1. Introduction

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. [7]).

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].

The problems stemming from assumptions (A1) and (A2) will be analyzed in detail in Sections 3.1 and 3.2.

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 [20]. (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 [13]. (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 [21]. (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 [13]. 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 [35], Omnès [36], 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. [37]). 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 [38]. 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. [34]), 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

The operator is unitary operator known as time development operator [29–31]. It can be easily shown that , where is the identity operator and .

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

With the use of the chain operator for each history we can define branch vectors (also known as chain kets or chain vectors) in [32–35, 38]:

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 [32] 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. [34]).

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 [39]. 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 [13]. 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 [13]. 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).

Figure 1: If the photon goes along path 1 then it always comes out at path 6. Destructively interfering branch vectors at path 5 are shown explicitly because these destructively interfering branch vectors are essential for arising of Afshar’s pseudo-paradox.

Figure 2: If the photon goes along path 2 then it always comes out at path 5. Destructively interfering branch vectors at path 6 are shown explicitly because these destructively interfering branch vectors are essential for arising of Afshar’s pseudo-paradox.

Figure 3: Coherent version of the setup with photon propagating along both paths 1 and 2. None of the interfering branch vectors is erased in order to illustrate the fact that there are 3 branch vectors constructively interfering and one branch vector destructively interfering at each of the detectors located at paths 5 and 6. The question is which two branch vectors from the quadruple set do annihilate each other and which two branch vectors remain? Branch vectors in blue come from path 1, while branch vectors in red come from path 2.

Figure 4: If one infers that there is destructive interference at path 4, then only one mathematical possibility for the output at path 5 and 6 is consistent. There is no which-way information. Branch vectors in blue come from path 1, while branch vectors in red come from path 2.

Figure 5: If one records the which-way information by entangling the photon paths with another degree of freedom there will be no destructive interference of the branch vectors at path 4 and the which-way information claims from the two single-path setups can be applied to this mixed setup also. In such a scenario the photon density matrix is not that of a pure state but mixed one. Red and blue branch vectors denote either photons with orthogonal polarizations (such as and ) or photons propagating at different times ( and ), in either case destructive interference at arm 4 cannot occur.

Figure 6: According to Unruh there is both which-way information in the coherent setup and one might infer existent destructive interference at path 4 if the destructive interference is not measured. Thus according to Unruh one can consistently infer the destructive interference, but not measure it. The antirealist logic goes by extrapolation from single-path setups to coherent double-path setup like this: verify that closing path 2 leads the photon to emerge always at path 6. This means that the photon path 6 is correlated with passage along path 1. Similarly one closes path 1 and verifies that photon passage along path 2 is correlated with emerging photon at path 5. Then according to Unruh the which-way information is guaranteed by the linearity of QM, and only trying to measure the interference by putting obstacle at path 4 destroys the which-way information. In other words one cannot measure both which-way information and destructive interference, yet one can infer the interference provided that it is not measured. This however implies that BS3 can distinguish the past of branch vectors coming along path 3, which is inconsistent with the postulates of standard QM.

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) [26]. 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 [13] 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 [13]. 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” [41], 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 [13].

If one infers the existent interference at path 4 and still claims that there is which-way information (as Unruh did in [13]) 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 [13]. 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 [26] 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 [42], 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 [13] 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 :