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 V2+D21, which engage visibility V and distinguishability D. Recently, equality V2+D2=P2—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 V2+D2=P2, 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 V2+D21. 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 4×4 Hermitian matrix of unit trace. Writing the eigenvalues of this matrix in decreasing order, λ1λ2λ3λ4, the maximal visibility is given by Vmax=12(λ3+λ4). 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 [39]. 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 [1018]. All this hints at the existence of a wide common ground for several quantum and classical phenomena [1924]. 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 Ic at point c on a screen of a two-slit interference setup (see Fig. 1) can be written as [1]

Ic=Ia+Ib+2|ϕa*ϕb|cos[arg(ϕa*ϕb)].

 

Fig. 1. Young-type array. By addressing multiple DoFs, e.g., polarization and path, one may have access to a richer domain of phenomena than in the standard, two-slit Young setup.

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Ik=|ϕ|k2 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 ϕa and ϕb, 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,z)=ua(r,z)ϕa(q)+ub(r,z)ϕb(q) [1]. Here, uk=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” W(F) with elements ϕi*ϕj, where i,j{a,b}. It is analogous to the polarization coherence matrix W(P) with elements Ek*El, (k,l{H,V}), a matrix that is introduced when dealing with a polarized field E=EHe^H+EVe^V. The degree of polarization can be defined with reference to either matrix. Thus, the degree of F-polarization reads

PF=14DetW(F)(TrW(F))2,
and similarly for W(P). Together with PF, one can define in terms of intensities both fringe visibility and the degree of distinguishability as
VF=IcmaxIcminIcmax+Icmin=2|Wab(F)|Waa(F)+Wbb(F),DF=|IaIb|Ia+Ib=|Waa(F)Wbb(F)|Waa(F)+Wbb(F).

By direct calculation one gets the statement of the PCT:

PF2=DF2+VF2.

Thus, the PCT holds true for any 2×2 Hermitian matrix ρ=ρij|ij|. Indeed, in terms of ρ’s eigenvalues,

λ±=12[(ρ11+ρ22)±(ρ11ρ22)2+4|ρ12|2],
we can define its “degree of polarization” in a natural way, as P(λ+λ)/(λ++λ). This leads to
P2=(ρ11ρ22)2(ρ11+ρ22)2+4|ρ12|2(ρ11+ρ22)2.

By defining V=2|ρ12|/(ρ11+ρ22) and D=|ρ11ρ22|/(ρ11+ρ22), as in Eq. (3), we recover the identity P2D2+V2. Alternatively, V could be replaced by the l1-norm of coherence [25], C=|ρ12|/(ρ11+ρ22), taking into account that “visibility” refers to a rather limited interferometric scenario, while |ρ12| more generally measures the amount of coherence that ρ contains.

Note that while P is a basis-independent quantity that reflects an intrinsic property of ρ, D and V are basis dependent and convey information about how ρ relates to the particular “reference frame” that is defined by the basis {|1,|2}. When ρ represents a density matrix, ρ11 and ρ22 have the meaning of probabilities and |ρ11ρ22| 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 ρ12, which vanishes in the reference frame or basis in which ρ 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 2×2 Hermitian matrix. To this end, we write ρ in terms of the Pauli matrices σk=1,2,3 and the identity matrix σ0:

ρ=12k=03Tr(ρ·σk)σk12k=03Skσk.

By setting ρ12=|ρ12|eiϕ, we get S0=ρ11+ρ22, S1=2|ρ12|cosϕ, S2=2|ρ12|sinϕ, and S3=ρ11ρ22. The degree of polarization is defined in terms of the Stokes parameters as P2=(S12+S22+S32)/S02. Introducing the foregoing expressions of Stokes parameters in this definition, we obtain

P2=S12+S22+S32S02=4|ρ12|2+(ρ11ρ22)2(ρ11+ρ22)2.

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

D2=(S3S0)2,V2=(S1S0)2+(S2S0)2,
the following identity appears as a reshaping of the standard definition of P given in Eq. (8):
P2D2+V2.

However, this reshaping depends on how we define D and V. The above definition is suggested by S32=(ρ11ρ22)2 and S12+S22=4|ρ12|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 UI that consists of three operations: beam splitting, phase shifting, and beam merging. These operations can be represented by the unitaries UBS=(σ1+σ3)/2 for beam splitting/merging, and Uϕ=exp(iϕσ3/2) for phase shifting. Consider the incoming state ρ(i)=(1/2)k=03Skσk. After being submitted to UI=UBSUϕUBS, state ρ(i) transforms into

ρ(f)=12k=03Sk(f)σk,
where S(f)=(S0,S1,S2cosϕ+S3sinϕ,S2sinϕ+S3cosϕ). If we now calculate the intensity Iϕ at the up-detector, we get, with σ3|±=±|±,
Iϕ=Tr(|++|·ρ(f))=12(1S2S0sinϕ+S3S0cosϕ)12(1+Vcos(ϕα)),
where we have changed ρ(f)ρ(f)/S0, so that it has unit trace, and defined α=tan1(S3/S2) and
V=[(S2S0)2+(S3S0)2]1/2.

This V is—up to relabeling of the Si—the same as that of Eq. (9). Moreover, maxima and minima of Iϕ are reached when cos(ϕα)=±1, so that IϕmaxIϕmin=V, in accordance with the standard definition of visibility. As for distinguishability, it can be defined in terms of the state ρw=UBSρ(i)UBS, which is associated with a two-way alternative. By projecting this state onto |±, we get the corresponding intensities: I±=Tr(|±±|ρw)=(1±S1). We can refer to these intensities and define distinguishability as

Dw=|I+I|=|S1|.

In Ref. [26], Dw 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 ρ(i) or to ρ(f); see Eq. (11). On noting that k=13Sk2S02, we get the inequality

Dw2+V21,
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 DF2+VF2=PF21 and D2+V2=P21 [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, DF, or Dw. 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 |+ 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 UI. We use the notation σi|±i=±|±i (i=1,2,3) for the eigenvectors of the Pauli matrices. The action of UI on the input-state |+3 is as follows:
|+3UBS|+1Uϕ12(|+3+eiϕ|3)UBS12(|+1+eiϕ|1)=12[(1+eiϕ)|+3+(1eiϕ)|3]|ψϕ.

We can measure intensities Iϕ,3±|ψϕ|±3|2 and Iϕ,1±|ψϕ|±1|2 by putting appropriate setups before a powermeter. Our readings should give

Iϕ,3±|ψϕ|±3|2=12(1±cosϕ),Iϕ,1±|ψϕ|±1|2=12.

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, |ψϕ, 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 Uϕ=exp(iϕσ3/2), let us implement it by Uϕ=exp(iϕσ1/2). Then, Eq. (11) still holds but now with S(f)=(S0,S1cosϕ+S2sinϕ,S1sinϕ+S2cosϕ,S3). As phase shifting is now defined with respect to the eigenstates of σ1, we measure the intensity Iϕ with respect to the “up” eigenvector of σ1, namely, |+1=(|+3+|3)/2. Thus, we have

Iϕ=Tr(|+1+|·ρ(f))=12(1+S1S0cosϕ+S2S0sinϕ)12(1+Vcos(ϕα)),
where now α=tan1(S2/S1) and
V=IϕmaxIϕmin=[(S1S0)2+(S2S0)2]1/2,
in accordance with Eq. (9), but differing from Eq. (13).

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 Ex(t)=a1(t)exp(i[ϕ1(t)ω¯t]) and Ey(t)=a2(t)exp(i[ϕ2(t)ω¯t]). When submitted to both a compensator that retards the y component by ϵ and a polarizer set at angle θ with respect to the x direction, the incoming beam turns into E(t,θ,ϵ)=Excosθ+eiϵEysinθ. The associated intensity I(θ,ϵ)=E(t,θ,ϵ)E*(t,θ,ϵ) is then given by

I(θ,ϵ)=Jxxcos2θ+Jyysin2θ+2JxxJyycosθsinθ|jxy|cos(βxyϵ).

Here, jxy=|jxy|exp(iβxy)=Jxy/JxyJxy, and the coherency matrix J is given by

J=(Ex*ExEy*ExEx*EyEy*Ey).

Equation (20) can be written as

I(θ,ϵ)=Ix+Iy+2IxIy|jxy|cos(βxyϵ),
with Ix=Jxxcosθ and Iy=Jyysinθ. This brings us into an interferometric-like scenario. Following Ref. [28], we consider the case in which θ is kept fixed while ϵ varies. Under these circumstances, the sinusoidal variation of I(θ,ϵ) reaches maxima and minima that satisfy
Imax(ϵ)Imin(ϵ)Imax(ϵ)+Imin(ϵ)=2|Jxy|sinθcosθJxxcos2θ+Jyysin2θ.

We now define visibility and distinguishability through

Vϵ=Imax(ϵ)Imin(ϵ)Imax(ϵ)+Imin(ϵ),Dϵ=|IxIy|Ix+Iy.

We then readily obtain

Vϵ2+Dϵ2=14(sin2θcos2θ)DetJ(Jxxcos2θ+Jyysin2θ)2.

This motivates the introduction of the following matrix, in place of the coherence matrix J:

Jϵ=(Jxxcos2θeiϵJxysinθcosθeiϵJyxsinθcosθJyysin2θ).

In terms of Jϵ, the RHS of Eq. (25) reads

14DetJϵ(TrJϵ)2Pϵ2,
so that Eq. (25) can be written as follows:
Dϵ2+Vϵ2=Pϵ2.

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

Imax(θ,ϵ)Imin(θ,ϵ)Imax(θ,ϵ)+Imin(θ,ϵ)=14DetJ(Jxx+Jyy)2.

Hence, if we define the left-hand side of the above equation as a visibility V(θ,ϵ), we have that

V(θ,ϵ)=P,
with P being defined with respect to the standard polarization coherence matrix 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, DF2+VF21 derives from Eq. (4) but, in spite of the similarity in notation, it is not the same as the inequality D2+V21 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 D2+V21.

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, 4×4 matrices:

G=(GaaHHGaaHVGabHHGabHVGaaVHGaaVVGabVHGabVVGbaHHGbaHVGbbHHGbbHVGbaVHGbaVVGbbHHGbbHV),
where Gkkjj=(Gkkjj)*=Ej(k)Ej*(k). Upper indices refer to polarization and lower indices refer to a second DoF, for instance, a spatial mode. By tracing over one DoF, one gets a 2×2 matrix that refers to the other DoF:
GS=TrP(G)=(GaaHH+GaaVVGabHH+GabVVGbaHH+GbaVVGbbHH+GbbVV),GP=TrS(G)=(GaaHH+GbbHHGaaHV+GbbHVGaaVH+GbbVHGaaVV+GbbVV).

We can probe the spatial coherence matrix GS=(Gij) (i,j{a,b}) with interferometric measurements that are insensitive to polarization. Double-slit interference should have a visibility VS=2|Gab|. The degree of spatial coherence—or spatial polarization—is a unitary invariant, |λaλb|, where λa and λb denote the eigenvalues of GS. On noting that

(λaλb)2=(GaaGbb)2+4|Gab|2DS2+VS2,
we see that maximal visibility and maximal distinguishability are given by
VmaxS=DmaxS=|λaλb|.

Let us now return to the full coherence matrix G. Its diagonal form reads GD=diag(λ1,λ2,λ3,λ4), with the eigenvalues taken in decreasing order: λ1λ2λ3λ4. Let us consider all possible unitaries U such that G=UGDU. The corresponding spatial matrices are given by GS=TrP(UGDU). The distinguishability associated to GS is DS=|(GaaHH+GaaVV)(GbbHH+GbbVV)|. As has been shown in Ref. [2], the maximal value of DS, which is also the maximal value of VS, is given by

VmaxS=λ1+λ2λ3λ4=12(λ3+λ4).

In polarization subspace, where the field is described by GP, we can define similar quantities and derive similar relationships. In particular, for the corresponding “visibility” VP we obtain

VmaxP=λ1+λ3λ2λ4=12(λ2+λ4).

Thus, VmaxPVmaxS. This is a striking result. Indeed, nothing prevents us from interchanging the roles of the two involved DoFs, thereby obtaining VmaxSVmaxP. 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 GD=diag(λ1,λ2,λ3,λ4) refers to the canonical, also called “computational” basis: {|aH,|aV,|bH,|bV}. In that case, GSD=TrP(GD)=diag((λ1+λ2),(λ3+λ4)) and the corresponding visibility is V=(λ1+λ2)(λ3+λ4). By submitting GD to unitary transformations, this visibility cannot increase, as proved in Ref. [2]. Note that given some non-diagonal G in the canonical basis, its diagonal form GD refers to another basis, the one constituted by the eigenvectors of G. Take, for example, a field whose coherency matrix reads

G=12(λ1+λ200λ1λ20λ3+λ4λ3λ400λ3λ4λ3+λ40λ1λ200λ1+λ2).

The eigenvalues of G are λ1,,λ4. As for the spatial matrix GS, we have GS=TrP(G)=1S/2, i.e., half the identity matrix. Thus, GS has null visibility associated to it. Matrix G is diagonal in the basis made by the maximally entangled Bell states, |ϕ±=(|aH±|bV)/2 and |Ψ±=(|aV±|bH)/2, so that we can write it also in the form

G=λ1|ϕ+ϕ+|+λ2|ϕϕ|+λ3|Ψ+Ψ+|+λ4|ΨΨ|.

That is, the matrix representing 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 VmaxS of Eq. (35) by submitting G to the unitary transformation that makes it diagonal in the canonical basis, namely,

U=|aHϕ+|+|aVϕ|+|bHΨ+|+|bVΨ|.

This gives

GD=UGU=λ1|aHaH|+λ2|aVaV|+λ3|bHbH|+λ4|bVbV|.

Associated to this GD we have now the visibility VmaxS of Eq. (35). This visibility has thus been obtained from the original field 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 VmaxS 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 D2+V21, 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 D2+V21 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.

 

Fig. 2. Mach–Zehnder-type array that serves to forge one qubit as a two-way superposition, while a second qubit (polarization) can be manipulated with optical elements (e.g., wave plates) that realize unitaries Uj=1,2. The input beam is prepared in a polarization state ρM(0) and acts as a “marker” for the path-qubit, which is prepared in state ρS(0) by means of the BS and the first mirror (M). The phase shifter (PS) allows us to generate interference patterns at the output detectors, in which intensities I(1) and I(2) can be recorded. While unitaries Uj=1,2 act only on the polarization qubit, they are activated by the path qubit.

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

ρS(0)=|ψψ|=|α|2σσ+|β|2σσ+αβ*σ+α*βσ,
where σ=|21| and σ=|12|. To mark the two ways by means of the second DoF, we apply a non-local unitary transformation containing two local unitaries: U1 and U2. The respective actions of these unitaries on the marker system are controlled by the first, interfering system [30]:
USM=σσU1+σσU2eiϕ.

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 ρS(0)ρM(0). After having been subjected to the action of USM, the two-party system is in the state

ρSM=USM(ρS(0)ρM(0))USM=|α|2σσρM(1)+|β|2σσρM(2)+α*βσeiϕρ˜M+β*ασeiϕρ˜M.

Here, ρM(k)=UkρM(0)Uk (k=1,2), and ρ˜M=U2ρM(0)U1. This process brings S from its initial state ρS(0) into a new state that is given by

ρS=TrM(ρSM)=|α|2σσ+|β|2σσ+α*βeiϕCσ+αβ*eiϕC*σ,
where C=TrMρ˜M. System S is thus in state ρS when it is ready to exit the two-way alternative of the interferometer. To obtain the state at the output of the interferometer, we submit ρS to the action of a BS that, for simplicity, we assume to be a symmetric one. As we saw before, this action is given by UBS=(σ1+σ3)/2=(σ+σ+σ3)/2. Hence,
ρSρSout=UBSρSUBS.

Having ρSout, we can calculate the intensity measured at, say, detector 1, by projecting ρSout with |11|=σσ:

I(1)=Tr(σσρSout)=12[|α|2+|β|2+2R(αβ*eiϕC)],
where R means the real part of its argument. A similar calculation gives the intensity I(2) at the second detector. From the above expression, we see that |C| is the visibility of the interference pattern. Indeed, setting α=β=1/2, i.e., considering a symmetrically prepared input state ρS(0)=|ψψ| with |ψ=(|1+|2)/2, we get I(1)=[1+|C|cos(ϕ+argC)]/2, so that Imax(1)Imin(1)=|C|.

As for distinguishability, we refer to the marker state

ρM=TrS(ρSM)=|α|2ρM(1)+|β|2ρM(2).

The above result suggests that we measure distinguishability by means of the (trace-) distance between the marker states ρM(1) and ρM(2). We have thus the following measures for visibility and distinguishability:

V=|TrMρ˜M|,D=12Tr|ρM(1)ρM(2)|,
where |A|AA is the “positive square root” of matrix A, and D is defined so that D[0,1]. As we see, even though the motivation to define V and D derives from considering two systems, the actual definitions of these quantities, as per Eq. (48), involve only one system, the marker system. Indeed, V and D are defined solely in terms of operators acting on the marker system: ρ˜M=U2ρM(0)U1 and ρM(k)=UkρM(0)Uk, with k=1,2.

We can now derive an equality connecting V and 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.,

ρM(0)=12(1+S·σ),
where we have assumed that ρM(0) is normalized and σ(σ1,σ2,σ3). It can be readily proved [31] that the trace-distance between two states (qubits) ρ1 and ρ2, i.e., D12=Tr|ρ1ρ2|/2, is simply given by half the Euclidean distance on the 3D space to which S belongs, in our case the Poincaré sphere:
D12=12Tr|ρ1ρ2|=12|S1S2|.

We are interested in calculating the trace-distance that defines distinguishability D of Eq. (48). The involved states are ρM(1)=U1ρM(0)U1 and ρM(2)=U2ρM(0)U2. Their corresponding Stokes vectors are thus S1=R1S and S2=R2S, where Ri=1,2 are the 3D rotations that are associated to the unitaries Ui=1,2. According to Eq. (50), to get D we need the Euclidean distance |R1SR2S|. Now, this distance is invariant under rotations, so that we can equally well write it as |R11(R1SR2S)|=|SR11R2S|. Rotation R11R2 can be specified through a rotation angle γ and a rotation axis n^. It is convenient to use the Euler–Rodrigues parameters e0=cos(γ/2) and e=sin(γ/2)n^. In terms of these parameters, we can write [32]

(R11R2)S=(e02e2)S+2(e·S)e+2e0(S×e).

This immediately gives |SR11R2S|=2|e2S(e·S)ee0(S×e)|, which can be used in Eqs. (48) and (50) to get

D2=e2S2(e·S)2.

The visibility is in turn given by V=|TrM(U2ρM(0)U1)|=|TrM(U1U2ρM(0))|. The unitary U1U2 is the SU(2) version of the 3D rotation R11R2, so that U1U2=cos(γ/2)+isin(γ/2)n^·σ. We get then

V=|TrM(U1U2ρM(0))|=12|Tr[(cos(γ2)1+isin(γ2)n^·σ)(1+S^·σ)]|=|cos(γ2)+isin(γ2)n^·S|=|e0+ie·S|.

On squaring the above expression we get V2=e02+(e·S)2, which together with Eq. (52) leads to our final result:

D2+V2=e02+e2S2=cos2(γ2)+P2sin2(γ2).

We have set S2P2 to highlight the connection with the PCT. Choosing γ=π we get D2+V2=P2, which can be understood as a variant of the PCT. On the other hand, because P21, the known inequality D2+V21 follows from Eq. (54) as well. Furthermore, when the marker system is prepared in an arbitrary pure state, P2=1 and D2+V2=1, as it was previously observed in Ref. [26]. When the marker system is instead prepared in a completely random (unpolarized) state, S2=0, i.e., S=0, and we have that D=0 [see Eq. (52)], while Eq. (54) reduces to V2=cos2(γ/2). In this case we reach full visibility (V=1) for γ=0. Several other cases can be similarly analyzed on noting that V and D can be written as, say, V2=e02+(e·S)2 and D2=(e×S)2. We have control over γ and n^ through the analogous parameters defining the local unitaries U1 and U2. It is thus possible to explore several instances of the relationship between visibility and distinguishability that Eq. (54) establishes. We can indeed vary S through different state preparations, and γ through different choices of U1 and U2. 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 D2+V21 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 D2+V21 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).

REFERENCES

1. J. H. Eberly, X.-F. Qian, and A. N. Vamivakas, “Polarization coherence theorem,” Optica 4, 1113–1114 (2017). [CrossRef]  

2. A. F. Abouraddy, “What is the maximum attainable visibility by a partially coherent electromagnetic field in Young’s double-slit interference?” Opt. Express 15, 18320–18331 (2017). [CrossRef]  

3. R. J. C. Spreeuw, “Classical wave-optics analogy of quantum-information processing,” Phys. Rev. A 63, 062302 (2001). [CrossRef]  

4. O. Gamel and D. F. V. James, “Causality and the complete positivity of classical polarization maps,” Opt. Lett. 36, 2821–2823 (2011). [CrossRef]  

5. A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherence matrix tomography,” Opt. Lett. 39, 2411–2414 (2014). [CrossRef]  

6. B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, L. L. Sánchez-Soto, and G. S. Agarwal, “Experimental violation of a Bell-like inequality with optical vortex beams,” New J. Phys. 17, 113046 (2015). [CrossRef]  

7. K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and E. A. Saleh, “Optical coherence matrix tomography,” Sci. Rep. 5, 15333 (2015). [CrossRef]  

8. D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116, 250404 (2016). [CrossRef]  

9. C. Okoro, H. E. Kondakci, A. F. Abouraddy, and K. C. Toussaint Jr., “Demonstration of an optical-coherence converter,” Optica 4, 1052–1058 (2017). [CrossRef]  

10. R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998). [CrossRef]  

11. A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052114 (2007). [CrossRef]  

12. C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010). [CrossRef]  

13. X.-F. Qian and J. H. Eberly, “Entanglement and classical polarization states,” Opt. Lett. 36, 4110–4112 (2011). [CrossRef]  

14. K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013). [CrossRef]  

15. X.-F. Qian, B. Little, J. C. Howell, and J. H. Eberly, “Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields,” Optica 2, 611–615 (2015). [CrossRef]  

16. M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92, 023833 (2015). [CrossRef]  

17. A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015). [CrossRef]  

18. N. Sandeau, H. Akhouayri, A. Matzkin, and T. Durt, “Experimental violation of Tsirelson’s bound by Maxwell fields,” Phys. Rev. A 93, 053829 (2016). [CrossRef]  

19. A. Luis, “Coherence, polarization, and entanglement for classical fields,” Opt. Commun. 282, 3665–3670 (2009). [CrossRef]  

20. B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010). [CrossRef]  

21. H. Chen, T. Peng, S. Karmakar, and Y. Shih, “Simulation of Bell states with incoherent thermal light,” New J. Phys. 13, 083018 (2011). [CrossRef]  

22. J. H. Eberly, “Shimony-Wolf states and hidden coherences in classical light,” Contemp. Phys. 56, 407–416 (2015). [CrossRef]  

23. J. H. Eberly, “Correlation, coherence and context,” Laser Phys. 26, 084004 (2016). [CrossRef]  

24. J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016). [CrossRef]  

25. T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,” Phys. Rev. Lett. 113, 140401 (2014). [CrossRef]  

26. B.-G. Englert, “Fringe visibility and which-way information: an inequality,” Phys. Rev. Lett. 77, 2154–2157 (1996). [CrossRef]  

27. S. Dürr and G. Rempe, “Can wave-particle duality be based on the uncertainty relation?” Am. J. Phys. 68, 1021–1024 (2000). [CrossRef]  

28. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

29. G. Jaeger, A. Shimony, and L. Vaidman, “Two interferometric complementarities,” Phys. Rev. A 51, 54–67 (1995). [CrossRef]  

30. B.-G. Englert, “Remarks on some basic issues in quantum mechanics,” Z. Naturforsch. 54a, 11–32 (1999).

31. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2007).

32. H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, 1980).

33. F. Gori, M. Santarsiero, and R. Borghi, “Vector mode analysis of a Young interferometer,” Opt. Lett. 31, 858–860 (2006). [CrossRef]  

References

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  • |

  1. J. H. Eberly, X.-F. Qian, and A. N. Vamivakas, “Polarization coherence theorem,” Optica 4, 1113–1114 (2017).
    [Crossref]
  2. A. F. Abouraddy, “What is the maximum attainable visibility by a partially coherent electromagnetic field in Young’s double-slit interference?” Opt. Express 15, 18320–18331 (2017).
    [Crossref]
  3. R. J. C. Spreeuw, “Classical wave-optics analogy of quantum-information processing,” Phys. Rev. A 63, 062302 (2001).
    [Crossref]
  4. O. Gamel and D. F. V. James, “Causality and the complete positivity of classical polarization maps,” Opt. Lett. 36, 2821–2823 (2011).
    [Crossref]
  5. A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherence matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
    [Crossref]
  6. B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, L. L. Sánchez-Soto, and G. S. Agarwal, “Experimental violation of a Bell-like inequality with optical vortex beams,” New J. Phys. 17, 113046 (2015).
    [Crossref]
  7. K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and E. A. Saleh, “Optical coherence matrix tomography,” Sci. Rep. 5, 15333 (2015).
    [Crossref]
  8. D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116, 250404 (2016).
    [Crossref]
  9. C. Okoro, H. E. Kondakci, A. F. Abouraddy, and K. C. Toussaint, “Demonstration of an optical-coherence converter,” Optica 4, 1052–1058 (2017).
    [Crossref]
  10. R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
    [Crossref]
  11. A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052114 (2007).
    [Crossref]
  12. C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
    [Crossref]
  13. X.-F. Qian and J. H. Eberly, “Entanglement and classical polarization states,” Opt. Lett. 36, 4110–4112 (2011).
    [Crossref]
  14. K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
    [Crossref]
  15. X.-F. Qian, B. Little, J. C. Howell, and J. H. Eberly, “Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields,” Optica 2, 611–615 (2015).
    [Crossref]
  16. M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92, 023833 (2015).
    [Crossref]
  17. A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
    [Crossref]
  18. N. Sandeau, H. Akhouayri, A. Matzkin, and T. Durt, “Experimental violation of Tsirelson’s bound by Maxwell fields,” Phys. Rev. A 93, 053829 (2016).
    [Crossref]
  19. A. Luis, “Coherence, polarization, and entanglement for classical fields,” Opt. Commun. 282, 3665–3670 (2009).
    [Crossref]
  20. B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
    [Crossref]
  21. H. Chen, T. Peng, S. Karmakar, and Y. Shih, “Simulation of Bell states with incoherent thermal light,” New J. Phys. 13, 083018 (2011).
    [Crossref]
  22. J. H. Eberly, “Shimony-Wolf states and hidden coherences in classical light,” Contemp. Phys. 56, 407–416 (2015).
    [Crossref]
  23. J. H. Eberly, “Correlation, coherence and context,” Laser Phys. 26, 084004 (2016).
    [Crossref]
  24. J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016).
    [Crossref]
  25. T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,” Phys. Rev. Lett. 113, 140401 (2014).
    [Crossref]
  26. B.-G. Englert, “Fringe visibility and which-way information: an inequality,” Phys. Rev. Lett. 77, 2154–2157 (1996).
    [Crossref]
  27. S. Dürr and G. Rempe, “Can wave-particle duality be based on the uncertainty relation?” Am. J. Phys. 68, 1021–1024 (2000).
    [Crossref]
  28. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).
  29. G. Jaeger, A. Shimony, and L. Vaidman, “Two interferometric complementarities,” Phys. Rev. A 51, 54–67 (1995).
    [Crossref]
  30. B.-G. Englert, “Remarks on some basic issues in quantum mechanics,” Z. Naturforsch. 54a, 11–32 (1999).
  31. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2007).
  32. H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, 1980).
  33. F. Gori, M. Santarsiero, and R. Borghi, “Vector mode analysis of a Young interferometer,” Opt. Lett. 31, 858–860 (2006).
    [Crossref]

2017 (3)

2016 (4)

D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116, 250404 (2016).
[Crossref]

N. Sandeau, H. Akhouayri, A. Matzkin, and T. Durt, “Experimental violation of Tsirelson’s bound by Maxwell fields,” Phys. Rev. A 93, 053829 (2016).
[Crossref]

J. H. Eberly, “Correlation, coherence and context,” Laser Phys. 26, 084004 (2016).
[Crossref]

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016).
[Crossref]

2015 (6)

J. H. Eberly, “Shimony-Wolf states and hidden coherences in classical light,” Contemp. Phys. 56, 407–416 (2015).
[Crossref]

X.-F. Qian, B. Little, J. C. Howell, and J. H. Eberly, “Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields,” Optica 2, 611–615 (2015).
[Crossref]

M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92, 023833 (2015).
[Crossref]

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, L. L. Sánchez-Soto, and G. S. Agarwal, “Experimental violation of a Bell-like inequality with optical vortex beams,” New J. Phys. 17, 113046 (2015).
[Crossref]

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and E. A. Saleh, “Optical coherence matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

2014 (2)

A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherence matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
[Crossref]

T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,” Phys. Rev. Lett. 113, 140401 (2014).
[Crossref]

2013 (1)

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

2011 (3)

2010 (2)

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[Crossref]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref]

2009 (1)

A. Luis, “Coherence, polarization, and entanglement for classical fields,” Opt. Commun. 282, 3665–3670 (2009).
[Crossref]

2007 (1)

A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052114 (2007).
[Crossref]

2006 (1)

2001 (1)

R. J. C. Spreeuw, “Classical wave-optics analogy of quantum-information processing,” Phys. Rev. A 63, 062302 (2001).
[Crossref]

2000 (1)

S. Dürr and G. Rempe, “Can wave-particle duality be based on the uncertainty relation?” Am. J. Phys. 68, 1021–1024 (2000).
[Crossref]

1999 (1)

B.-G. Englert, “Remarks on some basic issues in quantum mechanics,” Z. Naturforsch. 54a, 11–32 (1999).

1998 (1)

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
[Crossref]

1996 (1)

B.-G. Englert, “Fringe visibility and which-way information: an inequality,” Phys. Rev. Lett. 77, 2154–2157 (1996).
[Crossref]

1995 (1)

G. Jaeger, A. Shimony, and L. Vaidman, “Two interferometric complementarities,” Phys. Rev. A 51, 54–67 (1995).
[Crossref]

Abouraddy, A. F.

A. F. Abouraddy, “What is the maximum attainable visibility by a partially coherent electromagnetic field in Young’s double-slit interference?” Opt. Express 15, 18320–18331 (2017).
[Crossref]

C. Okoro, H. E. Kondakci, A. F. Abouraddy, and K. C. Toussaint, “Demonstration of an optical-coherence converter,” Optica 4, 1052–1058 (2017).
[Crossref]

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and E. A. Saleh, “Optical coherence matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherence matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
[Crossref]

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052114 (2007).
[Crossref]

Agarwal, G. S.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, L. L. Sánchez-Soto, and G. S. Agarwal, “Experimental violation of a Bell-like inequality with optical vortex beams,” New J. Phys. 17, 113046 (2015).
[Crossref]

Aiello, A.

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

Akhouayri, H.

N. Sandeau, H. Akhouayri, A. Matzkin, and T. Durt, “Experimental violation of Tsirelson’s bound by Maxwell fields,” Phys. Rev. A 93, 053829 (2016).
[Crossref]

Al Qasimi, A.

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016).
[Crossref]

Ali, H.

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016).
[Crossref]

Alonso, M. A.

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016).
[Crossref]

Baltanás, J. P.

D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116, 250404 (2016).
[Crossref]

Baumgratz, T.

T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,” Phys. Rev. Lett. 113, 140401 (2014).
[Crossref]

Borges, C. V. S.

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[Crossref]

Borghi, R.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref]

F. Gori, M. Santarsiero, and R. Borghi, “Vector mode analysis of a Young interferometer,” Opt. Lett. 31, 858–860 (2006).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

Cabello, A.

D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116, 250404 (2016).
[Crossref]

Chen, H.

H. Chen, T. Peng, S. Karmakar, and Y. Shih, “Simulation of Bell states with incoherent thermal light,” New J. Phys. 13, 083018 (2011).
[Crossref]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2007).

Cramer, M.

T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,” Phys. Rev. Lett. 113, 140401 (2014).
[Crossref]

Di Giuseppe, G.

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

Dürr, S.

S. Dürr and G. Rempe, “Can wave-particle duality be based on the uncertainty relation?” Am. J. Phys. 68, 1021–1024 (2000).
[Crossref]

Durt, T.

N. Sandeau, H. Akhouayri, A. Matzkin, and T. Durt, “Experimental violation of Tsirelson’s bound by Maxwell fields,” Phys. Rev. A 93, 053829 (2016).
[Crossref]

Eberly, J. H.

J. H. Eberly, X.-F. Qian, and A. N. Vamivakas, “Polarization coherence theorem,” Optica 4, 1113–1114 (2017).
[Crossref]

J. H. Eberly, “Correlation, coherence and context,” Laser Phys. 26, 084004 (2016).
[Crossref]

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016).
[Crossref]

J. H. Eberly, “Shimony-Wolf states and hidden coherences in classical light,” Contemp. Phys. 56, 407–416 (2015).
[Crossref]

X.-F. Qian, B. Little, J. C. Howell, and J. H. Eberly, “Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields,” Optica 2, 611–615 (2015).
[Crossref]

X.-F. Qian and J. H. Eberly, “Entanglement and classical polarization states,” Opt. Lett. 36, 4110–4112 (2011).
[Crossref]

Englert, B.-G.

B.-G. Englert, “Remarks on some basic issues in quantum mechanics,” Z. Naturforsch. 54a, 11–32 (1999).

B.-G. Englert, “Fringe visibility and which-way information: an inequality,” Phys. Rev. Lett. 77, 2154–2157 (1996).
[Crossref]

Fernández-Prieto, A.

D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116, 250404 (2016).
[Crossref]

Forbes, A.

M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92, 023833 (2015).
[Crossref]

Freire, M. J.

D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116, 250404 (2016).
[Crossref]

Frustaglia, D.

D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116, 250404 (2016).
[Crossref]

Gamel, O.

Giacobino, E.

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

Goldstein, H.

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, 1980).

Gori, F.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref]

F. Gori, M. Santarsiero, and R. Borghi, “Vector mode analysis of a Young interferometer,” Opt. Lett. 31, 858–860 (2006).
[Crossref]

Gutiérrez-Cuevas, R.

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016).
[Crossref]

Hor-Meyll, M.

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[Crossref]

Howell, J. C.

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016).
[Crossref]

X.-F. Qian, B. Little, J. C. Howell, and J. H. Eberly, “Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields,” Optica 2, 611–615 (2015).
[Crossref]

Hradil, Z.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, L. L. Sánchez-Soto, and G. S. Agarwal, “Experimental violation of a Bell-like inequality with optical vortex beams,” New J. Phys. 17, 113046 (2015).
[Crossref]

Huguenin, J. A. O.

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[Crossref]

Jaeger, G.

G. Jaeger, A. Shimony, and L. Vaidman, “Two interferometric complementarities,” Phys. Rev. A 51, 54–67 (1995).
[Crossref]

James, D. F. V.

Kagalwala, K. H.

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and E. A. Saleh, “Optical coherence matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherence matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
[Crossref]

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

Karmakar, S.

H. Chen, T. Peng, S. Karmakar, and Y. Shih, “Simulation of Bell states with incoherent thermal light,” New J. Phys. 13, 083018 (2011).
[Crossref]

Khoury, A. Z.

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[Crossref]

Kondakci, H. E.

C. Okoro, H. E. Kondakci, A. F. Abouraddy, and K. C. Toussaint, “Demonstration of an optical-coherence converter,” Optica 4, 1052–1058 (2017).
[Crossref]

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and E. A. Saleh, “Optical coherence matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

Konrad, T.

M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92, 023833 (2015).
[Crossref]

Leuchs, G.

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

Little, B.

Little, B. J.

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016).
[Crossref]

Losada, V.

D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116, 250404 (2016).
[Crossref]

Luis, A.

A. Luis, “Coherence, polarization, and entanglement for classical fields,” Opt. Commun. 282, 3665–3670 (2009).
[Crossref]

Lujambio, A.

D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116, 250404 (2016).
[Crossref]

Malhotra, T.

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016).
[Crossref]

Marquardt, C.

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

Matzkin, A.

N. Sandeau, H. Akhouayri, A. Matzkin, and T. Durt, “Experimental violation of Tsirelson’s bound by Maxwell fields,” Phys. Rev. A 93, 053829 (2016).
[Crossref]

McLaren, M.

M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92, 023833 (2015).
[Crossref]

Motka, L.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, L. L. Sánchez-Soto, and G. S. Agarwal, “Experimental violation of a Bell-like inequality with optical vortex beams,” New J. Phys. 17, 113046 (2015).
[Crossref]

Mukunda, N.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref]

Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2007).

Okoro, C.

Peng, T.

H. Chen, T. Peng, S. Karmakar, and Y. Shih, “Simulation of Bell states with incoherent thermal light,” New J. Phys. 13, 083018 (2011).
[Crossref]

Plenio, M. B.

T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,” Phys. Rev. Lett. 113, 140401 (2014).
[Crossref]

Qian, X.-F.

Rehacek, J.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, L. L. Sánchez-Soto, and G. S. Agarwal, “Experimental violation of a Bell-like inequality with optical vortex beams,” New J. Phys. 17, 113046 (2015).
[Crossref]

Rempe, G.

S. Dürr and G. Rempe, “Can wave-particle duality be based on the uncertainty relation?” Am. J. Phys. 68, 1021–1024 (2000).
[Crossref]

Saleh, B. E. A.

A. F. Abouraddy, K. H. Kagalwala, and B. E. A. Saleh, “Two-point optical coherence matrix tomography,” Opt. Lett. 39, 2411–2414 (2014).
[Crossref]

A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052114 (2007).
[Crossref]

Saleh, E. A.

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and E. A. Saleh, “Optical coherence matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

Sánchez-Soto, L. L.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, L. L. Sánchez-Soto, and G. S. Agarwal, “Experimental violation of a Bell-like inequality with optical vortex beams,” New J. Phys. 17, 113046 (2015).
[Crossref]

Sandeau, N.

N. Sandeau, H. Akhouayri, A. Matzkin, and T. Durt, “Experimental violation of Tsirelson’s bound by Maxwell fields,” Phys. Rev. A 93, 053829 (2016).
[Crossref]

Santarsiero, M.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref]

F. Gori, M. Santarsiero, and R. Borghi, “Vector mode analysis of a Young interferometer,” Opt. Lett. 31, 858–860 (2006).
[Crossref]

Shih, Y.

H. Chen, T. Peng, S. Karmakar, and Y. Shih, “Simulation of Bell states with incoherent thermal light,” New J. Phys. 13, 083018 (2011).
[Crossref]

Shimony, A.

G. Jaeger, A. Shimony, and L. Vaidman, “Two interferometric complementarities,” Phys. Rev. A 51, 54–67 (1995).
[Crossref]

Simon, B. N.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref]

Simon, R.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref]

Simon, S.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref]

Spreeuw, R. J. C.

R. J. C. Spreeuw, “Classical wave-optics analogy of quantum-information processing,” Phys. Rev. A 63, 062302 (2001).
[Crossref]

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
[Crossref]

Stoklasa, B.

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, L. L. Sánchez-Soto, and G. S. Agarwal, “Experimental violation of a Bell-like inequality with optical vortex beams,” New J. Phys. 17, 113046 (2015).
[Crossref]

Teich, M. C.

A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052114 (2007).
[Crossref]

Töppel, F.

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

Toussaint, K. C.

Vaidman, L.

G. Jaeger, A. Shimony, and L. Vaidman, “Two interferometric complementarities,” Phys. Rev. A 51, 54–67 (1995).
[Crossref]

Vamivakas, A. N.

J. H. Eberly, X.-F. Qian, and A. N. Vamivakas, “Polarization coherence theorem,” Optica 4, 1113–1114 (2017).
[Crossref]

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016).
[Crossref]

Velázquez-Ahumada, M. C.

D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116, 250404 (2016).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

Yarnall, T.

A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052114 (2007).
[Crossref]

Am. J. Phys. (1)

S. Dürr and G. Rempe, “Can wave-particle duality be based on the uncertainty relation?” Am. J. Phys. 68, 1021–1024 (2000).
[Crossref]

Contemp. Phys. (1)

J. H. Eberly, “Shimony-Wolf states and hidden coherences in classical light,” Contemp. Phys. 56, 407–416 (2015).
[Crossref]

Found. Phys. (1)

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
[Crossref]

Laser Phys. (1)

J. H. Eberly, “Correlation, coherence and context,” Laser Phys. 26, 084004 (2016).
[Crossref]

Nat. Photonics (1)

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[Crossref]

New J. Phys. (3)

A. Aiello, F. Töppel, C. Marquardt, E. Giacobino, and G. Leuchs, “Quantum-like nonseparable structures in optical beams,” New J. Phys. 17, 043024 (2015).
[Crossref]

B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, L. L. Sánchez-Soto, and G. S. Agarwal, “Experimental violation of a Bell-like inequality with optical vortex beams,” New J. Phys. 17, 113046 (2015).
[Crossref]

H. Chen, T. Peng, S. Karmakar, and Y. Shih, “Simulation of Bell states with incoherent thermal light,” New J. Phys. 13, 083018 (2011).
[Crossref]

Opt. Commun. (1)

A. Luis, “Coherence, polarization, and entanglement for classical fields,” Opt. Commun. 282, 3665–3670 (2009).
[Crossref]

Opt. Express (1)

A. F. Abouraddy, “What is the maximum attainable visibility by a partially coherent electromagnetic field in Young’s double-slit interference?” Opt. Express 15, 18320–18331 (2017).
[Crossref]

Opt. Lett. (4)

Optica (3)

Phys. Rev. A (6)

R. J. C. Spreeuw, “Classical wave-optics analogy of quantum-information processing,” Phys. Rev. A 63, 062302 (2001).
[Crossref]

M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92, 023833 (2015).
[Crossref]

N. Sandeau, H. Akhouayri, A. Matzkin, and T. Durt, “Experimental violation of Tsirelson’s bound by Maxwell fields,” Phys. Rev. A 93, 053829 (2016).
[Crossref]

A. F. Abouraddy, T. Yarnall, B. E. A. Saleh, and M. C. Teich, “Violation of Bell’s inequality with continuous spatial variables,” Phys. Rev. A 75, 052114 (2007).
[Crossref]

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[Crossref]

G. Jaeger, A. Shimony, and L. Vaidman, “Two interferometric complementarities,” Phys. Rev. A 51, 54–67 (1995).
[Crossref]

Phys. Rev. Lett. (4)

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[Crossref]

T. Baumgratz, M. Cramer, and M. B. Plenio, “Quantifying coherence,” Phys. Rev. Lett. 113, 140401 (2014).
[Crossref]

B.-G. Englert, “Fringe visibility and which-way information: an inequality,” Phys. Rev. Lett. 77, 2154–2157 (1996).
[Crossref]

D. Frustaglia, J. P. Baltanás, M. C. Velázquez-Ahumada, A. Fernández-Prieto, A. Lujambio, V. Losada, M. J. Freire, and A. Cabello, “Classical physics and the bounds of quantum correlations,” Phys. Rev. Lett. 116, 250404 (2016).
[Crossref]

Phys. Scripta (1)

J. H. Eberly, X.-F. Qian, A. Al Qasimi, H. Ali, M. A. Alonso, R. Gutiérrez-Cuevas, B. J. Little, J. C. Howell, T. Malhotra, and A. N. Vamivakas, “Quantum and classical optics—emerging links,” Phys. Scripta 91, 063003 (2016).
[Crossref]

Sci. Rep. (1)

K. H. Kagalwala, H. E. Kondakci, A. F. Abouraddy, and E. A. Saleh, “Optical coherence matrix tomography,” Sci. Rep. 5, 15333 (2015).
[Crossref]

Z. Naturforsch. (1)

B.-G. Englert, “Remarks on some basic issues in quantum mechanics,” Z. Naturforsch. 54a, 11–32 (1999).

Other (3)

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2007).

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, 1980).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

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Figures (2)

Fig. 1.
Fig. 1. Young-type array. By addressing multiple DoFs, e.g., polarization and path, one may have access to a richer domain of phenomena than in the standard, two-slit Young setup.
Fig. 2.
Fig. 2. Mach–Zehnder-type array that serves to forge one qubit as a two-way superposition, while a second qubit (polarization) can be manipulated with optical elements (e.g., wave plates) that realize unitaries Uj=1,2. The input beam is prepared in a polarization state ρM(0) and acts as a “marker” for the path-qubit, which is prepared in state ρS(0) by means of the BS and the first mirror (M). The phase shifter (PS) allows us to generate interference patterns at the output detectors, in which intensities I(1) and I(2) can be recorded. While unitaries Uj=1,2 act only on the polarization qubit, they are activated by the path qubit.

Equations (54)

Equations on this page are rendered with MathJax. Learn more.

Ic=Ia+Ib+2|ϕa*ϕb|cos[arg(ϕa*ϕb)].
PF=14DetW(F)(TrW(F))2,
VF=IcmaxIcminIcmax+Icmin=2|Wab(F)|Waa(F)+Wbb(F),DF=|IaIb|Ia+Ib=|Waa(F)Wbb(F)|Waa(F)+Wbb(F).
PF2=DF2+VF2.
λ±=12[(ρ11+ρ22)±(ρ11ρ22)2+4|ρ12|2],
P2=(ρ11ρ22)2(ρ11+ρ22)2+4|ρ12|2(ρ11+ρ22)2.
ρ=12k=03Tr(ρ·σk)σk12k=03Skσk.
P2=S12+S22+S32S02=4|ρ12|2+(ρ11ρ22)2(ρ11+ρ22)2.
D2=(S3S0)2,V2=(S1S0)2+(S2S0)2,
P2D2+V2.
ρ(f)=12k=03Sk(f)σk,
Iϕ=Tr(|++|·ρ(f))=12(1S2S0sinϕ+S3S0cosϕ)12(1+Vcos(ϕα)),
V=[(S2S0)2+(S3S0)2]1/2.
Dw=|I+I|=|S1|.
Dw2+V21,
|+3UBS|+1Uϕ12(|+3+eiϕ|3)UBS12(|+1+eiϕ|1)=12[(1+eiϕ)|+3+(1eiϕ)|3]|ψϕ.
Iϕ,3±|ψϕ|±3|2=12(1±cosϕ),Iϕ,1±|ψϕ|±1|2=12.
Iϕ=Tr(|+1+|·ρ(f))=12(1+S1S0cosϕ+S2S0sinϕ)12(1+Vcos(ϕα)),
V=IϕmaxIϕmin=[(S1S0)2+(S2S0)2]1/2,
I(θ,ϵ)=Jxxcos2θ+Jyysin2θ+2JxxJyycosθsinθ|jxy|cos(βxyϵ).
J=(Ex*ExEy*ExEx*EyEy*Ey).
I(θ,ϵ)=Ix+Iy+2IxIy|jxy|cos(βxyϵ),
Imax(ϵ)Imin(ϵ)Imax(ϵ)+Imin(ϵ)=2|Jxy|sinθcosθJxxcos2θ+Jyysin2θ.
Vϵ=Imax(ϵ)Imin(ϵ)Imax(ϵ)+Imin(ϵ),Dϵ=|IxIy|Ix+Iy.
Vϵ2+Dϵ2=14(sin2θcos2θ)DetJ(Jxxcos2θ+Jyysin2θ)2.
Jϵ=(Jxxcos2θeiϵJxysinθcosθeiϵJyxsinθcosθJyysin2θ).
14DetJϵ(TrJϵ)2Pϵ2,
Dϵ2+Vϵ2=Pϵ2.
Imax(θ,ϵ)Imin(θ,ϵ)Imax(θ,ϵ)+Imin(θ,ϵ)=14DetJ(Jxx+Jyy)2.
V(θ,ϵ)=P,
G=(GaaHHGaaHVGabHHGabHVGaaVHGaaVVGabVHGabVVGbaHHGbaHVGbbHHGbbHVGbaVHGbaVVGbbHHGbbHV),
GS=TrP(G)=(GaaHH+GaaVVGabHH+GabVVGbaHH+GbaVVGbbHH+GbbVV),GP=TrS(G)=(GaaHH+GbbHHGaaHV+GbbHVGaaVH+GbbVHGaaVV+GbbVV).
(λaλb)2=(GaaGbb)2+4|Gab|2DS2+VS2,
VmaxS=DmaxS=|λaλb|.
VmaxS=λ1+λ2λ3λ4=12(λ3+λ4).
VmaxP=λ1+λ3λ2λ4=12(λ2+λ4).
G=12(λ1+λ200λ1λ20λ3+λ4λ3λ400λ3λ4λ3+λ40λ1λ200λ1+λ2).
G=λ1|ϕ+ϕ+|+λ2|ϕϕ|+λ3|Ψ+Ψ+|+λ4|ΨΨ|.
U=|aHϕ+|+|aVϕ|+|bHΨ+|+|bVΨ|.
GD=UGU=λ1|aHaH|+λ2|aVaV|+λ3|bHbH|+λ4|bVbV|.
ρS(0)=|ψψ|=|α|2σσ+|β|2σσ+αβ*σ+α*βσ,
USM=σσU1+σσU2eiϕ.
ρSM=USM(ρS(0)ρM(0))USM=|α|2σσρM(1)+|β|2σσρM(2)+α*βσeiϕρ˜M+β*ασeiϕρ˜M.
ρS=TrM(ρSM)=|α|2σσ+|β|2σσ+α*βeiϕCσ+αβ*eiϕC*σ,
ρSρSout=UBSρSUBS.
I(1)=Tr(σσρSout)=12[|α|2+|β|2+2R(αβ*eiϕC)],
ρM=TrS(ρSM)=|α|2ρM(1)+|β|2ρM(2).
V=|TrMρ˜M|,D=12Tr|ρM(1)ρM(2)|,
ρM(0)=12(1+S·σ),
D12=12Tr|ρ1ρ2|=12|S1S2|.
(R11R2)S=(e02e2)S+2(e·S)e+2e0(S×e).
D2=e2S2(e·S)2.
V=|TrM(U1U2ρM(0))|=12|Tr[(cos(γ2)1+isin(γ2)n^·σ)(1+S^·σ)]|=|cos(γ2)+isin(γ2)n^·S|=|e0+ie·S|.
D2+V2=e02+e2S2=cos2(γ2)+P2sin2(γ2).

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