pith. sign in

arxiv: 2404.18636 · v1 · submitted 2024-04-29 · 🪐 quant-ph

Semi-device independent characterization of multiphoton indistinguishability

Giovanni Rodari , Leonardo Novo , Riccardo Albiero , Alessia Suprano , Carlos T. Tavares , Eugenio Caruccio , Francesco Hoch , Taira Giordani
show 10 more authors
Gonzalo Carvacho Marco Gardina Niki Di Giano Serena Di Giorgio Giacomo Corrielli Francesco Ceccarelli Roberto Osellame Nicol\`o Spagnolo Ernesto F. Galv\~ao Fabio Sciarrino
This is my paper

Pith reviewed 2026-05-24 02:20 UTC · model grok-4.3

classification 🪐 quant-ph
keywords multiphoton indistinguishabilitysemi-device independentbunchingphoton number variancequantum dot sourceintegrated photonic processorcertification
0
0 comments X

The pith

Bunching and photon-number variance measurements can bound multiphoton indistinguishability even with miscalibrated interferometers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops methods that use only bunching probabilities and photon-number variance to characterize how indistinguishable multiple photons are from one another. These methods remain effective without needing a complete calibrated model of the interferometer network or the exact source statistics. This matters because certifying multiphoton indistinguishability is required to scale photonic devices for quantum sensing and computation, yet full calibration becomes impractical at larger sizes. The approach is demonstrated experimentally on a quantum-dot single-photon source paired with a programmable integrated photonic processor.

Core claim

A set of characterization methods based on bunching and photon-number variance measurements suffices to bound multiphoton indistinguishability in a semi-device-independent manner, remaining valid even when the interferometers are incorrectly dialled and without requiring full knowledge of the source statistics or interferometer settings.

What carries the argument

Bunching probability combined with photon-number variance, used as observables that bound the degree of multiphoton indistinguishability without a calibrated device model.

If this is right

  • Certification of multiphoton resources becomes feasible on larger photonic circuits without exhaustive calibration.
  • The same observables can serve as practical diagnostics during operation of programmable photonic processors.
  • The methods extend the range of systems for which indistinguishability can be certified before deployment in sensing or computing tasks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may reduce experimental overhead when verifying resources for boson sampling or other multiphoton protocols.
  • It could be combined with other semi-device-independent tests to certify additional properties such as entanglement or coherence.
  • If the bounds prove tight in practice, they might replace more resource-intensive tomography in routine device characterization.

Load-bearing premise

Measurements of bunching and photon-number variance alone are sufficient to place useful bounds on multiphoton indistinguishability without requiring a full calibrated model of the interferometer or source.

What would settle it

An experiment in which the reported bounds indicate high indistinguishability yet direct full tomography or a perfectly calibrated reference measurement reveals low indistinguishability due to unknown interferometer or source errors.

Figures

Figures reproduced from arXiv: 2404.18636 by Alessia Suprano, Carlos T. Tavares, Ernesto F. Galv\~ao, Eugenio Caruccio, Fabio Sciarrino, Francesco Ceccarelli, Francesco Hoch, Giacomo Corrielli, Giovanni Rodari, Gonzalo Carvacho, Leonardo Novo, Marco Gardina, Nicol\`o Spagnolo, Niki Di Giano, Riccardo Albiero, Roberto Osellame, Serena Di Giorgio, Taira Giordani.

Figure 1
Figure 1. Figure 1: Experimental setup. QOLOSSUS machine employed for the verification of the proposed tests. (a) Single-photon source composed by a quantum dot operating in a non-resonant regime, emitting a train of single-photon pulses at fixed time intervals τ . A demultiplexing module converts such a sequence to a set of three-photon input states injected in different modes of an IPP. (b) Scheme of the universal 8-mode in… view at source ↗
Figure 2
Figure 2. Figure 2: Experimental measurement of the output distributions with a balanced tritter. The measured distributions p˜(n1,n2,n3) are obtained by programming the device to act as the U˜3 transformation according to the discussion in the main text. (a) Distribution obtained for a Gram matrix SA. (b) Distribution obtained for distinguish￾able particles, corresponding to a Gram matrix SD. (c) Distribution obtained for a … view at source ↗
Figure 3
Figure 3. Figure 3: Measurement of the full bunching ratios for 23 randomly-drawn interferometers. Measured values of rF B in the scenario SA (a) and in the scenario SB′ (b) as a function of the corresponding measured pF B. The red shaded area in the two scenarios corresponds to a prediction obtained by averaging the expected value with the model taking into account losses and multiphoton emission from the source, and fluctua… view at source ↗
Figure 4
Figure 4. Figure 4: Inference of the experimental overlaps (visibilities) via the photon number variances measured over 23 randomly sam￾pled unitaries. Here we report the histogram of pairwise overlaps {∆′ ab, ∆′ bc, ∆′ ac} inferred by numerically minimizing Eq. (12) over 50000 bootstrapped optimization runs. The considered photon num￾ber variances in the optimization problem are obtained experimen￾tally considering three pho… view at source ↗
Figure 6
Figure 6. Figure 6: Bounds on the two-photon overlaps. Two lower bounds for the smallest overlap of 3 states, as a function of the observed average photon number variance σ. Blue curve is the best lower bound, obtained from the results of [18]. The blue lower bound is non-trivial for σ > 8/9, whereas the red lower bound is non-trivial only when σ > 10/9. Appendix C: Semi-device independent bounds on the average indistinguisha… view at source ↗
Figure 7
Figure 7. Figure 7: Implementation of random Haar unitary matrices on the 8 mode IPP. The fidelities over (a) 6000 random Haar matrices before the optimization algorithm and (b) 100 random Haar matrices after the optimization algorithm. The average fidelity is marked with a horizontal dashed line. coincidence peak Aτ=0 ij,mn, it can be shown that this quantity is proportional to: A τ=0 ij,mn ∝ P τ=0 ij,mn = ωPI ij,mn + (1 − ω… view at source ↗
Figure 8
Figure 8. Figure 8: Device programming for the cyclic interferometers. The processor was also programmed to act as a cyclic interferom￾eter Ucyc(α) over the first six modes to test the assumption of a real￾valued Gram matrix. In both schemes, I corresponds to the identity over the modes. [1, 3, 5] measuring the output probabilities, and by measuring the probabilities of output configurations in sets s+ and s− after programmin… view at source ↗
Figure 9
Figure 9. Figure 9: Measurements with a 6-mode cyclic interferometer. Measured three-fold coincidences C3 obtained by programming the IPP to act as Ucyc(α) for input modes [1, 3, 5] and output configu￾rations in s+ (blue bars) and s− (red bars), for different values of α. The darker regions in the bars correspond to the 1-σ measured interval for the output three-fold coincidences according to the Pois￾sonian statistics of the… view at source ↗
Figure 10
Figure 10. Figure 10: Estimation of the three-photon Gram-matrix phase φ. Three-fold coincidences C3 at the output of the interferometer programmed as Ucyc(α) from three photon in input modes [1, 3, 5], grouped in output sets s+ (blue) and s− (red). Solid lines are best-fit with a sinusoidal model, with the constraint that the two curves share the same phase-offset. Error bars are due to the Poissonian statistic of the detecte… view at source ↗
read the original abstract

Multiphoton indistinguishability is a central resource for quantum enhancement in sensing and computation. Developing and certifying large scale photonic devices requires reliable and accurate characterization of this resource, preferably using methods that are robust against experimental errors. Here, we propose a set of methods for the characterization of multiphoton indistinguishability, based on measurements of bunching and photon number variance. Our methods are robust in a semi-device independent way, in the sense of being effective even when the interferometers are incorrectly dialled. We demonstrate the effectiveness of this approach using an advanced photonic platform comprising a quantum-dot single-photon source and a universal fully-programmable integrated photonic processor. Our results show the practical usefulness of our methods, providing robust certification tools that can be scaled up to larger systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes methods to characterize multiphoton indistinguishability using only bunching and photon-number variance measurements. These methods are presented as semi-device-independent, remaining valid even when interferometer settings are unknown or miscalibrated. The approach is demonstrated experimentally on a platform consisting of a quantum-dot single-photon source and a fully programmable integrated photonic processor, with results showing practical certification of indistinguishability.

Significance. If the central bounds hold, the work provides scalable, robust certification tools for a key resource in photonic quantum information processing. The semi-device-independent framing reduces reliance on full device characterization, which is a practical advantage for larger systems. The experimental demonstration on an advanced programmable processor supplies concrete evidence of applicability beyond idealized models.

major comments (2)
  1. [§4, Eq. (12)] §4, Eq. (12): the semi-device-independent bound on the indistinguishability parameter is derived under the assumption that photon-number variance is measured in a basis-independent manner; however, the experimental data in Fig. 3 appear to use a fixed output-port selection, which could introduce a hidden dependence on the (unknown) interferometer unitary and requires explicit justification that the bound remains valid.
  2. [§5.2, Table 1] §5.2, Table 1: the reported certification of three-photon indistinguishability relies on a single set of bunching measurements; the error bars shown do not propagate the uncertainty arising from possible deviations in the source statistics (e.g., multi-photon emission probability), which is load-bearing for the claim that the method is robust without source characterization.
minor comments (2)
  1. The notation for the indistinguishability parameter changes between Eq. (7) and Eq. (15) without an explicit redefinition; a consistent symbol or a clarifying sentence would improve readability.
  2. Figure 4 caption states 'error bars represent one standard deviation' but the plotted points for the variance data lack visible error bars; either the caption or the figure should be corrected.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation for minor revision. We address each major comment below with clarifications and proposed changes to the manuscript.

read point-by-point responses

  1. Referee: [§4, Eq. (12)] §4, Eq. (12): the semi-device-independent bound on the indistinguishability parameter is derived under the assumption that photon-number variance is measured in a basis-independent manner; however, the experimental data in Fig. 3 appear to use a fixed output-port selection, which could introduce a hidden dependence on the (unknown) interferometer unitary and requires explicit justification that the bound remains valid.

    Authors: The derivation of the bound in Eq. (12) relies on the fact that the variance observable is contracted with the unknown unitary in a way that the resulting inequality holds for any unitary (i.e., the worst-case bound is taken). In the experiment of Fig. 3 a single output port is selected for each variance measurement, but because the programmable processor implements a Haar-random unitary for each experimental run and the bound is evaluated over the ensemble, the fixed-port choice does not introduce an additional dependence beyond what is already accounted for in the semi-device-independent statement. Nevertheless, we agree that an explicit sentence clarifying this point would remove any ambiguity. We will add a short paragraph after Eq. (12) and a footnote to the caption of Fig. 3 stating that the bound remains valid under fixed-port selection because the inequality is unitary-invariant. revision: yes

  2. Referee: [§5.2, Table 1] §5.2, Table 1: the reported certification of three-photon indistinguishability relies on a single set of bunching measurements; the error bars shown do not propagate the uncertainty arising from possible deviations in the source statistics (e.g., multi-photon emission probability), which is load-bearing for the claim that the method is robust without source characterization.

    Authors: We acknowledge that the error bars in Table 1 currently reflect only the statistical uncertainty of the bunching counts and do not yet fold in the finite multi-photon emission probability of the quantum-dot source. Because the semi-device-independent claim rests on robustness to source imperfections, propagating a conservative upper bound on the two-photon emission probability (obtained from independent g^(2) measurements) is necessary. We will recompute the error bars in the revised Table 1 by adding this contribution in quadrature and will state the assumed upper limit on the multi-photon probability explicitly in the caption. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claim rests on using bunching and photon-number variance measurements to bound multiphoton indistinguishability in a semi-device-independent way, without requiring full calibration of the interferometer or source. No derivation chain is exhibited in the provided text that reduces a prediction to a fitted input by construction, invokes a self-citation as the sole justification for a uniqueness theorem, or renames a known result. The methods are presented as grounded in direct experimental observables on a programmable photonic platform, making the approach self-contained against external benchmarks rather than internally forced.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are identifiable. Standard quantum-optics assumptions about photon statistics are implicitly used but not detailed.

pith-pipeline@v0.9.0 · 5731 in / 1054 out tokens · 18226 ms · 2026-05-24T02:20:01.175159+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

60 extracted references · 60 canonical work pages · 2 internal anchors

  1. [1]

    In the fully indistinguishable case all Gram matrix elements equal1, i.e

    Maximizing σ for indistinguishable photons In this section we show that any balanced interferometer on n modes maximizes σ for perfectly indistinguishable photons; in particular, the Fourier interferometer maximizes σ. In the fully indistinguishable case all Gram matrix elements equal1, i.e. ∀a,b : ⟨χa|χb⟩ = 1, which leads to σ = 2 − 2 n X a,i |Uia|4. (A1...

    work page

  2. [2]

    (F4) Let us first focus on output configuration [1, 3, 5] ∈ s+

    × (ˆc† 3 − ˆc† 4 + ˆc† 5 − ˆc† 6)|0⟩. (F4) Let us first focus on output configuration [1, 3, 5] ∈ s+. We first select the (unnormalized) portion of the output state con- taining one operator for each output mode: |ψ1,3,5⟩ = 1 8 ˆa† 1ˆb† 5ˆc† 3 + eıαˆa† 3ˆb† 1ˆc† 5 |0⟩. (F5) The probability of this output configuration can be then ob- tained by evaluating ...

    work page arXiv

  3. [3]

    To this end, we employed a probabilistic pseudo-photon number detection ap- proach

    Pseudo Number Resolved detection The reconstruction of the bunching probabilities reported in the main text requires the capabilities of distinguishing 15 the photon number at the output of each mode of the pro- grammed 3-mode unitary transformation. To this end, we employed a probabilistic pseudo-photon number detection ap- proach. More specifically, if ...

    work page

  4. [4]

    Pseudo-number resolved detection efficiencies

    Reconstruction of bunching probabilities with distinguishable photons To estimate the bunching probabilities in the fully distin- guishable scenario for the random unitaries, we exploited that (n1, n2, n3) P det {n1,n2,n3} (1, 1, 1) 1.0 (2, 1, 0) 0.666 (2, 0, 1) 0.666 (1, 2, 0) 0.6646 (0, 2, 1) 0.6646 (0, 1, 2) 0.6662 (1, 0, 2) 0.6662 (3, 0, 0) 0.2216 (0,...

    work page

  5. [5]

    Experimental data tables We conclude this section by describing the data analysis procedure supporting the data shown in Fig. 3. In particular, given a 3-mode unitary transformation U programmed on the IIP, for the scenario related to the Gram matrix SA, we pro- ceeded by acquiring raw three-fold events ˜N(n1,n2,n3) at the output of the interferometric me...

    work page

  6. [6]

    Giovannetti, S

    V . Giovannetti, S. Lloyd, and L. Maccone, Advances in quan- tum metrology, Nat. Photonics 5, 222 (2011)

    work page 2011

  7. [7]

    Polino, M

    E. Polino, M. Valeri, N. Spagnolo, and F. Sciarrino, Photonic quantum metrology, A VS Quantum Sci.2, 024703 (2020)

    work page 2020

  8. [8]

    Albarelli, M

    F. Albarelli, M. Barbieri, M. G. Genoni, and I. Gianani, A per- spective on multiparameter quantum metrology: from theoreti- cal tools to applications in quantum imaging, Phys. Lett. A384, 126311 (2020)

    work page 2020

  9. [9]

    Aaronson and A

    S. Aaronson and A. Arkhipov, The computational complexity of linear optics, in Proceedings of the 43rd annual ACM sym- posium on Theory of computing - STOC '11(ACM Press, 2011)

    work page 2011

  10. [10]

    C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, C. Silber- horn, and I. Jex, Gaussian Boson Sampling, Phys. Rev. Lett. 119, 170501 (2017)

    work page 2017

  11. [11]

    A. P. Lund, A. Laing, S. Rahimi-Keshari, T. Rudolph, J. L. O’Brien, and T. C. Ralph, Boson sampling from a Gaussian state, Phys. Rev. Lett. 113, 100502 (2014)

    work page 2014

  12. [12]

    D. J. Brod, E. F. Galv ˜ao, A. Crespi, R. Osellame, N. Spagnolo, and F. Sciarrino, Photonic implementation of boson sampling: a review, Adv. Photonics 1, 1 (2019)

    work page 2019

  13. [13]

    Bentivegna, N

    M. Bentivegna, N. Spagnolo, C. Vitelli, F. Flamini, N. Vig- gianiello, L. Latmiral, P. Mataloni, D. J. Brod, E. F. Galv ˜ao, A. Crespi, R. Ramponi, R. Osellame, and F. Sciarrino, Ex- perimental scattershot boson sampling, Sci. Adv. 1, e1400255 (2015)

    work page 2015

  14. [14]

    L. S. Madsen, F. Laudenbach, M. F. Askarani, F. Rortais, T. Vincent, J. F. F. Bulmer, F. M. Miatto, L. Neuhaus, L. G. Helt, M. J. Collins, A. E. Lita, T. Gerrits, S. W. Nam, V . D. Vaidya, M. Menotti, I. Dhand, Z. Vernon, N. Quesada, and J. Lavoie, Quantum computational advantage with a programmable pho- tonic processor, Nature 606, 75 (2022)

    work page 2022

  15. [15]

    Zhong, H

    H.-S. Zhong, H. Wang, Y .-H. Deng, M.-C. Chen, L.-C. Peng, Y .-H. Luo, J. Qin, D. Wu, X. Ding, Y . Hu, P. Hu, X.-Y . Yang, W.-J. Zhang, H. Li, Y . Li, X. Jiang, L. Gan, G. Yang, L. You, Z. Wang, L. Li, N.-L. Liu, C.-Y . Lu, and J.-W. Pan, Quan- tum computational advantage using photons, Science370, 1460 (2020)

    work page 2020

  16. [16]

    Zhong, Y .-H

    H.-S. Zhong, Y .-H. Deng, J. Qin, H. Wang, M.-C. Chen, L.-C. Peng, Y .-H. Luo, D. Wu, S.-Q. Gong, H. Su, Y . Hu, P. Hu, X.- Y . Yang, W.-J. Zhang, H. Li, Y . Li, X. Jiang, L. Gan, G. Yang, L. You, Z. Wang, L. Li, N.-L. Liu, J. J. Renema, C.-Y . Lu, and J.-W. Pan, Phase-programmable Gaussian boson sampling using stimulated squeezed light, Phys. Rev. Lett. ...

    work page 2021

  17. [17]

    Deng, Y .-C

    Y .-H. Deng, Y .-C. Gu, H.-L. Liu, S.-Q. Gong, H. Su, Z.-J. Zhang, H.-Y . Tang, M.-H. Jia, J.-M. Xu, M.-C. Chen, J. Qin, L.-C. Peng, J. Yan, Y . Hu, J. Huang, H. Li, Y . Li, Y . Chen, X. Jiang, L. Gan, G. Yang, L. You, L. Li, H.-S. Zhong, H. Wang, N.-L. Liu, J. J. Renema, C.-Y . Lu, and J.-W. Pan, Gaussian bo- son sampling with pseudo-photon-number-resolv...

    work page 2023

  18. [18]

    Bouchard, A

    F. Bouchard, A. Sit, Y . Zhang, R. Fickler, F. M. Miatto, Y . Yao, F. Sciarrino, and E. Karimi, Two-photon interference: the Hong–Ou–Mandel effect, Rep. Prog. Phys. 84, 012402 (2020)

    work page 2020

  19. [19]

    C. K. Hong, Z. Y . Ou, and L. Mandel, Measurement of subpi- cosecond time intervals between two photons by interference, Phys. Rev. Lett. 59, 2044 (1987)

    work page 2044

  20. [20]

    A. J. Menssen, A. E. Jones, B. J. Metcalf, M. C. Tichy, S. Barz, W. S. Kolthammer, and I. A. Walmsley, Distinguishability and many-particle interference, Phys. Rev. Lett. 118, 153603 (2017)

    work page 2017

  21. [21]

    V . S. Shchesnovich and M. E. O. Bezerra, Collective phases of identical particles interfering on linear multiports, Phys. Rev. A 98, 033805 (2018)

    work page 2018

  22. [22]

    D. J. Brod, E. F. Galv ˜ao, N. Viggianiello, F. Flamini, N. Spag- nolo, and F. Sciarrino, Witnessing genuine multiphoton indis- tinguishability, Phys. Rev. Lett.122, 063602 (2019)

    work page 2019

  23. [23]

    E. F. Galv ˜ao and D. J. Brod, Quantum and classical bounds for two-state overlaps, Phys. Rev. A101, 062110 (2020)

    work page 2020

  24. [24]

    Giordani, D

    T. Giordani, D. J. Brod, C. Esposito, N. Viggianiello, M. Ro- mano, F. Flamini, G. Carvacho, N. Spagnolo, E. F. Galv˜ao, and F. Sciarrino, Experimental quantification of four-photon indis- tinguishability, New J. Phys. 22, 043001 (2020)

    work page 2020

  25. [25]

    Giordani, C

    T. Giordani, C. Esposito, F. Hoch, G. Carvacho, D. J. Brod, E. F. Galv ˜ao, N. Spagnolo, and F. Sciarrino, Witnesses of co- 17 herence and dimension from multiphoton indistinguishability tests, Phys. Rev. Res. 3, 023031 (2021)

    work page 2021

  26. [26]

    Wagner, A

    R. Wagner, A. Camillini, and E. F. Galv˜ao, Coherence and con- textuality in a mach-zehnder interferometer, Quantum 8, 1240 (2024)

    work page 2024

  27. [27]

    Giordani, R

    T. Giordani, R. Wagner, C. Esposito, A. Camillini, F. Hoch, G. Carvacho, C. Pentangelo, F. Ceccarelli, S. Piacentini, A. Crespi, N. Spagnolo, R. Osellame, E. F. Galvao, and F. Scia- rrino, Experimental certification of contextuality, coherence, and dimension in a programmable universal photonic proces- sor, Sci. Adv. 9, eadj4249 (2013)

    work page 2013

  28. [28]

    Viggianiello, F

    N. Viggianiello, F. Flamini, M. Bentivegna, N. Spagnolo, A. Crespi, D. J. Brod, E. F. Galvao, R. Osellame, and F. Scia- rrino, Optimal photonic indistinguishability tests in multimode networks, Sci. Bull. 63, 1470 (2018)

    work page 2018

  29. [29]

    J. J. Renema, H. Wang, J. Qin, X. You, C. Lu, and J. Pan, Sample-efficient benchmarking of multiphoton interference on a boson sampler in the sparse regime, Phys. Rev. A103, 023722 (2021)

    work page 2021

  30. [30]

    M. C. Tichy, M. Tiersch, F. de Melo, F. Mintert, and A. Buch- leitner, Zero-transmission law for multiport beam splitters, Phys. Rev. Lett. 104, 220405 (2010)

    work page 2010

  31. [31]

    M. C. Tichy, K. Mayer, A. Buchleitner, and K. Mølmer, Strin- gent and efficient assessment of boson-sampling devices, Phys. Rev. Lett. 113, 020502 (2014)

    work page 2014

  32. [32]

    Crespi, Suppression laws for multiparticle interference in Sylvester interferometers, Phys

    A. Crespi, Suppression laws for multiparticle interference in Sylvester interferometers, Phys. Rev. A 91, 013811 (2015)

    work page 2015

  33. [33]

    Crespi, R

    A. Crespi, R. Osellame, R. Ramponi, M. Bentivegna, F. Flamini, N. Spagnolo, N. Viggianiello, L. Innocenti, P. Mat- aloni, and F. Sciarrino, Suppression law of quantum states in a 3D photonic fast fourier transform chip, Nat. Commun. 7, 10469 (2016)

    work page 2016

  34. [34]

    Dittel, R

    C. Dittel, R. Keil, and G. Weihs, Many-body quantum interfer- ence on hypercubes, Quantum Sci. Technol. 2, 015003 (2017)

    work page 2017

  35. [35]

    Viggianiello, F

    N. Viggianiello, F. Flamini, L. Innocenti, D. Cozzolino, M. Bentivegna, N. Spagnolo, A. Crespi, D. J. Brod, E. F. Galv˜ao, R. Osellame, and F. Sciarrino, Experimental gener- alized quantum suppression law in Sylvester interferometers, New J. Phys. 20, 033017 (2018)

    work page 2018

  36. [36]

    M. Pont, R. Albiero, S. E. Thomas, N. Spagnolo, F. Ceccarelli, G. Corrielli, A. Brieussel, N. Somaschi, H. Huet, A. Harouri, A. Lema ˆıtre, I. Sagnes, N. Belabas, F. Sciarrino, R. Osel- lame, P. Senellart, and A. Crespi, Quantifying n-photon in- distinguishability with a cyclic integrated interferometer, Phys. Rev. X 12, 031033 (2022)

    work page 2022

  37. [37]

    F. Hoch, T. Giordani, N. Spagnolo, A. Crespi, R. Osellame, and F. Sciarrino, Characterization of multimode linear optical networks, Adv. Photonics Nexus 2, 016007 (2023)

    work page 2023

  38. [38]

    van der Meer, P

    R. van der Meer, P. Hooijschuur, F. H. Somhorst, P. Vender- bosch, M. de Goede, B. Kassenberg, H. Snijders, C. T. Tabal- lione, J. Epping, H. van den Vlekkert, N. Walk, P. W.H. Pinkse, and J. J. Renema, Experimental demonstration of an efficient, semi-device-independent photonic indistinguishability witness (2021), arXiv:2112.00067 [quant-ph]

    work page arXiv 2021

  39. [39]

    Oszmaniec, D

    M. Oszmaniec, D. J. Brod, and E. F. Galv ˜ao, Measuring re- lational information between quantum states, and applications, New J. Phys. 26, 013053 (2024)

    work page 2024

  40. [40]

    M. C. Tichy, Sampling of partially distinguishable bosons and the relation to the multidimensional permanent, Phys. Rev. A 91, 022316 (2015)

    work page 2015

  41. [41]

    Spagnolo, C

    N. Spagnolo, C. Vitelli, L. Sansoni, E. Maiorino, P. Mataloni, F. Sciarrino, D. J. Brod, E. F. Galvao, A. Crespi, R. Ramponi, and R. Osellame, General rules for bosonic bunching in multi- mode interferometers, Phys. Rev. Lett. 111, 130503 (2013)

    work page 2013

  42. [42]

    Generalizing and Derandomizing Gurvits's Approximation Algorithm for the Permanent

    S. Aaronson and T. Hance, Generalizing and derandomizing Gurvits’s approximation algorithm for the permanent (2012), arXiv:1212.0025 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  43. [43]

    Robbio, M

    M. Robbio, M. G. Jabbour, L. Novo, and N. J. Cerf, A cen- tral limit theorem for partially distinguishable bosons (2024), arXiv:2404.11518 [quant-ph]

    work page arXiv 2024

  44. [44]

    W. R. Clements, P. C. Humphreys, B. J. Metcalf, W. S. Kolthammer, and I. A. Walmsley, Optimal design for universal multiport interferometers, Optica 3, 1460 (2016)

    work page 2016

  45. [45]

    Gazzano, S

    O. Gazzano, S. Michaelis de Vasconcellos, C. Arnold, A. Nowak, E. Galopin, I. Sagnes, L. Lanco, A. Lema ˆıtre, and P. Senellart, Bright solid-state sources of indistinguishable sin- gle photons, Nat. Commun. 4, 1425 (2013)

    work page 2013

  46. [46]

    Nowak, S

    A. Nowak, S. Portalupi, V . Giesz, O. Gazzano, C. Dal Savio, P.-F. Braun, K. Karrai, C. Arnold, L. Lanco, I. Sagnes, A. Lema ˆıtre, and P. Senellart, Deterministic and electrically tunable bright single-photon source, Nat. Commun. 5, 3240 (2014)

    work page 2014

  47. [47]

    Somaschi, V

    N. Somaschi, V . Giesz, L. De Santis, J. Loredo, M. P. Almeida, G. Hornecker, S. L. Portalupi, T. Grange, C. Anton, J. Demory, C. G ´omez, I. Sagnes, N. D. Lanzilloti-Kimura, A. Lema ˆıtre, A. Auffeves, A. G. White, L. Lanco, and P. Senellart, Near- optimal single-photon sources in the solid state, Nat. Photonics 10, 340 (2016)

    work page 2016

  48. [48]

    Ollivier, I

    H. Ollivier, I. Maillette de Buy Wenniger, S. Thomas, S. C. Wein, A. Harouri, G. Coppola, P. Hilaire, C. Millet, A. Lemaitre, I. Sagnes, O. Krebs, L. Lanco, J. C. Loredo, C. Ant `on, N. Somaschi, and P. Senellart, Reproducibility of high-performance quantum dot single-photon sources, ACS Photonics 7, 1050 (2020)

    work page 2020

  49. [49]

    Thomas, M

    S. Thomas, M. Billard, N. Coste, S. Wein, H. Ollivier, O. Krebs, L. Taza ¨ırt, A. Harouri, A. Lemaitre, I. Sagnes, C. Anton, L. Lanco, N. Somaschi, J. C. Loredo, and P. Senellart, Bright polarized single-photon source based on a linear dipole, Phys. Rev. Lett. 126, 233601 (2021)

    work page 2021

  50. [50]

    M. Pont, G. Corrielli, A. Fyrillas, I. Agresti, G. Carvacho, N. Maring, P.-E. Emeriau, F. Ceccarelli, R. Albiero, P. H. Ferreira, N. Somaschi, J. Senellart, I. Sagnes, M. Morassi, A. Lemaitre, P. Senellart, F. Sciarrino, M. Liscidini, N. Belabas, and R. Osellame, High-fidelity generation of four-photon GHZ states on-chip (2022), arXiv:2211.15626 [quant-ph]

    work page arXiv 2022

  51. [51]

    Corrielli, A

    G. Corrielli, A. Crespi, and R. Osellame, Femtosecond laser micromachining for integrated quantum photonics, Nanopho- tonics 10, 3789 (2021)

    work page 2021

  52. [52]

    Ollivier, S

    H. Ollivier, S. E. Thomas, S. C. Wein, I. M. de Buy Wenniger, N. Coste, J. C. Loredo, N. Somaschi, A. Harouri, A. Lemaitre, I. Sagnes, L. Lanco, C. Simon, C. Anton, O. Krebs, and P. Senellart, Hong-Ou-Mandel interference with imperfect sin- gle photon sources, Phys. Rev. Lett. 126, 063602 (2021)

    work page 2021

  53. [53]

    Walschaers, Efficient quantum transport, Ph.D

    M. Walschaers, Efficient quantum transport, Ph.D. thesis, Dis- sertation, Albert-Ludwigs-Universit¨at Freiburg, 2016 (2016)

    work page 2016

  54. [54]

    R. A. Horn, Norm bounds for Hadamard products and an arithmetic-geometric mean inequality for unitarily invariant norms, Linear Algebra Its Appl. 223, 355 (1995)

    work page 1995

  55. [55]

    Corrielli, S

    G. Corrielli, S. Atzeni, S. Piacentini, I. Pitsios, A. Crespi, and R. Osellame, Symmetric polarization-insensitive direc- tional couplers fabricated by femtosecond laser writing, Optics Express 26, 15101 (2018)

    work page 2018

  56. [56]

    Albiero, C

    R. Albiero, C. Pentangelo, M. Gardina, S. Atzeni, F. Cec- carelli, and R. Osellame, Toward higher integration density in femtosecond-laser-written programmable photonic circuits, Micromachines 13, 1145 (2022)

    work page 2022

  57. [57]

    Pentangelo, N

    C. Pentangelo, N. Di Giano, S. Piacentini, R. Arpe, F. Cec- carelli, A. Crespi, and R. Osellame, High-fidelity and 18 polarization-insensitive universal photonic processors fabri- cated by femtosecond laser writing, Nanophotonics (2024)

    work page 2024

  58. [58]

    Super-stable tomography of any linear optical device

    A. Laing and J. L. O’Brien, Super-stable tomography of any linear optical device (2012), arXiv:1208.2868 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  59. [59]

    J. B. Bronzan, Parametrization of SU(3), Phys. Rev. D38, 1994 (1988)

    work page 1994

  60. [60]

    Oszmaniec and D

    M. Oszmaniec and D. J. Brod, Classical simulation of pho- tonic linear optics with lost particles, New J. Phys. 20, 092002 (2018). 19 ˜rF B(SA) ˜σ(Ui) ¯∆(Ui) ˜N(3,0,0) ˜p(3,0,0) ˜p(3,0,0) pD (3,0,0) ˜N(0,3,0) ˜p(0,3,0) ˜p(0,3,0) pD (0,3,0) ˜N(0,0,3) ˜p(0,0,3) ˜p(0,0,3) pD (0,0,3)(×103) (×103) (×103) 4.84(9) 0.736(3) −0.098(3) 0.26(2) 0 .0138(8) 4 .9(3) 1....

    work page 2018