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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 [

Simple arithmetic substitution of (

In 2004, Afshar and Cramer announced that Afshar had found an experimental way of showing that the principle of quantum complementarity is wrong [

Before we discuss the different viewpoints expressed in the complementarity debate, we would like to point out that under

All other

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 [

Introduction of an obstacle (absorber) at a place where the wavefunction of given quantum system is zero (

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

Some of the most notable positions expressed in the complementarity debate are as follows.

Quantum complementarity is violated; there are both which-way information in Afshar’s setup and detectable interference fringes, hence

Quantum complementarity is not violated in Afshar’s setup; it is true that

Quantum complementarity is not violated in Afshar’s setup,

Quantum complementarity is not violated in Afshar’s setup,

Quantum complementarity is not violated in Afshar’s setup,

Quantum complementarity is not violated in Afshar’s setup,

Quantum complementarity is not violated,

Quantum complementarity is not violated,

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

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

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

Let us take the closed quantum system to be described by a quantum state

In the following discussion we will assume that each ket

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 [

If the initial state at

Note that the superscripts in (

Although we have abbreviated the whole chain

In order to complete the sample space we could add one more history:

The time development of a quantum system in the histories perspective is given by the time-dependent Schrödinger equation (

The operator

With the use of time development operators, for the history

Notice that the adjoint is formed by replacing each

It is more natural to use

With the use of the

Due to the one-to-one mapping between

In the usual

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

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

The

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

Further, it can be seen that

In order to normalize the position kets, we might plug in arbitrary position ket

With these preliminary notes in mind, for a

Thus (

The quantum amplitudes

The

The possible countably infinite number of coarse-grained vectors for

Because for the construction of the coarse-grained Hilbert space

The benefits of the coarse-grained description are that

Unruh proposed a double Mach-Zehnder interferometric setup as a formal (and much easier to discuss) one-to-one model of Afshar’s setup [

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.

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.

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.

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.

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

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

In the first setup (Figure

In the second setup (Figure

If we now open both arms (Figure

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

Notice that if there are, say, two different polarization filters on paths 1 and 2 (e.g., right circular

A central feature of both Unruh’s argument [

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

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

Here, we defend the viewpoint that

If one infers the existent interference at path 4 and still claims that there is which-way information (as Unruh did in [

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 [

Lastly, we will show that putting obstacles at path 4 could not change the which-way information of the setup [

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

The second reason is a corollary of Reininger’s negative measurement in QM.

To measure the wavefunction where the amplitude is

Since we have 3 beam splitters in Unruh’s setup it is easy to be seen that there are exactly

If we work in a finer coarse graining, where we consider each interferometer arm, we can split each position vector

Mathematically the different coarse grainings require introduction of different standard Hilbert spaces to describe the system at each time

In order to easily calculate the quantum amplitudes contributed by each of the 8 quantum histories, we can use the chain operators

After applying the rule given by (

In (

Full derivation will be provided only for the first chain operator given by (

Substitution of the time development operators given by (

Further substitution of (

Since we are investigating setup in which we initially plug in the pure state

Here we remind that instead of using a single Greek subscript

One can multiply the operators within each chain operator

In the single-path experiment with beam splitter BS1 removed (Figure

Similarly in the single-path experiment with beam splitter BS1 replaced by fully silvered mirror (Figure

In the quantum coherent version of the setup without having polarization filters on paths 1 and 2 (Figure

Because of destructive interference at path 4, the constructive interference at both detectors

If we put right or left circular polarization filters (

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

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 [

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

Considering a mixture of single-pinhole setups Afshar argues that a photon that goes through

Action of a lens in a dual-slit setup—slits 1 and 2 create a two-slit image

Action of a lens in a dual-slit setup with different polarization filters (

Action of a lens in a dual-slit setup with different polarization filters (

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

Afshar argued that there is a major difference between Unruh’s setup and Afshar’s setup and compared

For Afshar’s setup

If a lens is used, after the interference has occurred, to direct the

The branch vectors that will be responsible for which-way information in

Note that the absolute value of the coefficients in front of each

If the eight interfering branch vectors from Qureshi’s calculation are denoted as

Here each

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

If one accepts the standard QM axioms as

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

The photon quantum probability amplitude behind a slit (aperture) could be calculated with the use of Huygens-Fresnel principle:

For the terms in the exponent we can further approximate

With the use of the Heaviside theta function

The photon

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.,

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:

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

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

Each slit wavefunction is given by

The results for the incoherent setup are shown in Figure

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

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 [

We have shown that a proper understanding of QM cannot be reached by relying on classical intuition such as

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

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.

Indeed the calculation of

In both coherent and incoherent setups one can consider

Similarly, in both coherent and incoherent setups one can consider

With the above remarks in mind, the correct calculation of

The reader is referred to [

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 [

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 (

The first one is the Schrödinger equation (

The second one is called the Guiding Equation:

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

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

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. [

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

The very same principle of

The proposed explanations by [

Due to the general applicability of Feynman’s sum-over-histories approach we consider the current work superior to previous work [

At the end, we will briefly assess each of the positions (P1)–(P8) formulated in Section

Positions (P1) [

Position (P3) [

Position (P4) [

Position (P5) [

Position (P6) [

Position (P7) [

The position (P8) [

N. Bohr was the most prominent proponent of a philosophical position known as

In 1926, Schrödinger developed a wave equation that describes the evolution of quantum probability amplitudes in space and time [

Al-Khalili reported [

The author would like to thank