## Abstract

Light is neither wave nor particle, but both, according to Bohr’s complementarity principle, which was first devised to qualitatively characterize quantum phenomena. Later, quantification was achieved through inequalities such as ${V}^{2}+{D}^{2}\le 1$, which engage visibility $V$ and distinguishability $D$. Recently, equality ${V}^{2}+{D}^{2}={P}^{2}$—the polarization coherence theorem (PCT)—was established, incorporating polarization $P$ and addressing both quantum and classical coherences. This shows that Bohr’s complementarity is not restricted to quantum phenomena. We derive an extension of the PCT that also applies to quantum and classical light fields carrying intertwined, dichotomic observables, such as polarization and two-path alternative. We discuss how constraints critically depend on the chosen measurement strategy. This may prompt various experiments to exhibit complementary features that possibly lurk behind hidden coherences.

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

## 1. INTRODUCTION

Young’s double slit experiment, the very archetype of interferometric phenomena, is still giving rise to new insights concerning the nature of light. Indeed, relying on Young’s array, two theorems have been recently reported [1,2]. One of them, the polarization coherence theorem (PCT) [1], establishes the equality ${V}^{2}+{D}^{2}={P}^{2}$, which relates visibility ($V$) and distinguishability ($D$) with polarization ($P$) of light beams. These quantities were previously known to be constrained only by inequalities such as ${V}^{2}+{D}^{2}\le 1$. The second theorem [2] establishes the maximal visibility that can be achieved in a double-slit array when a partially coherent and partially polarized beam is submitted to general unitary transformations that engage polarization and spatial degrees of freedom (DoFs). Such a beam can be described by means of a $4\times 4$ Hermitian matrix of unit trace. Writing the eigenvalues of this matrix in decreasing order, ${\lambda}_{1}\ge {\lambda}_{2}\ge {\lambda}_{3}\ge {\lambda}_{4}$, the maximal visibility is given by ${V}_{\mathrm{max}}=1-2({\lambda}_{3}+{\lambda}_{4})$. Previous work did not address unitary transformations (unitaries) in full generality. Reference [2] stressed the symmetrical role played by polarization and spatial DoFs. Each of these DoFs represents a two-state system, so that there is no intrinsic mathematical difference between them, a fact that was also pointed out in connection to the PCT [1]. As Eberly and coworkers have repeatedly stressed, polarization is a concept that applies beyond its original domain, being generally a two-party property. Thus, we may refer to polarized spatial modes. Reciprocally then, we should also be entitled to define visibility when dealing with polarization modes (also referred to as “spin” modes, to avoid confusion). Concepts that were originally introduced with a specific case in mind have thus revealed themselves as being applicable in quite different scenarios. It can also occur that these scenarios merge into a new entity, one whose properties can be revealed only when dealing with the whole and that disappear from our sight as soon as this sight embraces just one part of the entity. This is the case when, say, two DoFs become entangled. By observing just one of the entangled parties, it can occur that only randomness can be recorded. We can then be fooled into attributing pure randomness to a phenomenon that only when viewed in its entirety would reveal its intrinsic and complete, hidden coherence.

It is somewhat surprising that this kind of insight has been achieved only recently in the realm of classical optics. Indeed, several phenomena that have been newly exhibited in the classical scenario are nothing but a remake of well-known quantum phenomena [3–9]. The complete randomness that is observed in, e.g., each of two maximally entangled spin-1/2 particles is but one prominent example that directly relates to hidden coherence in classical fields. Similarly, techniques that were routinely employed in quantum state tomography have recently found their counterpart in coherence matrix tomography [7]. Even Bell violations—once seen as the test of quantumness *par excellence*—have also been used as an entanglement witness with classical light beams [10–18]. All this hints at the existence of a wide common ground for several quantum and classical phenomena [19–24]. These phenomena might have been addressed independently from one another only because of historical reasons, and not because they radically differ from each other in regard to their physical content. In addition to entanglement and the mimicking of tomographic techniques, classical optics can also profit from various insights that the quantum approach has produced when addressing different kinds of measurement. One of our main concerns here is to exhibit the decisive role played by our choice of a specific measurement strategy, and how it may co-determine the very nature of the phenomenon under study. So as hidden coherences may be exposed by adopting the appropriate measurement strategy, also some of the aforementioned, recent results can show features that remain hidden until the involved systems are submitted to well-designed measurements. As we shall see, the scope of the PCT depends on the associated type of measurement to which the involved observables are submitted. This is also the case with the maximal visibility that can be achieved in connection with one of two correlated DoFs. The very concepts of visibility and distinguishability may vary when changing the interferometric scenario. A case in point is the extension of the PCT that can be achieved by changing its original scenario, as is shown in this work.

This paper has been organized as follows. Section 2 discusses the recently reported PCT, in connection with general two-state systems. It is shown how the content of this theorem may depend on the measuring setup. Section 3 addresses another recent result: the achievable maximal visibility in a double-slit interferometer. Section 4 reports the main result of this work: an extended PCT. Some concluding remarks deal with the quantum–classical interplay.

## 2. PCT AND ITS AFTERMATHS

The intensity ${I}_{c}$ at point $c$ on a screen of a two-slit interference setup (see Fig. 1) can be written as [1]

${I}_{k}=\u27e8{|\mathrm{\varphi}|}_{k}^{2}\u27e9$ is the intensity of light coming only through slit $k=a,b$, with angular brackets denoting ensemble average. An interference pattern arises upon variation of the relative phase between amplitudes ${\mathrm{\varphi}}_{a}$ and ${\mathrm{\varphi}}_{b}$, which superpose on the screen after propagation from $a$ and $b$, respectively. The light field can be described by means of the expression $F({r}_{\perp},z)={u}_{a}({r}_{\perp},z){\mathrm{\varphi}}_{a}(q)+{u}_{b}({r}_{\perp},z){\mathrm{\varphi}}_{b}(q)$ [1]. Here, ${u}_{k=a,b}$ are spatial mode functions and $q$ stands for unspecified DoFs, upon which the field amplitudes may depend. One can define a “mode polarization coherence matrix” ${\mathcal{W}}^{(F)}$ with elements $\u27e8{\mathrm{\varphi}}_{i}^{*}{\mathrm{\varphi}}_{j}\u27e9$, where $i,j\in \{a,b\}$. It is analogous to the polarization coherence matrix ${\mathcal{W}}^{(P)}$ with elements $\u27e8{E}_{k}^{*}{E}_{l}\u27e9$, $(k,l\in \{H,V\})$, a matrix that is introduced when dealing with a polarized field $\mathbf{E}={E}_{H}{\widehat{\mathbf{e}}}_{H}+{E}_{V}{\widehat{\mathbf{e}}}_{V}$. The degree of polarization can be defined with reference to either matrix. Thus, the degree of $F$-polarization reads

By direct calculation one gets the statement of the PCT:

Thus, the PCT holds true for any $2\times 2$ Hermitian matrix $\rho ={\sum}_{{\rho}_{ij}}|i\u27e9\u27e8j|$. Indeed, in terms of $\rho $’s eigenvalues,

By defining $V=2|{\rho}_{12}|/({\rho}_{11}+{\rho}_{22})$ and $D=|{\rho}_{11}-{\rho}_{22}|/({\rho}_{11}+{\rho}_{22})$, as in Eq. (3), we recover the identity ${P}^{2}\equiv {D}^{2}+{V}^{2}$. Alternatively, $V$ could be replaced by the ${l}_{1}$-norm of coherence [25], $C=|{\rho}_{12}|/({\rho}_{11}+{\rho}_{22})$, taking into account that “visibility” refers to a rather limited interferometric scenario, while $|{\rho}_{12}|$ more generally measures the amount of coherence that $\rho $ contains.

Note that while $P$ is a basis-independent quantity that reflects an intrinsic property of $\rho $, $D$ and $V$ are basis dependent and convey information about how $\rho $ relates to the particular “reference frame” that is defined by the basis $\{|1\u27e9,|2\u27e9\}$. When $\rho $ represents a density matrix, ${\rho}_{11}$ and ${\rho}_{22}$ have the meaning of probabilities and $|{\rho}_{11}-{\rho}_{22}|$ is a measure of our “which-way” knowledge, also called “predictability” [26,27]. The very concept of “way”—or “path”—presupposes a reference frame. Similar considerations can be made concerning ${\rho}_{12}$, which vanishes in the reference frame or basis in which $\rho $ is diagonal. All these remarks apply to both the classical and the quantum domains. We just need to interchange intensities and probabilities in going from one domain to the other.

It is useful to rewrite the above results in terms of Stokes parameters, which can be introduced whenever one deals with a $2\times 2$ Hermitian matrix. To this end, we write $\rho $ in terms of the Pauli matrices ${\sigma}_{k=\mathrm{1,2},3}$ and the identity matrix ${\sigma}_{0}$:

By setting ${\rho}_{12}=|{\rho}_{12}|{e}^{-i\varphi}$, we get ${S}_{0}={\rho}_{11}+{\rho}_{22}$, ${S}_{1}=2|{\rho}_{12}|\mathrm{cos}\text{\hspace{0.17em}}\varphi $, ${S}_{2}=2|{\rho}_{12}|\mathrm{sin}\text{\hspace{0.17em}}\varphi $, and ${S}_{3}={\rho}_{11}-{\rho}_{22}$. The degree of polarization is defined in terms of the Stokes parameters as ${P}^{2}=({S}_{1}^{2}+{S}_{2}^{2}+{S}_{3}^{2})/{S}_{0}^{2}$. Introducing the foregoing expressions of Stokes parameters in this definition, we obtain

We see then, [see Eq. (6)], that by defining

However, this reshaping depends on how we define $D$ and $V$. The above definition is suggested by ${S}_{3}^{2}={({\rho}_{11}-{\rho}_{22})}^{2}$ and ${S}_{1}^{2}+{S}_{2}^{2}=4{|{\rho}_{12}|}^{2}$, where the right-hand sides (RHSs) connect with an interferometric scenario in the way given by Eq. (3).

#### A. Forerunner of the PCT: Distinguishability as Predictability

It will be useful to briefly discuss an approach [26] that came close to the PCT. Let us consider the most basic interferometric transformation ${U}_{I}$ that consists of three operations: beam splitting, phase shifting, and beam merging. These operations can be represented by the unitaries ${U}_{\mathrm{BS}}=({\sigma}_{1}+{\sigma}_{3})/\sqrt{2}$ for beam splitting/merging, and ${U}_{\varphi}=\mathrm{exp}(i\varphi {\sigma}_{3}/2)$ for phase shifting. Consider the incoming state ${\rho}^{(i)}=(1/2)\sum _{k=0}^{3}{S}_{k}{\sigma}_{k}$. After being submitted to ${U}_{I}={U}_{\mathrm{BS}}{U}_{\varphi}{U}_{\mathrm{BS}}$, state ${\rho}^{(i)}$ transforms into

where ${\mathbf{S}}^{(f)}=({S}_{0},{S}_{1},{S}_{2}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\varphi +{S}_{3}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\varphi ,-{S}_{2}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\varphi +{S}_{3}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\varphi )$. If we now calculate the intensity ${I}_{\varphi}$ at the up-detector, we get, with ${\sigma}_{3}|\pm \u27e9=\pm |\pm \u27e9$,This $V$ is—up to relabeling of the ${S}_{i}$—the same as that of Eq. (9). Moreover, maxima and minima of ${I}_{\varphi}$ are reached when $\mathrm{cos}(\varphi -\alpha )=\pm 1$, so that ${I}_{\varphi}^{\mathrm{max}}-{I}_{\varphi}^{\mathrm{min}}=V$, in accordance with the standard definition of visibility. As for distinguishability, it can be defined in terms of the state ${\rho}_{w}={U}_{\mathrm{BS}}{\rho}^{(i)}{U}_{\mathrm{BS}}^{\u2020}$, which is associated with a two-way alternative. By projecting this state onto $|\pm \u27e9$, we get the corresponding intensities: ${I}_{\pm}=\mathrm{Tr}(|\pm \u27e9\u27e8\pm |{\rho}_{w})=(1\pm {S}_{1})$. We can refer to these intensities and define distinguishability as

In Ref. [26], ${D}_{w}$ was dubbed “predictability,” because of the quantum context in which it was introduced. We have added the label $w$ to differentiate this distinguishability (or predictability) from the one previously defined, denoted by $D$. The latter referred either to ${\rho}^{(i)}$ or to ${\rho}^{(f)}$; see Eq. (11). On noting that $\sum _{k=1}^{3}{S}_{k}^{2}\le {S}_{0}^{2}$, we get the inequality

which was established by different authors (see, e.g., Ref. [26] and citations therein). Even though the above inequality has the same mathematical origin as both ${D}_{F}^{2}+{V}_{F}^{2}={P}_{F}^{2}\le 1$ and ${D}^{2}+{V}^{2}={P}^{2}\le 1$ [see Eqs. (4) and (10)] they have different meanings. Depending on the two beams we want to distinguish from one another, we have a corresponding measure: $D$, ${D}_{F}$, or ${D}_{w}$. Similar considerations can be made concerning visibility. Our definition of visibility refers to some intensity pattern. This intensity, in turn, corresponds to a projection onto some reference state, such as the up-state $|+\u27e9$ in Eq. (12). We obtain different patterns by varying the states that we project on. To make this point clearer, let us consider the basic interferometric transformation ${U}_{I}$. We use the notation ${\sigma}_{i}{|\pm \u27e9}_{i}=\pm {|\pm \u27e9}_{i}$ ($i=\mathrm{1,2},3$) for the eigenvectors of the Pauli matrices. The action of ${U}_{I}$ on the input-state ${|+\u27e9}_{3}$ is as follows:We can measure intensities ${I}_{\varphi ,3}^{\pm}\equiv |\u27e8{\psi}_{\varphi}{|\pm \u27e9}_{3}{|}^{2}$ and ${I}_{\varphi ,1}^{\pm}\equiv |\u27e8{\psi}_{\varphi}{|\pm \u27e9}_{1}{|}^{2}$ by putting appropriate setups before a powermeter. Our readings should give

Hence, in one case we get an interference pattern, while in the other case we get a flat response. This difference does not come from the measured state, $|{\psi}_{\varphi}\u27e9$, but from the measuring setup. As a further illustration of the role played by the measuring strategy, let us consider the following case.

Instead of implementing phase shifting by ${U}_{\varphi}=\mathrm{exp}(i\varphi {\sigma}_{3}/2)$, let us implement it by ${U}_{\varphi}=\mathrm{exp}(i\varphi {\sigma}_{1}/2)$. Then, Eq. (11) still holds but now with ${\mathbf{S}}^{(f)}=({S}_{0},{S}_{1}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\varphi +{S}_{2}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\varphi ,-{S}_{1}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\varphi +{S}_{2}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\varphi ,{S}_{3})$. As phase shifting is now defined with respect to the eigenstates of ${\sigma}_{1}$, we measure the intensity ${I}_{\varphi}$ with respect to the “up” eigenvector of ${\sigma}_{1}$, namely, ${|+\u27e9}_{1}=({|+\u27e9}_{3}+{|-\u27e9}_{3})/\sqrt{2}$. Thus, we have

#### B. Variant of the PCT: Visibility Equals the Degree of Polarization

A more drastic difference that stems from diverging measuring strategies comes about in the following case [28]. Let us consider a transversal, quasi-monochromatic beam, the associated analytic signals of which read ${E}_{x}(t)={a}_{1}(t)\mathrm{exp}(i[{\varphi}_{1}(t)-\overline{\omega}t])$ and ${E}_{y}(t)={a}_{2}(t)\mathrm{exp}(i[{\varphi}_{2}(t)-\overline{\omega}t])$. When submitted to both a compensator that retards the $y$ component by $\u03f5$ and a polarizer set at angle $\theta $ with respect to the $x$ direction, the incoming beam turns into $E(t,\theta ,\u03f5)={E}_{x}\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta +{e}^{i\u03f5}{E}_{y}\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta $. The associated intensity $I(\theta ,\u03f5)=\u27e8E(t,\theta ,\u03f5){E}^{*}(t,\theta ,\u03f5)\u27e9$ is then given by

Here, ${j}_{xy}=|{j}_{xy}|\mathrm{exp}(i{\beta}_{xy})={J}_{xy}/\sqrt{{J}_{xy}{J}_{xy}}$, and the coherency matrix $\mathbf{J}$ is given by

Equation (20) can be written as

We now define visibility and distinguishability through

We then readily obtain

This motivates the introduction of the following matrix, in place of the coherence matrix $\mathbf{J}$:

In terms of ${\mathbf{J}}_{\u03f5}$, the RHS of Eq. (25) reads

Alternatively, we may take maxima and minima of $I(\theta ,\u03f5)$ with respect to both $\theta $ and $\u03f5$. In such a case one gets [28]

Hence, if we define the left-hand side of the above equation as a visibility ${V}_{(\theta ,\u03f5)}$, we have that

with $P$ being defined with respect to the standard polarization coherence matrix $\mathbf{J}$. We thus obtain quite different identities, depending on the intensities we are referring to. It is therefore important to make clear the context in which a given relationship holds true. For example, as we have seen, ${D}_{F}^{2}+{V}_{F}^{2}\le 1$ derives from Eq. (4) but, in spite of the similarity in notation, it is not the same as the inequality ${\mathcal{D}}^{2}+{\mathcal{V}}^{2}\le 1$ that was derived in Ref. [26]. The latter defines visibility and distinguishability by making reference to two DoFs, in contrast to the single DoF that was invoked to establish Eq. (4). Similar considerations hold for the maximal visibility that was discussed in Ref. [2]. We address next this issue and then return to the derivation of inequality ${\mathcal{D}}^{2}+{\mathcal{V}}^{2}\le 1$.## 3. MAXIMAL VISIBILITY

Abouraddy [2] has recently answered the question about the maximal visibility that can be achieved in a double-slit interferometer, with a partially coherent and partially polarized field. Such a field has two accessible DoFs. It is assumed that we can apply arbitrary unitaries engaging the two DoFs. The addressed fields can be represented by Hermitian, non-negative, trace-one, $4\times 4$ matrices:

We can probe the spatial coherence matrix ${G}_{S}=({G}_{ij})$ ($i,j\in \{a,b\}$) with interferometric measurements that are insensitive to polarization. Double-slit interference should have a visibility ${V}_{S}=2|{G}_{ab}|$. The degree of spatial coherence—or spatial polarization—is a unitary invariant, $|{\lambda}_{a}-{\lambda}_{b}|$, where ${\lambda}_{a}$ and ${\lambda}_{b}$ denote the eigenvalues of ${G}_{S}$. On noting that

Let us now return to the full coherence matrix $\mathbf{G}$. Its diagonal form reads ${\mathbf{G}}^{D}=\mathrm{diag}({\lambda}_{1},{\lambda}_{2},{\lambda}_{3},{\lambda}_{4})$, with the eigenvalues taken in decreasing order: ${\lambda}_{1}\ge {\lambda}_{2}\ge {\lambda}_{3}\ge {\lambda}_{4}$. Let us consider all possible unitaries $U$ such that $\mathbf{G}=U{\mathbf{G}}^{D}{U}^{\u2020}$. The corresponding spatial matrices are given by ${G}_{S}={\mathrm{Tr}}_{P}(U{\mathbf{G}}^{D}{U}^{\u2020})$. The distinguishability associated to ${G}_{S}$ is ${D}^{S}=|({G}_{aa}^{HH}+{G}_{aa}^{VV})-({G}_{bb}^{HH}+{G}_{bb}^{VV})|$. As has been shown in Ref. [2], the maximal value of ${D}^{S}$, which is also the maximal value of ${V}^{S}$, is given by

In polarization subspace, where the field is described by ${G}_{P}$, we can define similar quantities and derive similar relationships. In particular, for the corresponding “visibility” ${V}^{P}$ we obtain

Thus, ${V}_{\mathrm{max}}^{P}\le {V}_{\mathrm{max}}^{S}$. This is a striking result. Indeed, nothing prevents us from interchanging the roles of the two involved DoFs, thereby obtaining ${V}_{\mathrm{max}}^{S}\le {V}_{\mathrm{max}}^{P}$. It is a matter of convention that we assign the meaning of polarization to the first or to the second DoF. Of course, the predicted value of maximal visibility cannot depend on our arbitrary choice of nomenclature, but it does depend on how we physically implement the production of a field such as that given by Eq. (31). To interchange the original roles given to polarization and spatial DoF, we must modify our original optical setup, and this has an impact on the maximal visibility we can achieve, in accordance with Eqs. (35) and (36).

We can gain additional insight in the above state of affairs by looking at the demonstration of Eq. (35) given in Ref. [2]. It rests on the implicit assumption that the diagonal form ${\mathbf{G}}^{D}=\mathrm{diag}({\lambda}_{1},{\lambda}_{2},{\lambda}_{3},{\lambda}_{4})$ refers to the canonical, also called “computational” basis: $\{|aH\u27e9,|aV\u27e9,|bH\u27e9,|bV\u27e9\}$. In that case, ${G}_{S}^{D}={\mathrm{Tr}}_{P}({\mathbf{G}}^{D})=\mathrm{diag}(({\lambda}_{1}+{\lambda}_{2}),({\lambda}_{3}+{\lambda}_{4}))$ and the corresponding visibility is $V=({\lambda}_{1}+{\lambda}_{2})-({\lambda}_{3}+{\lambda}_{4})$. By submitting ${\mathbf{G}}^{D}$ to unitary transformations, this visibility cannot increase, as proved in Ref. [2]. Note that given some non-diagonal $\mathbf{G}$ in the canonical basis, its diagonal form ${\mathbf{G}}^{D}$ refers to another basis, the one constituted by the eigenvectors of $\mathbf{G}$. Take, for example, a field whose coherency matrix reads

The eigenvalues of $\mathbf{G}$ are ${\lambda}_{1},\dots ,{\lambda}_{4}$. As for the spatial matrix ${G}_{S}$, we have ${G}_{S}={\mathrm{Tr}}_{P}(\mathbf{G})={\mathbb{1}}_{S}/2$, i.e., half the identity matrix. Thus, ${G}_{S}$ has null visibility associated to it. Matrix $\mathbf{G}$ is diagonal in the basis made by the maximally entangled Bell states, $|{\mathrm{\varphi}}^{\pm}\u27e9=(|aH\u27e9\pm |bV\u27e9)/\sqrt{2}$ and $|{\mathrm{\Psi}}^{\pm}\u27e9=(|aV\u27e9\pm |bH\u27e9)/\sqrt{2}$, so that we can write it also in the form

That is, the matrix representing $\mathbf{G}$ in the Bell basis is a diagonal matrix, while the matrix representing it in the canonical basis is the one given in Eq. (37). As already said, a field having this matrix has no visibility. We can get the visibility ${V}_{\mathrm{max}}^{S}$ of Eq. (35) by submitting $\mathbf{G}$ to the unitary transformation that makes it diagonal in the canonical basis, namely,

This gives

Associated to this ${\mathbf{G}}^{D}$ we have now the visibility ${V}_{\mathrm{max}}^{S}$ of Eq. (35). This visibility has thus been obtained from the original field $\mathbf{G}$ of Eq. (37), by submitting this field to the action of the non-local $U$ given by Eq. (39). As already said, by interchanging the roles of polarization and spatial mode—with a corresponding change of optical setup—we would get a visibility ${V}_{\mathrm{max}}^{S}$ given by Eq. (36).

A quite different but related interplay between two DoFs occurs when one DoF is used as a which-way “marker,” a topic we address next.

## 4. EXTENDED PCT

As we said before, in deriving the inequality ${\mathcal{D}}^{2}+{\mathcal{V}}^{2}\le 1$, two DoFs were involved [26]. In contrast to this approach, Eq. (4) and its related inequality involve a single DoF. Thus, we cannot say that ${\mathcal{D}}^{2}+{\mathcal{V}}^{2}\le 1$ follows from Eq. (4). It instead follows from another relationship, which we address here. This relationship holds irrespective of the quantum or classical framework that one may have in mind. Originally, the above inequality was derived in connection with wave–particle duality [26,29]. However, we need to consider only a pair of two-state systems, i.e., two “qubits,” in an interferometric scenario, such as the one shown in Fig. 2. To fix ideas, we could think of one system as representing a two-way alternative, i.e., the two paths of the interferometer, while the other system represents an “internal” DoF, such as polarization. The latter should play the role of a “marker”: the more effectively it distinguishes one path from the other, the more deleterious its effect on the capability of the first system to produce an interference pattern with high visibility.

Initially, the first system, $S$, is prepared by means of a beam splitter (BS) in a coherent superposition of the two-way alternative: $|\psi \u27e9=\alpha |1\u27e9+\beta |2\u27e9$. The corresponding coherence matrix, or projector, is thus

We have included a phase factor on the second term to account for the relative phase that the interfering system acquires along its way toward detection. Initially, system and marker are in a product state ${\rho}_{S}^{(0)}\otimes {\rho}_{M}^{(0)}$. After having been subjected to the action of ${U}_{SM}$, the two-party system is in the state

Here, ${\rho}_{M}^{(k)}={U}_{k}{\rho}_{M}^{(0)}{U}_{k}^{\u2020}$ ($k=\mathrm{1,2}$), and ${\tilde{\rho}}_{M}={U}_{2}{\rho}_{M}^{(0)}{U}_{1}^{\u2020}$. This process brings $S$ from its initial state ${\rho}_{S}^{(0)}$ into a new state that is given by

Having ${\rho}_{S}^{\mathrm{out}}$, we can calculate the intensity measured at, say, detector 1, by projecting ${\rho}_{S}^{\mathrm{out}}$ with $|1\u27e9\u27e81|={\sigma}^{\u2020}\sigma $:

As for distinguishability, we refer to the marker state

The above result suggests that we measure distinguishability by means of the (trace-) distance between the marker states ${\rho}_{M}^{(1)}$ and ${\rho}_{M}^{(2)}$. We have thus the following measures for visibility and distinguishability:

We can now derive an equality connecting $\mathcal{V}$ and $\mathcal{D}$, in the following way. The marker system $M$ can be considered to be a two-state system. This is because $S$ is a two-state system, so that it effectively correlates with a two-dimensional subspace of a second system, according to Schmidt’s theorem [31]. To fix ideas, let us assume that $M$ represents polarization. A polarized state can be generally represented in terms of its associated Stokes vector, as in Eq. (7), i.e.,

where we have assumed that ${\rho}_{M}^{(0)}$ is normalized and $\mathit{\sigma}\equiv ({\sigma}_{1},{\sigma}_{2},{\sigma}_{3})$. It can be readily proved [31] that the trace-distance between two states (qubits) ${\rho}_{1}$ and ${\rho}_{2}$, i.e., ${\mathcal{D}}_{12}=\mathrm{Tr}|{\rho}_{1}-{\rho}_{2}|/2$, is simply given by half the Euclidean distance on the 3D space to which $\mathbf{S}$ belongs, in our case the Poincaré sphere:We are interested in calculating the trace-distance that defines distinguishability $\mathcal{D}$ of Eq. (48). The involved states are ${\rho}_{M}^{(1)}={U}_{1}{\rho}_{M}^{(0)}{U}_{1}^{\u2020}$ and ${\rho}_{M}^{(2)}={U}_{2}{\rho}_{M}^{(0)}{U}_{2}^{\u2020}$. Their corresponding Stokes vectors are thus ${\mathbf{S}}_{1}={\mathcal{R}}_{1}\mathbf{S}$ and ${\mathbf{S}}_{2}={\mathcal{R}}_{2}\mathbf{S}$, where ${\mathcal{R}}_{i=\mathrm{1,2}}$ are the 3D rotations that are associated to the unitaries ${U}_{i=\mathrm{1,2}}$. According to Eq. (50), to get $\mathcal{D}$ we need the Euclidean distance $|{\mathcal{R}}_{1}\mathbf{S}-{\mathcal{R}}_{2}\mathbf{S}|$. Now, this distance is invariant under rotations, so that we can equally well write it as $|{\mathcal{R}}_{1}^{-1}({\mathcal{R}}_{1}\mathbf{S}-{\mathcal{R}}_{2}\mathbf{S})|=|\mathbf{S}-{\mathcal{R}}_{1}^{-1}{\mathcal{R}}_{2}\mathbf{S}|$. Rotation ${\mathcal{R}}_{1}^{-1}{\mathcal{R}}_{2}$ can be specified through a rotation angle $\gamma $ and a rotation axis $\widehat{\mathbf{n}}$. It is convenient to use the Euler–Rodrigues parameters ${e}_{0}=\mathrm{cos}(\gamma /2)$ and $\mathbf{e}=\mathrm{sin}(\gamma /2)\widehat{\mathbf{n}}$. In terms of these parameters, we can write [32]

This immediately gives $|\mathbf{S}-{\mathcal{R}}_{1}^{-1}{\mathcal{R}}_{2}\mathbf{S}|=2|{\mathbf{e}}^{2}\mathbf{S}-(\mathbf{e}\xb7\mathbf{S})\mathbf{e}-{e}_{0}(\mathbf{S}\times \mathbf{e})|$, which can be used in Eqs. (48) and (50) to get

The visibility is in turn given by $\mathcal{V}=|{\mathrm{Tr}}_{M}({U}_{2}{\rho}_{M}^{(0)}{U}_{1}^{\u2020})|=|{\mathrm{Tr}}_{M}({U}_{1}^{\u2020}{U}_{2}{\rho}_{M}^{(0)})|$. The unitary ${U}_{1}^{\u2020}{U}_{2}$ is the SU(2) version of the 3D rotation ${\mathcal{R}}_{1}^{-1}{\mathcal{R}}_{2}$, so that ${U}_{1}^{\u2020}{U}_{2}=\mathrm{cos}(\gamma /2)+i\text{\hspace{0.17em}}\mathrm{sin}(\gamma /2)\widehat{\mathbf{n}}\xb7\mathit{\sigma}$. We get then

On squaring the above expression we get ${\mathcal{V}}^{2}={e}_{0}^{2}+{(\mathbf{e}\xb7\mathbf{S})}^{2}$, which together with Eq. (52) leads to our final result:

We have set ${\mathbf{S}}^{2}\equiv {\mathcal{P}}^{2}$ to highlight the connection with the PCT. Choosing $\gamma =\pi $ we get ${\mathcal{D}}^{2}+{\mathcal{V}}^{2}={\mathcal{P}}^{2}$, which can be understood as a variant of the PCT. On the other hand, because ${\mathcal{P}}^{2}\le 1$, the known inequality ${\mathcal{D}}^{2}+{\mathcal{V}}^{2}\le 1$ follows from Eq. (54) as well. Furthermore, when the marker system is prepared in an arbitrary pure state, ${\mathcal{P}}^{2}=1$ and ${\mathcal{D}}^{2}+{\mathcal{V}}^{2}=1$, as it was previously observed in Ref. [26]. When the marker system is instead prepared in a completely random (unpolarized) state, ${\mathbf{S}}^{2}=0$, i.e., $\mathbf{S}=\mathbf{0}$, and we have that $\mathcal{D}=0$ [see Eq. (52)], while Eq. (54) reduces to ${\mathcal{V}}^{2}={\mathrm{cos}}^{2}(\gamma /2)$. In this case we reach full visibility ($\mathcal{V}=1$) for $\gamma =0$. Several other cases can be similarly analyzed on noting that $\mathcal{V}$ and $\mathcal{D}$ can be written as, say, ${\mathcal{V}}^{2}={e}_{0}^{2}+{(\mathbf{e}\xb7\mathbf{S})}^{2}$ and ${\mathcal{D}}^{2}={(\mathbf{e}\times \mathbf{S})}^{2}$. We have control over $\gamma $ and $\widehat{\mathbf{n}}$ through the analogous parameters defining the local unitaries ${U}_{1}$ and ${U}_{2}$. It is thus possible to explore several instances of the relationship between visibility and distinguishability that Eq. (54) establishes. We can indeed vary $\mathbf{S}$ through different state preparations, and $\gamma $ through different choices of ${U}_{1}$ and ${U}_{2}$. A thorough analysis of the implications that Eq. (54) may have for various physical realizations goes beyond the scope of the present work and will be reported elsewhere.

## 5. CONCLUDING REMARKS

Inequalities like ${\mathcal{D}}^{2}+{\mathcal{V}}^{2}\le 1$ were originally derived in the context of wave–particle duality, namely, the context of Bohr’s complementarity principle. According to this principle, neither of the two mutually exclusive notions of wave and particle can provide full account of fundamental physical phenomena, among which light is a prominent member. A complete picture should embrace both particle-like and wave-like properties. The extent to which one property excludes the other requires adequate quantification before it can be submitted to experimental test. This quantification has been proposed in relatively recent times, by means of the aforementioned inequalities. However, even though the narrative of most proposals has been couched in quantum terms, their respective mathematical treatments do not actually invoke any quantum principle or axiom. It should thus come as no surprise that said inequalities also apply to classical fields. Concerning the physical phenomenon that light represents, a truly quantum feature shows up only when photons are detected, for example, when an interferometric pattern is gradually generated, photon by photon. If we are instead just dealing with, say, two DoFs such as polarization and path (momentum), it is immaterial whether we assign these DoFs to photons or to classical light beams. That is, when the phenomenon under study concerns only some relationship between these two DoFs, this relationship can be experimentally confirmed equally well by using photons or classical light beams. In this respect, the derivation of the constraint ${\mathcal{D}}^{2}+{\mathcal{V}}^{2}\le 1$ that was presented in this work should be compared with others, e.g., those of Refs. [26,27], which are seemingly framed by the quantum axiomatic. The latter approach strongly supports the wrong idea that the derived results exclusively belong to the quantum realm.

People have introduced measures, such as visibility, distinguishability, indistinguishability, and predictability, with the aim of providing the complementarity principle with quantitative support. In doing so, they have occasionally given the same name to different concepts. For this reason, two different constraints may be derived, apparently engaging the same measures. Such state of affairs has been illustrated in this work by addressing two different PCTs, both of them referring to visibility, distinguishability, and the degree of polarization. The latter should be generally understood as a two-party property, as repeatedly stressed by Eberly and collaborators. A degree of polarization can thus be defined whenever one deals with a two-state system. The same observation applies regarding concepts such as visibility and distinguishability. If, in addition, we consider correlated two-state systems, entanglement enters the picture and properties of one system may depend on changes suffered by its partner system. Hidden coherences can then be exposed under different circumstances. As we have seen, a “marker system” can have strong influence on the achievable visibility of the interference pattern that its partner system may produce. In all of this, it is the vector-space structure that plays the decisive and fundamental role. By addressing a single vector space we have access to a limited domain of physical phenomena. By addressing several vector spaces that integrate a more complex, tensor-product structure, we have access to a richer domain of physical phenomena, possibly containing new features that belong to the whole but not to its parts (see, e.g., Ref. [33]). However, we should not wrongly attribute the appearance of new features to our dealing with quantum phenomena, in cases in which we are addressing just the vector-space structure of these phenomena and nothing else. It is our hope that the results of this work will help identify such cases and so avoid drawing a quantum–classical borderline at the wrong place.

## Funding

DGI-PUCP (441).

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