Experimental quantification of genuine four-photon indistinguishability

TL;DR

This paper presents an efficient four-photon experiment in a linear-optical setup that accurately quantifies genuine four-photon indistinguishability, advancing tools for multiparticle quantum resource characterization.

Contribution

It introduces a heralding-free interferometer design and applies a new theoretical framework to measure and bound multiphoton indistinguishability in a resource-efficient manner.

Findings

High agreement between experimental results and theory

Effective bounds on two-photon overlaps

Novel technique for multiphoton interference characterization

Abstract

Photon indistinguishability plays a fundamental role in information processing, with applications such as linear-optical quantum computation and metrology. It is then necessary to develop appropriate tools to quantify the amount of this resource in a multiparticle scenario. Here we report a four-photon experiment in a linear-optical interferometer designed to simultaneously estimate the degree of indistinguishability between three pairs of photons. The interferometer design dispenses with the need of heralding for parametric down-conversion sources, resulting in an efficient and reliable optical scheme. We then use a recently proposed theoretical framework to quantify genuine four-photon indistinguishability, as well as to obtain bounds on three unmeasured two-photon overlaps. Our findings are in high agreement with the theory, and represent a new resource-effective technique for the characterization of multiphoton interference.

Tables

Table 1. Table 1: Summary of the measured overlaps and inferred bounds for c 𝟏 subscript 𝑐 1 \bm{c_{1}} . Only state XXXX displays a significant non-zero value for c 1 subscript 𝑐 1 c_{1} . For the other states, in which at least one photon is distinguishable from the others, the inferred value for c 1 subscript 𝑐 1 c_{1} is statistically compatible with c 1 = 0 subscript 𝑐 1 0 c_{1}=0 . For these states we have reported the value of the upper bound in Eq. ( 3 ). The overlap values have been retrieved from the distributions shown in Fig.  3 . All the errors are derived from Poissonian uncertainties associated to photon counts and then propagated via Monte Carlo methods.

State	$c_1$	$r_{\mathrm{AB}}$	$r_{\mathrm{BC}}$	$r_{\mathrm{CD}}$
XXXX	$[0.34\left(1\right),\mathrm{\hspace{0.17em}0.64}\left(1\right)]$	$0.826\left(6\right)$	$0.640\left(8\right)$	$0.872\left(4\right)$
XXXY	$0.00\left(2\right)$	$0.802\left(8\right)$	$0.780\left(8\right)$	$0.00\left(2\right)$
XXYY	$0.01\left(1\right)$	$0.832\left(6\right)$	$0.01\left(1\right)$	$0.802\left(6\right)$
XXYZ	$0.00\left(1\right)$	$0.834\left(6\right)$	$0.00\left(1\right)$	$0.01\left(1\right)$
XYYZ	$0.01\left(2\right)$	$0.01\left(2\right)$	$0.68\left(1\right)$	$0.04\left(2\right)$
XYWZ	$0.00\left(2\right)$	$0.02\left(2\right)$	$0.00\left(2\right)$	$0.00\left(2\right)$

Table 2. Table 2: Bounding unmeasured overlaps. We report the lower bounds (LBs) and upper bounds (UBs) for the unmeasured overlaps starting from those observed in the experiment for each state. We adopt two different models for describing the multiphoton state, the first expressed in Eq. ( 2 ) and the second in the form | A ⟩ ⊗ | B ⟩ ⊗ | C ⟩ ⊗ | D ⟩ tensor-product ket 𝐴 ket 𝐵 ket 𝐶 ket 𝐷 \left|{A}\right\rangle\otimes\left|{B}\right\rangle\otimes\left|{C}\right\rangle\otimes\left|{D}\right\rangle . The bounds reported without uncertainty correspond to the cases for which the model gives trivial bounds, i.e. 0 for LBs and 1 for UBs.

State	Mixed classical state model (i)			Separable pure state model (ii)
	$r_{\mathrm{AC}}$	$r_{\mathrm{AD}}$	$r_{\mathrm{BD}}$	$r_{\mathrm{AC}}$	$r_{\mathrm{AD}}$	$r_{\mathrm{BD}}$
XXXX	$[0.46\left(1\right),\mathrm{\hspace{0.17em}0.81}\left(1\right)]$	$[0.34\left(1\right),\mathrm{\hspace{0.17em}0.94}\left(1\right)]$	$[0.511\left(9\right),\mathrm{\hspace{0.17em}0.768}\left(9\right)]$	$[0.23\left(1\right),\mathrm{\hspace{0.17em}0.955}\left(5\right)]$	$[0.017\left(4\right),\mathrm{\hspace{0.17em}0.976}\left(4\right)]$	$[0.28\left(1\right),\mathrm{\hspace{0.17em}0.925}\left(6\right)]$
XXXY	$[0.58\left(1\right),\mathrm{\hspace{0.17em}0.98}\left(1\right)]$	$[0,\mathrm{\hspace{0.17em}0.40}\left(2\right)]$	$[0,\mathrm{\hspace{0.17em}0.20}\left(2\right)]$	$[0.34\left(1\right),\mathrm{\hspace{0.17em}0.9991}\left(8\right)]$	$[0,\mathrm{\hspace{0.17em}1}]$	$[0,\mathrm{\hspace{0.17em}1}]$
XXYY	$[0,\mathrm{\hspace{0.17em}0.17}\left(1\right)]$	$[0,\mathrm{\hspace{0.17em}0.37}\left(1\right)]$	$[0,\mathrm{\hspace{0.17em}0.20}\left(1\right)]$	$[0,\mathrm{\hspace{0.17em}1}]$	$[0,\mathrm{\hspace{0.17em}1}]$	$[0,\mathrm{\hspace{0.17em}1}]$
XXYZ	$[0,\mathrm{\hspace{0.17em}0.14}\left(1\right)]$	$[0,\mathrm{\hspace{0.17em}1}]$	$[0,\mathrm{\hspace{0.17em}0.97}\left(2\right)]$	$[0,\mathrm{\hspace{0.17em}1}]$	$[0,\mathrm{\hspace{0.17em}1}]$	$[0,\mathrm{\hspace{0.17em}1}]$
XYYZ	$[0,\mathrm{\hspace{0.17em}0.33}\left(1\right)]$	$[0,\mathrm{\hspace{0.17em}1}]$	$[0,\mathrm{\hspace{0.17em}0.36}\left(2\right)]$	$[0,\mathrm{\hspace{0.17em}1}]$	$[0,\mathrm{\hspace{0.17em}1}]$	$[0,\mathrm{\hspace{0.17em}1}]$
XYWZ	$[0,\mathrm{\hspace{0.17em}0.94}\left(3\right)]$	$[0,\mathrm{\hspace{0.17em}1}]$	$[0,\mathrm{\hspace{0.17em}0.96}\left(2\right)]$	$[0,\mathrm{\hspace{0.17em}1}]$	$[0,\mathrm{\hspace{0.17em}1}]$	$[0,\mathrm{\hspace{0.17em}1}]$

Equations

$$p_b^{ij}=\frac{1+r_{ij}}{2}.$$

$$\rho =c_1{\rho }^1+\sum _{s>1}{c}_s{\rho }_s^{\perp },$$

$$r_{AB}+r_{BC}+r_{CD}-2\le c_1\le \min (r_{AB},r_{BC},r_{CD}).$$

$$r_{AC}\in$$

$$r_{BD}\in$$

$$r_{AD}\ge$$

$$r_{AD}\le$$

$$r_{AC}-r_{AB}-r_{CD},r_{CD}-r_{AB}-r_{BC}).$$

$$r_{AC}\le {\left(\sqrt{r_{AB}{r}_{BC}}+\sqrt{(1-r_{AB})(1-r_{BC})}\right)}^2$$

$$r_{AC}\ge {\left(\sqrt{r_{AB}{r}_{BC}}-\sqrt{(1-r_{AB})(1-r_{BC})}\right)}^2,$$

Full text

I Introduction

Photon indistinguishability is a key concept in quantum physics, which can be characterized via two-particle interference Mandel (1999). From the first experimental observation carried out using entangled photons produced by a parametric down-conversion source Hong et al. (1987), the quantum signature of this phenomenon, known as Hong-Ou-Mandel (HOM) effect, has proved to be essential for tasks such as characterization of single photon sources, practical protocols of quantum communication Chrapkiewicz et al. (2016); Ricci et al. (2004); Irvine et al. (2004); Carvacho et al. (2018); Pirandola et al. (2015); Nagali et al. (2009) and quantum computation. Recent proposals use two-photon interference for the realization of two-qubit gates, essential elements for photon-based quantum computing schemes Knill et al. (2001). Multiphoton interference is at the core of the computational complexity of linear optical networks, as exemplified by the Boson Sampling problem Aaronson and Arkhipov (2011); Broome et al. (2013); Spring et al. (2013); Tillmann et al. (2013); Crespi et al. (2013); Spagnolo et al. (2013a); Bentivegna et al. (2015); Carolan et al. (2015); Tillmann et al. (2015); Wang et al. (2017); Loredo et al. (2017); He et al. (2017); Wang et al. (2018a, b); Paesani et al. (2018); Zhong et al. (2018, 2019); Brod et al. (2019a). This problem, solved naturally by multiphoton interference, shows that simulating the dynamics of indistinguishable photons is likely to be hard for classical computers, which also suggests that certification of genuine multiphoton interference is a difficult problem Aaronson and Arkhipov (2014); Tichy et al. (2014); Aolita et al. (2015); Crespi (2015); Walschaers et al. (2016); Bentivegna et al. (2016); Wang and Duan (2016); Dittel et al. (2017); Spagnolo et al. (2014); Carolan et al. (2014); Bentivegna et al. (2014); Crespi et al. (2016); Viggianiello et al. (2018a); Agresti et al. (2019); Giordani et al. (2018); Viggianiello et al. (2018b); Flamini et al. (2019).

Due to the fundamental role of such quantum effects in quantum information and quantum optics, recently much effort has been devoted to increasing the number of interfering particles, and to the development of new methods to characterize the amount of indistinguishability of multiphoton states produced by single-photon sources. These methods aim to find effective strategies in terms both of physical and computational resources needed to address the problem. Many techniques are based on bosonic coalescence Spagnolo et al. (2013a); Carolan et al. (2014), i.e. the tendency of indistinguishable photons to come out from the same port of an interferometer, which can be viewed as generalizations of the original two-photon HOM effect to the case of multiple ports and photons Tichy et al. (2012); Spagnolo et al. (2013b); Tichy et al. (2014); Carolan et al. (2015); Menssen et al. (2017); Crespi et al. (2016); Viggianiello et al. (2018a); Shchesnovich and Bezerra (2017); Agne et al. (2017); Viggianiello et al. (2018b).

Here we present an approach for quantifying indistinguishability in multiphoton experiments. To this aim we focus on extracting information from sets of two-photon HOM experiments realizable in a single inteferometer with a suitable design. We describe each possible experiment by a graph, with nodes representing photons and edges representing two-photon HOM tests Brod et al. (2019b). As we discuss in the following, this method provides a simple and efficient design capable of quantifying genuine multiphoton indistinguishability. Indeed, a test based on logical constraints, theoretically introduced in Ref. Brod and Galvão (2019), results in lower bounds for the amount of such quantity. In this work we experimentally demonstrate the effectiveness of this approach, furnishing an extensive experimental study of genuine indistinguishability in a four-photon state. We also demonstrate other useful predictions of the model, which greatly simplifies the experimental effort for the complete characterization of pairwise overlaps of multiphoton states Brod and Galvão (2019). Simply put, the theory allows us to infer precise bounds on pairwise overlaps not accessible or measured directly by the experiment, based on information on the overlaps that were actually measured. The predictions we obtain on the unmeasured overlaps inferred from experimental data represent the first application of such techniques for a multiphoton state. By slightly changing the experimental set-up, we were able to directly confirm the theoretical prediction for one of the previously unmeasured overlaps.

Our results confirm the feasibility of the method when applied to an actual physical system. In particular, the special design of the interferometer allows to perform the experiment with a parametric down-conversion source without the need for heralding. This represents a demonstration of a practical approach to the characterization of multiphoton sources, which promises to decrease the experimental effort required to benchmark future deterministic single-photon sources Michler et al. (2000); Ding et al. (2016); Somaschi et al. (2016) in the regime of high number of photons Loredo et al. (2017); Wang et al. (2017); He et al. (2017); Wang et al. (2018a).

II Quantifying four-photon indistinguishability

In the recent Refs. Brod et al. (2019b); Brod and Galvão (2019), an approach was theoretically proposed to characterize multiphoton indistinguishability, based on two-photon HOM tests and constraints imposed by logic and quantum theory. Here we review the main concepts of this approach, as applied to the case of interest for the experimental implementation we report in Section III.

Consider a weighted, undirected, linear graph $P_{4}$ with four vertices $A,B,C$ and $D$, and three edges $r_{AB},r_{BC}$ and $r_{CD}$ (see Fig. 1). Vertices correspond to single-photon states, and the edge weights to the overlap $r_{ij}=Tr(\rho_{i}\rho_{j})$ between the states of photons $i$ and $j$. Each overlap $r_{ij}$ can be estimated experimentally via the probability of bunching $p_{b}^{ij}$ in a HOM test between photons $i$ and $j$, as they are related by Garcia-Escartin and Chamorro-Posada (2013):

[TABLE]

Moreover, it is possible to experimentally estimate multiple overlaps using a single interferometer, by measuring the probability of bunching at each output port. In Fig. 1 we show such an interferometer, which we can use to estimate the three overlaps for the $P_{4}$ graph.

We discuss the quantification of multiphoton indistinguishability for states of two different types undergoing the test represented by the graph of Fig. 1. These two models allow us to quantify genuine indistinguishability, as well as to infer bounds on pairwise overlaps that have not been measured. The first model we consider consists of (generally mixed) four-photon states of the form:

[TABLE]

where $\rho^{1}$ is a state of 4 perfectly indistinguishable photons, and $\rho_{s}^{\perp}$ are pure states where every pair of photons is either perfectly identical or orthogonal (i.e. distinguishable), and at least two photons are orthogonal. In the rest of the work we will refer to this model as (i). Following the approach of Ref. Brod et al. (2019b), we identify the degree of multiphoton indistinguishability of state (2) with the value of $c_{1}$, as it is the probability associated with preparing perfectly indistinguishable photons. States of the form (2) above were called “classical” in Ref. Brod and Galvão (2019), as there exists some basis in which they are diagonal and thus display no coherences.

In Ref. Brod et al. (2019b) it was shown that the HOM probabilities of bunching could be used, in a suitable family of interferometers, to obtain non-trivial bounds for $c_{1}$. These interferometers, which perform multiple HOM tests, can be described as star graphs, where HOM tests are represented by a central vertex connected to all the others. As shown in Supplementary Information, a similar argument can be made for the interferometer in Fig. 1, associated with graph $P_{4}$ (the possibility to find bounds for $c_{1}$ is implicit in the more general results of Ref. Brod and Galvão (2019)). More explicitly, we show that $c_{1}$ is bounded by:

[TABLE]

Then, the measurement of $r_{AB}$, $r_{BC}$ and $r_{CD}$ via the three HOM tests in the interferometer in Fig. 1 enables us to quantify the degree of indistinguishability.

We are interested in inferring the values of the overlaps which were not measured by our interferometer: $r_{AC},r_{BD}$, and $r_{AD}$. In particular, it is easy to check that $c_{1}$ in Eq. (2) is a lower bound for the estimate of all overlaps $r_{ij}$, hence the lower bound in Eq. (3) also applies to them. More stringent bounds can be found by applying the results of Ref. Brod and Galvão (2019) to the four-photon state (2) undergoing measurements at the output of the interferometer in Fig. 1, obtaining (see Supplementary Information):

[TABLE]

We will now retrieve corresponding bounds for states described by a second model, which we label (ii). We assume that our four-photon state is pure and separable, i.e. of the form $\left|{\mathrm{A}}\right\rangle\otimes\left|{\mathrm{B}}\right\rangle\otimes\left|{\mathrm{C}}\right\rangle\otimes\left|{\mathrm{D}}\right\rangle$. We do not constrain the pure states describing different photons to be either identical or completely distinguishable. Ref.  Brod and Galvão (2019) obtains general bounds for unmeasured overlaps of such product states. In the case of a four-photon state it is possible to show that:

[TABLE]

and, if $r_{AB}+r_{BC}>1$,

[TABLE]

otherwise $r_{AC}\geq 0$. By permuting indices $B\to A$, $C\to B$, and $D\to C$, the inequalities above also give upper and lower bounds for $r_{BD}$. We can then prove bounds on the unknown overlap $r_{AD}$ by applying the same reasoning above, but using $r_{AB}$ and the inferred range for $r_{BD}$ (alternatively, using $r_{CD}$ and the inferred range for $r_{AC}$). These bounds are discussed in detail in Supplementary Information.

We note that the two models (i) and (ii) we consider are different approximations of the four-photon input state. The ideal case of four perfectly indistinguishable photons corresponds respectively to $c_{1}=1$ in the state described by Eq. (2) and to state $\left|{\psi}\right\rangle\otimes\left|{\psi}\right\rangle\otimes\left|{\psi}\right\rangle\otimes\left|{\psi}\right\rangle$ in our product-state model. The aim of this analysis is to apply these two descriptions to a four-photon state produced by a non-deterministic single-photon source, with the goal of quantifying the indistinguishability and providing estimations of unmeasured overlaps.

III Experimental implementation

In this section we describe the experimental apparatus we use to quantify four-photon indistinguishability (Fig. 2), and also to obtain bounds for the unmeasured two-state overlaps.

To generate our input states we employ double-pair emission from a Type-II spontaneous parametric down-conversion (SPDC) source, where all four photons are spectrally filtered by means of 3 nm band-pass filters. In the first stage after photon generation, photons in input modes (B,C) propagate through a pair of beam splitters, implemented as a sequence of a half wave-plate and a polarizing beam splitter. This set of optical elements implements the first layer of the interferometer shown in Fig. 1. In the second stage, all 6 modes are coupled in single-mode fibers, and propagate through polarization control stages and a set of delay lines to modulate their degree of indistinguishability. In the third stage all photons, in superposition over the six modes, are coupled to three single-mode fiber beam splitters (SMFBSs), corresponding to the second layer of the interferometer, where Hong-Ou-Mandel tests are performed. Finally, the output modes of each FBS are connected to multimode fiber beam splitters to enable approximate photon-number resolving detection. Four-photon coincidence events are then recorded via 12 single-photon avalanche photodiodes and a coincidence detection apparatus.

An important experimental characteristic of the interferometer in Fig. 1 is that it enables four-photon experiments without the need for heralding. To understand this feature, let us start by considering only the input states corresponding to the generation of two photon pairs, which are described by the occupation numbers $\{(1,1,0,0,1,1),(2,2,0,0,0,0),(0,0,0,0,2,2)\}$. The structure of interferometer in Fig. 1 is such that all output configurations accessible to input state $(1,1,0,0,1,1)$ are different from those associated to the other two (double pair) inputs produced by the source. This observation makes it possible to unambiguously post-select only those output events corresponding to the desired four-photon input state $(1,1,0,0,1,1)$ with no need for heralding (see Supplementary Information for further details).

The complete set of probabilities associated to all four-photon output configurations is shown in Fig. 3. We have identified six interesting input configurations, corresponding to different situations of pairwise indistinguishability, which we labelled as {XXXX, XXXY, XXYY, XXYZ, XYYZ, XYWZ} (where identical letters denote ideally indistinguishable photons). The XXXX configuration corresponds to the fully indistinguishable case, while in the other cases the number of distinguishable photons increases up to the fully distinguishable state XYWZ. We obtained fine control over pairwise distinguishability by tuning the temporal delays.

For each class of input state we collected $N\geq 10^{4}$ events. Fig. 3 shows the values of the distributions corresponding to the experimental data $p^{\mathrm{m}}$, compared against the expected distributions $p^{\mathrm{e}}$ that take into account partial indistinguishability and multi-pair contributions (see Supplementary Information for more details). In particular, to estimate $p^{\mathrm{e}}$ we considered probabilistic mixtures of states in which pairs of photons can only be either identical or perfectly distinguishable, as prescribed by model (i). The agreement between measured $p^{\mathrm{m}}$ and expected distributions $p^{\mathrm{e}}$ is quantified using the Total Variation Distance (TVD), defined as $\mathrm{TVD}=1/2\sum_{i}|p^{\mathrm{m}}_{i}-p^{\mathrm{e}}_{i}|$ (see Fig. 3). The comparison with the data confirms that $p^{\mathrm{e}}$ reproduces very well the distribution associated to the state when projected on the basis of the occupation numbers in the output modes.

III.1 Determining genuine multiphoton indistinguishability

In Section II we have seen that for classical states it is possible to use measured overlaps to bound the amount of genuine multiphoton indistinguishability, as characterized by coefficient $c_{1}$ in Eq. (3). For the XXXX configuration, the obtained bounds for $c_{1}$ are $0.34\pm 0.01\leq c_{1}\leq 0.640\pm 0.008$, indicating the presence of genuine four-photon indistinguishability by 31 standard deviations. As expected, the other input states do not display any level of genuine four-photon indistinguishability (as $c_{1}=0$). Indeed, configurations different from XXXX have been measured to confirm the effectiveness of the method. The reported data show that each state with at least one photon distinguishable from the others does not display any statistically significant component of genuine multiphoton indistinguishability. The resulting bounds for $c_{1}$ for these states are reported in Table 1.

In Fig. 4, each configuration has been represented in the space spanned by the three relative delays $(\Delta x_{\mathrm{AB}}$, $\Delta x_{\mathrm{BC}}$, $\Delta x_{\mathrm{CD}})$. The plot shows the predicted surfaces corresponding to the lower bound for $c_{1}$, defined in Eq. (3), calculated for Gaussian overlaps ($r_{\mathrm{AB}}$, $r_{\mathrm{BC}}$, $r_{\mathrm{CD}}$) with respect to the relative delays in the ideal case (pink surface) and by considering partial photon indistinguishability corresponding to the adopted source (green surface). For each of the two scenarios, all points enclosed within the corresponding surface lead to a non-trivial lower bound for $c_{1}$, which characterizes genuine multiphoton indistinguishability. The parameters adopted for the calculation of the two surfaces are inferred from the widths and visibilities of the HOM dips (further details on HOM measurements and on the estimation of the surfaces are given in Supplementary Information). The ratio between the volumes enclosed in the two surfaces, $V_{a}$ for the pink one and $V_{b}$ for the green, is $V_{b}/V_{a}\sim 0.525$, thus confirming that the region corresponding to a non-trivial lower bound for $c_{1}$ is smaller when experimental imperfections are taken into account.

III.2 Determining unmeasured overlaps

Our experimental apparatus measures a set of three two-photon overlaps: $r_{AB},r_{BC}$ and $r_{CD}$. The theoretical results described in Section II allow us to obtain bounds on the unmeasured overlaps $r_{AC},r_{BD}$ and $r_{AD}$. As discussed previously, this can be done assuming two different models, (i) and (ii), for the states we prepare. The first model assumes that the state is classical, i.e. a coherence-free mixture of states for which every pair of photons is either distinguishable or identical to each other. The second model assumes that the four photons are in a tensor product of pure states with any degree of two-photon indistinguishability. It is worth noting that both models could reproduce well the behaviour of the generated four-photon states in the basis of mode occupation numbers, and can thus be employed to infer predictions on the four-photon state indistinguishability. In Table 2 we report the calculated upper and lower bounds for unmeasured overlaps, using the two models. As we can see, for all cases where unmeasured overlaps were expected to be non-zero, we obtained lower bounds consistent with this expectation. This includes positive values for the three unmeasured two-photon overlaps for input XXXX, as well as a positive value for $r_{AC}$ for input XXXY.

The interferometer we have implemented allows, with a slight change in the set-up, to verify directly the prediction of Table 2 regarding the overlap $r_{AD}$. This is possible by swapping photons A with C and B with D at the input of the interferometer, by acting on the polarization degree of freedom of the source (see Fig. 5). This operation results in measurements corresponding to the graph C-D-A-B, all performed with input configuration XXXX. The new measurements for overlaps $r_{AB},r_{CD}$ resulted in values $r_{\mathrm{CD}}=0.86(1)$, $r_{\mathrm{AB}}=0.842(8)$. Interestingly, we can now estimate also the previously unmeasured $r_{AD}$ [$r_{\mathrm{AD}}=0.65(1)$]. The new measurements of overlaps $r_{AB}$ and $r_{CD}$ are compatible with the values obtained with the previous set-up, while the new measured value for $r_{AD}$ is compatible with the previously derived upper and lower bounds.

IV Discussion

Multiphoton indistinguishability is a promising resource for quantum information processing, and so it is crucial to identify genuine instances of it, as well as characterize it efficiently Brod et al. (2019b); Brod and Galvão (2019). Here we have experimentally demonstrated an approach to quantify genuine multiphoton indistinguishability in a four-photon experiment. We also characterize the degree of two-photon indistinguishability without the need for direct measurements of all two-photon experiments. A fundamental advantage of our approach is the use of a single interferometric set-up to effectively implement multiple Hong-Ou-Mandel tests. The experimental data, besides this direct characterization of the methodology introduced in Refs. Brod et al. (2019b); Brod and Galvão (2019), can be used to bound all the unmeasured two-photon overlaps. Our interferometer design allows for the use of multiple parametric down-conversion single-photon sources, without the need for heralding. These features showcase the promise of our approach for the characterization of future multiphoton sources. In particular, the method could find applications in the case of deterministic sources Michler et al. (2000); Ding et al. (2016); Somaschi et al. (2016); Huber et al. (2018); Basso Basset et al. (2019), which have been recently identified as a promising route to obtain larger photon numbers with high purity Loredo et al. (2017); Wang et al. (2017); He et al. (2017); Wang et al. (2018a).

Acknowledgements.

We acknowledge funding from the European Research Council (ERC) Advanced Grant CAPABLE (Composite integrated photonic platform by femtosecond laser micro-machining, Grant Agreement No. 742745), from Sapienza Progetto di Ateneo H2020-ERC QUMAC (MAChine learning via hybrid integrated QUANTUM photonics), and from Instituto Nacional de Ciência e Tecnologia de Informação Quântica (INCT-IQ/CNPq). G.C. acknowledges Becas Chile and Conicyt.