# Polarization-orbital angular momentum duality assisted entanglement
observation for indistinguishable photons

Nijil Lal<EMAIL_ADDRESS>Physical Research Laboratory, Ahmedabad 380009,
India  Sarika Mishra  Physical Research Laboratory, Ahmedabad 380009, India
Indian Institute of Technology, Gandhinagar 382355, India  Anju Rani  Physical
Research Laboratory, Ahmedabad 380009, India  Indian Institute of Technology,
Gandhinagar 382355, India  Anindya Banerji  Physical Research Laboratory,
Ahmedabad 380009, India  Chithrabhanu Perumangattu  Centre for Quantum
Technologies, National University of Singapore, 3 Science Drive 2, S117543,
Singapore  R. P. Singh  Physical Research Laboratory, Ahmedabad 380009, India

###### Abstract

Duality in the entanglement of identical particles manifests that entanglement
in only one variable can be revealed at a time. We demonstrate this using
polarization and orbital angular momentum (OAM) variables of indistinguishable
photons generated from parametric down conversion. We show polarization
entanglement by sorting photons in even and odd OAM basis, while sorting them
in two orthogonal polarization modes reveals the OAM entanglement. The duality
assisted observation of entanglement can be used as a verification for the
preservation of quantum indistinguishability over communication channels.
Indistinguishable photons entangled in complementary variables could also
evoke interest in distributed quantum sensing protocols and remote
entanglement generation.

††preprint: APS/123-QED

##  I Introduction

Complementarity is a unique manifestation of quantum mechanics, like
entanglement. Introduced as a concept by Niels Bohr [1] and developed through
further rigorous scientific discussion, complementarity broadly states that
objects possess mutually exclusive properties such that the full knowledge of
one property precludes full knowledge of the conjugate one. The wave-particle
duality [2] and Heisenberg’s uncertainty principle [3] are closely associated
with the complementarity principle. Welcher Weg experiments demonstrate the
complementarity between distinguishability and interference visibility
implying that quantum interference will take place only if the measurement
does not distinguish between the interfering pathways [4, 5, 6, 7]. In other
words, indistinguishability leads to quantum interference. It is interesting
to note that interference is related to coherence, which is a wave property,
while distinguishability is associated with localized variables, which is a
particle-like property [8, 9]. Indistinguishability of photons evoke great
interest in quantum information protocols [10, 11]. In fact, many studies have
aimed towards reducing the distinguishability in entangled systems [12, 13,
14] to achieve higher visibility of quantum interference.

Indistinguishability requires perfect overlap in spatio-temporal position,
energy, polarization etc. which can be demonstrated through two-photon
interference experiments. Indistinguishable photons could be generated in the
output of a Hong-Ou-Mandel interferometer [15]. A source of highly
indistinguishable and entangled photon pairs is crucial to various quantum
information applications [16, 17, 18]. However, it is not possible to observe
the entanglement between such indistinguishable photons unless we sort and
separate them in terms of a physical variable, such as their position,
momentum, polarization, orbital angular momentum (OAM) etc. For a general case
of degenerate, non-collinear type-II spontaneous parametric down-conversion
(SPDC) output, we can write the identical photons in terms of their different
degrees of freedom as,

$\left|\Psi\right\rangle=\frac{1}{\sqrt{2}}\left(\left|H,\mathbf{k}_{i}\right\rangle\left|V,\mathbf{k}_{s}\right\rangle+\left|V,\mathbf{k}_{i}\right\rangle\left|H,\mathbf{k}_{s}\right\rangle\right)$
(1)

where the state is written in terms of polarization and linear momentum. This
can also be written as a polarization entangled state,

$\left|\Psi\right\rangle=\frac{1}{\sqrt{2}}\left(\left|H\right\rangle_{k_{i}}\left|V\right\rangle_{k_{s}}+\left|V\right\rangle_{k_{i}}\left|H\right\rangle_{k_{s}}\right)$
(2)

using their linear momentum as a label to differentiate the subsystems. It is
also possible to express the same state as,

$\left|\Psi\right\rangle=\frac{1}{\sqrt{2}}\left(\left|k_{i}\right\rangle_{H}\left|k_{s}\right\rangle_{V}+\left|k_{s}\right\rangle_{H}\left|k_{i}\right\rangle_{V}\right).$
(3)

which is entangled in linear momentum and the individual subsystems are
labelled by their polarization. This complimentary behaviour of two
independent degrees of freedom of identical particles is called duality in
entanglement [19, 20, 21]. To demonstrate this, we use another degree of
freedom, namely orbital angular momentum. Considering a type-II SPDC process,
the generated photon pairs are independently entangled in polarization and
OAM, following the birefringence properties of the crystal and OAM
conservation ($l_{p}=l_{1}+l_{2}$) [22, 23, 24]. The corresponding state can
be expressed as,

$\displaystyle|\Psi\rangle=$
$\displaystyle\frac{1}{2}(|H,l_{1},\bm{k}_{i}\rangle|V,l_{2},\bm{k}_{s}\rangle+|H,l_{2},\bm{k}_{i}\rangle|V,l_{1},\bm{k}_{s}\rangle$
(4)
$\displaystyle+|V,l_{1},\bm{k}_{i}\rangle|H,l_{2},\bm{k}_{s}\rangle+|V,l_{2},\bm{k}_{i}\rangle|H,l_{1},\bm{k}_{s}\rangle).$

For a collinear output, the linear momentum labelling given in Eq. (4) becomes
unavailable since $k_{i}=k_{s}=k$ and it becomes impossible to observe the
entanglement. However, one can write,

$\displaystyle|\Psi\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|H\rangle_{l_{1}}|V\rangle_{l_{2}}+|V\rangle_{l_{1}}|H\rangle_{l_{2}}\right)\otimes|\bm{k}\rangle$
(5)
$\displaystyle\equiv\frac{1}{\sqrt{2}}\left(|l_{1}\rangle_{H}|l_{2}\rangle_{V}+|l_{2}\rangle_{H}|l_{1}\rangle_{V}\right)\otimes|\bm{k}\rangle.$

In most protocols involving the entanglement of orbital angular momentum of
photons, the infinite dimensional OAM spectrum in the output of SPDC is
restricted to a two-dimensional basis by the post-selection of the twin-
photons. Due to this post-selection, a large amount of generated photons which
belong to the other states in the infinite dimensional OAM basis are lost. A
method to avoid this loss is to use an alternate basis defined by the even and
odd states of OAM [25, 26].

In this work, we propose that the OAM of twisted photons defined in their
even-odd basis can be used to separate the otherwise completely
indistinguishable photons in the collinear output. For a pump beam carrying an
odd OAM value, the SPDC photons will be generated in pairs of even and odd OAM
states, following the conservation of OAM. For pump OAM, $l_{p}=1$, in a
collinear type-II SPDC process where the idler-signal pairs are generated in
orthogonal polarization states, the output OAM state can be written as,

$\displaystyle|\Psi\rangle_{SPDC}$
$\displaystyle=\sum_{m=-\infty}^{+\infty}c_{m,1-m}|m\rangle_{H}|1-m\rangle_{V}$
(6)
$\displaystyle=c_{0,1}|0\rangle_{H}|1\rangle_{V}+c_{1,0}|1\rangle_{H}|0\rangle_{V}$
$\displaystyle\hskip
8.5359pt+c_{2,-1}|2\rangle_{H}|-1\rangle_{V}+c_{-1,2}|-1\rangle_{H}|2\rangle_{V}+...$
$\displaystyle=\frac{1}{\sqrt{2}}(|E\rangle_{H}|O\rangle_{V}+|O\rangle_{H}|E\rangle_{V}).$

Here, the twin photons are indistinguishable in every other degree of freedom,
including their spatial position, except for their polarization and OAM.
However, it is not possible to make individual measurements on these photons
unless we separate them under some label. We use their polarization state to
label these individual photons and observe the entanglement in the even-odd
basis of the OAM. In the same way, the state in Eq. (6) can be written by
labelling them as even and odd OAM states,

$|\Psi\rangle_{SPDC}=\frac{1}{\sqrt{2}}(|H\rangle_{E}|V\rangle_{O}+|V\rangle_{E}|H\rangle_{O}).$
(7)

Hence, depending upon whether the polarization or OAM has been used for the
labelling, we can observe the entanglement in the other degree of freedom. It
can be seen that the distinguishability of the associated particles reveal the
entanglement. Experimentally, this means that the method that we use to
distinguish signal and idler photons dictates the degree of freedom in which
the photons are entangled.

Sorting of photons based on the polarization can be achieved using a simple
polarizing beam splitter which separates H and V, revealing the entanglement
in even-odd OAM states. To observe the polarization entanglement, we use an
even-odd OAM sorter. The conventional and commonly used method for selecting
and measuring the OAM component of photons is the phase-flattening technique.
The spiral phase corresponding to a desired OAM state is flattened using
spiral phase plates or holograms displayed on spatial light modulators. The
resulting fundamental Gaussian mode is then coupled to a single mode fiber for
measurement. However, such projective measurements based on phase-flattening
have shortcomings in terms of efficiency and dependence on pump
characteristics [27]. Moreover phase flattening projects the photon into one
of the OAM states and hence cannot be used in even-odd sorting. Using the two-
dimensional even-odd basis for the twin-photon OAM states, the efficiency of
entanglement protocols could be increased.

Figure 1a illustrates a basic set up for an even-odd OAM sorter that involves
a Mach-Zehnder interferometer with a Dove prism in each arms [28, 29]. A dove
prism flips the OAM from $+l$ to $-l$ during the internal reflection. When two
Dove prisms are kept in the two arms of an interferometer, it introduces an
OAM dependent relative phase $2l\alpha$ where $\alpha$ is the relative
rotation of the Dove prisms. The two Dove prisms are oriented perpendicular to
one another and hence $\alpha=\pi/2$. This introduces a phase $l\pi$ between
the two arms of the interferoemeter.

(a)

(b)

Figure 1: Interferometric sorter for even and odd OAM states of light using
Dove prism in (a) Mach-Zehnder arrangement and (b) Folded Mach-Zehnder
arrangement. In both figures, the orientation of the two Dove prisms are
perpendicular to each other.

The relative phase difference would turn out to be odd multiples of $\pi$ for
all odd OAM orders and even multiples of $\pi$ for even OAM orders. As a
result, the constructive interference will take place in different output
ports for even and odd OAM values. To overcome the stability concerns, the
Mach-Zehnder arrangement in Fig. 1a could be reconfigured as a more robust
polarizing Sagnac interferometer [30, 31]. However, a polarizing
interferometer destroys the indistinguishability in polarization. In this
experiment, we adopt a folded Mach-Zehnder arrangement [Fig. 1b] to set up a
robust interferometer without affecting the indistinguishability as well as
the duality.

##  II Methodology

We take a periodically-poled type-II KTP (ppKTP) crystal to observe the
duality in entanglement of polarization and OAM of twin photons generated in
SPDC. In type-II SPDC, the idler and signal photons will have perpendicular
polarizations along ordinary and extra-ordinary axes of the crystal. The
output polarization state can be written as,

$\Psi=\frac{1}{\sqrt{2}}\ket{H}_{i}\ket{V}_{s}\pm\ket{V}_{i}\ket{H}_{s}.$ (8)

Before going for the collinear indistinguishable photons, we consider the non-
collinear generation of photon-pairs where the two photons are emitted along
directions following the phase-matching condition,

${\mathbb{k}_{i}}+{\mathbb{k}_{s}}\approx{\mathbb{k}_{p}}$

where $\mathbb{k}$ is the linear momentum vector. The down-converted output
forms a cone of correlated signal-idler photon pairs following the non-
collinear phase matching condition.

Figure 2: Experimental setup to observe the entanglement in a non-collinear
type-II SPDC from a ppKTP crystal ($\chi^{(2)}$). Polarization measurements
are carried out using a combination of half-wave plate (HWP) and a polarizer
(P) in each arm. $\mathrm{L_{FC}}$ \- Aspheric lens associated with the fiber
coupler (FC), SMF - Single mode fiber, SPCM - Single photon counting module,
CC - Coincidence counter.

The experimental schematic is given in Fig. 2. We use a Toptica TopMode (405
nm) laser as the pump which is loosely focused within the crystal using the
lens, $\mathrm{L_{1}}$ (f=50 cm) such that the paraxial approximation for OAM
conservation is valid. A band-pass filter (BF, 810 $\pm$ 5 nm) blocks the
residual pump while transmitting the down-converted output.

Figure 3: The collimated SPDC output images obtained using an EMCCD kept at
different distances from the crystal plane. A 10 cm lens placed after the SPDC
output collimates the diverging cone of photon pairs.

With this spectral filtering, we consider our source to be giving degenerate
photon pairs in the setups discussed in this work. The diverging cone of SPDC
photons is collimated using lens, $\mathrm{L_{2}}$ (f=10 cm), for the overall
length of the experiment [Fig. 3]. The low intensity photon distributions are
imaged using an EMCCD (Andor iXon3). Signal and idler photons are then coupled
to single mode fibers (SMF) through fiber coupling systems, FC (Thorlabs
CFC-5X-B), consisting of a fiber launcher and an aspherical lens
($\mathrm{L_{FC}}$, f = 4.6 mm). These photons are detected at single photon
counting modules, SPCM (Excelitas AQRH-16-FC), whose output is given to a
coincidence counter, CC (ID Quantique ID800 TDC), to obtain the coincidence
counts. To observe maximum entanglement, maximum overlap between the two
polarization modes has to be ensured.

Figure 4: The crystal position and tilt are adjusted such that the SPDC output
corresponding to both the polarizations are spatially overlapping.
Diametrically opposite regions (in red circles) corresponds to the entangled
photons.

The crystal position and tilt for pump incidence are adjusted such that good
spatial overlap between the H-polarized cone and V-polarized cone is achieved
[Fig. 4]. The correlated photons will be falling along diametrically opposite
points following the phase matching conditions. Two regions, marked in red
circles, are selected and coupled to the detector system.

The polarization projections are done using the combination of half wave plate
(HWP) and a polarizer (P) kept in each arm of the SPDC output [Fig. 2].

Figure 5: Polarization correlations corresponding to the spatially separated
photons in the non-collinear down conversion pair. Error bars indicate
statistical uncertainty of one standard deviation.

The half wave plate, $\mathrm{HWP_{1}}$, in the idler arm is kept at fixed
angles of rotation, $\theta_{1}$ = 0$\degree$, 45$\degree$, 90$\degree$ and
135$\degree$ and the second half wave plate kept in the signal arm
($\mathrm{HWP_{2}}$) is rotated as $\theta_{2}$ to observe the polarization
correlation. The experimentally observed polarization correlations,
corresponding to the projections in HV basis as well as DA basis, are given as
the visibility curves in Fig. 5. The normalized coincidences are plotted along
y-axis with the variation of $\theta_{2}$ for (blue) $\theta_{1}$ =
0$\degree$, (red) $\theta_{1}$ = 45$\degree$, (green) $\theta_{1}$ =
90$\degree$ and (purple) $\theta_{1}$ = 135$\degree$ where $\theta_{1}$ and
$\theta_{2}$ correspond to the rotation of $\mathrm{HWP_{1}}$ and
$\mathrm{HWP_{2}}$ respectively in Fig. 2. The observed visibilities in two
bases are 96.4$\pm$0.2% (HV basis) and 94.1$\pm$0.3% (DA basis) respectively.
The Bell parameter is estimated to be, S = 2.69 $\pm$ 0.03.

The non-collinear down-converted photon pairs discussed above are labelled by
their spatial positions, and hence distinguishable. A collinear output can be
assumed to give indistinguishable photon pairs since they are not separated in
spatial modes.

Figure 6: Variation of the spatial distribution of the SPDC output from a
type-II ppKTP crystal with different temperature values for quasi-phase
matching. We observe that collinear output is obtained for a temperature, 40
$\degree$C.

For a periodically poled crystal, the phase matching will be governed by the
temperature of the crystal [32, 33] and hence the crystal is mounted on a
temperature controlling oven. By varying the temperature, one can achieve
collinear phase matching condition as shown in Fig. 6. For our crystal, the
collinear output is obtained at 40 $\degree$C.

###  II.1 Observation of polarization entanglement through OAM sorting

We use a double Mach-Zehnder OAM sorting interferometer to separate the
photons on the basis of their OAM and reveal the entanglement in polarization.
A double Mach-Zehnder type interferometer could be understood as a normal
Mach-Zehnder interferometer, folded back such that the input and output beam
splitters become the same. Such a configuration will have the stability of a
common path interferometer, since both the arms see same optical components,
and the ease of inserting independent components in the interfering arms as in
a Mach-Zehnder interferometer.

The collinear correlated pairs of photons having even and odd OAM orders are
sent to a double Mach-Zehnder interferometer containing two Dove prisms which
are kept in the individual paths as given in Fig. 7. The Dove prisms are kept
such that their relative orientation is perpendicular to each other. The
individual photons undergo single photon interference within the
interferometric sorter. Photons carrying an odd OAM will constructively
interfere in the odd port (O) whereas photons having even OAM will show up in
the even port (E). The polarization projections are done using the combination
of a half wave plate and a polarizer kept in each output port.

Figure 7: Schematic to sort the even-odd states of OAM from a collinear SPDC
with pump carrying OAM ($l_{p}=1$). The Dove prisms within the double Mach-
Zehnder interferometer is kept orthogonal to each other. Half wave plate (HWP)
along with polarizer (P) corresponds to the polarization projectors. O refers
to the constructive port for odd OAM and E labels the constructive port for
even OAM.

At first, an alignment laser beam (810 nm, Thorlabs) is used to verify the
sorting of even and odd OAM modes. The collinear output is then sent along the
same path. Coincidences are maximized in the detectors kept in ports E and O,
and polarization projection measurements are carried out to observe the
entanglement visibility.

###  II.2 Observation of OAM entanglement through polarization sorting

The entanglement in the even-odd basis of orbital angular momentum can be
observed by separating the indistinguishable photons with their polarization
as a label.

Figure 8: Schematic to observe entanglement in the even-odd basis of OAM by
sorting photons in polarization. Indistinguishable photons in the type-II
down-converted pairs are sorted in polarization using a polarizing beam
splitter (PBS).

A polarizing beam splitter (PBS) separates the H and V polarized photons in
the transmitted and reflected ports respectively as given in Fig. 8. OAM
projections are done using a spatial light modulator (SLM) through phase-
flattening and coupling to single mode fibers. Additional lenses are used to
image the crystal plane to the SLM, as well as the SLM plane to the coupling
fibre tip for effective coupling. The combination of two lenses,
$\mathrm{L_{2}}$ and $\mathrm{L_{3}}$ in each arm, image the modes generated
in the crystal plane onto the SLM planes. The lens after the SLM,
$\mathrm{L_{4}}$ along with the aspheric lens within the fiber coupler,
$\mathrm{L_{FC}}$, image the modes generated after phase-flattening at the SLM
onto the fiber tip. The lenses are chosen such that the spatial mode sizes
match the mode field diameter of the fiber coupling system. OAM projections in
the even-odd basis are carried out with the help of identical spatial light
modulators kept in the two arms. The holograms corresponding to superpositions
of even orders and odd orders act as the counterparts of D and A projections
in the HV basis.

##  III Results and Discussion

While passing through the OAM sorting setup given in Fig. 7, the down
converted photons alternately choose between the even and odd ports depending
upon their OAM value. Before making polarization entanglement measurements on
these photons, the action of even-odd sorting within our setup needs to be
verified.

Figure 9: Verification of sorting of even and odd OAM states in the folded
Mach-Zehnder sorter. The top two rows correspond to the outputs in the even
and odd ports when pumped with a Gaussian ($l_{p}$ =0) and the bottom two rows
correspond to that for a pump carrying $l_{p}$ =1 OAM.

The OAM state of the photons in the output ports of the sorter are measured
using the standard technique involving phase-flattening through SLM and
coupling to single mode fibers. The singles in each port corresponding to
different OAM values are represented as a grayscale chart in Fig. 9. The
values are normalized with respect to the number photons in the Gaussian mode
when pumped with a Gaussian beam, being the largest among all. When pumped
with a Gaussian beam, photons are down converted in pairs of odd-odd or even-
even pairs, following the conservation of OAM, and thus the photon pairs end
up in the same port. It can be easily seen from the chart that photons
carrying even and odd OAM values line up in the corresponding ports and their
intensities are defined by the OAM spectrum of the SPDC output. For a pump
carrying OAM, $l_{p}=1$, the pairs are generated in even-odd pairs and they go
to different ports. This is evident from how the corresponding intensity
values are distributed between the two ports. For example, 0 in even port and
1 in odd port show similar intensity since they are generated together and so
on. Moreover, the stark complementary behaviour in the intensity corresponding
to even and odd ports for a partiular OAM value shows the effective sorting in
the setup.

Figure 10 shows the polarization correlations between the even and odd output
ports of the sorter [Fig. 7]. The indistinguishable photons are efficiently
sorted under the label of their orbital angular momentum and polarization
correlations are observed in both HV as well as DA basis.

Figure 10: Polarization correlations corresponding to projections in the
output ports of the even-odd sorted collinear SPDC output. Error bars indicate
statistical uncertainty of one standard deviation.

The normalized coincidences are plotted along y-axis with the variation of
$\theta_{2}$ for (blue) $\theta_{1}$ = 0$\degree$, (red) $\theta_{1}$ =
45$\degree$, (green) $\theta_{1}$ = 90$\degree$ and (purple) $\theta_{1}$ =
135$\degree$. The observed visibilities are 77.5$\pm$0.3% (HV basis) and
71.6$\pm$0.3% (DA basis).

The Bell parameter is estimated to be, S = 2.11 $\pm$ 0.03. It can be seen in
the plot that the minima corresponding to different visibility profiles are
not going completely to zero. This is due to the possible leakage of even OAM
modes into the odd port (and vice versa). The reduced visibility can be
understood as a manifestation of the imperfections in the sorting
interferometer. Hence, with an improved interferometric sorter, it is possible
to obtain near unity visibility.

The OAM correlations between the H and V output ports of the polarizing beam
splitter [Fig. 8] in the even-odd basis of OAM is given in Fig. 11. The
indistinguishable photons are efficiently sorted under the label of their
polarization and OAM visibility is observed in both EO as well as
$\mathrm{D_{EO}A_{EO}}$ basis. The normalized coincidences are plotted along
y-axis with the variation of $\theta_{2}$ for (blue) $\theta_{1}$ =
0$\degree$, (red) $\theta_{1}$ = 45$\degree$, (green) $\theta_{1}$ =
90$\degree$ and (purple) $\theta_{1}$ = 135$\degree$.

Figure 11: OAM correlations corresponding to two orthogonal polarization
projections in the collinear SPDC output. Error bars indicate statistical
uncertainty of one standard deviation.

A calculation of visibility gives 92.7$\pm$0.3% (EO basis) and 80.9$\pm$0.3%
($\mathrm{D_{EO}A_{EO}}$ basis). The Bell parameter is estimated to be, S =
2.46 $\pm$ 0.08.

While one is required to alight multiple interferometers along with sorters to
undertake a general set of projective measurements in the true linear and
diagonal even-odd basis [25], we have carried out the measurements through OAM
projections using SLM. The use of SLM however introduces efficiency
constraints and limitations in exploting all the available OAM modes. Since
our aim is to demonstrate the duality of entanglement in our setup, we have
made the measurements in a basis defined by $l\in\\{1,2\\}$ where 1 and 2
represents the odd and even OAM states respectively. However, for practical
applications, the projective measurements in the even-odd basis needs to be
done as given in Ref. [25] in order to explore all the available photons. The
difference between the visibility of E and O curves in Fig. 11 can be easily
understood as the result of our choice of a reduced OAM basis consisting of
only 1 and 2. Equal visibility could be achieved by doing the measurements in
the full even-odd basis as mentioned above.

##  IV Conclusions

In this paper, we demonstrate the duality in entanglement of a collinear,
indistinguishable pair of photons generated in a spontaneous parametric down
conversion process. We show polarization entanglement for indistinguishable
photons by sorting the photon OAM using a double Mach-Zehnder even-odd sorter.
This method can increase the availability of entangled photons since we are
not eliminating any photon from the generated output in contrast to the case
of limiting them to two-dimensional OAM bases such as ($+l,-l$) or ($0,l$).
All the down-converted photons are sorted using an even-odd sorter in order to
observe the polarization entanglement of otherwise indistinguishable collinear
photons. Similarly, we demonstrate OAM entanglement by sorting photons using a
simple polarizing beam splitter and executing OAM projections on the photon
pairs in the even-odd basis.

The entanglement studies of systems that display duality can give identical
results in both the variables. This could evoke great interest in studying
entanglement unaffected by the mutual interaction of particles. By further
improving the efficiency of the sorter and incorporating all available OAM
modes in even-odd projections, the entanglement measures estimated in the
paper could be improved and could be shown to be more identical than obtained.
In addition, duality assisted observation of entanglement can be used as a
test for verification of indistinguishability of photons in quantum
information processing. The measurement of entanglement in the complimentary
variables could also reveal any change in the indistinguishability of photons
over communication channels which could have arisen due to possible
eavesdropping. The indistinguishable photons entangled in complementary
variables may also find applications in distributed quantum sensing through
phase estimation as well as remote entanglement generation in a quantum
network.

## Acknowledgement

We are grateful to Dr. Ali Anwar and Mr. Pranav Tiwari for their valuable
suggestions and discussion.

## References

  * Bohr [1928] N. Bohr, The quantum postulate and the recent development of atomic theory, Nature 121, 580 (1928).
  * Wootters and Zurek [1979] W. K. Wootters and W. H. Zurek, Complementarity in the double-slit experiment: Quantum nonseparability and a quantitative statement of bohr’s principle, Phys. Rev. D 19, 473 (1979).
  * Kraus [1987] K. Kraus, Complementary observables and uncertainty relations, Phys. Rev. D 35, 3070 (1987).
  * Greenberger and Yasin [1988] D. M. Greenberger and A. Yasin, Simultaneous wave and particle knowledge in a neutron interferometer, Phys. Lett. A 128, 391 (1988).
  * Herzog  _et al._ [1995] T. J. Herzog, P. G. Kwiat, H. Weinfurter, and A. Zeilinger, Complementarity and the quantum eraser, Phys. Rev. Lett. 75, 3034 (1995).
  * Pittman  _et al._ [1996] T. B. Pittman, D. V. Strekalov, A. Migdall, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, Can two-photon interference be considered the interference of two photons?, Phys. Rev. Lett. 77, 1917 (1996).
  * Bera  _et al._ [2015] M. N. Bera, T. Qureshi, M. A. Siddiqui, and A. K. Pati, Duality of quantum coherence and path distinguishability, Phys. Rev. A 92, 012118 (2015).
  * Mandel [1991] L. Mandel, Coherence and indistinguishability, Opt. Lett. 16, 1882 (1991).
  * Qian and Agarwal [2020] X.-F. Qian and G. S. Agarwal, Quantum duality: A source point of view, Phys. Rev. Res. 2, 012031 (2020).
  * Lo Franco and Compagno [2018] R. Lo Franco and G. Compagno, Indistinguishability of elementary systems as a resource for quantum information processing, Phys. Rev. Lett. 120, 240403 (2018).
  * Das  _et al._ [2020] S. Das, G. Paul, and A. Banerji, Hyper-hybrid entanglement, indistinguishability, and two-particle entanglement swapping, Phys. Rev. A 102, 052401 (2020).
  * Branning  _et al._ [1999] D. Branning, W. P. Grice, R. Erdmann, and I. A. Walmsley, Engineering the indistinguishability and entanglement of two photons, Phys. Rev. Lett. 83, 955 (1999).
  * Torres  _et al._ [2005] J. P. Torres, M. W. Mitchell, and M. Hendrych, Indistinguishability of entangled photons generated with achromatic phase matching, Phys. Rev. A 71, 022320 (2005).
  * Killoran  _et al._ [2014] N. Killoran, M. Cramer, and M. B. Plenio, Extracting entanglement from identical particles, Phys. Rev. Lett. 112, 150501 (2014).
  * Hong  _et al._ [1987] C. K. Hong, Z. Y. Ou, and L. Mandel, Measurement of subpicosecond time intervals between two photons by interference, Phys. Rev. Lett. 59, 2044 (1987).
  * Müller  _et al._ [2014] M. Müller, S. Bounouar, K. D. Jöns, M. Glässl, and P. Michler, On-demand generation of indistinguishable polarization-entangled photon pairs, Nat. Photonics 8, 224 (2014).
  * He  _et al._ [2013] Y.-M. He, Y. He, Y.-J. Wei, D. Wu, M. Atatüre, C. Schneider, S. Höfling, M. Kamp, C.-Y. Lu, and J.-W. Pan, On-demand semiconductor single-photon source with near-unity indistinguishability, Nat. Nanotechnol. 8, 213 (2013).
  * Wolfgramm  _et al._ [2008] F. Wolfgramm, X. Xing, A. Cerè, A. Predojević, A. M. Steinberg, and M. W. Mitchell, Bright filter-free source of indistinguishable photon pairs, Opt. Express 16, 18145 (2008).
  * Bose and Home [2013] S. Bose and D. Home, Duality in entanglement enabling a test of quantum indistinguishability unaffected by interactions, Phys. Rev. Lett. 110, 140404 (2013).
  * Bhatti  _et al._ [2015] D. Bhatti, J. von Zanthier, and G. S. Agarwal, Entanglement of polarization and orbital angular momentum, Phys. Rev. A 91, 062303 (2015).
  * Moreva  _et al._ [2015] E. Moreva, G. Brida, M. Gramegna, S. Bose, D. Home, and M. Genovese, Bell measurements as a witness of a dualism in entanglement, Phys. Rev. A 91, 062117 (2015).
  * Mair  _et al._ [2001] A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Entanglement of the orbital angular momentum states of photons, Nature 412, 313 (2001).
  * Anwar  _et al._ [2020] A. Anwar, N. Lal, S. Prabhakar, and R. P. Singh, Selective tuning of Hilbert spaces in states encoded with spatial modes of light, New J. Phys. 22, 113020 (2020).
  * Lal  _et al._ [2020] N. Lal, A. Banerji, A. Biswas, A. Anwar, and R. P. Singh, Photon statistics of twisted heralded single photons, J. Mod. Opt. 67, 126 (2020).
  * Perumangatt  _et al._ [2017] C. Perumangatt, N. Lal, A. Anwar, S. Gangi Reddy, and R. Singh, Quantum information with even and odd states of orbital angular momentum of light, Physics Letters A 381, 1858 (2017).
  * Perumangatt  _et al._ [2015] C. Perumangatt, A. Abdul Rahim, G. R. Salla, S. Prabhakar, G. K. Samanta, G. Paul, and R. P. Singh, Three-particle hyper-entanglement: teleportation and quantum key distribution, Quantum Inf. Process. 14, 3813 (2015).
  * Qassim  _et al._ [2014] H. Qassim, F. M. Miatto, J. P. Torres, M. J. Padgett, E. Karimi, and R. W. Boyd, Limitations to the determination of a laguerre–gauss spectrum via projective, phase-flattening measurement, J. Opt. Soc. Am. B 31, A20 (2014).
  * Leach  _et al._ [2004] J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon, Phys. Rev. Lett. 92, 013601 (2004).
  * Deng  _et al._ [2007] L.-P. Deng, H. Wang, and K. Wang, Quantum CNOT gates with orbital angular momentum and polarization of single-photon quantum logic, J. Opt. Soc. Am. B 24, 2517 (2007).
  * Perumangatt  _et al._ [2016] C. Perumangatt, N. Lal, A. Anwar, S. G. Reddy, and R. P. Singh, A stabilized polarization controlled orbital angular momentum sorter, in  _13th International Conference on Fiber Optics and Photonics_ (Optical Society of America, 2016) p. Tu4A.46.
  * Slussarenko  _et al._ [2010] S. Slussarenko, V. D’Ambrosio, B. Piccirillo, L. Marrucci, and E. Santamato, The Polarizing Sagnac Interferometer: a tool for light orbital angular momentum sorting and spin-orbit photon processing, Opt. Express 18, 27205 (2010).
  * Fiorentino  _et al._ [2005] M. Fiorentino, C. E. Kuklewicz, and F. N. C. Wong, Source of polarization entanglement in a single periodically poled $\mathrm{KTiOPO_{4}}$ crystal with overlapping emission cones, Opt. Express 13, 127 (2005).
  * Steinlechner  _et al._ [2014] F. Steinlechner, M. Gilaberte, M. Jofre, T. Scheidl, J. P. Torres, V. Pruneri, and R. Ursin, Efficient heralding of polarization-entangled photons from type-0 and type-II spontaneous parametric downconversion in periodically poled $\mathrm{KTiOPO_{4}}$, J. Opt. Soc. Am. B 31, 2068 (2014).