moment of arrival at the beam splitter BS3 are “indistinguishable” in the standard Hilbert space (Figure 6).
One of Unruh’s objections to the above argument was that quantum states propagating along definite trajectories do not exist in any standard QM interpretation . This is wrong, because quantum states propagating along a given history are well-defined mathematical constructions known as branch vectors and could be found in all modern QM texts (cf. [32, 34, 36]). In  it was explained that each quantum history must be represented by a sequence of events (operators) and a branch vector evolving in time along a given quantum history could be represented as a sequence of state vectors. In addition, Unruh claimed that which-way information is present at the detectors even though we know destructive interference occurs at path 4. Such a claim implicitly assumes at least 3 events separated in time: (1) passage through one of the interferometer arms in Unruh’s setup, (2) destructive interference at an intermediate time point at path 4, and (3) detection at the detectors. Therefore if Unruh is to be consistent, he should not talk only about state vectors , instead the discussion should be based on the notions of history projection operators in history Hilbert space or chain operators and branch vectors in standard Hilbert space, as explained in Section 2.
3.2. Rebuttal of Auxiliary Assumption (A2)
Lastly, we will show that putting obstacles at path 4 could not change the which-way information of the setup [24–26]. This is necessary because all researchers who believe in (A1) are eventually forced to accept (A2) in order to cover up lurking paradoxes.
The first reason in favor of the conclusion that putting obstacles at path 4 could not change the which-way information of the setup is that mathematical theorems follow analytically from the axioms, and they are true if the axioms are true. Because the calculation of the interference pattern is a mathematical theorem, it must be true if the axioms of standard QM are true. The truth (or validity) of calculated interference pattern cannot be further proved by measurement. Indeed, if the measurement does not agree with the calculation and one believes in the truth of the axioms of standard QM, then one should conclude that either the measurement apparatus is broken or corrupt or the noise in the measurement was intolerably high.
The second reason is a corollary of Reininger’s negative measurement in QM. Reininger’s Negative Measurement. If one observes that a quantum particle is absent at place where the wavefunction amplitude was nonzero, the observation collapses the wavefunction, so that now the quantum particle is with probability 1 at all other locations, except the observed empty one. Corollary. It is possible to perform a nondestructive measurement without collapsing the quantum particle wavefunction only in regions where the quantum particle wavefunction amplitude is zero, because both before and after the measurement the quantum particle is with probability 1 at all other locations, except the observed empty one, that is, there has been no change of the quantum state. Thus, QM permits measurements of the quantum particle, which do not collapse the quantum particle wavefunction, and these could be done where the quantum particle wavefunction amplitude is zero.
To measure the wavefunction where the amplitude is zero is, for example, somewhere outside of the interferometer behind mirrors 1–4. If we put obstacle outside of the interferometer we do not expect to change the which-way information of the photon. It is just plain obvious. The fact that the space of path 4 happens to be inside the interferometer does not make it privileged in this respect. If the wavefunction there is zero, putting obstacle does not collapse the wavefunction and does not change anything to the which-way information. Such a philosophical argument might be met with suspicion by antirealists, but it is necessarily (mathematically) true. In the next section we will prove rigorously that the which-way information claims follow from the mathematics of the setup and particularly depends on whether there is inferred destructive interference at path 4 or not. This effectively rebuts the auxiliary assumption (A2) formulated in Section 1.
4. Complementarity in Unruh’s Setup
Since we have 3 beam splitters in Unruh’s setup it is easy to be seen that there are exactly alternative quantum histories for a photon to travel from to one of the two detectors or . We assume that the detectors are located within the corresponding path 6 and path 5 and trap the photon within a QED cavity. Thus the detection itself will not require separate projector different from or , since the coarse graining is such that and . Using Unruh’s original coarse graining  with an orthonormal position basis , we can write the 8 possible quantum histories as history projection operators in history Hilbert space :
If we work in a finer coarse graining, where we consider each interferometer arm, we can split each position vector into two halves and (before and after the mirror reflection, resp.), so that we obtain higher-dimensional Hilbert spaces describing the system at each time , which contain basis vectors defined by 3-dimensional generalization of (2.15). From the newly defined finer coarse-grained Hilbert spaces we can construct new history Hilbert space , where we can discuss finer coarse-grained histories:
Mathematically the different coarse grainings require introduction of different standard Hilbert spaces to describe the system at each time and require different history Hilbert spaces. However since the different coarse grainings refer to the same underlying quantum reality and there is a one-to-one mapping between histories with different coarse grainings (e.g., ), hereafter we will refer to such corresponding histories with the same name (e.g., we will call both and simply “”). This will not cause confusion because the use of different coarse-grainings does not lead to differences in the calculated quantum amplitudes for the different quantum histories.
In order to easily calculate the quantum amplitudes contributed by each of the 8 quantum histories, we can use the chain operators given by (2.7) instead of the expressions for . If we consider times for which the photon travels: () from the laser apparatus to BS1, () from BS1 to M1 or M2, () from M1 or M2 to BS2, () from BS2 to M3 or M4, () from M3 or M4 to BS3, and () from BS3 to or , we can write time development operators constructed from mirror reflection or beam splitting as follows:
After applying the rule given by (2.7) and after simplification we obtain the following chain operators in :
In (4.4)–(4.11) we use a coarse graining such that each path is divided into and halves, respectively before and after the corresponding mirror, for example, path 1 is considered as path before the mirror M1 and as path after the mirror M1. It is easy to verify that the above chain operators indeed propagate an input along the specified histories. One has to read the operators from right to left and keep in mind that the operator transforms input into output , that is, since . However it might be necessary to prove that these chain operators are indeed equivalent to the chain operators constructed by (2.7) for each history given by (2.3).
Full derivation will be provided only for the first chain operator given by (4.4):
Here we remind that instead of using a single Greek subscript for each history projection operator (quantum history) we use chain of subscripts (e.g., describes the history of a photon passing via path 2, then path 4 and emerging at path 6). To the history can be ascribed branch vector .
One can multiply the operators within each chain operator and obtain just a single operator that transforms the input state directly into output state at or . The obtained operator however describes photon “jump” from to or and therefore cannot capture the concept of photon path (history). The latter operator could be useful only if we are interested to get the final output of the operator without being interested in intermediate events such as passage along the interferometer arms.
In the single-path experiment with beam splitter BS1 removed (Figure 1) the branch vectors travel along path 1 and there is destructive interference of branch vectors and at and constructive interference of branch vectors and at :
Similarly in the single-path experiment with beam splitter BS1 replaced by fully silvered mirror (Figure 2) the branch vectors travel along path 2 and there is destructive interference of branch vectors and at and constructive interference of branch vectors and at :
In the quantum coherent version of the setup without having polarization filters on paths 1 and 2 (Figure 4), the destructive interference at path 4 implies
Because of destructive interference at path 4, the constructive interference at both detectors and is due to branch vectors coming from both slits. The calculated interference (even without measuring it by wire grid) erases the which-way information exactly as the complementarity principle requires.
If we put right or left circular polarization filters ( or ) on path 1 and path 2 labeling the which-way information (Figure 5), then the equations for each single slit setup hold:
Due to the which-way information labels however, there is no interference at path 4 (i.e., there is nonzero probability to detect photon at path 4). What this analysis shows is that it is impossible both to have satisfied destructive quantum interference at path 4 and to have which-way information, which requires destructive self-interferences of branch vectors coming from path 1 at the opposite detector (path 5) and from path 2 at the opposite detector (path 6). For example, the branch vector cannot annihilate both vectors and at once and this follows from the fact that
This exhausts the proof of the statement that there is no which-way information if one infers (calculates) quantum interference at path 4 and decides the Georgiev-Unruh debate [13, 24–26]. Knowing (mathematically proving) that there is destructive interference destroys the which-way information even without measuring the destructive interference. Mathematical knowledge (theorems) follows via rigorous deduction from the QM postulates (axioms). Thus Unruh ignored the fact that knowledge can be not only synthetic (i.e., experimental), but can be also analytic (true by virtue of mathematical proof and therefore not subject to experimental testing once one accepts the standard QM postulates).
5. Afshar’s Setup
In Afshar’s setup, light generated by a laser passes through two closely spaced circular pinholes. After the dual pinholes, a lens refocuses the light so that each image of a pinhole is received by a separate photodetector. A wire grid could be placed at the dark fringes before the lens (or at the focal plane of the lens after the lens), where the photon wavefunction is zero due to destructive interference. The corollary from the Reininger’s negative measurement however states that placing the wire grid at a place where the photon wavefunction is zero cannot alter the photon wavefunction and cannot change the results from the setup. Therefore, we will consider only cases without wire grid on the photon path.
Considering a mixture of single-pinhole setups Afshar argues that a photon that goes through pinhole 1 impinges only on detector and, similarly, if it goes through pinhole 2 impinges only on detector (Figures 7–9). Exactly as in Unruh’s setup, Afshar investigates a statistical mixture of single-pinhole setups and after that draws nonsequitur conclusions for the quantum coherent setup with both pinholes open. According to Afshar, there is an one-to-one correspondence between pinholes and the corresponding images even when the light coherently passes through both pinholes (notice the complete analogy with Unruh’s setup). While in classical optics this is a straightforward conclusion, in quantum coherent setups we will shortly prove that each image of a pinhole in the coherent dual pinhole setup is counterintuitively assembled by light coming from both pinholes at once.
First, we will explain generally why Afshar’s setup is equivalent to Unruh’s setup. In Afshar’s setup we have two pinholes, a lens, and detectors that record photons streaming away from the pinhole images created at the image plane of the lens. If one opens only pinhole 1 the light will go only to image 1, and if one opens only pinhole 2 the light will go only to image 2. One may, analogously to Unruh’s setup, inconsistently postulate which-way information in the quantum coherent setup. However, one should note that, in the single pinhole experiments, at the image plane of the lens the zero light intensity outside the central Airy disc of the pinhole image is a result of destructive quantum interference. Thus in the case of open pinhole 1 the branch vectors from pinhole 1 will constructively interfere at image 1, whereas there will be destructively interfering branch vectors contributed by pinhole 1 at image 2 because image 2 resides outside the central Airy maximum of image 1. The absolute value for each of these destructively interfering branch vectors contributed by pinhole 1 at image 2 will be further calculated to be exactly when both pinholes are open or if only pinhole 1 is open (the difference is due to different normalization procedures for the single- and double-pinhole setups). The situation with only pinhole 1 open is therefore analogous to the case in which photons travel only along path 1 in Unruh’s setup (Figure 1). Similar argument shows that the situation with only pinhole 2 open is analogous to the case in which photons travel only along path 2 in Unruh’s setup (Figure 2). However, if both pinholes are open and some of the branch vectors coming from pinhole 2 cross-interfere with branch vectors coming from pinhole 1 at the dark fringes at the focal plane of the lens, there will remain a contribution by pinhole 2 at image 1 that will compensate exactly the decrease in the quantum probability amplitude contributed by pinhole 1 at image 1 caused by the destructive interference at the dark fringes. Thus arises the very same problem as in Unruh’s setup: how do we decide which branch vectors will annihilate, and which will remain to contribute to the detected image intensity? If one postulates the existent interference at the focal plane of the lens then the annihilation between and at the dark fringes (where denotes branch vectors passing through pinhole 1 and denotes branch vectors passing through pinhole 2) will be equivalent to the interference at path 4 of Unruh’s setup, and the final observed intensities at the detectors cannot be claimed to come only from one of the pinholes. Instead the branch vectors constructively interfering at the bright fringes come from both pinholes 1 and 2 and go to both images 1 and 2. In the case with both pinholes open there is no which-way information in Afshar experiment. Counterintuitively each image of a pinhole is assembled from light coming by half from both pinholes (passing through the bright fringes).
Afshar argued that there is a major difference between Unruh’s setup and Afshar’s setup and compared “Unruh’s experiment to one in which the wings are removed from an airplane. The experimenter might find that a wingless airplane cannot fly, but such an experiment would not prove anything about the flight capabilities of an intact plane.” In the following exposition however we will prove mathematically that Afshar’s thesis is wrong and that Unruh’s and Afshar’s setups are indeed equivalent. An exact calculation for Afshar’s setup is adduced by Qureshi for spin particle . Qureshi shows that the quantum state at the overlap region where the dark interference fringes occur can be written as where is the mass of the particle, , , is the amplitude contribution from pinhole 1, is the amplitude contribution from pinhole 2, is the width of the wave packets, and is the slit separation.
For Afshar’s setup so the sinh terms cancel out at the dark fringes and what is left at the bright fringes is
If a lens is used, after the interference has occurred, to direct the part into detector and the part into detector , one easily sees that the branch vectors from each slit evolve into a superposition of two parts that go to both detectors.
The branch vectors that will be responsible for which-way information in mixed setups and make possible the one-to-one correspondence , are hidden in the erased sinh terms. If we express the hyperbolic functions with the help of exponential ones, namely, and , after simple algebraic work one may recover the four zeroed sinh components at the dark fringes in the form
Note that the absolute value of the coefficients in front of each branch vector in (5.2) and (5.3) (going from one of the slits to bright or dark fringe and then to one of the detectors) becomes exactly as obtained in the analysis of Unruh’s setup [24–26].
If the eight interfering branch vectors from Qureshi’s calculation are denoted as , where arise from the cosh terms and arise from the sinh terms, then the one-to-one mapping with the eight branch vectors defined in Section 4 is
Here each is given with the output state only; however it should be understood as a branch vector propagating along certain quantum history and hence is mathematically defined by a sequence of states. This is implied in the requirement for or and sinh or cosh origin of each . Purely algebraic reasoning (e.g., , etc.) treats each only as a complex number  and fails to grasp the quantum history concept defined in Section 2. Treating each as a branch vector associated with a certain quantum history explicitly verifies that and terms in Qureshi’s calculation have the same meaning as path 1 and path 2 in Unruh’s setup; cosh and sinh terms have the meaning of path 3 and path 4, and and terms have the meaning of detection at or .
6. Fresnel Diffraction Integrals for Coherent and Incoherent Afshar’s Setups
In the previous sections we have presented a general description of Afshar’s setup and have shown its equivalence with Unruh’s setup. Furthermore, we have resolved the pseudo-paradox using a thorough mathematical description using Feynman’s sum over histories. Here, we will provide exact description of Afshar’s setup using Fresnel diffraction integrals and will calculate the resulting probability density functions using computer calculations with Wolfram’s Mathematica 7. The rationale is to address some of the misconceptions propagated in the work of Drezet, who defended the complementarity principle using a trivially true fact that one cannot detect the photon twice . In Drezet’s opinion if one detects a photon at the image plane, one cannot infer for the same photon what the probability density function was at the focal plane of the lens. Further, Drezet gives examples of various wavefunctions at the focal plane, which do have interference minima and argues that one needs the full information at the focal plane in order to discriminate between the various alternatives. Such a naive argument is nevertheless erroneous. Since one has the axioms of standard QM as given knowledge, the issue of Unruh’s and Afshar’s arguments is not to experimentally prove/disprove the axioms of standard QM. Instead the debate is focused on the question whether the principle of complementarity fails provided that the standard axioms of QM are true.
If one accepts the standard QM axioms as true then Drezet argument appears to be a straw man argument. One can calculate the photon probability density functions both at the focal plane and at the image plane using Fresnel diffraction integrals. Moreover, one can argue that some of the wavefunctions provided by Drezet for the focal plane of the lens will not evolve into two resolved peaks at the image plane (they are not solutions of the Fresnel diffraction integrals). Thus they are ruled out a priori due to theoretical considerations (i.e., the geometry of the lens setup), and no measurement is needed. Instead, one should be concerned with the consistency between the calculated probability density functions and the inferred which-way information claims. The fact that the calculated probability density functions will be exactly the ones that could be measured follows directly from the premise that QM axioms are true. Therefore further discussion on the possible discrepancy between calculated and measured probability density functions is relevant only if one wants to disprove standard QM as adequate description of the world.
After clarifying that all philosophical issues on the complementarity debate presume that standard QM axioms are true, we will go on and calculate the photon probability density functions for coherent or incoherent double-slit setups. Using the results from the computer calculations we will illustrate the which-way information claims and link them with the quantum histories discussed in Sections 4 and 5.
The photon quantum probability amplitude behind a slit (aperture) could be calculated with the use of Huygens-Fresnel principle: where the distance , the wave number , , are the coordinates at the plane of the slit, is the aperture function (describing the transmission through the slit), are the coordinates at the detection plane, and is the distance behind the slit. The previous expression is quite general and is based on the approximation inherent in the scalar theory and the assumption (cf. ).
For the terms in the exponent we can further approximate while for the denominator we can use just :
With the use of the Heaviside theta function defined by (2.16) we can compactly express for a double slit as follows: where both slits extend from to along -axis (height), and slit 1 extends from to and slit 2 extends from to along -axis (width). Essentially this is just a neat way to say that the function for any region within one of the slits and outside the slits.
The photon probability density function was numerically calculated according to the Born rule using Wolfram’s Mathematica 7. In our calculations we have used laser emitting photons with nm and two square pinholes (with side of 250 μm) separated by distance of 2 mm. Further we have positioned the slit at a distance of from a convex lens with focal length m. The image plane will thus be also located at a distance behind the lens. The parameters of the setup closely resemble the setup reported by Afshar [8–10].
Here we report the calculation of photon probability density function as the photon passes through the double slit, refracts in the lens, and reproduces the double-slit image at the image plane in two scenarios: (1) coherent setup—there are no polarization filters at the slits—and (2) incoherent setup—there are different polarization filters at each slit (e.g., and ).
For the coherent setup the Born rule requires that we add each slit wavefunction and then square the sum in order to obtain the probability density function: For the wavefunction in the coherent setup we obtain the general expression given in (6.3), which can be also written as where
The probability to find a photon either at image 1′ or at image 2′ cannot be ascribed to the contribution of branch vectors coming from only one of the slits (Figure 7). Instead we have equal contributions from branch vectors passing through both slits and a bright fringe in accordance with the analysis presented in Section 5. Because the quantum histories that pass through a dark fringe cancel each other, there is no one-to-one correspondence between slits 1 or 2 and images 1′ or 2′.
In the incoherent setup if we put different polarization filters at each slit, we will entangle the photon path through a slit with the photon polarization. Thus we will have new higher dimensional Hilbert space (as discussed in Section 3), and in the region behind the slit the photon amplitudes passing through each slit will not interfere. For the incoherent setup the Born rule requires that we square the slit wavefunctions before we add them:
Each slit wavefunction is given by
The results for the incoherent setup are shown in Figure 8. There are no interference maxima or minima behind the slits or the focal plane of the lens. If we plot the evolution of the probabilities due to each polarized branch vector separately, we can see that each branch vector goes from slit 1 or slit 2 to the corresponding image 1′ or image 2′ and there is complete which-way information (Figure 9). Actually the plot shows the resultant squared sum of two branch vectors for each slit: one branch vector that corresponds to region where dark fringe occurs in the coherent setup, and one branch vector that corresponds to region where bright fringe occurs in the coherent setup. The destructively interfering branch vectors from each slit at the opposite image are not shown, analogously to Figure 5.
After calculation of the probability density functions for the coherent and incoherent setups, we see that at the image plane there could be formed two types of images. In incoherent setup the images at the image plane contain which-way information; however there is no interference before or after the lens (Figures 8 and 9). In contrast, in coherent setup there is interference before and after the lens and the images do not contain which-way information. We have calculated that the interference is there due to standard QM postulates, and we have shown that this interference implies that the images at the image plane are assembled from branch vectors coming from both slits. The central issue is that the erasure of which-way information happens even without inserting wire grid in front of the photons. We have proved this rigorously using Feynman’s sum-over-histories approach. It is equivalent to standard QM formalism, but has the advantage to visualize things so that they can be explained even to a reader who does not want to follow in detail the mathematical equations but who has some geometrical intuition.
Here we point out that other commentators on Afshar’s setup claimed that only inserting the wire grid erases the which-way information through diffraction, otherwise without wire grid there is which-way information at the image [11, 12, 14]. Furthermore, they construct various arguments in order to explain why there is no interference at the focal plane (i.e., why the visibility of the interference is ). Here we have proved that the interference in the coherent setup is a consequence of the standard QM axioms and that even without wire grids on the photon path exactly as Afshar insists. However contrary to Afshar’s beliefs, the mere fact that there is interference (no matter if it is measured or not) is sufficient to erase the which-way information. The mechanism is not associated with putative diffraction from the wire grid that one needs to detect the interference, instead the mechanism is inherent in the way one sums and annihilates the branch vectors coming from the two slits.
7. Computing Visibility and Distinguishability
We have shown that a proper understanding of QM cannot be reached by relying on classical intuition such as auxiliary assumption (A1). It is exactly the case that mathematically one cannot extrapolate conclusions from incoherent setups to coherent setups even in cases when the observable intensities did not differ between the incoherent and the coherent setup. In incoherent setups the thing that does not happen is the cross-interference of branch vectors passing through the first slit with branch vectors passing through the second slit, which matters if one makes which-way information claims. Therefore to start from existent which-way information in coherent version of Afshar’s setup is to start from inconsistent position and then try to defend the principle of complementarity with fallacious arguments [12–14, 18–20]. Whatever the details of the arguments are, they are all mistaken for the fact that they accept the existence of which-way information in the coherent lens setup with no wire grid on the photon path.
One of the most interesting things in the debate is the proper computation of the distinguishability and visibility in the setup. Afshar claimed he has violated the Englert-Greenberger duality relation . Because and are pure states in Afshar’s and Unruh’s setups, visibility and distinguishability are defined by (1.2) and (1.3). Simple arithmetic work gives us
Since the duality relation is a mathematically true statement (theorem) then it cannot be disproved by experiment and certainly means that Afshar’s arguments, through which he violates the duality relation, are inconsistent.
From this does not follow that any argument that defends complementarity is correct. The wrong proof of correct theorem is still wrong and even worse. Kastner went even further and agreed that but insisted that we are summing from two different planes—focal plane and image plane [18, 19]. In contrast, the current work clearly shows that if and at the focal plane of the lens, then the same and holds at the image plane of the lens, and one can never get .
Indeed the calculation of and depends on distinguishability of the states and and the existence of interference pattern in case of indistinguishable states and . The calculation of and in Unruh’s and Afshar’s setup is different for quantum coherent setup (indistinguishable states and ) and incoherent (mixed state) setup (distinguishable states and ).
In both coherent and incoherent setups one can consider , , , at (keep in mind that for Afshar’s setup subscripts 1 and 2 denote passage through slits 1 and 2, subscripts 3 and 4 denote passage through bright fringe or dark fringe, and subscripts 5 and 6 denote detection at or ). The most tricky part of the computation is to be aware that three histories , , and contribute exactly amplitude to detector and only contributes . In the coherent setup the amplitude contributed by the history can be incorrectly attributed to the history and hence one can erroneously compute and instead of the correct and . The possible error for the coherent setup is easy to make if we forget that destructive interference occurs at path 4 and annihilates with . In incoherent setup there is no interference at path 4, therefore and annihilate at path 6, and only vectors and from slit 1 contribute to the final amplitude at .
Similarly, in both coherent and incoherent setups one can consider , , , at . In coherent setup annihilates with at path 4 and both and contribute to the amplitude left at . In incoherent setup and annihilate at path 5, and only vectors and from slit 2 contribute to the final amplitude at .
With the above remarks in mind, the correct calculation of and is straightforward. In incoherent setups it is exactly and . At detector we have and and at detector we have and . In coherent setups the existence of interference in the overlap region makes and for all times after the interference has occurred. At detector we have and , and at detector we have and . This happens because at the dark fringes some of the 8 branch vectors annihilate each other; however these annihilated branch vectors are needed in order to obtain which-way information. It is now clear why even though there are no wire grids on photon paths, the which-way information is erased. Unruh wrongly believed that BS3 or the lens can “undo” the interference . Instead, if interference has occurred it cannot be “undone.” One can restore only the appearance of the two-slit image (the probability density functions at the image plane might look similar in coherent and incoherent setups) but cannot restore the which-way information.
The reader is referred to  for a typical example how for coherent setup instead of the correct and one starts with and inverted, that is, and . Then one needs to “invent” some arguments to invert and values in case when there are wires on the photon paths.
Afshar’s pseudo-paradox results from quantum coherent overlap of 2 branch vectors from one path that destructively self-interferes with 2 branch vectors from another path that constructively self-interferes [24–26]. The smallest possible number of alternative histories for arising of such “paradox” is probably and the proof is purely combinatorial. Therefore Unruh’s setup is the simplest possible way to formalize Afshar’s pseudo-paradox. In Afshar’s setup there are infinite number of quantum histories and at the first glance the discussion of which-way information may seem more complicated. The solution is to use very coarse coarse graining leaving only 8 quantum histories and then to construct one-to-one mapping between branch vectors in Unruh’s setup and branch vectors in Afshar’s setup as was done in Section 5.
8. Bohmian QM and Complementarity
At the end we would like to point out that the problems with the existence of which-way information can be also formulated even in Bohmian QM, although in this case the problematic statement should not be formulated as related to the photon hidden trajectories, which are calculated from (8.1). In Bohmian QM the state of a system of particles is described by its wavefunction , a complex (or spinor) valued function on the space of possible configurations of the system, together with its actual configuration defined by the actual positions of its particles (the actual positions are hidden). The theory is then defined by two evolution equations.
The first one is the Schrödinger equation (1.4) for , where is the nonrelativistic (Schrödinger) Hamiltonian, containing the masses of the particles and a potential energy term.
The second one is called the Guiding Equation: for , the simplest first-order evolution equation for the positions of the particles that is compatible with the Galilean (and time reversal) covariance of the Schrödinger evolution [44, 45].
The essence of the above equations is that one first needs to solve the Schrödinger equation and then obtain the probability distribution using the Born rule (e.g., as we did in Figure 7 for coherent version of Afshar’s setup). Only after having this information one can plug in the data into the Guiding equation (see the occurrence of the Born rule in the denominator). In the coherent setup the plotted hidden trajectories will have to obey the condition that they do not cross each other, and therefore as correctly described by Drezet the hidden trajectories of a photon passing through slit 1 will always go to the opposite detector , and the hidden trajectories of a photon passing slit 2 will always go to the opposite detector .
However we can easily reformulate the main problem so that it has nothing to do with the hidden trajectories, that is, we may ask through which slit pass the branch vectors that contribute to the Bohmian quantum potentials at each detector? For example, suppose that Unruh was correct and each quantum amplitude at one of the detectors was contributed only from branch vectors coming from the corresponding slit. Then in Bohmian QM it would appear that the photon passing through slit 1 is guided solely by the Bohmian quantum potential contributed by branch vectors coming from slit 2 at at the moment when the photon is absorbed by , and analogously it would appear that the photon passing through slit 2 is guided solely by the Bohmian quantum potential contributed by branch vectors coming from slit 1 at at the moment when the photon is absorbed by . The original problem that leads to the pseudo-paradox is thus present also in Bohmian QM: which branch vectors are responsible for the photon to be able to arrive at or ? If, for example, only was present at then the photon guided through slit 2 would arrive at due to the fact that was present at . In such case the presence of would be the only cause for the photon to enter . The Guiding equation did not solve the problem; just we had to formulate the problem as related to the Bohmian quantum potential and not related to the hidden trajectories. Contrary to Drezet’s beliefs (cf. ), the problem is not changed at all in Bohmian QM, what is changed is the language that we use in order to formulate the problem.
Thus the essence of the debate is solely based on studying the solutions of the Schrödinger equation and the way we manipulate the branch vectors. It is irrelevant whether we will add Guiding equation to the QM formalism or not, hence Drezet’s discussion on Bohmian QM did not help clarify anything. Moreover, if we could derive a moral, it is that in order to be mathematically rigorous we should always be careful if somewhere along the argument we have not changed fundamentally the meaning of what is understood as a photon path.
In this work we have concisely summarized all notable positions discussing the principle of complementarity in Unruh’s and Afshar’s setups. Then we have provided complete mathematical description of those two setups using Feynman’s sum-over-histories approach. A major result that we have obtained is that the image plane of a lens does not always provide the which-way information, contrary to widespread belief in the opposite (cf. [13, 14, 20, 46]). If the photon passes coherently through both slits, the destructive interference of branch vectors at the focal plane of the lens erases the which-way information. With the use of Feynman’s sum-over-histories approach we have shown that the principle of complementarity is derived from the requirement for mathematical consistency in the process of summation/annihilation of branch vectors. A photon that has pure state density matrix in coherent double-slit setup and forms interference pattern at the focal plane of the lens, has the corresponding which-way information erased (branch vectors from both slits contribute half the intensity at each detector at the image plane). In contrast, a photon that has mixed state density matrix in incoherent double-slit setup (with different polarization filters at the slits) does not form interference at the focal plane of the lens and keeps the which-way information from the slits. This implies that , even if we consider measurements of identically prepared photons both at the focal plane and at the image plane of the lens.
Indeed, philosophically it should not matter whether we detect at the image plane or at the focal plane of the lens. Let us consider Drezet’s argument that we cannot detect a photon twice in the context of single-photon double-slit experiment. If one does not believe in a principle known as contrafactual definiteness, then there is no reason for one to believe that each photon detected at the focal plane of a lens forms interference pattern. A single photon forms a spot at the photographic film and not interference pattern. In order to obtain interference pattern one needs to run many single-photon trials and detect many photons prepared in the same way. Only after that using the axioms of standard QM one could argue that all these photons were in the same quantum state and each formed interference pattern individually. This assumes contrafactual definiteness because even if a single photon is detected as a spo