1. Introduction

Wave-particle duality is widely seen as a genuine quantum feature. However, when it comes to quantify it, wave-particle duality is formulated in terms of two complementary features that can be exhibited with both quantum and classical setups: an interference pattern and which-way information [1–10]. Light interferometry, for instance, can be implemented with both quantum and classical resources, single photons and optical beams, respectively. Quantifiers such as visibility ($\mathcal {V}$) and distinguishability ($\mathcal {D}$) can be obtained either from probability measurements or else from intensity measurements. Let us consider the latter, to be specific. In a two-path interferometer, one performs three intensity measurements, $I_a$, $I_b$ and $I_c$. These measurements occur at three locations in, say, a Young setup or in a Mach-Zehnder interferometer (MZI). In either case we can define an abstract vector space that is spanned by two vectors, $| {\varphi _a}\rangle$ and $|{\varphi _b}\rangle$. Here, we use Dirac notation without implying any quantum interpretation [11]. Vectors in this space can be written as $|{\varphi _c}\rangle=c_a|{\varphi _a}\rangle+c_b|{\varphi _b}\rangle$. The coefficients $c_{i=a,b}$ may carry physical information about propagation properties. For instance, $c_i=d_i\exp {(i\phi _i)}$, where $d_i=\text {const.}$ and $\phi _i=k_i z$ corresponds to the accumulation of a phase proportional to path-length $z$, without amplitude variation. Intensity measurements should be expressible in terms of the $c_i$, e.g., $I_{i}=\langle |c_i|^{2}\rangle$, where angular brackets denote some averaging procedure; for example, a statistical average. It is then useful to construct a $2\times 2$ matrix $M$ with elements $\langle c_{i}^{\ast }c_j \rangle$, where $i,j \in \left \{a,b\right \}$. We can then define distinguishability in terms of $I_{i=a,b}$, as

As for visibility, it refers to the contrast between the bright and dark fringes that make up the interferometric pattern, a pattern that can be recorded as variations of the intensity that is associated to the state $|{\varphi _c}\rangle=c_a|{\varphi _a}\rangle+c_b|{\varphi _b}\rangle$. Consider, for example, a MZI. Each of its two beam splitters (BSs) produces the transformations $|{\varphi _a}\rangle\rightarrow (|{\varphi _a}\rangle+|{\varphi _b}\rangle)/\sqrt {2}$ and $|{\varphi _b}\rangle\rightarrow (|{\varphi _a}\rangle-|{\varphi _b}\rangle)/\sqrt {2}$. Moreover, a relative phase-shift $\delta$ between $|{\varphi _a}\rangle$ and $|{\varphi _b}\rangle$ accumulates, due to propagation from the first to the second BS. Hence, if we take $|{\varphi _c}\rangle$ as the state produced by the first BS, the intensity measured at one output port of the second BS is given by [12]

On varying $\delta$, one obtains maximal and minimal intensities. Visibility can then be defined as

It has been proved [1–5] that $\mathcal {V}$ and $\mathcal {D}$ are mutually constrained by the inequality

From the hermitian matrix $M$, we can also obtain the “degree of polarization”, which can be defined in terms of the eigenvalues $\lambda _{\pm }$ of $M$, as follows:

On using $\lambda _{\pm }=2^{-1}\left ((M_{aa}+M_{bb})\pm \sqrt {(M_{aa}-M_{bb})^{2}+4|M_{ab}|^{2}}\right )$, we get

From Eqs.  (1), (3) and (6), we get the so-called polarization-coherence theorem (PCT) [6–9]:

Let us next consider a pure, two-qubit state $|\Phi \rangle$. Its two constituent qubits can be entangled. In this case, entanglement may be quantified using concurrence [13]: $\mathcal {C}(\Phi )=|\langle \Phi |\widetilde {\Phi }\rangle |$. Here, $|\widetilde {\Phi }\rangle =(\sigma _y \otimes \sigma _y)|\Phi ^{\ast }\rangle$, where $|\Phi ^{\ast }\rangle$ is the complex-conjugate of $|\Phi \rangle$ in the computational basis $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle \}$, and $\sigma _y$ is the Pauli matrix whose eigenvectors are $|{0}\rangle\pm i|{1}\rangle)/\sqrt {2}$. The two reduced density matrices, $\rho _1=\operatorname {Tr}_2|\Phi \rangle \langle \Phi |$ and $\rho _2=\operatorname {Tr}_1|\Phi \rangle \langle \Phi |$, have common eigenvalues $\lambda _{\pm }$. It can be shown that $\mathcal {C}(\Phi )=2\sqrt {\lambda _{+}\lambda _{-}}$. From this and $\mathcal {P}=|\lambda _{+}-\lambda _{-}|$, owing to $\lambda _{+}+\lambda _{-}=1$, it follows that $\mathcal {C}^{2}+\mathcal {P}^{2}=1$. We can thus write (7) in the following form, which holds for both $\rho _1$ and $\rho _2$ [14]:

The above, so-called “triality relation” has also been derived in a fully quantum framework [6]. It is worth noticing that a similar triality has been established for multipath interference [15]. In this case, the involved quantifiers are: a generalized path predictability instead of $\mathcal {D}$, Hilbert-Schmidt coherence $\mathcal {C}_{HS}$ instead of $\mathcal {V}$, and $I$-concurrence [16] instead of $\mathcal {C}$. In the multipath case, fringe contrast can increase together with path-distinguishability [17], so that visibility is not a good quantifier of wave-particle duality in that case. $\mathcal {C}_{HS}$ is instead a good quantifier and coincides with $\mathcal {V}$ in the two-path case [15]. While interference requires coherent superposition and this property is faithfully measured using $\mathcal {C}_{HS}$, wave features are not necessarily involved in state-superposition. Fringe contrast is more directly involved with the constructive/destructive interference of waves, making $\mathcal {V}$ a good quantifier of wavelike features. Unfortunately though, $\mathcal {V}$ can enter complementarity relations only in the two-path case. Experimentally, the difference between $\mathcal {V}$ and $\mathcal {C}_{HS}$ is quite tangible. While $\mathcal {V}$ is measured by comparing brightness and darkness, which reflect constructive and destructive wave interference, measuring $\mathcal {C}_{HS}$ requires performing state-tomography [18], which hardly shows a direct connection with wave features. Despite its relative simplicity, the two-path scenario seems therefore more appropriate for addressing fundamental issues, such as Bohr’s complementarity (including delayed-choice versions [19]), than other, more involved configurations.

When dealing with two qubits, one of them can play the role of a “quanton” and the other the role of a “marker”. The two qubits may be attached to one and the same physical object. The defining feature of a quanton is that it can exhibit wave-particle duality. Operationally, this means that the quanton should be capable of playing a double role. First, when submitted to an interferometric setup, the quanton must give rise to an interference pattern. This requires from the quanton that it gathers information from the various paths of the interferometer, thereby “sensing” the whole setup. Second, the quanton should be localizable, at least to some degree, on one of the paths. To this end, the quanton can be chosen so that it carries an “internal” degree of freedom, the purpose of which is to serve as a which-way marker. Notice that this operational meaning of wave-particle duality applies not only to neutrons and photons, but to classical light beams as well. In the latter case, one should perhaps better refer to “wave-ray duality”. Anyhow, it turns out that the more effective the marker is, the less visible is the interferogram. On following this approach, Englert [4] derived inequality Eq. (4). However, in this inequality and in related versions of it, only one of the two qubits was effectively engaged [2,3,7]. In this work, we report experimental results that exhibit duality through constraints that simultaneously involve the two qubits.

The PCT Eq. (7) and the triality relation Eq. (8) have been experimentally exhibited using all-optical setups [20,21]. These and other constraints shed light on the meaning of “duality”, a meaning that becomes clearer once it is expressed in quantitative terms. By referring to measurable quantities, one can give an operational definition of duality and so go beyond the quantum framework, in which duality was originally introduced. Constraints that effectively involve two qubits, such as the ones we address here, are rather contrived as compared to those involving only one qubit. Even so, one can perform experimental tests of said constraints with all-optical setups, as we show in what follows.

2. Two-qubit extension of the PCT

Let us consider a Mach-Zehnder interferometer. We can submit to this interferometer either polarized photons or else polarized light beams. We take polarization as the which-way marker. The marker can be made effective by applying a polarization-transformation $U$ that is conditioned on the path followed by the quanton (see Fig. 1). Any $U \in SU(2)$ can be implemented with three birefringent plates: two quarter-wave plates (QWPs) and one half-wave plate (HWP) [22]. The initial state of quanton and marker, $\rho ^{(i)}_{QM}$, is submitted to the interferometer. The action of the interferometer is represented by a unitary transformation $U_{QM}$ that contains the path-conditioned action of $U$. Following [4], we define visibility as $\mathcal {V}=\left | \operatorname {Tr} \left (U\rho ^{(i)}_{M}\right )\right |$, with $\rho ^{(i)}_{M}=\operatorname {Tr}_Q\rho ^{(i)}_{QM}$. Distinguishability is defined as the trace-distance between the two marker states, $\rho _M^{(i)}$ and $\rho _M^{U}=U\rho _M^{(i)}U^{\dagger }$. That is, $\mathcal {D}=\operatorname {Tr} \left |\rho _M^{U}-\rho _M^{(i)}\right |/2$, where $|M|\equiv \sqrt {M^{\dagger }M}$ for matrix $M$.

 

Fig. 1. Mach-Zehnder-type setup to deal with two qubits. The path-qubit, associated to the two-way alternative, plays the role of the “system” or “quanton”. A quanton’s internal qubit (spin, polarization, etc.) serves as a which-way “marker”. The unitary $U$ acts on the marker-qubit. BS1 and BS2 are beam splitters. An unbalanced BS1 can produce path states of the form $|\psi \rangle =\alpha |1\rangle +\beta |2\rangle$. The initial, system-marker state $\rho _{S}^{(i)}\otimes \rho _{M}^{(i)}$ is available after BS1 and submitted to a non-local unitary (see text). By setting $U$ to the identity one can address the single-qubit case.

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In the case of polarization transformations $U \in SU(2)$, they can be visualized as rotations of the Stokes vector. A convenient parametrization is given in this case by $U=\exp {(i \gamma \hat {\mathbf {n}} \cdot \boldsymbol {\sigma }/2)}$, where $\gamma$ is the rotation angle, $\hat {\mathbf {n}}$ is a unit vector along the rotation axis, and $\boldsymbol {\sigma }$ is the three-vector of Pauli matrices. One can show [23] that, in this case,

Equation (9) implies Eq. (7) for $\gamma =\pi$, and hence also Eq. (4). As we see, these constraints are established by addressing the two-dimensional polarization space. While the original goal in [4] was to employ polarization as a marker for the quanton, it turns out that polarization ended up being a surrogate for the quanton. The wave-particle quantifiers refer therefore to the two-dimensional space used to describe polarization, not the path. This is because the beam-splitter BS1 in Figure (1) is a balanced one and, therefore, the two paths have equal weights. If we replace BS1 by an unbalanced beam-splitter and remove $U$, we have the two-way alternative as a quanton without marker. It is then possible to obtain Eq. (7) and, hence, also Eq. (4) [24]. Having an unbalanced beam-splitter and $U$ in place, we can activate both quanton and marker. Next, we show what happens in such a case.

Let us consider an initial, quanton-marker state given by $\rho _{QM}^{(i)}=\rho _{Q}^{(i)}\otimes \rho _{M}^{(i)}$, with $\rho _{Q}^{(i)}=|\psi ^{(i)}_Q\rangle \langle \psi ^{(i)}_Q|$ and $|\psi ^{(i)}_Q\rangle =\alpha |1\rangle +\beta |2\rangle$. He have thus assumed that the quanton is initially prepared in a pure state. As for the marker, its initial state can be generally written in the form $\rho _{M}^{(i)}= \left (\sigma _0+\mathbf {S}^{(i)} \cdot \boldsymbol {\sigma }\right )/2$, where $\sigma _0$ stands for the unit matrix. The length of the Stokes vector $\mathbf {S}^{(i)}$ is the degree of polarization: $\mathcal {P}\equiv |\mathbf {S}^{(i)}|\leq 1$. Pure states correspond to $\mathcal {P}=1$. State $\rho _{QM}^{(i)}$ is assumed to be available after BS1 (see Fig. 1) and is then submitted to the unitary transformation

Before entering BS2, the two-qubit system is in the state $\rho _{QM}=U_{QM}\rho _{QM}^{(i)}U_{QM}^{\dagger }$. Single-qubit states are obtained by partial tracing. The state of the quanton is thus $\rho _Q=\operatorname {Tr}_M \rho _{QM}$, and the state of the marker is given, explicitly, by

Hence, $\rho _M$ is the weighted sum of two states, each one being related to one arm of the interferometer. We can then define distinguishability $\mathcal {D}$ as

If $|\alpha |^{2}=|\beta |^{2}=1/2$, then $\mathcal {D}$ reduces to the previously defined distinguishability [4]: $\mathcal {D}=\operatorname {Tr} \left |\rho _M^{U}-\rho _M^{(i)}\right |/2$. The definition of $\mathcal {D}$ in Eq. (12) takes into account both the biased path-choice and the application or not of the polarization transformation.

In order to quantify visibility, let us consider the intensity measured at one of the two output ports of the second beam-splitter (BS2):

We define visibility as we did before, i.e., as

From Eq. (14), we get

From Eq. (12), we readily obtain [24]

A direct calculation gives $|\mathbf {S}|^{2}=\left \{|\alpha |^{4}+|\beta |^{4}-2|\alpha |^{2}|\beta |^{2}\left [(e_0^{2}-\mathbf {e}^{2}+2\mathbf {e}^{2}\cos ^{2}\varphi )\right ]\right \}\mathcal {P}^{2}$, and

Equation (18) generalizes the PCT [7] to the two-qubit case. On setting $\alpha =\beta =1/\sqrt {2}$, we recover the previous results we have discussed. Indeed, in that case we must pick the first option in Eq. (18), setting $D_w=0$ and $V_w=1$. We then obtain $\mathcal {D}^{2}+\mathcal {V}^{2}=e_0^{2}+\mathcal {P}^{2} \mathbf {e}^{2}$. On substituting $e_0=\cos (\gamma /2)$ and $\mathbf {e}^{2}=\sin ^{2}(\gamma /2)$, we get Eq. (9), the one-qubit extension of the PCT. Our generalization of the PCT contains visibilities and distinguishabilities that refer to two different qubits. Such a generalization differs from other, recently reported ones [25–28]. Notice that, while $D_w$ and $V_w$ refer to the path-qubit only, $\mathcal {D}$ and $\mathcal {V}$ refer to both the path-qubit and the polarization-qubit.

The generalized PCT, given by Eq. (18), is a rather involved relationship and, at first sight, hardly accessible to experimental tests. However, linear optics provides versatile tools that allow us to implement intricate transformations using various degrees of freedom. We have exploited the versatility of all-optical setups to experimentally exhibit the generalized, two-qubit version of the PCT. Next, we discuss our setup and the results of our measurements.

3. All-optical setup and experimental results

We performed experimental tests of constraint Eq. (18) using classical light; a polarized beam from a HeNe laser source (633 nm). One qubit was realized by the polarization of the beam and the other by its path. Our two-qubit states spanned therefore a vector space whose basis can be taken to be the tensor product $\{|H\rangle, |V\rangle \} \otimes \{|1\rangle, |2\rangle \}$. Our setup is shown in Fig (2). Panel (a) contains the stage where the pure (path) quanton-state $|\psi _Q^{(i)}\rangle$ is generated. Panel (b) contains the stage used for generation of the mixed (polarization) marker-state $\rho _M^{(i)}$. Panel (c) corresponds to the interferometric stage.

 

Fig. 2. Experimental setup. Part a): generation of the path state $|\psi _Q^{(i)}\rangle$. Part b): generation of the mixed polarization state $\rho _M^{(i)}$. Part c): Interferometric arrangement. H: half-wave plate. Q: quarter-wave plate. LDPBS: lateral displacement polarizing beam-splitter. PBS: polarizing beam-splitter. BS: 50:50 beam-splitter. LDBS: lateral displacement 50:50 beam-splitter. M: mirror. The two figures were made using the 3DOptix simulation software.

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We produced the path-state $|\psi _Q^{(i)}\rangle =\alpha |1\rangle + \beta |2\rangle$ as follows. First, a HWP set to $\delta +\pi /4$ (denoted $H(\delta + \pi /4)$), transformed the vertically polarized light produced by the HeNe laser into the coherent superposition $\cos (2\delta )|H\rangle +\sin (2\delta )|V\rangle$. By means of a lateral displacement polarizing beam splitter (LDPBS), the incoming, linearly polarized light was separated into a horizontally polarized beam on path $|1\rangle$, and a vertically polarized beam on path $|2\rangle$. In this way, the two-qubit state $\cos (2\delta )|H\rangle |1\rangle +\sin (2\delta )|V\rangle |2\rangle$ was generated. From it, we obtained the target path-state, $|\psi _Q^{(i)}\rangle =\cos (2\delta )|1\rangle +\sin (2\delta )|2\rangle$, by changing the polarization of beam $|1\rangle$ from horizontal to vertical. This was done with a HWP set to $\pi /4$.

Next, we generated the mixed polarization state $\rho _M^{(i)}$ on each of the two beams, using the elements shown in Fig. (2), panel (b). In this way, we produced the initial, quanton-marker state $\rho _{QM}^{(i)}=\rho _{Q}^{(i)}\otimes \rho _{M}^{(i)}$. To produce $\rho _M^{(i)}$, we submitted the beams to a HWP set to $\pi /2 - \theta$. This transformed the polarization states to $\sin (2\theta )|{H}\rangle+\cos (2\theta )|{V}\rangle$. A polarizing beam splitter (PBS) then separated the horizontal and vertical polarization components of each beam into a pair of transmitted and reflected beams, respectively. After the BS (see Fig. (2), panel (b)), the original components of each path were incoherently recombined. As a result, we had two beams, for paths $|1\rangle$ and $|2\rangle$ respectively, in the mixed polarization state $\rho _M^{(i)} = \sin ^{2} (2\theta ) |H\rangle \langle H| + \cos ^{2} (2\theta ) |V\rangle \langle V|$.

Finally, the two beams were submitted to an interferometric arrangement, shown in Fig. (2), panel (c). A QWP set to $\pi /4 - \varphi /2$ transformed the mixed polarization state, so that its Stokes vector made an angle $\varphi$ with the $\hat {\mathbf {z}}$ axis of the Poincaré sphere. Two HWPs, $H(0)H(-\gamma /4)$, set on path $|1\rangle$, implemented the unitary transformation $U =\exp {(- i \gamma \hat {\mathbf {n}} \cdot \boldsymbol \sigma /2)}$, where $\hat {\mathbf {n}}=(0,0,1)$ and $\boldsymbol {\sigma } \equiv (\sigma _z, \sigma _x, \sigma _y)$. This corresponds to a rotation by an angle $\gamma$ around the $\hat {\mathbf {z}}$ axis of the Poincaré sphere (for the Bloch sphere, $\boldsymbol {\sigma } \equiv (\sigma _x, \sigma _y, \sigma _z)$). The HWP set to $0$ served an additional purpose. It compensated the relative $\pi$-phase that the lateral displacement beam splitter (LDBS) introduces between horizontal and vertical components of the polarized beam that propagates along path $|2\rangle$. The LDBS was used to superpose the beams in mixed polarization states and so achieve interference between them. The choice $\hat {\mathbf {n}}=\hat {\mathbf {z}}$ implies no essential loss of generality. If one wants to implement different transformations $U =\exp {(- i \gamma \hat {\mathbf {n}} \cdot \boldsymbol \sigma /2)}$, the only change needed is the replacement of $H(0)H(-\gamma /4)$ by two QWPs and one HWP [22] on path $|1\rangle$, plus a phase-compensating $H(0)$ on path $|2\rangle$. As can be seen in Fig. (2), panel (c), this could be easily done.

To test Eq. (18), we obtained the quantities $\mathcal {V}$, $\mathcal {D}$, $D_w$ and $V_w$ from the measured parameters that enter their respective definitions. We repeated 5 times the measurement of each parameter. The intensity measurements were performed with a photomultiplier (Thorlabs PM100D). To measure fringe visibility $\mathcal {V}$, maximum and minimum intensities were produced by varying the separation between the two mirrors shown on panel (c). This separation controls the relative phase $\phi$ between the interfering beams in the LDBS (see Eq. (13)). To obtain $\mathcal {D}$, we set a QWP, a HWP and a polarizer at the output of the LDBS shown in Fig. (2), panel (c). At the output of the LDBS, we performed Stokes polarimetry for each beam within the interferometer. From this, we got the quantities $\alpha$, $\beta$, $\boldsymbol S^{(i)}$ and $\boldsymbol S^{(u)} = \mathcal {R} \boldsymbol S^{(i)}$.

We tested (18) by considering three cases, with a corresponding choice of values for the parameters $(\delta, \theta, \varphi, \gamma )$. In the first case, we varied $\gamma$ in the range $0 < \gamma < 2 \pi$, and kept fixed $\delta, \theta, \varphi$. The vector $\boldsymbol S^{(i)}$ then lies on the $XY$-plane and is parallel to the positive $\hat {\mathbf {x}}$ axis, whereas $\boldsymbol S^{(u)}$ results from rotating $\boldsymbol S^{(i)}$ by an angle $\gamma$ about the $\hat {\mathbf {z}}$ axis. In the second case, we varied $\delta$ in the range $0 < \delta < \pi /4$, keeping $\theta, \varphi, \gamma$ constant. In the third case, we varied $\varphi$ in the range $0 < \varphi < \pi$, keeping constant $\delta, \theta, \gamma$. Vector $\boldsymbol S^{(u)}$ results from rotating $\boldsymbol S^{(i)}$ by an angle $\gamma = \pi$ about the $\hat {\mathbf {z}}$ axis. Both vectors made an angle $\varphi$ with respect to the $\hat {\mathbf {z}}$ axis. The plane that contains these vectors rotates the same angle about the $\hat {\mathbf {z}}$ axis. Figures (3–5) show our measurement results and the corresponding Stokes vectors, $\boldsymbol S^{(i)}$ and $\boldsymbol S^{(u)}$, for each case. As we can see, there is good agreement between theoretical results and experimental outputs.

 

Fig. 3. First case: $\delta = \pi /3, \theta = \pi /16, \varphi = \pi /2, 0 < \gamma < 2\pi$. Left panel: Experimental test of the two-qubit extension of the PCT, which is embodied in the behavior of $\mathcal {V}^{2} +\mathcal {D}^{2}$. Right panel: Geometrical representation of the Stokes vectors $\boldsymbol S^{(i)}$ and $\boldsymbol S^{(u)} = \mathcal {R} \boldsymbol S^{(i)}$.

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Fig. 4. Second case: similar to the first case, but with parameter choice $0 < \delta < \pi /4, \theta = \pi /16, \varphi = \pi /2, \gamma = \pi /2$.

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Fig. 5. Third case: similar to the first case, but with parameter choice $\delta = \pi /6, \theta = \pi /18, 0 < \varphi < \pi, \gamma = \pi$.

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4. Summary and conclusions

The quantification of wave-particle duality by means of visibility and distinguishability led to the establishment of various constraints involving these quantities. A first, simple constraint: $\mathcal {D}^{2}+\mathcal {V}^{2} \leq 1$, gradually evolved into more sophisticated expressions that included additional measurable quantities, such as concurrence and polarization. Moreover, the inclusion of a which-way marker, an internal degree of freedom carried by quantons – physical entities capable of exhibiting wave-particle duality – allowed to formulate generalized constraints. Among these, there are constraints in which $\mathcal {D}$ and $\mathcal {V}$ turn out to be related to both quanton and marker, while additional visibilities and distinguishabilities show up, which are related only to the quanton or only to the marker. All this discloses a rather intricate connection among various degrees of freedom and coherences that belong to the physical objects being addressed. The disclosed connections are neither quantal nor classical. They are properties that derive from the linear vector-space structure that is shared by some quantum and classical phenomena. Light, in its various manifestations, classical and quantal, lends itself as an almost ideal means to exhibit the aforementioned connections. In this work, we addressed constraints that, at first sight, seemed to be hardly accessible to experimental tests. However, linear optics proved to be an adequate tool to accomplish the task, in our case using classical light beams. Our results showed very good agreement between theory and experiment. With slight modifications, our setup would also serve to run tests with single photons. Due to its applicability to both quantum and classical phenomena, in which what matters is the involved degrees of freedom, the tested constraints can been seen as a manifestation of generalized wave-particle duality.