## Abstract

Fundamentally contradictory but inescapably joined dual attributes, wave and particle, remain a conceptually unsettling element at the heart of quantum mechanics. It was a career-long unanswered challenge for Bohr to rationalize quantum duality. The conceptual dilemma it presents has remained an open issue, a topic of continued discussion, ever since. Here we report the discovery of an experimentally manageable route to control the weirdness of duality. Ironically, entanglement, the other conceptually challenging weirdness of quantum theory, will be shown to be in control of duality. We establish a simple identity through which entanglement prescribes quantitatively the degree of duality, of combined waveness and particleness, that can be recorded in any one-quantum two-path coherence experiment.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. MOST WEIRD

In the 1920s, the joining of two incompatible words as a label, wave mechanics, was an accurate indication that the newly emerging physics would be very unusual. The identification of particles as waves was a contradiction of all previous experience and was the first counterintuitive weirdness of quantum theory to appear. One can argue that it is still the central weirdness. In the years after his invention of complementarity in 1928 [1], as an answer to this strange duality of de Broglie [2], Bohr was never satisfied that he had found the right resolution of its paradoxical character [3]. Quantum duality remains as weird today as in the 1920s. Academic, philosophical, and historical discussion of the conceptual dilemma of duality has never stopped [4,5]. Attention to it by the physics community has moved through several stages of increasing sophistication, but after 90 years of reflection, there has been no resolution. Here we describe a resolution. It is noteworthy that our resolution also applies to quantum duality’s classical analog, which is the ray-wave dichotomy of optics. This is appropriate, since optical physics and single-particle quantum mechanics are both theories founded on coherence and interference in linear vector spaces, and they share the issues inherent in duality.

## 2. INTERFERENCE COHERENCE

The two-path experimental scenario leading to the first quantification of the coherence of light is well known. It was invented by Thomas Young for his famous demonstration of optical interference [6]. It is still well suited for discussions of coherence because the specific context for a Young-type observation can be modified to expose different coherences. The role of context is illustrated in Fig. 1, but its importance is often not recognized [7]. This is especially true in the discussion of so-called “hidden” coherences, a number of which have been reported and analyzed recently [8–14].

The basis for our analysis will be the classical optical field. The results are new, and we also report a (classical) tomographic examination that provides experimental confirmation. An optical field has more than 3 degrees of freedom, but 3 degrees of freedom allow us to utilize a sufficiently general theory of vector-mode coherence [15]. Classical optics is a suitable ground for discussing these issues, but all results to be developed below have one-quantum counterparts, as should become clear.

## 3. DUALITY BALANCE

In 1979, after 50 years of discussion and debate about wave and particle interpretations of duality [3,4,9,16–31], Wootters and Zurek [32] drew attention to the fact that duality need not represent an absolute incompatibility. As they indicated, Young-type scenarios can allow an amount of particleness to be given up for an increase in waveness, and vice versa, implying a kind of “duality balance” between wave and particle (or wave and ray) interpretations.

Many theoretical examinations since 1979 confirm that both particleness and waveness are open to analysis and fully support the balance implied by Wootters and Zurek. There is agreement on a central result, an inequality that quantifies the “amount” of duality present, the sum ${V}^{2}+{D}^{2}$ of two commonly accepted markers, visibility $V$ for waveness and which-path distinguishability $D$, for particleness. The multiply rederived and accepted quantification of their combination is this:

with more than one interpretation of $V$ and $D$.One naturally asks, if previous treatments have arrived at a quantification formula, what can remain to be found? Here we are pointing out that two oversights are built into the interpretation of inequalities such as Eq. (1). They are valid as they stand, but are still short of the complete story. To examine the oversights, and also prepare the way for our experimental setup, we first rederive Eq. (1) as a standard but incomplete quantification.

## 4. VECTOR MODE COHERENCE

Our field’s complex amplitude depends on 3 degrees of freedom: (i) its two-dimensional transverse coordinate ${r}_{\perp}$, hereafter denoted simply as $r$; (ii) time $t$; and (iii) intrinsic polarization $\widehat{s}$, which we will call by the shorthand term spin. The required field coherence function in the Young source plane (cf. Fig. 1) is made from the fields at the two slit locations, $a$ and $b$. Those two fields have arbitrary amplitudes and unit spins: ${\overrightarrow{E}}_{a}={\widehat{s}}_{a}{E}_{a}({r}_{a},t)$ and ${\overrightarrow{E}}_{b}={\widehat{s}}_{b}{E}_{b}({r}_{b},t)$. Thus, the two fields received at point ${r}_{c}$ on the detection screen are

where $A$ and $B$ are the usual purely imaginary propagation coefficients, and the field arguments are the same as at the source except for the time delays due to travel to the screen point ${r}_{c}$ (following Wolf [33], Section 3.1).There are three detectable signals (intensities) at the screen, one from each slit separately and one from the combination of the two fields at ${r}_{c}$. The total intensity is the sum of the individual intensities and the cross term

The visibility $V$ of the measured fringes has its usual expression, based on the extreme $\pm 1$ values of the phase cosine function, which leads to

The which-path distinguishability $D$, i.e., the degree to which the light field at screen $c$ is coming from only one of the two slits, $a$ or $b$, is taken to be given by the standard expression

This quantity is also called predictability by Jaeger*et al.*[18]. By combining Eqs. (7) and (8), we recover Eq. (1), the agreed duality inequality:

## 5. CALLING OUT OVERSIGHTS

In a summary [16] of his attempts to get control of the duality concept, Bohr focused on two important criteria in any analysis (see [4]). These were *exclusivity* and *completeness*, and awareness of them guides attention to two oversights not previously called out, to the best of our knowledge. These oversights cause one to overlook weaknesses in the usual interpretations of Eq. (1).

First, one should note lack of completeness, because the Young scenario allows for the simultaneous vanishing of $V$ and $D$, while a finite signal is nevertheless present. For example, if beams $a$ and $b$ are equally strong, the standard measure of distinguishability vanishes: $D=0$. And if the equally strong beams are also oppositely polarized, there can be no interference and no fringes (cf. Fig. 1). Thus the absence of fringes can accompany the absence of distinguishability. A nonzero signal must be characterized by something more, in addition to $V=0$ and $D=0$, a clear indication of incompleteness. Something is missing.

Another oversight is to fail to note that a recently identified coherence theorem connects $V$ and $D$ in a strict equality with degree of polarization $P$ (see [34]):

However, because $P$ is bounded as $0\le P\le 1$, by changing $P$ one can reduce $D$ without increasing $V$, or the opposite–they do not need to trade tightly with each other. Again, something seems to be missing, and this is right.What is missing is attention to entanglement (quantum or classical [35]), confirming a suggestion made 20 years ago about quantum weirdness by Knight [36]. Quantification of coherence in our own recent optical research [7,13,37–40] has exposed connections to entanglement. Other recent work has subjected coherence to quantification and examination of relations to entanglement [41–43]. The unifying quantum-classical aspect is the vector space foundation common to quantum states and classical wave field theories. We have already demonstrated the intimate connection of entanglement and polarization [13]. They are constrained in Young-type two-slit configurations by an additional identity:

where $C$ is the entanglement measure called concurrence [44]. That is, $P$ and $C$ are Young-type perfect opposites. Gaining some of ${P}^{2}$ comes at the expense of losing an equal amount of ${C}^{2}$, which is an unexpected possibility of rearrangement, since $P$ and $C$ have such different traditional origins and interpretations. As De Zela has correctly emphasized [45], this unusual kind of pairing is significant, not coincidental.Now one immediately sees that by controlling $P$ via Eq. (11), entanglement automatically controls duality. That is, when we eliminate ${P}^{2}$ between Eqs. (10) and (11), the duality inequality becomes an equality:

The completeness, and simultaneously the exclusivity, needed by duality (and complementarity) is now found in the control by entanglement. Weaker or stronger entanglement forces more or less duality. The combination greatly extends (actually completes) the previous inequality [Eq. (1)] into a tight equality, the identity, We believe this identity is the needed resolution Bohr was seeking. It fully contains the essential weirdness of duality in its fundamental two-slit or two-path context. Importantly, in Eq. (13), nothing is extra, and nothing is missing. The addition of ${C}^{2}$ supplies the quantity that has been missing, both from the result [Eq. (1)] and from the earlier discussion, namely, entanglement, and in the amount needed via concurrence. Still, as a relation among three observables in a Young-type experiment, Eq. (13) is open to test. In the following section we provide the results of that test.## 6. DESCRIPTION OF EXPERIMENT

Here we report tomographic observation of all three elements of Eq. (13): visibility, distinguishability, and entanglement. Our setup and laboratory results are sketched below. We engage a specific Young-analog context. By use of a monochromatic laser, the temporal dependence can be factored out completely, leaving

where we have departed slightly from the notation in Eqs. (2)–(6), and have shifted the meaning of the coefficients $A$ and $B$ in order to take the space-mode dependent functions ${u}_{a}(r,z)$ and ${u}_{b}(r,z)$, which are the replacements here for ${E}_{ac}$ and ${E}_{bc}$ in Eqs. (2) and (3), to be unit-normalized. We will see that the important influence of the mutual coherence in the experiment will come from the two spin components, determined by their relative angle $\theta $ and phase $\xi $, i.e., we will haveThe preparation stage is designed to generate optical beams of the form given in Eq. (14). Specifically, a single-mode laser operating at 795 nm, with linear $\widehat{x}$ polarization, passes through a half-wave plate (HWP1) and changes into an arbitrary linearly polarized state $\widehat{s}$. A polarizing beam splitter (PBS) splits the beam into $\widehat{x}$ and $\widehat{y}$ polarizations. The transmitted channel contains the signal ${\mathrm{Au}}_{a}({r}_{\perp}){\widehat{s}}_{a}$ in path $a$. It then passes through a 50/50 beam splitter (BS1) that directs it to a mirror mounted on a translation stage (TS). The reflected channel (labeled $b$) of the PBS passes through a half-wave plate (HWP2) to become ${\mathrm{Bu}}_{b}({r}_{\perp}){\widehat{s}}_{b}$, where ${\widehat{s}}_{b}={e}^{i\xi}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta \widehat{x}+\mathrm{sin}\text{\hspace{0.17em}}\theta \widehat{y}$. The two spatial functions ${u}_{a}$ and ${u}_{b}$ obviously have no overlap. Then the light field is characterized exactly by Eq. (14).

Here the parameter ratio $|A/B|$ is controlled by HWP1 and ${\gamma}_{\mathrm{exp}}=\mathrm{cos}\text{\hspace{0.17em}}\theta {e}^{i\xi}$ is controlled by the combination of HWP2 in channel $b$ (in terms of $\theta $) and the TS in channel $a$ (in terms of phase $\xi $). The output 50/50 beam splitter (BS2) combines channels $a$ and $b$ to produce interference. The light beam then enters the measurement stage so that $V$, $D$, and $C$ can be registered.

Fringe visibility $V$ can be simply achieved by continuously registering the intensity at the output of BS2 while moving the TS. The systematic maximum visibility that is achievable for our MZI is 98.1%, obtained by producing equal intensities of the two channels and measuring interference intensities with a polarizer placed right after BS2. All measurements of visibilities in other arrangements are corrected by this systematic maximum. The which-way distinguishability $D$ can be obtained straightforwardly with intensity measurements by blocking one of the two channels, $a$ or $b$. Measurement of entanglement, quantified by concurrence $C$, between the spatial degree of freedom $\{{u}_{a},{u}_{b}\}$ and polarization space $\{\widehat{x},\widehat{y}\}$, is realized by a tomography setup, as shown in Fig. 2. It is a joint measurement of the Stokes-like parameters in both degrees of freedom [46].

## 7. CONFIRMATION BY TOMOGRAPHY MEASUREMENT

Each degree of freedom has two effective “states,” i.e., ${u}_{a}$, ${u}_{b}$ in the spatial domain and $\widehat{x}$, $\widehat{y}$ in the spin polarization domain. In reality, the two paths may not be perfectly coherent, i.e., the contents of the modes are not perfectly pure. These cases result in a matrix description of the light beam in the basis $\{{u}_{a}\widehat{x},{u}_{a}\widehat{y},{u}_{b}\widehat{x},{u}_{b}\widehat{y}\}$, given as

Then, following the standard two-qubit tomography procedure [46], the output beams are measured jointly in the Stokes-like basis of both degrees of freedom to recover each matrix element in Eq. (16). In principle, one needs to project the light onto the following 36 joint bases, i.e., $\{{u}_{a},{u}_{b},({u}_{a}\pm {u}_{b})/\sqrt{2},({u}_{a}\pm i{u}_{b})/\sqrt{2}\}\otimes \{\widehat{x},\widehat{y},(\widehat{x}\pm \widehat{y})/\sqrt{2},(\widehat{x}\pm i\widehat{y})/\sqrt{2}\}$, six bases in each degree of freedom. The projections of light in the six spin polarization state bases are standard, by the use of quarter-wave and half-wave plates combined with PBSs, as shown in the tomography measurement section of the setup in Fig. 2.

The projection in the spatial basis $\{{u}_{a},{u}_{b}\}$ is realized straightforwardly by blocking one of the two channels of the MZI. The projection onto the bases $({u}_{a}\pm {u}_{b})/\sqrt{2}$ can be realized by leaving both channels open. BS2 is a unitary rotation operator of the two incoming paths. It operates as an effective projection with one output port in the basis $({u}_{a}+{u}_{b})/\sqrt{2}$ and the other port in $({u}_{a}-{u}_{b})/\sqrt{2}$. Finally, the projection in the basis $({u}_{a}\pm i{u}_{b})/\sqrt{2}$ is achieved with the combination of BS2 and the TS that can introduce an additional $\pi /2$ phase in path $a$.

With the above arrangements, one is able to realize measurements in all joint bases. Then the coherence matrices [Eq. (16)] for all 13 beams can be reconstructed in the standard basis ${u}_{a}\widehat{x}$, ${u}_{a}\widehat{y}$, ${u}_{b}\widehat{x}$, ${u}_{b}\widehat{y}$. The moduli of the measured matrix elements are shown, respectively, in Fig. 3.

The completed duality or complementarity identity ${V}^{2}+{D}^{2}+{C}^{2}=1$ in Eq. (13) calls to mind a unit sphere in a space with axes that are the independent degrees of freedom $V\text{-}D\text{-}C$. The measured $VDC$ values, as well as the sum ${V}^{2}+{D}^{2}+{C}^{2}$, are shown in Table 1, and they do fit on the surface of one octant of a unit sphere. This is shown in Fig. 4. One sees that the three-way duality identity [Eq. (13)] that connects visibility-distinguishability-entanglement is confirmed.

## 8. OVERVIEW

Our central result is, in the first case, the derivation of the equation ${V}^{2}+{D}^{2}=1-{C}^{2}$, showing how entanglement controls the amount of duality available. Then the formulation of the identity ${V}^{2}+{D}^{2}+{C}^{2}=1$ in Eq. (13) is automatic. It was followed by its experimental validation. As we pointed out, previously proposed quantitative duality measures engaging single inequalities such as Eq. (1) cannot embody completeness or exclusivity. A fully developed examination [13,34,48,49] of polarization coherence (both ordinary and generalized) relative to entanglement (both quantum and classical), as in Eq. (11), has only more recently been attempted. Finally, although we suggest that ${V}^{2}+{D}^{2}+{C}^{2}=1$ represents the simplest possible closing of a 90-year discussion of duality, we do not conclude that physics is finished with complementarity. Recent work shows that it can also be formulated for more than one entity. The first steps, which have already been taken [12,18,20,26,27], open a window onto multi-entity coherences that are not yet well explored or even conventionally named.

We conclude with two remarks: (A) It is fascinating that the two elements of quantum theory most often said to be conceptually mysterious or weird, duality and entanglement, are intimately related, and in control of each other, as our classical derivation of Eq. (13) implies. (B) The fully quantum single-photon version of our derivation is easy to imagine and will be presented elsewhere [50]. Its resolution of the duality paradox also shows that duality is controlled by entanglement (and vice versa), just as is described here.

## Funding

National Science Foundation (NSF) (PHY-1203931, PHY-1505189, PHY-1539859); Army Research Office (ARO) (W911NF-16-1-0162); Office of Naval Research (ONR) (N00014-14-1-0260).

## Acknowledgment

We are pleased to acknowledge communication with colleagues Iwo Bialynicki-Birula, Paul Brumer, Steven Cundiff, Justin Dressel, B.-G. Englert, Andrew Jordan, Peter Knight, Peter Milonni, Michael Raymer, Lukasz Rudnicki, and Juan P. Torres.

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