# Quantum-inspired protocol for measuring the degree of similarity between
spatial shapes

Daniel F. Urrego<EMAIL_ADDRESS>ICFO – Institut de Ciencies
Fotoniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels, Barcelona, Spain.  Juan P. Torres  ICFO – Institut de Ciencies
Fotoniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels, Barcelona, Spain.  Department of Signal Theory and
Communications, Universitat Politecnica de Catalunya, 08034 Barcelona, Spain

(September 3, 2024)

###### Abstract

We put forward and demonstrate experimentally a quantum-inspired protocol that
allows to quantify the degree of similarity between two spatial shapes
embedded in two optical beams without the need to measure the amplitude and
phase across each beam. Instead the sought-after information can be retrieved
measuring the degree of polarization of the combined optical beam, a
measurement that is much easier to implement experimentally. The protocol
makes use of non-separable optical beams, whose main trait is that different
degrees of freedom (polarization and spatial shape here) can not be described
independently.

##  I Introduction

Entanglement is a genuine quantum feature that describes the quantum
correlations that exist between physically separated systems whose origin
cannot be explained with classical physics concepts. E. Schrodinger
Schrödinger (1935) defined entanglement as “the characteristic trait of
quantum mechanics, the one that enforces its entire departure from classical
lines of thought". For the sake of simplicity, let us consider a pure state
$|\Psi\rangle$ composed of two subsystems, $A$ and $B$, with finite dimensions
$d$. Entanglement between subsystems $A$ and $B$ is related to the idea of
separability. One can always write the quantum state as a Schmidt
decomposition Peres (1995)

$|\Psi\rangle_{AB}=\sum_{i=1}^{d}\sqrt{\lambda_{i}}|u_{i}\rangle_{A}|v_{i}\rangle_{B},$
(1)

where $|u_{i}\rangle_{A}$ and $|v_{i}\rangle_{B}$ constitute orthonormal basis
in the Hilbert spaces of subsystems $A$ and $B$, and $\lambda_{i}$ are real
and positive coefficients normalized so that $\sum_{i}\lambda_{i}=1$. If the
Schmidt decomposition contains a single term, i.e., $\lambda_{1}=1$ and
$\lambda_{i}=0$ for $i\neq 1$, the state is said to be separable, or not
entangled. Otherwise the quantum state is entangled. A maximally entangled
state corresponds to the case $\lambda_{i}=1/d$ for $i=1\dots d$.

The idea of separability also plays an important role in classical optics
Spreeuw (1998), although now we consider different degrees of freedom of an
individual system. One can generate optical beams where the different degrees
of freedom that describe the beam cannot be considered separatelyMcLaren  _et
al._ (2015); Zhao  _et al._ (2020). For instance, considering the polarization
and spatial shape degrees of freedom, one can generate Vallés  _et al._
(2014); McLaren  _et al._ (2015); Chille  _et al._ (2016); Ndagano  _et al._
(2016); Eberly  _et al._ (2017) an optical beam with electric field $E({\bf
r})$,

$\vec{E}({\bf r})=\Big{[}\sqrt{I_{H}}\,\Psi_{H}({\bf
r})\,\hat{H}+\sqrt{I_{V}}\,\Psi_{V}({\bf r})\,\hat{V}\Big{]}\exp(ikz),$ (2)

where ${\bf r}=(x,y)$ is the transverse position, $z$ is the direction of
propagation, $k$ is the wavenumber and ${\hat{H}}$ and ${\hat{V}}$ refer to
horizontal and perpendicular polarizations. $I_{H,V}$ are the intensities of
the two orthogonal polarizations, and $\Psi_{H}({\bf r})$ and $\Psi_{V}({\bf
r})$ are the normalized spatial shapes of each polarization component.

The analysis of the similarities and differences between Eqs. (1) and (2), and
its implications and interpretations, has driven intense research in the last
two decades Spreeuw (2001); Kaltenbaek  _et al._ (2009); Simon  _et al._
(2010); Qian and Eberly (2011); De Zela (2014); Kagalwala  _et al._ (2013);
Svozilík  _et al._ (2015). Some researchers refer to the non-separability of
Eq. (2) as nonquantum or classical entanglement Qian  _et al._ (2015, 2017),
although others claim that the use of the word entanglement in a non-quantum
context might be misleading Karimi and Boyd (2015). Non-separable optical
beams have been tested using Bell’s inequality experimental scenarios,
obtaining similar results to its quantum counterpart Kagalwala  _et al._
(2013); Borges  _et al._ (2010); Stoklasa  _et al._ (2015); Vallés  _et al._
(2014). However, the interpretation of these results seems to be different
when using single-photon detections and field intensity measurements
Markiewicz  _et al._ (2019), leaving no place for the idea of violation of
Bell’s inequality in the classical case. How far one can go in this analogy is
a matter of discussion and controversy in the scientific community.

The analogies and differences between classical and quantum separability
allows one to bring questions and solutions from the quantum domain into the
classical one Kwiat  _et al._ (2000); Collins and Popescu (2002); Grover and
Sengupta (2002); Bhattacharya  _et al._ (2002); Erkmen and Shapiro (2006);
Banaszek  _et al._ (2007). For instance, by inspecting how quantum principal
component analysis achieves exponential speedup, Tang Tang (1995) describes an
analog classical protocol that is only polynomially slower than its quantum
counterpart. Leaving out the implications for interpretations and determining
what is uniquely quantum mechanical, quantum ideas can serve as a source of
inspiration for new classical solutions, the so-called quantum-inspired
protocols Kaltenbaek  _et al._ (2008); Lavoie  _et al._ (2009); Mazurek  _et
al._ (2013); Ogawa  _et al._ (2015); Altmann  _et al._ (2018); Urrego  _et
al._ (2020). These classical protocols based on non-separable optical beams
have been used to encode information in optical communications Milione  _et
al._ (2015a, b), to characterize quantum channels Ndagano  _et al._ (2017), to
perform state transfer Hashemi Rafsanjani  _et al._ (2015), to do stimulated
emission depletion (STED) microscopy Török and Munro (2004) and for sensing
and metrology Töppel  _et al._ (2014); Berg-Johansen  _et al._ (2015).

Here we demonstrate a quantum inspired protocol that aims at measuring the
degree of similarity between two spatial shapes without measuring the beam
profiles. The direct measurement of the profile of a light beam is always a
cumbersome measurement since it implies measuring amplitudes and phases across
the whole beam. Instead we obtain the sought-after information by measuring
only the degree of polarization of an optical beam, a measurement easily done
with the help of half-wave plates and polarizing beam splitters. The protocol
considered is inspired by the analysis of the first-order coherence of
subsystems of a bipartite quantum entangled state. We will see that the
protocol proposed here is an example where classical physics seems to mimic
certain aspects of a truly quantum effect such it is entanglement.

##  II Description of the protocol for measuring the degree of similarity
between two spatial shapes by measuring the degree of polarization

The protocol we put forward here originates from the following observation.
When detecting the intensity of the entire non-uniform optical beam, its state
of polarization can be described by the coherence matrix Mandel and Wolf
(1995); De Zela (2014)

$J=\left(\begin{tabular}[]{cc}$I_{H}$&$\sqrt{I_{H}I_{V}}\,\gamma$\\\
$\sqrt{I_{H}I_{V}}\,\gamma^{*}$&$I_{V}$\end{tabular}\right),$ (3)

where $\gamma=\int d\textbf{r}\,\Psi_{H}({\bf r})\Psi^{*}_{V}({\bf r})$ is the
overlap between the spatial shapes $\Psi_{H}({\bf r})$ and $\Psi_{V}({\bf r})$
of the horizontal and vertical components. The condition for a completely
polarized wave is Mandel and Wolf (1995) $\text{det}\leavevmode\nobreak\
J=I_{H}I_{V}\,(1-|\gamma|^{2})=0$ which implies that the two spatial shapes
should be equal. In general, the degree of polarization $P$ of the beam is

$P=\Big{[}1-\frac{4\,\text{det}\leavevmode\nobreak\
J}{(\text{Tr}\leavevmode\nobreak\
J)^{2}}\Big{]}^{1/2}=\Big{[}1-\frac{4I_{H}I_{V}\,(1-|\gamma|^{2})}{(I_{H}+I_{V})^{2}}\Big{]}^{1/2}.$
(4)

If we can measure the degree of polarization and determine the overlap
$\gamma$ we can determine the degree of similarity between two spatial shapes
without measuring the amplitude and phase at each location of the two spatial
shapes. When $I_{H}=I_{V}$, the degree of polarization is equal to the
overlap, i.e., $P=|\gamma|$.

This analysis is reminiscent of what happens with a two-photon quantum
entangled state of the form

$|\Psi\rangle_{AB}=\alpha|w_{1}\rangle_{A}|H\rangle_{B}+\beta|w_{2}\rangle_{A}|V\rangle_{B},$
(5)

where $|w_{1}\rangle$ and $|w_{2}\rangle$ designate two polarization states of
photon $A$, in general non-orthogonal, and the coefficients $\alpha$ and
$\beta$ are the weights of the two components of the state $|\Psi\rangle_{AB}$
with $|\alpha|^{2}+|\beta|^{2}=1$. The quantum state of photon $B$ is

$\rho_{B}=\left(\begin{tabular}[]{cc}$|\alpha|^{2}$&$\alpha^{*}\beta\,\langle
w_{1}|w_{2}\rangle$\\\ $\alpha\beta^{*}\,\langle
w_{2}|w_{1}\rangle$&$|\beta|^{2}$\end{tabular}\right),$ (6)

Notice the remarkable similarity between Eqs. (3) and (6). The equivalent
expression of the the degree of polarization given in Eq. (4) is the purity
$p$ of the state

$p=1-|\alpha|^{2}||\beta|^{2}(1-|\gamma|^{2}),$ (7)

where we have defined again $\gamma=\langle w_{1}|w_{2}\rangle$. For arbitrary
single-photon states, the purity of the quantum state is a measure of the
degree of first-order coherence of the state Titulaer and Glauber (1966).

Figure 1: Sketch of the protocol. We want to determine is an input spatial
shape has an equal counterpart in a database that contains many spatial shapes
for comparison. In a more general scenario, we want to quantify the degree of
similarity of the input shape with all of the spatial shapes contained in the
database.

Let us describe now the theoretical basis of the protocol we put forward here.
We consider that we need to determine if an input spatial shape has an equal
counterpart in a database that contains a myriad of spatial shapes (see Fig.
1). The input spatial shape has to be compared with each one of the spatial
shapes contained in the database. The input spatial shape, given by
$\Psi_{H}({\bf r})$, is embedded in an optical beam with horizontal
polarization. A spatial shape of the database, given by $\Psi_{V}({\bf r})$,
is embedded in an optical beam with vertical polarization. The two light beams
enter a polarizing beam splitter (PBS) that projects into diagonal
$\hat{D}=\frac{1}{\sqrt{2}}(\hat{H}+\hat{V})$ and antidiagonal
$\hat{A}=\frac{1}{\sqrt{2}}(\hat{H}-\hat{V})$ polarizations. The total
intensity detected by a bucket detector is

$I(\phi)=\frac{1}{2}\Big{(}I_{H}+I_{V}+2\sqrt{I_{H}I_{V}}|\gamma|\cos(\phi+\theta)\Big{)},$
(8)

where $\gamma=|\gamma|\exp(i\theta)$ and the parameter $\phi$ is a
controllable phase introduced between the two optical beams with orthogonal
polarizations. As the phase $\phi$ is varied, the intensity $I(\phi)$ varies
accordingly, reaching maxima $I_{max}$ and minima $I_{min}$ of intensity. The
visibility is

$\text{V}=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}=\frac{2\sqrt{I_{H}I_{V}}}{I_{H}+I_{V}}|\gamma|.$
(9)

When $I_{H}=I_{V}$ the visibility gives the overlap, i.e. $V=|\gamma|$ that is
the degree of polarization. When the intensities of the beams with orthogonal
polarizations are different, we can make use of the concept of
distinguishability Eberly  _et al._ (2017)

$D=\frac{|I_{H}-I_{V}|}{I_{H}+I_{V}}.$ (10)

The distinguishability is zero when the two beams have the same intensity, and
it reach a value of $1$ when $I_{H}\gg I_{v}$ or $I_{V}\gg I_{H}$. Using Eq.
(9) for the visibility and Eq. (10) for the distinguishability, we can write
that the modulus of the overlap ($|\gamma|$) is

$|\gamma|=\frac{\text{V}}{\sqrt{1-\text{D}^{2}}}.$ (11)

By measuring the distinguishability $D$ and the visibility $V$ we obtain the
overlap $|\gamma|$ that quantifies the degree of similarity between the input
spatial shape and the spatial shapes contained in the database. Notice that we
also determine the degree of polarization of the optical beam, that is

$P=\Big{[}1-(1-D^{2})\,(1-|\gamma|^{2})\Big{]}^{1/2}.$ (12)

The visibility and distinguishability can be easily measured.

##  III Experimental demonstration of the protocol

###  III.1 Experimental setup

Figure 2 shows the experimental setup used to quantify, by measuring the
degree of polarization, the degree of similarity between two spatial shapes.
We use a Gaussian beam generated with a HeNe laser
($\lambda=633\leavevmode\nobreak\ \text{nm}$). The beam is given a diagonal
polarization with the help of a polarizer and a half wave plate
($\text{HWP}_{1}$). It is expanded and collimated using a telescope (lenses
$\text{L}_{1}$ and $\text{L}_{2}$). With the help of a polarization beam
splitter ($\text{PBS}_{1}$), a mirror and $\text{HWP}_{2}$, the beam is split
into two parallel propagating beams with vertical and horizontal
polarizations. Each beam impinges on half of the screen of a spatial light
modulator (SLM, Hamamatsu X10768-01, 792 $\times$ 600 pixels with a pixel
pitch of 20 µm) prearranged with a hologram. The two modulated beams reflected
from the SLM are recombined with the help of $\text{PBS}_{2}$ and two HWPs
($\text{HWP}_{3}$ and $\text{HWP}_{4}$). The angles of $\text{HWP}_{3}$ and
$\text{HWP}_{4}$ tune the values of the intensities between the two
orthogonally polarized beams. The resulting beam contains in one polarization
the input spatial shape and in the orthogonal polarization one spatial shape
of the database. The beam leaves $\text{PBS}_{2}$ through one output port and
the other output port is blocked and neglected. The beam is sent through a
telescope ($\text{L}_{3}$ and $\text{L}_{4}$) for collimation.

Figure 2:  Experimental set-up. PBS: Polarization beam splitter; M: mirrors;
HWP: half-wave plates; PZ: piezo-electric; L: lenses; Pol: Polarizer; PD:
Photodetector: BB: beam blocker.

In the detection stage, the beam traverse a polarization-sensitive Mach-
Zehnder interferometer (MZI), where beams with orthogonal polarizations
acquire a phase difference by varying the length of one arm of the
interferometer with a linear translation stage with piezo-electronics.
$\text{HWP}_{5}$ and $\text{HWP}_{6}$ flip the polarization to its orthogonal.
The output beam of the MZI passes through $\text{HWP}_{7}$ set at
$22.5^{\circ}$ and $\text{PBS}_{7}$ to project into diagonal polarization.
Light transmitted by $\text{PBS}_{5}$ is focused into a photodiode, while
light coming from the reflecting output of the $\text{PBS}_{5}$ is neglected.

The data acquired from the photodiode is processed by a lock-in amplifier
system which has as reference the signal of an optical chopper (600 Hz) placed
before $\text{PBS}_{1}$. The integration time of the lock-in amplifier is set
at 30 ms. A data acquisition system (DAQ6008) logs 1000 samples of the output
signal from the lock-in amplifier while the piezo is moving.

###  III.2 Results and Discussions

As first experiment, to show a first proof of the feasibility of the protocol
proposed, we generate using the two halves of the SLM, two optical beams:

$\displaystyle\vec{E}_{1}(\rho,\varphi)=U(\rho)\,\exp(i\varphi)\hat{H},$ (13)
$\displaystyle\vec{E}_{2}(\rho,\varphi)=U(\rho)\,\Big{[}\cos(\theta/2)\,\,\exp(i\varphi)+\sin(\theta/2)\,\exp{-i\varphi}\Big{]}\hat{V},$
(14)

where $(\rho,\varphi)$ are cylindrical coordinates. $U(\rho)\,\exp(i\varphi)$
corresponds to an optical beam with orbital angular momentum (OAM) per photon
of $\hbar\omega$. $U(\rho)$ is the resulting radial profile when the phase
imprinted in the SLM is of the form
$\text{Mod}\big{(}2\pi/\Delta+tan^{-1}(y/x),\pi\big{)}$ Janicijevic and
Topuzoski (2008), where $\Delta$ is the period of the grating that allows us
to separate the diffraction orders. W consider only the first diffraction
order. The parameter $\theta$ is an angle to control the amplitude of the two
modes with OAM per photon of $+\hbar\omega$ and $-\hbar\omega$ embedded with
vertical polarization. It can be easily shown that the overlap is
$|\gamma|=|\cos(\theta/2)|$.

Figure 3: Overlap for three different values of the distinguishability for the
spatial shapes shown in Eqs. (13) and (14). The solid line is the theoretical
prediction and the dots are experimental results. The error bars indicate the
uncertainty due to propagation of errors in the measurement.

Figure 3 shows the overlap $|\gamma|$ as a function of the parameter $\theta$.
The solid line is the theoretical prediction and the dots correspond the
experimental results. For each point the overlap is obtained measuring the
visibility and the distinguishability. The visibility is obtained by measuring
the interference pattern when the phase difference varies. The
distinguishability is obtained by measuring the intensity of each arm of the
interferometer. The circle (orange), square (green), and diamond (purple)
points are the experimental outcomes when the degree of distinguishability
takes values of $D\approx 0$, $D\approx 0.23$, and $D\approx 0.54$,
respectively. When $\theta=0$ the two spatial profiles are equal, so the
overlap gets its maximum value of $1$. When $\theta$ increases, the overlap
decreases until reaching a value of $0$ for $\theta=\pi$. These results show
that the protocol to determine $|\gamma|$ is insensitive to differences in
intensity between the two optical beams embedded with different spatial beam
profiles.

For the demonstration of the protocol described above, we perform a second set
of experiments. We consider the objects shown in Fig. 4(a), that are similar,
although not identical, to the objects used by Xie et al. Xie  _et al._ (2017)
to demonstrate parameter sensing making use of the complex orbital angular
momentum spectrum of light beams. The objects are emulated with the help of a
spatial light modulator (SLM), that is divided in two halves to allow for the
presence of two different spatial shapes. The input spatial shape is generated
by a disk with an opening angle $\delta$ oriented an angle $\varphi$ measured
from the axis $y=0$. The spatial shapes included in the database are generated
with disks with opening angle $\beta$ oriented an angle $\alpha$. In the area
of the SLM that corresponds to the opening angle, the pixels of the SLM
introduce a zero phase. Everywhere else the phase introduced by the pixels of
the SLM is $\pi$, so we can assume that the result of illumination of the
object it to multiply the incoming illumination beam by $+1$ or $-1$ depending
if the corresponding position of the pixel is inside the opening angle or not.
Therefore that overlap is

$|\gamma|=\Big{|}\frac{A_{+}-A_{-}}{A_{+}+A_{-}}\Big{|},$ (15)

where $A_{+}$ is the area where both beams have the same phase, and $A_{-}$ is
the area where they have different phases. Since the total area is
$A_{T}=A_{+}+A_{-}$, we have

$|\gamma|=\Big{|}1-2\frac{A_{-}}{A_{T}}\Big{|}.$ (16)

Fig. 4(b) shows an example of the procedure used to determine the overlap. The
first row shows the input spatial shape that is a disk with opening angle
$3\pi/8$. The second row shows a spatial shape of the database with opening
angle $\pi/8$ that is rotated in steps of $\Delta\alpha=\pi/8$. The areas with
the same or different value are shown in the third row. In the example,
$A_{-}/A_{T}$ oscillates between a maximum value of $A_{-}/A_{T}=0.5$ and a
minimum value of $A_{-}/A_{T}=0.25$. The fourth row shows the value of the
overlap for each step of the scan.

Figure 4: (a) Input spatial shape and an example of a spatial shape contained
in the database. (b) As example, the input spatial shape is a disk with
opening angle $\beta=\frac{3\pi}{8}$ oriented a certain angle as shown in the
first row. The second row corresponds to a spatial shape of the database. The
opening angle is $\beta=\pi/8$ and is rotated in steps of
$\Delta\alpha=\frac{\pi}{8}$ in the interval $\\{0,2\pi\\}$. The third row
shows with different colors the areas where both spatial shapes coincide (the
same value $+1$ or $-1$) and where they do not coincide. The last row shows
the value calculated of the overlap.

Fig. 5 shows the experimental results obtained for three different input
spatial shapes. The solid line is the theoretical prediction using Eq. (16),
and the dots are the data obtained experimentally. The error bars indicate the
uncertainty of the measurement. Fig. 5(a) shows the case when
$\delta=\frac{\pi}{8}$ and $\phi=\frac{25\pi}{16}$. Fig. 5(b) shows the case
when $\delta=\frac{3\pi}{8}$ and $\phi=\frac{7\pi}{8}$. Finally, Fig. 5(c)
shows the case when $\delta=\frac{9\pi}{8}$ and $\phi=\frac{\pi}{4}$.

It can be seen that the overlap is $|\gamma|=1$ when the two spatial shapes
are equal and have the same orientation, as shown in Fig. 5(a). Otherwise, the
overlap never reach the maximum value of $1$. In all cases one observes a
similar pattern as function of $\alpha$. There is a rising and falling edge
varying the angle $\alpha$. It can be shown that the distance between the
midpoints of the rising and falling edges is the value of $\delta$ (this is
shown for the sake of clarity in Fig. 5). The experimental and theoretical
values of $\delta$ agree well for the cases represented in Figs. 5(a) and
5(c). For the case represented in Fig. 5(b) there is a slight discrepancy. We
obtain an experimental estimation of $\delta$ of $5\pi/16$, when the
theoretical value is $3\pi/8$. This might due to the fact that we are varying
the angle $\alpha$ in steps of $\pi/16$, that is precisely the discrepancy we
observe in this case.

---

---

---

Figure 5: Overlap as function of the angle $\alpha$. (a) Input spatial shape
with $\delta=\frac{\pi}{8}$ and $\varphi=\frac{25\pi}{16}$, (b) Input spatial
shape with $\delta=\frac{3\pi}{8}$ and $\varphi=\frac{7\pi}{8}$; (c) Input
spatial shape with $\delta=\frac{9\pi}{8}$ and $\varphi=\frac{\pi}{4}$. The
spatial shape coming from the database has an opening angle of
$\beta=\frac{\pi}{8}$ in all cases. We vary the angle $\alpha$ in steps of
$\Delta\alpha=\frac{\pi}{16}$ in the interval $\\{0,2\pi\\}$

##  IV Conclusions

We have put forward and demonstrated experimentally a protocol to quantify the
degree of similarity between two spatial shapes embedded into two orthogonally
polarized optical beams. The protocol avoids measuring the amplitude and phase
at each point of of both beams, a measurement that is generally cumbersome to
do. Instead it measures the degree of polarization of the combined optical
beam, a measurement that is generally much easier to do. The measurement of
the degree of polarization only requires measuring the power imbalance between
two arms of a Mach-Zehnder interferometer and the visibility of intensity
fringes as function of the path difference. The measurement is insensitive to
a global phase difference between the spatial shapes and to intensity
differences between them.

The key enabling tool of the protocol is the use of non-separable beams, where
different degrees of freedom can not be considered independently. In our case
the two degrees of freedom considered are polarization and spatial shape. The
non-separability is also an important feature of quantum entanglement, where
the quantum state of entangled separate entities can not be separated in
quantum states corresponding to each entity.

The protocol presented here is thus another example of a quantum-inspired
technology. The inspiration comes from the observation that measuring the
purity (first-order coherence) of a single photon of a bipartite two-photon
state is a quantifier of the overlap between the two states of the companion
photon with whom the single photon measured can be correlated. This specific
feature of quantum entangled states can be mimicked classically making use of
non-separable optical beams. This is a first step toward the implementation of
a new kind of image comparator, an image comparison analysis tool where images
are compared without the need to measure and analyze its shapes. This is an
interesting practical consequence of the theory of coherence of non-separable
and non-uniform optical beams.

## Acknowledgements

This work is part of the R$\&$D project CEX2019-000910-S, funded by the
Ministry of Science and innovation (MCIN/ AEI/10.13039/501100011033/), from
Fundació Cellex, Fundació Mir-Puig, and from Generalitat de Catalunya through
the CERCA program. We acknowledge suport from the project 20FUN02 “POLight”,
that has received funding from the EMPIR programme co-financed by the
Participating States and from the European Union’s Horizon 2020 research and
innovation programme. JPT also acknowledges financial support from project
QUISPAMOL (PID2020-112670GB-I00) funded by MCIN/AEI /10.13039/501100011033.
DFU acknowledge financial support (PRE2018-085072) financed by
MCIN/AEI/10.13039/501100011033 and FSE “El FSE invierte en tu futuro”

## Disclosures

The authors declare no conflicts of interest.

## References

  * Schrödinger (1935) E. Schrödinger, Mathematical Proceedings of the Cambridge Philosophical Society 31, 555 (1935).
  * Peres (1995) A. Peres,  _Quantum Theory: Concepts and Methods_, Fundamental Theories of Physics (Springer Netherlands, 1995).
  * Spreeuw (1998) R. J. C. Spreeuw, Foundations of Physics 28, 361 (1998).
  * McLaren  _et al._ (2015) M. McLaren, T. Konrad, and A. Forbes, Phys. Rev. A 92, 023833 (2015).
  * Zhao  _et al._ (2020) B. Zhao, X.-B. Hu, V. Rodríguez-Fajardo, A. Forbes, W. Gao, Z.-H. Zhu, and C. Rosales-Guzmán, Applied Physics Letters 116, 091101 (2020), https://doi.org/10.1063/1.5142163 .
  * Vallés  _et al._ (2014) A. Vallés, V. D’Ambrosio, M. Hendrych, M. Mičuda, L. Marrucci, F. Sciarrino, and J. P. Torres, Phys. Rev. A 90, 052326 (2014).
  * Chille  _et al._ (2016) V. Chille, S. Berg-Johansen, M. Semmler, P. Banzer, A. Aiello, G. Leuchs, and C. Marquardt, Optics Express 24, 12385 (2016).
  * Ndagano  _et al._ (2016) B. Ndagano, H. Sroor, M. McLaren, C. Rosales-Guzmán, and A. Forbes, Opt. Lett. 41, 3407 (2016).
  * Eberly  _et al._ (2017) J. H. Eberly, X.-F. Qian, and A. N. Vamivakas, Optica 4, 1113 (2017).
  * Spreeuw (2001) R. J. C. Spreeuw, Phys. Rev. A 63, 062302 (2001).
  * Kaltenbaek  _et al._ (2009) R. Kaltenbaek, J. Lavoie, and K. J. Resch, Phys. Rev. Lett. 102, 243601 (2009).
  * Simon  _et al._ (2010) B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, Phys. Rev. Lett. 104, 023901 (2010).
  * Qian and Eberly (2011) X.-F. Qian and J. H. Eberly, Opt. Lett. 36, 4110 (2011).
  * De Zela (2014) F. De Zela, Phys. Rev. A 89, 013845 (2014).
  * Kagalwala  _et al._ (2013) K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, Nature Photonics 7, 72 (2013).
  * Svozilík  _et al._ (2015) J. c. v. Svozilík, A. Vallés, J. Peřina, and J. P. Torres, Phys. Rev. Lett. 115, 220501 (2015).
  * Qian  _et al._ (2015) X.-F. Qian, B. Little, J. C. Howell, and J. H. Eberly, Optica 2, 611 (2015).
  * Qian  _et al._ (2017) X.-F. Qian, A. N. Vamivakas, and J. H. Eberly, Opt. Photon. News 28, 34 (2017).
  * Karimi and Boyd (2015) E. Karimi and R. W. Boyd, Science 350, 1172 (2015).
  * Borges  _et al._ (2010) C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, Phys. Rev. A 82, 033833 (2010).
  * Stoklasa  _et al._ (2015) B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, L. L. Sánchez-Soto, and G. S. Agarwal, New Journal of Physics 17, 113046 (2015).
  * Markiewicz  _et al._ (2019) M. Markiewicz, D. Kaszlikowski, P. Kurzyński, and A. Wójcik, npj Quantum Information 5, 5 (2019).
  * Kwiat  _et al._ (2000) P. G. Kwiat, J. R. Mitchell, P. D. D. Schwindt, and A. G. White, Journal of Modern Optics 47, 257 (2000).
  * Collins and Popescu (2002) D. Collins and S. Popescu, Phys. Rev. A 65, 032321 (2002).
  * Grover and Sengupta (2002) L. K. Grover and A. M. Sengupta, Phys. Rev. A 65, 032319 (2002).
  * Bhattacharya  _et al._ (2002) N. Bhattacharya, H. B. van Linden van den Heuvell, and R. J. C. Spreeuw, Phys. Rev. Lett. 88, 137901 (2002).
  * Erkmen and Shapiro (2006) B. I. Erkmen and J. H. Shapiro, Phys. Rev. A 74, 041601 (2006).
  * Banaszek  _et al._ (2007) K. Banaszek, A. S. Radunsky, and I. A. Walmsley, Optics Communications 269, 152 (2007).
  * Tang (1995) E. Tang, Phys. Rev. Lett. , 060503 (1995).
  * Kaltenbaek  _et al._ (2008) R. Kaltenbaek, J. Lavoie, D. N. Biggerstaff, and K. J. Resch, Nature Physics 4, 864 (2008).
  * Lavoie  _et al._ (2009) J. Lavoie, R. Kaltenbaek, and K. J. Resch, Opt. Express 17, 3818 (2009).
  * Mazurek  _et al._ (2013) M. D. Mazurek, K. M. Schreiter, R. Prevedel, R. Kaltenbaek, and K. J. Resch, Scientific Reports 3, 1582 (2013).
  * Ogawa  _et al._ (2015) K. Ogawa, S. Tamate, T. Nakanishi, H. Kobayashi, and M. Kitano, Phys. Rev. A 91, 013846 (2015).
  * Altmann  _et al._ (2018) Y. Altmann, S. McLaughlin, M. J. Padgett, V. K. Goyal, A. O. Hero, and D. Faccio, Science 361 (2018), 10.1126/science.aat2298.
  * Urrego  _et al._ (2020) D. F. Urrego, D. Lopez-Mago, V. V. na Hernández, and J. P. Torres, Opt. Express 28, 12661 (2020).
  * Milione  _et al._ (2015a) G. Milione, T. A. Nguyen, J. Leach, D. A. Nolan, and R. R. Alfano, Opt. Lett. 40, 4887 (2015a).
  * Milione  _et al._ (2015b) G. Milione, M. P. J. Lavery, H. Huang, Y. Ren, G. Xie, T. A. Nguyen, E. Karimi, L. Marrucci, D. A. Nolan, R. R. Alfano, and A. E. Willner, Opt. Lett. 40, 1980 (2015b).
  * Ndagano  _et al._ (2017) B. Ndagano, B. Perez-Garcia, F. S. Roux, M. McLaren, C. Rosales-Guzman, Y. Zhang, O. Mouane, R. I. Hernandez-Aranda, T. Konrad, and A. Forbes, Nature Physics 13, 397 (2017).
  * Hashemi Rafsanjani  _et al._ (2015) S. M. Hashemi Rafsanjani, M. Mirhosseini, O. S. Magaña Loaiza, and R. W. Boyd, Phys. Rev. A 92, 023827 (2015).
  * Török and Munro (2004) P. Török and P. Munro, Opt. Express 12, 3605 (2004).
  * Töppel  _et al._ (2014) F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, New Journal of Physics 16, 073019 (2014).
  * Berg-Johansen  _et al._ (2015) S. Berg-Johansen, F. Töppel, B. Stiller, P. Banzer, M. Ornigotti, E. Giacobino, G. Leuchs, A. Aiello, and C. Marquardt, Optica 2, 864 (2015).
  * Mandel and Wolf (1995) L. Mandel and E. Wolf,  _Optical Coherence and Quantum Optics_ (Cambridge University Press, 1995).
  * Titulaer and Glauber (1966) U. M. Titulaer and R. J. Glauber, Physical Review 145, 1041 (1966).
  * Janicijevic and Topuzoski (2008) L. Janicijevic and S. Topuzoski, Journal of the Optical Society of America B 25, 2659 (2008).
  * Xie  _et al._ (2017) G. Xie, H. Song, Z. Zhao, G. Milione, Y. Ren, C. Liu, R. Zhang, C. Bao, L. Li, Z. Wang, K. Pang, D. Starodubov, B. Lynn, M. Tur, and A. E. Willner, Opt. Lett. 42, 4482 (2017).