# Symmetry allows for distinguishability
in totally destructive many-particle interference

Julian Münzberg<EMAIL_ADDRESS>Institut für Experimentalphysik,
Universität Innsbruck, Technikerstr. 25, 6020 Innsbruck, Austria  Christoph
Dittel  Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-
Herder-Str. 3, 79104 Freiburg, Germany  EUCOR Centre for Quantum Science and
Quantum Computing, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Str. 3,
79104 Freiburg, Germany  Maxime Lebugle  Eulitha, Studacherstrasse 7b, 5416
Kirchdorf, Switzerland  Andreas Buchleitner  Physikalisches Institut, Albert-
Ludwigs-Universität Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany
EUCOR Centre for Quantum Science and Quantum Computing, Albert-Ludwigs-
Universität Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany
Alexander Szameit  Institut für Physik, University of Rostock, Albert-
Einstein-Str. 23, 18059 Rostock, Germany  Gregor Weihs  Robert Keil  Institut
für Experimentalphysik, Universität Innsbruck, Technikerstr. 25, 6020
Innsbruck, Austria

###### Abstract

We investigate, in a four photon interference experiment in a laser-written
waveguide structure, how symmetries control the suppression of many-body
output events of a $J_{x}$ unitary. We show that totally destructive
interference does not require mutual indistinguishability between all, but
only between symmetrically paired particles, in agreement with recent
theoretical predictions. The outcome of the experiment is well described by a
quantitative simulation which accounts for higher order emission of the photon
source, imbalances in the scattering network, partial distinguishability, and
photon loss.

Suggested keywords

††preprint: APS/123-QED

##  I Introduction

Many-particle interference lies at the very heart of many quantum information
and computations schemes with photons [1, 2, 3], since coherent superpositions
of many-particle states can accommodate a level of complexity which is out of
reach for deterministic classical simulations [4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16]. A necessary condition for quantum interference on the many-
particle level is the indistinguishability of distinct many-particle
transition amplitudes, thus lifting wave particle duality [17] from the single
particle to the many particle level [18].

Importantly, such indistinguishability of amplitudes relies on two distinct
aspects of the specific physical setting – the superimposed amplitudes’
topologies, and the indistinguishabilities of the involved particles. The
topology, determined by the external potential seen by the particles,
determines the actual number of particles which potentially populate the
contributing many-particle transition amplitudes, and thereby defines the
subsets of particles which ought to be mutually indistinguishable for many-
particle interference to impact on the experimentally accessible counting
statistics.

In the paradigmatic setting of two-particle interference due to Hong, Ou, and
Mandel (HOM) [19], where two photons, with controllable degree of
(in-)distinguishability, each enter one of the input modes of a balanced
beamsplitter, the topological aspect remains trivial, since only two two-
particle transition amplitudes (both photons reflected, or both photons
transmitted at one single, symmetric potential barrier) are coupled, and the
experimental phenomenology thus entirely hinges on the indistinguishability of
the incoming photons. Consequently, the famous strict suppression of the
coincident output event, with one particle detected in each output port,
reliably certifies that indistinguishability.

A generalisation of the HOM setting to larger numbers of modes and particles
is possible, giving rise, under suitable symmetry requirements for the
(single-particle) unitary which defines the non-interacting many-particle
scattering process, to so-called suppression laws [20, 12, 21, 22, 23, 24]
which identify those transition events which are strictly suppressed by
destructive many-particle interference. These have been validated
experimentally [25, 26, 27, 28, 29, 30, 31, 32], and found a theoretical
formulation [33, 34] which – by algebraic considerations, hence applicable for
arbitrary system sizes – anchors them to the symmetry properties of the many-
particle input state and of the scattering unitary under permutations. Since
symmetry considerations reduce the complexity of the general many-particle
interference problem, the experimental validation of said suppression rules
was suggested as a viable certification protocol for bona-fide many-particle
interference phenomena, as it directly assesses the granular features of many-
particle quantum interference, in contrast, e.g., to mean-field samplers [35,
27, 30, 36, 37].

However, because of the, in general, non-trivial imprint of the transition
amplitudes’ topology it is important to realise that many-body interference
effects do not necessarily require mutual indistinguishability of all involved
particles [38]. Specifically for many-particle suppression rules, this implies
that many-particle transmission events can be perfectly suppressed even for
pairwise perfectly distinguishable subsets of particles, if only the topology
of the superimposed transition amplitudes remains invariant under their
exchange, which is ultimately controlled by the symmetry properties of the
injected many-particle state [39].

Here we experimentally unfold this refinement of the physics underlying many-
particle suppression laws, by implementing the evolution of four photons under
the so-called $J_{x}$ unitary transformation, for different mutual
distinguishabilities between the particles. So far, this scattering scenario
has only been investigated experimentally with photon pairs, making it
impossible to distinguish between the effect of symmetry and full
indistinguishability [22, 29]. However, by using four photons we here show
that the suppression effect persists as long as the initial state exhibits the
required symmetry, even if two pairs of indistinguishable particles are made
fully distinguishable. Our findings are in agreement with recent theoretical
insights and open new perspectives in the characterization and validation of
many-body indistinguishability.

This work is structured as follows: In Sec. II we recap the suppression law
for the $J_{x}$ unitary and introduce the experimentally investigated input
states differing by the particles’ indistinguishability. Depending on these
states, the scenario is expected to show either suppression or no suppression.
Section III describes the experimental setup and outlines the measurement
procedure. We present the experimental results in Sec. IV and conclude in Sec.
V. Details on further aspects of the measurement procedure and the theoretical
model describing our experiment are deferred to the Appendix.

##  II  Suppression law for the $J_{x}$ unitary

Let us consider a set of $N$ bosons initially prepared in a multi-mode Fock
state with respect to the spatial input modes of an optical scattering device.
The latter implements a unitary transformation which redistributes the
particles over its $n$ modes, with the many-particle output state analysed by
projective measurements of the individual modes’ occupation numbers. Given
that the unitary is invariant under mode permutations, and that the input
state satisfies a related permutation-symmetry, the general suppression laws
formulated in [33, 34] specify which input-output combinations are suppressed
by totally destructive many-particle interference.

The so-called $J_{x}$ unitary satisfies such permutation symmetry, and,
according to the general suppression laws, exhibits a large number of
suppressed output events for input states which are mirror symmetric with
respect to the central mode. While a detailed derivation of the $J_{x}$
suppression law can be found in [34, 39], in the following we summarize the
main ingredients relevant for our work.

The single-particle $J_{x}$ unitary
$U^{\mathrm{J}}(t)=e^{\mathrm{i}J_{x}t/\hbar}$ is generated by the angular
momentum operator $J_{x}$ in $x$ direction with matrix elements [22, 40, 29]

$\left[J_{x}\right]_{k,j}=\frac{\hbar}{2}\left(\sqrt{k(n-k)}\delta_{j,k+1}+\sqrt{j(n-j)}\delta_{j,k-1}\right),$
(1)

where $\delta_{j,k}$ is the Kronecker delta. For an evolution time $t=\pi/2$,
the resulting unitary $U^{\mathrm{J}}(\pi/2)\equiv U^{\mathrm{J}}$ appears, up
to a phase factor, mirror symmetric with respect to the central mode [34, 39],

$U^{\mathrm{J}}_{k,\pi^{\mathrm{J}}(j)}=U^{\mathrm{J}}_{k,j}\exp\left(\mathrm{i}\pi\left[k-j+\frac{n-1}{2}\right]\right),$
(2)

with

$\pi^{\mathrm{J}}(j)=n+1-j$ (3)

the mirror-symmetric permutation of modes $j\in\\{1,\dots,n\\}$, and
$\mathscr{P}^{\mathrm{J}}$ the corresponding single-particle permutation
operator.111Note that in Eq. (2), we attribute the columns and rows of
$U^{\mathrm{J}}$ to the input and output modes, respectively. In the
literature this is sometimes assumed the other way around. Given that the
unitary evolution and the measurement of the particles’ output mode occupation
only act upon their external degrees of freedom – i.e., their mode indices –
we can trace out all remaining – _internal_ – degrees of freedom which,
potentially, render them partially distinguishable. Note that in our photonic
setting, the internal degrees of freedom include, e.g., the photons’
polarization, spectral properties, and arrival times. This procedure results
in the reduced external $N$-particle density operator $\rho_{\mathrm{E}}$ (see
[39, 18] and Appendix A for details). For all mirror-symmetric input states
with uncorrelated internal degrees of freedom (see Appendices A.1 and A.3) and
an external state $\rho_{\mathrm{E}}$ satisfying

$[(\mathscr{P}^{\mathrm{J}})^{\otimes N},\rho_{\mathrm{E}}]=0,$ (4)

we retrieve the following suppression law: All output states with an odd
number of particles in even output modes are suppressed [34].

Note that Eq. (4) does not necessarily require the mutual indistinguishability
of all constituent particles. In the case of internal product states only
particles occupying modes which belong to the same cycle $c$ of
$\pi^{\mathrm{J}}$ – and thus define those subsets of particles which mutually
interfere to induce the suppression of the output events predicted by (4) –
must share the same internal state for (4) to hold.222In other terms, the
cycle structure underlying a given input state’s permutation symmetry defines
the orders – two, three, $\ldots\,N$ particles – of multi-particle
interference contributions to the predicted event suppression.

In our experiment, we demonstrate the relation between the input state’s
symmetry [Eq. (4)] and the suppression of many-body output events for the
interference of $N=4$ photons on a $J_{x}$ unitary with $n=7$ modes. In cycle
notation, the mode permutation (3) then reads
$\pi^{\mathrm{J}}=(1\,7)(2\,6)(3\,5)(4)$, with cycles $c_{1}=(1\,7)$,
$c_{2}=(2\,6)$, $c_{3}=(3\,5)$, and $c_{4}=(4)$.

Figure 1:  Experimentally investigated input states of the $J_{x}$ unitary.
(a) Photons are input to the odd modes of the unitary. Four different symmetry
scenarios with respect to the mirror axis (central mode) are illustrated.
Suppression is expected only for the upper two scenarios. The particles’
internal states are illustrated by their coloring. In (b) the occupation of
the permutation cycles $c_{j}$ is shown. The photonic waveguide structure
implementing the $J_{x}$ unitary is illustrated in (c).

We investigate four different cases of the particles’ indistinguishability
structure illustrated in Fig. 1(a). These scenarios correspond to the
particles on input being mutually indistinguishable, inter-cycle
distinguishable, intra-cycle distinguishable, and mutually distinguishable.
Even though the second scenario features distinguishable particles in modes
belonging to distinct cycles, it remains invariant under $\pi^{\mathrm{J}}$,
and, thus, gives rise to the above suppression law [compare Fig. 1(b)]. In
contrast, the latter two input conformations cannot give rise to the
suppression effect here under scrutiny. The counting statistics observed in
our experimental analysis unambiguously reveals the symmetry-induced minimal
requirements on the mutual indistinguishability only of sub-sets of the
incoming N-particle state for interference-induced output event suppression.

##  III  Experimental setup

Figure 2:  Schematic illustration of the four-photon SPDC source, of the
multiphoton interference, and of the detection setup. A pulsed Ti:Sa laser was
frequency doubled by second harmonic generation in a BiBO crystal in collinear
configuration. The residual pump light was filtered by dichroic mirrors, while
the up-converted light was focused into a BBO crystal, where pairs of photons
were generated in a type-I SPDC process. The emitted and spectrally filtered
photon pairs were collected via four polarization maintaining single mode
fibers labelled as channel (Ch.) a-d. Three freespace optical delay stages
were used to adjust the temporal delay between photons from different
channels. The light was coupled to the waveguide chip via a fiber array. The
output of the waveguide structure was collected with multi-mode fibers and
measured using APDs and a coincidence counter (Coin. Counter).

The many-particle interference experiment was performed on an integrated
linear optics platform. For the implementation of the $J_{x}$ unitary, we used
an array of seven evanescently coupled modes in a $\mathrm{f}\mathrm{s}$
laser-written waveguide structure in fused-silica [41, 42]. The evanescent
coupling between the waveguides was chosen according to the nearest neighbor
coupling structure from Eq. (1), with an interaction length corresponding to
an evolution time $t=\pi/2$. A schematic representation is shown in Fig. 1(c).
See Appendix B for details on the waveguide fabrication.

The four-photon input state was generated by double-pair emission of a type-I
spontaneous parametric down-conversion (SPDC) source [43]. The experimental
setup is illustrated in Fig. 2. A $200\text{\,}\mathrm{f}\mathrm{s}$-pulsed
Ti:Sa laser with a repetition rate of
$76\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ and a center wavelength of
$795\text{\,}\mathrm{n}\mathrm{m}$ was frequency doubled in a Bismuth Borate
(BiBO) crystal. The resulting up-converted V-polarized light with an optical
power of $\approx$150\text{\,}\mathrm{m}\mathrm{W}$$ was focused into a Barium
Borate (BBO) crystal, where photon pairs were created in a type-I non-
collinear SPDC process. The emitted H-polarized photon pairs at a center
wavelength of $795\text{\,}\mathrm{n}\mathrm{m}$ were collected into four
polarization maintaining single-mode fibers after spectral filtering using
$\approx$3\text{\,}\mathrm{n}\mathrm{m}$$ FWHM bandpass filters, which reduces
spectral correlations and improves the indistinguishability between photons
from different pairs. The four fibers, labelled channels a-d, were oriented
such that the channel pairs a and b, as well as c and d (also called mode
pairs), collect photons from opposite spots of the SPDC emission cone. Hence,
by momentum conservation, each mode pair collects both photons originating
from the same SPDC event. The collection of two photon pairs within their
coherence time (i.e. temporally overlapping wave packets) can arise from two
different types of events: either the photon pairs were collected by the same
mode pair, e.g. both channel a and b collect two photons, or by different mode
pairs, such that each channel a-d collects one photon.

In the experiment, we connected channels a and b to input modes 1 and 7 of the
$J_{x}$ unitary (i.e. to the modes of cycle $c_{1}$) and channels c and d to
modes 3 and 5 (of cycle $c_{3}$), respectively. This connectivity was used for
all four investigated scenarios. Hence, the photon collection procedure
resulted in the input mode occupations

$\displaystyle\vec{R}_{1}$ $\displaystyle=(2,0,0,0,0,0,2),$ (5a)
$\displaystyle\vec{R}_{2}$ $\displaystyle=(1,0,1,0,1,0,1),$ (5b)
$\displaystyle\vec{R}_{3}$ $\displaystyle=(0,0,2,0,2,0,0),$ (5c)

with $\vec{R}_{j}$ listing the number of photons in each input mode. Note
that, in the ideal case of perfect indistinguishability between all particles,
the states corresponding to all three possible input configurations satisfy
the required mirror symmetry [see Eq. (3)]. In the experiment, however, there
are residual spectral correlations, rendering photons from different pairs
partially distinguishable [44]. Considering this, the described connectivity
also ensures that photons from the same SPDC event always occupy modes
belonging to the same cycle, such that the symmetry conditions are best
satisfied in the presence of experimental imperfections.

To control the temporal delays between the photons and, thus, their mutual
distinguishability, we used free-space optical delay stages in three of the
four channels. The four scenarios shown in Fig. 1(a) were implemented by
appropriately adjusting the photons’ time delays. For coupling light into the
quasi-transverse-magnetic mode of the waveguide structure and collecting the
outcome, we used a commercially available polarization maintaining and multi-
mode V-groove fiber array, respectively. The fibers of the output array were
connected to seven avalanche photodiodes (APDs) with an average detection
efficiency of 65%.

After having optimized the temporal overlap of photons from different channels
in HOM-type measurements [see Appendix C for details], we recorded all
fourfold coincidence events with a time tagging device. Only non-bunching
output events (at most one photon ends up in an output mode) are
experimentally accessible, since the used APDs are non-number-resolving. In
parallel to fourfold coincidences, we collected single channel count rates, as
well as twofold and threefold coincidences between all output modes [see
Appendices D and E].

##  IV Results

For each scenario shown in Fig. 1(a), we recorded fourfold coincidences
between distinct output modes. Dependent on the scenario, the average event
rate ranges between 0.0023 and $0.0045\text{\,}\mathrm{H}\mathrm{z}$, with a
total number of collected events, $N_{\mathrm{total}}$, between 566 and 1154
[see Tab. 1]. The experimental results are shown in Fig. 3 together with the
theoretical predictions obtained from a simulation of the experiment.
Experimental and simulated results are both normalized.

Figure 3:  Experimental and simulated output statistics of all fourfold
coincidences between distinct output modes. Panel (a) and (b) show the output
statistics obtained for a _mutually indistinguishable_ and for an _inter-cycle
distinguishable_ input state, respectively, for which all output events in the
grey shaded area are ideally suppressed. The output statistics obtained for an
_intra-cycle distinguishable_ and for a _mutually distinguishable_ input state
is shown in (c) and (d), respectively. Here no suppression effect is expected,
even in the ideal case. Green bars correspond to experimental (Exp) data, and
the error bars indicate one standard deviation of the Poissonian counting
statistics. The bars to the right of the experimental data correspond to a
simulation of the experiment, with the bars’ dark blue, blue, light blue, and
grey part indicating the contribution stemming from the input mode occupation
$\vec{R}_{1}$, $\vec{R}_{2}$, $\vec{R}_{3}$ [see Eq. (5)], and from the six-
photon background of the photon source, respectively.

In Fig. 3(a) and (b), all output events in the grey shaded area are suppressed
in the ideal case, as predicted by the suppression law. While our data is
substantially biased by higher-order multi-photon-pair emission (grey bars in
Fig. 3; $\approx 34\%$ of all registered four-photon events can be attributed
to the six-photon background), we can confirm a clear reduction of all ideally
suppressed events in the case of _mutually indistinguishable_ and _inter-cycle
distinguishable_ particles [see Fig. 3(a) and (b)], to be contrasted with the
cases of _intra-cycle_ and _mutually distinguishable_ particles [see Fig. 3(c)
and (d)], where no suppression effect is expected nor observed.

In order to benchmark the suppression, we use the degree of suppression
violation $\mathcal{D}=N_{\mathrm{forbidden}}/N_{\mathrm{total}}$ [30] with
$N_{\mathrm{forbidden}}$ the number of recorded four-photon coincidences
corresponding to ideally suppressed output events, and $N_{\mathrm{total}}$
the total number of recorded events. Note that in the ideal case
$\mathcal{D}=0$. However, we expect a non-vanishing value of $\mathcal{D}$ due
to fabrication errors of the waveguide structure, imperfections in the
photons’ indistinguishability (caused by their spectro-temporal properties),
and higher-order multi-pair emission.

In our theoretical simulation of the experiment we account for the above
experimental imperfections. Pertinent to photon pairs from SPDC are their
spectral correlations arising from energy and momentum conservation in the
generation process, such that the combined internal state of all particles
does not factorize into individual internal states of the particles [44]. Not
only does this spectral entanglement influence the ratio between spontaneous
and stimulated emission in the SPDC process and, thereby, the photon number
distribution [45, 46, 47], it also strongly affects the many-particle
interference for configurations with more than one photon occupying the same
initial mode. The established method of calculation using the so-called
partial indistinguishability matrix [48] is applicable to this situation, but
computationally costly, as it requires a double summation over all
permutations of particle orderings. The more efficient formalism from Ref.
[49], however, cannot be used here as it relies on factorizable internal
states. Adopting the formalism of [39], we here derive a new framework for
many-particle interference of correlated photon pairs, which avoids the costly
summation over all particle orderings and replaces it by a faster summation
over inequivalent orderings only. This framework is the first, to our
knowledge, which simultaneously incorporates correlated internal states,
multiple occupancy of external input modes, as well as the correct weights
between stimulated and spontaneous emission in the various multi-pair
contributions of the SPDC, and is described in detail in Appendix A.

The experimentally measured and theoretically predicted degrees of suppression
violation are summarized in Table 1.

Table 1:  For the four investigated scenarios illustrated in Fig 1(a), the measured ($\mathcal{D}_{\mathrm{exp}}$) and simulated ($\mathcal{D}_{\mathrm{th}}$) degree of suppression violation is listed together with the total number $N_{\mathrm{total}}$ of recorded four-photon coincidences.  | $N_{\mathrm{total}}$ | $\mathcal{D}_{\mathrm{exp}}$ | $\mathcal{D}_{\mathrm{th}}$
---|---|---|---
_mutually indistinguishable_ | 862 | $0.300\pm 0.019$ | 0.214
_inter-cycle distinguishable_ | 566 | $0.274\pm 0.022$ | 0.226
_intra-cycle distinguishable_ | 921 | $0.408\pm 0.021$ | 0.414
_mutually distinguishable_ | 1154 | $0.403\pm 0.019$ | 0.415

As quantified by the degree of suppression violation for both _mutually
indistinguishable_ and _inter-cycle distinguishable_ particles there is a
significantly smaller fraction of events in the grey area of Fig. 3 compared
to the cases of _mutually distinguishable_ and _intra-cycle distinguishable_
particles which violate the mirror symmetry (4). This confirms the presence of
destructive many-particle interference in the two symmetric scenarios alone.

Comparison of theoretical and experimental results [see Fig. 3] yields
reasonable agreement. Yet, for the cases mutually indistinguishable and inter-
cycle distinguishable [see Fig. 3(a) and (b)], the simulation predicts a
significantly lower degree of violation than measured. We attribute these
deviations mainly to the following systematic errors and limitations of the
model: 1) The experimental unitary reconstruction is subject to imprecisions,
and the theoretically assumed phase relations may not be perfectly realised
[see Appendix B]. 2) In addition to the substantial contributions from six
generated photons on the statistics of four-photon outputs, there is also an
eight-photon background (i.e. quadruple pair productions) that we estimate to
comprise about 5% of all detected events. This background is not accounted
for, due to the computational overhead. However, if we assume a uniform
distribution of this eight-photon background among all fourfold outputs
(suppressed and unsuppressed), we estimate that $\mathcal{D}_{\mathrm{th}}$
would increase from initially 0.214 to 0.226 for the _mutually
indistinguishable_ scenario. 3) Inaccuracies in the estimated generation
probabilities and losses before the unitary would lead to a different
weighting of the contributions of the input states. That in turn may shift the
probability distribution of the output events. 4) Finally, the
indistinguishabilties of photons from the same pair, as well as of photons
from different pairs, are estimated via HOM measurements [see Appendix F]. We
noticed a drift in the indistinguishability during the measurement, which we
correct for in the simulation, but this correction has its own precision
limits [see Appendix E]. Errors in the estimated indistinguishabilities lead
to errors in the probability distribution of output events.

The optimal degree of suppression achievable in the experiment is mostly
limited by the unwanted multi-photon background of the photon source. From a
simulation excluding the six-photon background, we retrieve a theoretically
predicted degree of suppression violation of 0.102 (compared to 0.214 with
background) for the mutually indistinguishable case. It is possible to reduce
this background by decreasing the SPDC pump power, which, however, also
decreases the source brightness, leading to a longer integration time to
achieve the same experimental quality of the output statistics. Since our
integration time per configuration was 3-4 days, this is only possible if one
improves the transmission efficiency at the same time, e.g. via enhanced
coupling from the fiber to the chip. Alternatively, one can replace the non-
deterministic SPDC source by a quasi-deterministic source e.g. by actively
demultiplexing photons from a semiconductor quantum dot (QD) [50, 51, 8, 52].
Resonantly excited semiconductor QDs feature almost zero multi-photon emission
[53, 54, 55] and higher source brightness. Hence, they can be used to further
optimise the suppression and, at the same time, increase the precision of the
experiment.

##  V  Conclusion

In this work, we experimentally demonstrated that totally destructive many-
particle interference (aka event suppression on output) does not necessarily
require mutual indistinguishability of all involved particles. Instead, merely
the input state’s symmetry matters, such that only particles occupying modes
belonging to the same cycle of the underlying permutation-symmetry must be
indistinguishable. As a figure of merit, we used the degree of suppression
violation $\mathcal{D}_{\mathrm{exp}}$, which is significantly smaller for
input states satisfying the symmetry required for the suppression as compared
to input states violating this symmetry (and, thereby, destroying the
necessary indistinguishability of those particles populating interfering many-
particle amplitudes which, by the input state’s cycle structure, contribute to
the predicted suppression event). In order to simulate our experiment we
developed a general theoretical model designed to describe many-particle
interference of photons with arbitrarily correlated spectral properties. Our
model allows for the multifold occupation of input modes, incorporates
quantitatively correct weights between the various emission orders in
realistic SPDC photon sources with imperfections, and predicts the
experimentally observed four-photon statistics with reasonable accuracy.

Our results clearly demonstrate that the observation of many-body suppression
does not necessarily suffice to conclude that all involved constituents are
mutually indistinguishable. This refines the suggestion that fully destructive
many-particle interference across highly symmetric scattering devices can
unambiguously certify a boson sampling device: In general, the output event
suppression by multi-particle interference only certifies the symmetry of the
input state with respect to a permutation of modes, and the
indistinguishability of those particles pertaining to subgroups identified by
the diverse cycles of that very permutation. Only when this permutation
consists of a single cycle of length $N$, as in the case of the $N\times N$
Fourier unitary [21, 39], does suppression suffice to certify the particles’
mutual indistinguishability.

###### Acknowledgements.

We acknowledge useful discussion with Stefan Frick. G.W., R.K., and J.M.
acknowledge support from the Austrian Science Fund (FWF Projects No. I2562,
P30459, and F7114). A.S. thanks the Deutsche Forschungsgemeinschaft (Grants SZ
276/9-2, SZ 276/12-1, SZ 276/20-1, SZ 276/21-1, BL 574/13-1) and the Alfried
Krupp von Bohlen and Halbach Foundation for financial support. M.L.
acknowledges funding from the Marie Curie Actions within the Seventh Framework
Programme for Research of the European Commission, under the Initial Training
Network PICQUE, Grant No. 608062. C.D. acknowledges the Georg H. Endress
foundation for financial support.

##  Appendix A  Many-particle interference from SPDC photons

In this section, we present our theoretical model to describe many-particle
interference experiments with photon pairs collected from a probabilistic
multi-pair SPDC source. Note that our model is not limited to the present
setup, but can be used in a wide variety of experimental settings to calculate
the input-output probabilities of pairwise correlated and partially
distinguishable multi-photon states under any unitary transformation $U$.
Similar considerations have been made for two photons fed into a Sylvester
interferometer [30], where the authors consider fabrication errors in the
unitary, multiphoton emission, and partial distinguishability of the photons.
Here, our description generalizes this approach to multiple partially
distinguishable photon pairs.

The model is divided into the following steps: We construct the $N$-photon
input state (with, in general, $N>4$) consisting of $N/2$ photon pairs
obtained from SPDC and calculate the generation probabilities of these states
in Sec. A.1. The particles’ internal states are calculated from the product of
reconstructed joint spectral amplitudes of the emitted photon pairs. Loss
before the unitary is modeled by extending the optical modes of the unitary
with additional ancillary modes and unbalanced beamsplitters, with lost
photons being coupled into these ancillary modes [see Sec. A.2]. Afterwards,
for each so-constructed and loss-weighted input state, input-output
probabilities are calculated according to our formalism presented in Sec. A.3.
This requires the matrix representation of the many-particle density operator,
which accounts for partial distinguishability of the photons as calculated
from the overlaps of the permuted internal states from Sec. A.1.

In general, the relative phases between all input states must be taken into
account in the calculation of the output probabilities, since these phases
result in an interference of the probability amplitudes corresponding to the
different input states [56, 57]. However, observing the time traces of
recorded two-fold coincidence counts shows that these relative phases
fluctuate [see Appendix D] much faster than the integration time of the
experiments. Therefore, the experimental data results from an incoherent
mixture of these input states, such that we can consider an incoherent
superposition of the input-output probabilities of all contributing input
states weighted by their generation probabilities and losses. The resulting
probabilities are then renormalized, accounting for the relative output
transmissivities and detector efficiencies, which were obtained in the unitary
reconstruction process [see Appendix B].

###  A.1  Model of the SPDC source

We consider the four-mode SPDC source [cf. Fig. 2] as two independent, pulsed
pair sources, which produce two quantum states $\ket{\Psi^{(\mathrm{ab})}}$
and $\ket{\Psi^{(\mathrm{cd})}}$ in the two mode pairs of interest. The
overall state reads [45]

$\ket{\Psi}=\ket{\Psi^{(\mathrm{ab})}}\otimes\ket{\Psi^{(\mathrm{cd})}}.$ (6)

The state of each parametric process can be written as a sum over the emission
of $P$ photon pairs, with the state generated by the first source being [47]

$\ket{\Psi^{(\mathrm{ab})}}=c^{(\mathrm{ab})}\sum_{P=0}^{\infty}\frac{\left(p^{(\mathrm{ab})}\right)^{P/2}}{P!}\sqrt{\mathcal{N}^{(\mathrm{ab})}_{P}}\ket{\Psi^{(\mathrm{ab})}_{P}}.$
(7)

Here, $c^{(\mathrm{ab})}\lesssim 1$ is a normalization constant and
$\left(c^{(\mathrm{ab})}\right)^{2}p^{(\mathrm{ab})}\approx p^{(\mathrm{ab})}$
is the pair generation probability per pump pulse. The state of $P$ photon
pairs produced in a pump pulse is given in second quantization by an integral
over the particles’ internal states, i.e. their frequencies [47],

$\displaystyle\ket{\Psi^{(\mathrm{ab})}_{P}}=\frac{1}{\sqrt{\mathcal{N}^{(\mathrm{ab})}_{P}}}\int\mathrm{d}\vec{\omega}\,\Phi^{(\mathrm{ab})}(\omega_{1},\omega_{1}^{\prime})\cdots\Phi^{(\mathrm{ab})}(\omega_{P},\omega_{P}^{\prime})\times$
$\displaystyle\hat{a}_{\mathrm{a}}^{\dagger}(\omega_{1})\hat{a}_{\mathrm{b}}^{\dagger}(\omega_{1}^{\prime})\cdots\hat{a}_{\mathrm{a}}^{\dagger}(\omega_{P})\hat{a}_{\mathrm{b}}^{\dagger}(\omega_{P}^{\prime})\ket{0},$
(8)

with $\Phi^{(\mathrm{ab})}(\omega,\omega^{\prime})$ the JSA of a single photon
pair, and
$\mathrm{d}\vec{\mathcal{\omega}}=\mathrm{d}\omega_{1}\mathrm{d}\omega^{\prime}_{1}\ldots\mathrm{d}\omega_{P}\mathrm{d}\omega^{\prime}_{P}$,
and $\hat{a}^{\dagger}_{\mathrm{a}}(\omega)$ the creation operator of a photon
with frequency $\omega$ in mode $\mathrm{a}$. The normalization coefficient
$\mathcal{N}^{(\mathrm{ab})}_{P}$ is dictated by the commutation rules of the
ladder operators and can be calculated via a summation over all intra-mode
permutations:

$\displaystyle\mathcal{N}_{P}^{(\mathrm{ab})}=\int\mathrm{d}\vec{\omega}\,\left(\Phi^{(\mathrm{ab})}(\omega_{1},\omega_{1}^{\prime})\cdots\Phi^{(\mathrm{ab})}(\omega_{P},\omega_{P}^{\prime})\right)^{*}\times$
$\displaystyle\sum\limits_{\pi,\sigma\in
S_{P}}\Phi^{(\mathrm{ab})}(\omega_{\pi(1)},\omega_{\sigma(1)}^{\prime})\cdots\Phi^{(\mathrm{ab})}(\omega_{\pi(P)},\omega_{\sigma(P)}^{\prime}).\quad$
(9)

Here, $S_{P}$ is the symmetric group of $P$ elements (corresponding to the $P$
particles in each of the two modes). Equation (7) can be interpreted as an
interpolation between a two-mode squeezed vacuum state
($\mathcal{N}^{(\mathrm{ab})}_{P}=P!^{2}$) and purely accidental multipair
generation ($\mathcal{N}^{(\mathrm{ab})}_{P}=P!$). The former arises from an
uncorrelated JSA
($\Phi(\omega,\omega^{\prime})=\phi(\omega)\phi(\omega^{\prime})$) and a
maximal contribution of stimulated emission, and leads to perfect, heralded
HOM visibility between photons from different pairs [45], while the latter
corresponds to a maximally correlated JSA
($\Phi(\omega,\omega^{\prime})=\delta(\omega+\omega^{\prime}-\omega_{\mathrm{p}})\Phi(\omega,\omega^{\prime})=\Phi(\omega,\omega-\omega_{\mathrm{p}})$
with the pump frequency $\omega_{\mathrm{p}}$) and purely spontaneous
emission, producing zero HOM visibility. Note that the difference between
these two extremes takes no effect for single-pair emission, i.e. $P=1$.

The probability of the first source to generate the state
$\ket{\Psi^{(\mathrm{ab})}_{P}}$ of $P$ pairs is derived from Eq. (7):

$p_{P}^{(\mathrm{ab})}=\left(c^{(\mathrm{ab})}\right)^{2}\,\frac{\left(p^{(\mathrm{ab})}\right)^{P}}{{P!}^{2}}\mathcal{N}^{(\mathrm{ab})}_{P}.$
(10)

The joint probability to generate $P$ pairs in channels a and b, and $Q$ pairs
in channels c and d is then

$p_{P,Q}=\left(c^{(\mathrm{ab})}c^{(\mathrm{cd})}\right)^{2}\left(p^{(\mathrm{ab})}\right)^{P}\left(p^{(\mathrm{cd})}\right)^{Q}\frac{\mathcal{N}_{P}^{(\mathrm{ab})}\mathcal{N}_{Q}^{(\mathrm{cd})}}{{P!}^{2}\,{Q!}^{2}}.$
(11)

From a preliminary characterization of the source, we obtain
$p^{(\mathrm{ab})}=0.026$ and $p^{(\mathrm{cd})}=0.033$ [see Appendix F]. In
order to calculate $\mathcal{N}^{(\mathrm{ab}),(\mathrm{cd})}_{P}$, we
construct JSAs $\Phi^{(\mathrm{ab}),(\mathrm{cd})}(\omega,\omega^{\prime})$
that reproduce our experimentally measured HOM and heralded HOM visibilities
[see Appendix E]. From this we obtain numeric values of
$\mathcal{N}^{(\mathrm{ab})}_{2}\approx\mathcal{N}^{(\mathrm{cd})}_{2}\approx
3.18$ and
$\mathcal{N}^{(\mathrm{ab})}_{3}\approx\mathcal{N}^{(\mathrm{cd})}_{3}\approx
21.65$. Note that in order to account for the distinguishability between the
particles induced by different time-of-arrivals in the experimentally
investigated cases of _inter-cycle distinguishable_ and _intra-cycle
distinguishable_ particles, we include an additional phase
$e^{\mathrm{i}\omega\tau}$ with a sufficiently large temporal delay $\tau$
between the SPDC source channels. With Eq. (11), we calculate the generation
probability of states up to six photons, which is summarized in Table 2. We
find a 32 times enhanced generation probability of four-photon states compared
to six-photon states. Despite this seemingly low rate of created six-photon
states, they nevertheless contribute significantly to the measured counting
statistics due a combinatorial advantage over four-photon states in the lossy
setup: To yield a four-photon coincidence from an initial six-photon state, up
to two photons may be lost before detection. There are $\binom{6}{2}=15$
possible combinations of lost photons, which significantly increases the
relative weight of six-photon states in the lossy setup. From the simulation,
we predict that 33.6% to 35.2% (depending on the measured scenario) of all
registered non-bunching output events originate from an initial six-photon
creation.

Table 2:  All states consisting of up to six photons, that possibly lead to a fourfold coincidence after evolving under the unitary transformation $U_{\mathrm{exp}}$. $\vec{R}_{(\mathrm{a,b,c,d})}$ is the mode occupation list of the four SPDC channels, $N$ is the number of photons in that state, $\vec{R}_{J_{x}}$ is the mode occupation list for the $n=7$ input modes of the $J_{x}$ unitary (excluding the additional unoccupied ancillary input modes), $p_{\mathrm{gen}}$ is the generation probability calculated via Eq. (11), and $p_{\mathrm{gen,\,norm}}$ is the generation probability normalized to all listed states. Note that despite the very low generation probabilities of the six-photon states, they contribute significantly to the final output probability due to a combinatorial advantage in the lossy setup.  $\vec{R}_{(\mathrm{a,b,c,d})}$ | $N$ | $\vec{R}_{J_{x}}$ | $p_{\mathrm{gen}}$ | $p_{\mathrm{gen,\,norm}}$
---|---|---|---|---
(2, 2, 0, 0) | 4 | $\vec{R}_{1}$ | 0.000493 | 0.230
(1, 1, 1, 1) | 4 | $\vec{R}_{2}$ | 0.000788 | 0.368
(0, 0, 2, 2) | 4 | $\vec{R}_{3}$ | 0.000797 | 0.372
(3, 3, 0, 0) | 6 | (3, 0, 0, 0, 0, 0, 3) | 0.000010 | 0.004
(2, 2, 1, 1) | 6 | (2, 0, 1, 0, 1, 0, 2) | 0.000016 | 0.007
(1, 1, 2, 2) | 6 | (1, 0, 2, 0, 2, 0, 1) | 0.000020 | 0.010
(0, 0, 3, 3) | 6 | (0, 0, 3, 0, 3, 0, 0) | 0.000020 | 0.009

###  A.2  Photon loss

Loss processes before the fabricated $J_{x}$ unitary are summarized in the
channel transmission factors $\eta_{j}$ (see Appendix F for their values and
the method of characterization). Their combined impact on the dynamics can be
modelled by extending $U_{\mathrm{exp}}$ with seven uncoupled ancillary modes
and prepending to it a unitary of seven unbalanced beamsplitters (one for each
mode) feeding the lost photons into these ancillary modes with probability
$1-\eta_{j}$. This extended unitary $\mathcal{U}$ is then used in the
subsequent calculations.

###  A.3  Partial distinguishability of correlated photon pairs

Let us consider a state of $N$ partially distinguishable photons, which occupy
external states $\ket{\vec{E}}=\ket{E_{1},\dots,E_{N}}$ of an $n$-mode
scattering network with unitary $\mathcal{U}$. The $N$-particle state is
determined by its mode occupation list $\vec{R}=(R_{1},\dots,R_{n})$ or
equivalently by its mode assignment list $\vec{E}=(E_{1},\dots,E_{N})$, with
$E_{\alpha}$ the mode occupied by the $\alpha$th particle [33]. The particles’
internal state is modelled by frequencies
$\ket{\vec{\omega}}=\ket{\omega_{1},\dots,\omega_{N}}$, yielding the combined
state

$\ket{\vec{E}}\otimes\ket{\vec{\omega}}=\ket{E_{1},\dots,E_{N}}\otimes\ket{\omega_{1},\dots,\omega_{N}}.$
(12)

The combined state of a correlated photon pair from SPDC in the mode pair a, b
is described by an integral over the particles’ internal states, which are
correlated according to the JSA $\Phi(\omega_{1},\omega_{2})$. In the first
quantization without symmetrization, the combined state is given by

$\int\mathrm{d}\omega_{1}\mathrm{d}\omega_{2}\,\Phi(\omega_{1},\omega_{2})\ket{\mathrm{a,b}}\ket{\omega_{1},\omega_{2}}.$
(13)

To correctly include the exchange symmetry of bosons, the two-photon state
needs to be symmetric under exchange of particles. After the required
symmetrization, the state of one created photon pair reads

$\ket{\Psi_{1}^{(\mathrm{ab})}}=\int\mathrm{d}\omega_{1}\mathrm{d}\omega_{2}\,\Phi(\omega_{1},\omega_{2})\ket{\vec{R}\,:\,\vec{\omega}}$
(14)

with
$\ket{\vec{R}\,:\,\vec{\omega}}=\left(\ket{\mathrm{a,b}}\ket{\omega_{1},\omega_{2}}+\ket{\mathrm{b,a}}\ket{\omega_{2},\omega_{1}}\right)/\sqrt{2}$
and the normalization condition of the JSA is
$\int\mathrm{d}\omega_{1}\mathrm{d}\omega_{2}\,\left|\Phi(\omega_{1},\omega_{2})\right|^{2}=1$.

Next, we consider a state of $P$ photon pairs emitted from the first source
into mode a and b, as well as $Q$ photon pairs emitted from a second source
into mode c and d. The total number of photons is $N=2(P+Q)$. We can describe
the state by symmetrizing the expression

$\int\mathrm{d}\,\vec{\omega}\Phi(\vec{\omega})\ket{\vec{E}}\ket{\vec{\omega}},$
(15)

with the total JSA given by the product of JSAs corresponding to all photon
pairs,

$\Phi(\vec{\omega})=\prod_{i=1}^{P}\Phi^{(\mathrm{ab})}(\omega_{2i-1},\omega_{2i})\prod_{j=P+1}^{P+Q}\Phi^{(\mathrm{cd})}(\omega_{2j-1},\omega_{2j}),$
(16)

as well as

$\ket{\vec{E}}=\ket{\underbrace{\mathrm{a,b}\dots,\mathrm{a,b}}_{\mathrm{a,b};\,P\,\mathrm{times}},\underbrace{\mathrm{c,d}\dots,\mathrm{c,d}}_{\mathrm{c,d};\,Q\,\mathrm{times}}},$
(17)

and $\mathrm{d}\vec{\omega}=\mathrm{d}\omega_{1}\dots\mathrm{d}\omega_{N}$.
Symmetrizing the above equation yields the state

$\ket{\Psi_{P,Q}}\propto\sum_{\pi\in
S_{N}}\ket{\vec{E}_{\pi}}\otimes\int\mathrm{d}\vec{\omega}\,\Phi(\vec{\omega})\ket{\vec{\omega}_{\pi}},$
(18)

where we use the shorthands
$\ket{\vec{E}_{\pi}}=\ket{E_{\pi(1)},\dots,E_{\pi(N)}}$ and
$\ket{\vec{\omega}_{\pi}}=\ket{\omega_{\pi(1)},\dots,\omega_{\pi(N)}}$. The
calculation of Eq. (18) can be simplified by reducing the summation over all
permutations $S_{N}$ to a summation over permutations leading to inequivalent
external states $\ket{\vec{E}_{\pi}}$, that is, to permutations which only
permute particles across different modes. To this end, one can decompose
$\pi\in S_{N}$ as $\pi=\xi\mu$ with $\xi\in S_{\vec{R}}$ and $\mu\in\Sigma$.
Here, $S_{\vec{R}}=S_{R_{1}}\otimes\cdots\otimes S_{R_{N}}$ is a Young
subgroup of $S_{N}$, and $\Sigma$ the right transversal of $S_{\vec{R}}$ in
$S_{N}$ [39, 18]. This results in

$\ket{\Psi_{P,Q}}\propto\sum_{\mu\in\Sigma,\xi\in
S_{\vec{R}}}\ket{\vec{E}_{\xi\mu}}\otimes\int\mathrm{d}\vec{\omega}\,\Phi(\vec{\omega})\ket{\vec{\omega}_{\xi\mu}}.$
(19)

We utilize $\ket{\vec{E}_{\xi\mu}}=\ket{\vec{E}_{\mu}}$ and write
$\vec{\omega}$ as $\vec{w}_{\xi^{\prime}}$ with $\xi^{\prime}\in S_{\vec{R}}$,
and $\vec{w}\in X_{\vec{w}}=\\{\vec{w}_{\xi}\,|\,\xi\in S_{\vec{R}}\\}$ (such
that $\\{\vec{\omega}\\}=\cup_{\vec{w}}X_{\vec{w}}$ and
$\int\mathrm{d}\vec{\omega}=\int\mathrm{d}\vec{w}\sum_{\xi^{\prime}\in
S_{\vec{R}}}$). This yields

$\ket{\Psi_{P,Q}}\propto\sum_{\mu\in\Sigma}\ket{\vec{E}_{\mu}}\otimes\int\mathrm{d}\vec{w}\sum_{\xi,\xi^{\prime}\in
S_{\vec{R}}}\,\Phi(\vec{w}_{\xi^{\prime}})\ket{\vec{w}_{\xi^{\prime}\xi\mu}}.$
(20)

Since $S_{\vec{R}}$ forms a group, we can substitute
$\xi^{\prime}\xi=\xi^{\prime\prime}$ and sum over $\xi^{\prime\prime}$ instead
of $\xi^{\prime}$,

$\ket{\Psi_{P,Q}}\propto\sum_{\mu\in\Sigma}\ket{\vec{E}_{\mu}}\otimes\int\mathrm{d}\vec{w}\sum_{\xi,\xi^{\prime\prime}\in
S_{\vec{R}}}\,\Phi(\vec{w}_{\xi^{\prime\prime}(\xi)^{-1}})\ket{\vec{w}_{\xi^{\prime\prime}\mu}}.$
(21)

Next, we use the same trick as above and write $\vec{w}_{\xi^{\prime\prime}}$
as $\vec{\omega}$ with $\int\mathrm{d}\vec{w}\sum_{\xi^{\prime\prime}\in
S_{\vec{R}}}=\int\mathrm{d}\vec{\omega}$, and instead of summing over $\xi$,
we sum over $(\xi)^{-1}$ (this substitution can be done since $S_{\vec{R}}$
forms a group), which results in

$\ket{\Psi_{P,Q}}\propto\sum_{\mu\in\Sigma}\ket{\vec{E}_{\mu}}\otimes\int\mathrm{d}\vec{\omega}\left(\sum_{\xi\in
S_{\vec{R}}}\Phi(\vec{\omega}_{\xi})\right)\ket{\vec{\omega}_{\mu}}.$ (22)

We identify the not yet normalized coefficients (in the parentheses)

$\bar{C}(\vec{\omega})=\sum_{\xi\in S_{\vec{R}}}\Phi(\vec{\omega}_{\xi}).$
(23)

Normalizing Eq. (22) then leads to [39, 18]

$\ket{\Psi_{P,Q}}=\frac{1}{\sqrt{R}}\sum_{\mu\in\Sigma}\ket{\vec{E}_{\mu}}\otimes\ket{\Omega_{\mu}},$
(24)

with $R=N!/|S_{\vec{R}}|=N!/(P!Q!)^{2}$ and

$\ket{\Omega_{\mu}}=\int\mathrm{d}\vec{\omega}\,C(\vec{\omega})\ket{\vec{\omega}_{\mu}}.$
(25)

The internal state coefficients $C(\vec{\omega})$ take into account the
possible orderings within a mode and are calculated from

$C(\vec{\omega})=\frac{\sum_{\xi\in
S_{\vec{R}}}\Phi(\vec{\omega}_{\xi})}{\sqrt{\int\mathrm{d}\vec{\omega^{\prime}}\left|\sum_{\xi\in
S_{\vec{R}}}\Phi(\vec{\omega^{\prime}}_{\xi})\right|^{2}}},$ (26)

with the normalization factor in the denominator such that
$\int\mathrm{d}\vec{\omega}|C(\vec{\omega})|^{2}=1$.

The many-particle state (24) describes the initial state of our experiment.
Since the unitary transformation $\mathcal{U}$ and the measurement of the
particles’ output mode occupation doesn’t act upon the particles’ internal
degrees of freedom, we can trace them out, resulting in the reduced external
many-particle state

$\displaystyle\rho_{\mathrm{E}}$
$\displaystyle=\mathrm{Tr}_{\mathrm{I}}(\ket{\Psi_{P,Q}}\bra{\Psi_{P,Q}})$
$\displaystyle=\sum_{\mu,\nu\in\Sigma}[\rho_{\mathrm{E}}]_{\mu,\nu}\ket{\vec{E}_{\mu}}\bra{\vec{E}_{\nu}},$
(27)

with

$[\rho_{\mathrm{E}}]_{\mu,\nu}=\frac{1}{R}\langle\Omega_{\nu}|\Omega_{\mu}\rangle.$
(28)

The transition probability to obtain the output mode occupation $\vec{S}$ from
the input mode occupation $\vec{R}$ is then obtained by [39]

$p_{\vec{R}\rightarrow\vec{S}}=S\sum_{\mu,\nu\in\Sigma}[\rho_{\mathrm{E}}]_{\mu,\nu}\prod_{\alpha=1}^{N}\mathcal{U}_{F_{\alpha},E_{\mu(\alpha)}}\mathcal{U}^{*}_{F_{\alpha},E_{\nu(\alpha)}},$
(29)

with $S=N!/|S_{\vec{S}}|$, $|S_{\vec{S}}|=\prod_{j=1}^{n}S_{j}!$ and $\vec{F}$
the output mode assignment list (defined similar to the input mode assignment
list $\vec{E}$). With Eq. (29), we can calculate all relevant input-output
probabilities for each input state listed in Table 2. Note that an output
fourfold coincidence can originate from any of the four-photon input states
$\vec{R}_{1}$, $\vec{R}_{2}$, $\vec{R}_{3}$ with no photon being lost.
However, fourfold coincidences can also arise from a six-photon input state
with up to two photons ending up in ancillary modes and/or up to two photons
lost after the unitary and/or up to three photons occupying the same output
mode (collision event). All these cases are subsumed as six-photon background
in Fig. 3.

##  Appendix B  Waveguide fabrication and unitary characterization

Figure 4:  Reconstructed amplitudes of the $J_{x}$ unitary. We launch single
photons into each input mode of the waveguide structure and measure the
relative output intensity in each output mode. From this, we reconstruct the
amplitudes of the $J_{x}$ unitary, $|[U_{\mathrm{recon}}]_{k,j}|^{2}$,
following the algorithm from Ref. [58]. Panel (h) shows the fidelities
$F_{j}=\left(\sum_{k=1}^{7}\sqrt{|[U_{\mathrm{recon}}]_{k,j}|^{2}\cdot|[U^{\mathrm{J}}]_{k,j}|^{2}}\right)^{2}$
of the reconstructed amplitudes, which benchmark how well the physical unitary
agrees with the ideal $J_{x}$ unitary.

We fabricated the waveguide chip implementing the $J_{x}$ unitary in fused
silica (Corning 7980, ArF grade) using the femtosecond laser direct-write
approach [41]. Due to nonlinear absorption, the transparent material is
modified within the focal region, producing a local increase of the refractive
index. A Coherent RegA 9000 amplifier seeded by a Mira Ti:Sa femtosecond laser
oscillator was used. After amplification, the pulses with
$150\text{\,}\mathrm{f}\mathrm{s}$ pulse duration centred at
$800\text{\,}\mathrm{n}\mathrm{m}$ had an energy of
$450\text{\,}\mathrm{n}\mathrm{J}$ at a repetition rate of
$100\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$. Waveguides were permanently
inscribed in the bulk while moving the sample at a constant speed of
$60\text{\,}\mathrm{m}\mathrm{m}\,\mathrm{m}\mathrm{i}\mathrm{n}^{-1}$ (high-
precision positioning stages ALS 130, Aerotech Inc. with a positioning error
of $\pm$0.1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$). The beam was
focused with a $20\times$ objective, producing a mode field diameter of the
guided mode of the order of $18\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$
x $20\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ at
$815\text{\,}\mathrm{n}\mathrm{m}$. Fan-in and fan-out sections were arranged
in a three-dimensional geometry to permit coupling to fiber arrays with
standard spacing of $127\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ while
minimizing cross-talk.

Prior to the experiment, we characterized the unitary using single photons.
The experimentally reconstructed amplitudes of the unitary matrix are compared
to the ideal amplitudes of the $J_{x}$ unitary in Fig. 4. We reconstructed the
unitary by sending single photons from the SPDC source into each input mode
separately. For each input, we monitored the single-photon count rate in each
of the seven output modes using APDs. By performing a least squares
optimization, we reconstructed the amplitudes
$|[U_{\mathrm{recon}}]_{k,j}|^{2}$ according to the procedure in [58]. From
this data we additionally retrieved the relative output transmissivities
(including detector efficiencies), which we used in our simulation of the
four-photon output probabilities (theory bars in Fig. 3). The phases of the
matrix elements, $[\mathrm{Arg}(U_{\mathrm{recon}})]_{k,j}$, were not
reconstructed, however, from our experience with similar waveguide structures
we expect that the fabricated structure closely matches the ideal phases of
the $J_{x}$-lattice. According to experience, the propagation constant of the
laser-written waveguides (which ultimately determines this phase) is much less
sensitive to fabrication errors than the coupling rate between neighboring
waveguides (which only influences the amplitudes of the unitary matrix). A
two-photon interference experiment conducted on the sample supports our
hypothesis, since its results agree well with the theoretical predictions,
which are based on the ideal phases of the $J_{x}$ unitary. Therefore, we
model the matrix elements of the unitary probed in our experiment as
$[U_{\mathrm{exp}}]_{k,j}=|[U_{\mathrm{recon}}]_{k,j}|\cdot
e^{\mathrm{i}[\mathrm{Arg}(U^{\mathrm{J}})]_{k,j}}$.

##  Appendix C  Procedure to temporally overlap the SPDC channels

We adjusted the optical delays between the four inputs to the unitary by
performing two-photon HOM-type measurements on the waveguide structure. First,
we launched channels c and d into input modes 3 and 5, respectively. We
monitored coincidences between output modes 1 and 4, while scanning the
optical delay stage. The corresponding output state is almost perfectly
suppressed for indistinguishable photons, while showing a reasonably high
output probability for distinguishable photons. In the same fashion, we
optimize source channel b relative to channel d, and thereafter channel a
relative to channel b, while monitoring for each delay scanning multiple
output combinations at the same time that exhibit good contrast in the
detected coincidence rate between distinguishable and indistinguishable
photons.

##  Appendix D  Phase fluctuation from twofold coincidence data

Figure 5:  Selected two-fold coincidence counts and single-photon count rates
during the measurement of the _mutually indistinguishable_ particle scenario.
Counts are averaged over an interval of $60\text{\,}\mathrm{s}$ for a total
measurement time of 88 hours. (a) Time traces of two output states which have
approximately equal contribution from input state
$\vec{R_{4}}=(0,0,1,0,1,0,0)$ and $\vec{R_{5}}=(1,0,0,0,0,0,1)$ according to
the theoretical predictions. The observed coincidence rate fluctuates heavily
suggesting that the relative phase of these two contributions fluctuates. (b)
Time traces of single-photon count rates of all seven output channels. (c)
Selected time traces of output states which are dominated by contributions of
input state $\vec{R_{4}}$. These output states are ideally suppressed
according to the suppression law. We attribute the slow drift in the observed
coincidence rates to a change in the indistinguishability of the photons
caused by changes of the optical path length. Almost no fast fluctuations in
the coincidence rate are observed since mostly one input state contributes.
(d) Selected time traces of output states, which are dominated by
contributions of input state $\vec{R_{5}}$. Neither strong fast fluctuations
nor significant slow drifts of the coincidence rates are observed.

In general, one needs to take into account the relative phases of all
contributing input states in the calculation of the final output probability
distribution. By applying and averaging over a series of phases in one of the
input modes one can realize a mixed state, which allows for simpler addition
of probabilities. This can be realized via a combination of quarter-, and
half-wave plates oriented such that the input state is kept unchanged except
for an additional phase factor [57, 38]. In our experiment the mixed state is
automatically obtained via time integration, as the relative phases fluctuate
rapidly during the measurement. This can be shown by looking at the time
traces of twofold coincidence counts for the scenario of _mutually
indistinguishable_ photons. These time traces were recorded parallel to the
four-photon coincidences of the main experiment. Figure 5(a) depicts the
coincidence rate of two output states with approximately equal contribution
from input states $\vec{R_{4}}=(0,0,1,0,1,0,0)$ and
$\vec{R_{5}}=(1,0,0,0,0,0,1)$ [see the two rightmost states in Fig. 7]. The
coincidence rates fluctuate strongly from 50 to
$250\text{\,}\mathrm{H}\mathrm{z}$, while the single-photon count rates keep
approximately constant [Fig. 5(b)]. This suggests that the relative phase of
the contributing inputs fluctuated, e.g. through temperature drifts of the
optical fibers from the source to the unitary.

##  Appendix E  Estimation of average HOM visibility from twofold coincidences
and JSA reconstruction

Figure 6:  Calculated fidelity for an output probability distribution of all
twofold, non-bunching coincidence events as a function of the
indistinguishabilities $V_{\mathrm{ab}}$ and $V_{\mathrm{cd}}$. Figure 7:
Output probability distribution of all twofold, non-bunching coincidence
events. For the simulated result, we used optimized time-averaged
indistinguishabilities of $V_{\mathrm{ab}}=0.94$ and $V_{\mathrm{cd}}=0.90$.
The fidelity is $F=0.991$. The experimental and simulated degree of
suppression violation are $D_{\mathrm{exp}}=0.198$ and
$D_{\mathrm{sim}}=0.202$, respectively. Experimental error bars from the
counting statistics are too small to be visible. The inset shows the
discretized JSA of channels a and b (the JSA of channels c and d is similar).

For the investigated scenario of _mutually indistinguishable_ particles we
observed a slow change in the recorded twofold coincidence counts over time
[compare Fig. 5(c) and (d)]. This effect is most pronounced for output states,
which receive contributions mostly from the input state $\vec{R_{4}}$. We
attribute this drift to relative optical path length changes in the pertinent
channels (here c and d) of the setup from the SPDC source to the $J_{x}$
unitary, which leads to a degradation of the indistinguishability from the
initially optimized situation. This may arise from the asymmetric response to
temperature changes of the freespace delay in channel c and the fiber in
channel d, and changes the output probability distribution (counting
statistics), e.g. it may increase the probability to observe an ideally
suppressed output event due to a degrading of the particles’
indistinguishability.

We estimate the time-averaged degraded indistinguishability of photons from
the same pair from the total experimental twofold coincidence counts, which we
collect in parallel to the fourfold coincidences of the main experiment. We
compare the experimentally obtained output distribution of all non-bunching
two-photon events with the simulated output distribution for varying
indistinguishabilities $V_{\mathrm{ab}}$ and $V_{\mathrm{cd}}$, i.e.,
indistinguishability between photons in channels a and b, as well as channels
c and d. Additionally, we keep the indistinguishability between photons
originating from different pairs at a fixed value of 59.1% (retrieved from a
separate heralded HOM measurement, see Appendix F).

For each simulated output distribution, we calculate the fidelity according to

$F=\left(\sum_{i}\sqrt{p_{\mathrm{exp},i}\,p_{\mathrm{sim},i}}\right)^{2},$
(30)

with the sum running over all non-bunching two-photon outputs. We consider
input states $\vec{R_{4}}$ and $\vec{R_{5}}$, as well as a background of four-
and six-photon states, where two and four photons are lost before the unitary,
respectively (these are the dominant contributions under the considerations of
losses). The background resulting from initial states of four- and six-photons
depends on the indistinguishability between photons from different pairs and
contributes around 22.5% to the measured final output events. As shown in Fig.
6, the fidelity reaches its maximum value for indistinguishabilties of
$V_{\mathrm{ab}}=0.94$ and $V_{\mathrm{cd}}=0.90$. These values should
therefore best represent the time-averaged indistinguishabilties in the
experiment. For the above estimated time-averaged indistinguishabilities, the
normalized output distribution of all non-bunching two-fold coincidence events
is plotted in Fig. 7. It reaches a fidelity of $F=0.991$.

We finally calculate a JSA that reproduces the measured and estimated
indistinguishabilities by following the procedure in [59]. First a phase
matching function is calculated in the paraxial approximation of the involved
spatial modes of the pump laser and the photon-collecting fibers. We apply a
Gaussian pump spectral envelope and spectral filter functions
($3\text{\,}\mathrm{n}\mathrm{m}$ FWHM bandwidth, as in the setup) to obtain a
discretized JSA on a frequency grid of size $17\times 17$. The JSA of SPDC
channel a and b is plotted in the inset of Fig. 7.

Instead of modelling the reduction of indistinguishability by a path-length
drift directly in the time-domain, we mimick this effect by mutually shifting
the central transmission frequencies of the filters (thus reducing spectral
instead of temporal overlaps). The estimated time-averaged
indistinguishabilities are reproduced with an up-converted pump spectral width
of $0.4\text{\,}\mathrm{n}\mathrm{m}$ FWHM, as well as pairwise filter offsets
of $0.625\text{\,}\mathrm{n}\mathrm{m}$ between channels a and b and
$0.8\text{\,}\mathrm{n}\mathrm{m}$ between channels c and d. The internal
state of multiple pairs is then calculated as a tensor product of the two-
photon JSAs [cf. Eq. (16)].

##  Appendix F  Source parameters and photon loss

To properly simulate the source and subsequent channel losses in the
experiment, we needed to estimate the pair generation probability, the
collection efficiencies of the source, as well as the indistinguishabilities
of photons from different channels.

A preliminary characterisation of the source lets us obtain the pair
generation probability and the collection efficiencies of the four source
channels. We attached each source channel directly to an APD, which resulted
in typical single-photon count rates of
$160-$370\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ at a pump power of
$\approx$150\text{\,}\mathrm{m}\mathrm{W}$$. Additionally, we measured
coincidence rates of $31\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ between
channels a and b, as well as $25\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$
between channels c and d. From these rates, we reconstructed pair generation
probabilities per pump pulse of $p^{(\mathrm{ab})}=2.6\%$ and
$p^{(\mathrm{cd})}=3.3\%$ for channels a and b, and channels c and d,
respectively. Additionally, we obtained collection efficiencies of 19%, 18%,
10%, and 22% for the four channels a-d. In the source characterization, we
used detectors with a detection efficiency of 68%. The collection efficiencies
were corrected for this detection efficiency.

These channel collection efficiencies as well as the incoupling efficiencies
to the seven input modes of the unitary, which includes fiber-to-chip coupling
and propagation loss on the chip, are combined to input mode transmissivity
factors $\eta_{j}$ for $j=1,\ldots,7$. We extracted the incoupling
efficiencies from the unitary-reconstruction data [cf. Appendix B], comparing
the total transmissivity across all output modes for the relevant input modes
and assuming an on-chip propagation loss of
$0.5\text{\,}\mathrm{d}\mathrm{B}\mathrm{/}\mathrm{c}\mathrm{m}$ in all
straight parts of the waveguide structure. For the four occupied input modes
in the experiment, we calculate input mode transmissivity factors
$\eta_{1}=0.055$, $\eta_{3}=0.034$, $\eta_{5}=0.107$, and $\eta_{7}=0.065$.

We performed a heralded HOM-interference experiment on a fiber beamsplitter,
where we interfered photons from channels b and d, using a and c as additional
heralding channels [45]. From the experiment, we attained an
indistinguishability of 59.1% for photons originating from different pairs. In
a standard HOM experiment with photons from the same pair, we retrieved
maximal visibilities of 96.1% for source channels a and b, and 98.2% for
source channels c and d. In the actual experiment, these indistinguishabilties
vary over time, probably due to optical path length drifts, which reduces the
temporal overlap of photons from different channels [see Appendix E].

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  *[HOM]: Hong, Ou, and Mandel
  *[SPDC]: spontaneous parametric down-conversion
  *[BiBO]: Bismuth Borate
  *[BBO]: Barium Borate
  *[APDs]: avalanche photodiode
  *[QD]: quantum dot
  *[QDs]: quantum dot
  *[JSA]: joint spectral amplitude
  *[JSAs]: joint spectral amplitude
  *[APD]: avalanche photodiode