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arxiv: 2606.23231 · v1 · pith:TNOOLCKYnew · submitted 2026-06-22 · 🪐 quant-ph

Efficient Energy-Constrained Semi-Device-Independent QRNG with an Integrated Heterodyne Receiver

Mattia Sabatini , Carles Roch i Carceller , Tommaso Bertapelle , Andrea Peri , Giulio Gualandi , Mat\'ias Rub\'en Bola\~nos , Yoann Pi\'etri , Giacomo Corrielli
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Paolo Villoresi Giuseppe Vallone Roberto Osellame Armin Tavakoli Marco Avesani
This is my paper

Pith reviewed 2026-06-26 08:03 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum random number generatorsemi-device-independentheterodyne detectioncoherent statesphoton-number constraintsentropy certificationintegrated photonicscontinuous-variable QRNG
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The pith

A four-state coherent-state constellation measured by heterodyne detection certifies 0.223 bits of randomness per shot under photon-number constraints.

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

The paper shows how to certify randomness in a quantum random number generator while assuming only that the source states obey a photon-number limit. It implements the protocol with four coherent states arranged symmetrically in phase space and detects them with an integrated heterodyne receiver. Semidefinite relaxation then produces tight lower bounds on the Shannon entropy extractable from the measurement outcomes. The experiment reaches 0.223 certified bits per measurement, the highest rate reported for any continuous-variable semi-device-independent QRNG. Entropy-accumulation techniques further allow finite-size certification without requiring independent and identically distributed rounds.

Core claim

We realize the protocol using a four-state coherent-state constellation symmetrically distributed in phase space and measured by heterodyne detection, certifying 0.223 bit per measurement, which is the highest value reported to date for a continuous-variable semi-device-independent QRNG. The implementation combines a low-loss integrated photonic heterodyne receiver with a simple transmitter assembled from commercial components, providing a practical and high-speed architecture for semi-device-independent randomness generation.

What carries the argument

Photon-number-constrained semi-device-independent QRNG protocol, in which a bound on the mean photon number of the prepared states is combined with heterodyne detection outcomes and semidefinite relaxation to compute lower bounds on extractable Shannon entropy.

If this is right

  • Finite-size randomness can be certified without assuming independent and identically distributed rounds by combining the bounds with entropy accumulation.
  • The photon-number constraint is experimentally verifiable and therefore practical for photonic hardware.
  • The integrated heterodyne receiver and commercial transmitter together form a high-speed, low-loss architecture suitable for real-world deployment.
  • The achieved rate of 0.223 bits per measurement exceeds all previously reported values for continuous-variable semi-device-independent QRNGs.

Where Pith is reading between the lines

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

  • The same photon-number bounding approach could be applied to other continuous-variable protocols to obtain device-independent-style security with modest assumptions.
  • Integration of the receiver on a photonic chip suggests a path toward compact, mass-producible QRNG modules.
  • Relaxing the four-state constellation to higher-order modulations might increase the certified rate while preserving the same constraint framework.

Load-bearing premise

The photon-number constraints on the source states are sufficient, when combined with semidefinite relaxation techniques, to compute tight lower bounds on the certifiable Shannon entropy.

What would settle it

An experiment that records the heterodyne statistics of the four-state constellation while enforcing the stated photon-number bound and finds that the resulting semidefinite program returns an entropy lower bound materially below 0.223 bits per measurement.

Figures

Figures reproduced from arXiv: 2606.23231 by Andrea Peri, Armin Tavakoli, Carles Roch I Carceller, Giacomo Corrielli, Giulio Gualandi, Giuseppe Vallone, Marco Avesani, Mat\'ias Rub\'en Bola\~nos, Mattia Sabatini, Paolo Villoresi, Roberto Osellame, Tommaso Bertapelle, Yoann Pi\'etri.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
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Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
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Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

Semi-device-independent QRNG frameworks represent a particularly attractive approach, combining strong security guarantees with high randomness generation rates while relying only on reduced and practical physical assumptions. A recently proposed approach based on photon-number constraints is particularly suited to photonic implementations, where these assumptions can be easily assessed experimentally. Here, we experimentally demonstrate a quantum random number generator within this framework, enabling the direct computation of lower bounds on the certifiable Shannon entropy via semidefinite relaxation techniques. When combined with entropy accumulation methods, this approach enables finite-size randomness certification without assuming independent and identically distributed rounds. We realize the protocol using a four-state coherent-state constellation symmetrically distributed in phase space and measured by heterodyne detection, certifying 0.223 bit per measurement, which is the highest value reported to date for a continuous-variable semi-device-independent QRNG. The implementation combines a low-loss integrated photonic heterodyne receiver with a simple transmitter assembled from commercial components, providing a practical and high-speed architecture for semi-device-independent randomness generation.

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

1 major / 2 minor

Summary. The manuscript experimentally realizes a continuous-variable semi-device-independent QRNG protocol using a four-state coherent-state constellation symmetrically placed in phase space, measured via an integrated low-loss heterodyne receiver. It certifies 0.223 bits of randomness per measurement—the highest reported for this class—by imposing experimentally verifiable photon-number constraints, computing lower bounds on conditional Shannon entropy via semidefinite relaxation, and applying entropy accumulation to obtain finite-size guarantees without an i.i.d. assumption.

Significance. If the reported entropy bounds are tight, the result would establish a practical, high-rate architecture for semi-DI randomness generation that combines commercial transmitter components with integrated photonic detection, while relaxing the need for full device characterization. The combination of photon-number constraints with SDP relaxation and entropy accumulation is a methodological strength that could generalize to other photonic QRNG implementations.

major comments (1)
  1. [Abstract/methods] Abstract and methods: the central claim of 0.223 bit/measurement rests on the semidefinite relaxation producing a sufficiently tight lower bound on the conditional Shannon entropy given only the photon-number constraint and the four-state constellation. No duality-gap bound, comparison against an exact SDP or numerical optimization on the same constraint set, and no sensitivity analysis under small violations of the photon-number assumption are provided; if the relaxation gap is non-negligible the certified value would be overestimated.
minor comments (2)
  1. The description of the integrated heterodyne receiver would benefit from explicit loss figures, bandwidth, and how the photon-number constraint is experimentally bounded (e.g., mean photon number or support).
  2. Clarify the precise SDP formulation (objective, constraints, and relaxation hierarchy) used to obtain the entropy lower bound; a supplementary file with the solver input would improve reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract/methods] Abstract and methods: the central claim of 0.223 bit/measurement rests on the semidefinite relaxation producing a sufficiently tight lower bound on the conditional Shannon entropy given only the photon-number constraint and the four-state constellation. No duality-gap bound, comparison against an exact SDP or numerical optimization on the same constraint set, and no sensitivity analysis under small violations of the photon-number assumption are provided; if the relaxation gap is non-negligible the certified value would be overestimated.

    Authors: We agree that the manuscript does not supply a duality-gap certificate, a direct comparison to an exact solver or finer discretization, or a sensitivity study for photon-number violations. These omissions leave open the possibility that the reported lower bound on conditional entropy is not tight. In the revised manuscript we will add (i) a numerical comparison of the current SDP relaxation against a discretized convex optimization over the same constraint set and (ii) a first-order sensitivity analysis showing the change in the certified entropy when the photon-number bound is relaxed by a few percent. If the gap remains negligible, the 0.223 bit/measurement figure will be retained with the new supporting data; otherwise the certified value will be adjusted downward. revision: yes

Circularity Check

0 steps flagged

No circularity: SDP-based entropy lower bounds are independent of the certified quantity

full rationale

The paper's central claim is an experimental certification of 0.223 bits/measurement obtained by feeding measured heterodyne statistics (under an experimentally assessed photon-number constraint) into a semidefinite relaxation that produces a lower bound on conditional Shannon entropy, followed by entropy accumulation. This chain does not reduce any quantity to a fitted parameter by construction, nor does it rely on a self-citation whose content is itself the target result. The SDP relaxation is a standard numerical method whose output is not definitionally identical to the input data; any gap between the relaxation and the true minimum is a question of bound tightness rather than circularity. No self-definitional, fitted-input, or uniqueness-imported steps appear in the derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies insufficient detail to enumerate concrete free parameters, axioms, or invented entities. The framework rests on photon-number constraints whose experimental accessibility is asserted but not quantified here.

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discussion (0)

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Reference graph

Works this paper leans on

52 extracted references · 1 canonical work pages

  1. [1]

    dimension×DV polarization – – 0.03 23·10 −6

  2. [2]

    overlap×DV time-bin 0.22 – 50 11

  3. [3]

    overlap×DV on-off keying 0.33 – 50 16.5

  4. [4]

    overlap×DV on-off keying 0.0061 – 1 0.006

  5. [5]

    overlap✓DV time-bin – 1.299 ‡ 0.526 0.683 ∗

  6. [6]

    energy×DV time-bin 0.23 – 10 2.3 ∗

  7. [7]

    energy✓DV on-off keying – 0.1 12.5 1.25

  8. [8]

    energy×CV BPSK 0.09 – 1250 113

  9. [9]

    energy×CV ternary coherent states 0.0117 – 100 1.165

  10. [10]

    Comparative analysis of Semi-DI QRNG implementations with untrusted source and measurement

    energy✓CV BPSK – 0.12 (0.18 ∗) 1250 145.5 This work energy✓CV QPSK 0.10 0.22 ‡ 250 55.6 TABLE I. Comparative analysis of Semi-DI QRNG implementations with untrusted source and measurement. The com- parison includes the underlying protocol assumptions, such as the bound on the quantum states and the requirement of an independent and identically distributed...

    2023

  11. [11]

    Ac´ ın and L

    A. Ac´ ın and L. Masanes, Certified randomness in quan- tum physics, Nature540, 213 (2016)

    2016

  12. [12]

    Y.-Q. Nie, L. Huang, Y. Liu, F. Payne, J. Zhang, and J.-W. Pan, The generation of 68 gbps quantum random number by measuring laser phase fluctuations, Review of Scientific Instruments86, 063105 (2015)

    2015

  13. [13]

    Bruynsteen, T

    C. Bruynsteen, T. Gehring, C. Lupo, J. Bauwelinck, and X. Yin, 100-gbit/s integrated quantum random number generator based on vacuum fluctuations, PRX Quantum 4, 010330 (2023)

    2023

  14. [14]

    Pironio, A

    S. Pironio, A. Ac´ ın, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, Random numbers certi- fied by bell’s theorem, Nature464, 1021 (2010)

    2010

  15. [15]

    Bierhorst, E

    P. Bierhorst, E. Knill, S. Glancy, Y. Zhang, A. Mink, S. Jordan, A. Rommal, Y.-K. Liu, B. Christensen, S. W. Nam, M. J. Stevens, and L. K. Shalm, Experimentally generated randomness certified by the impossibility of superluminal signals, Nature556, 223 (2018)

    2018

  16. [16]

    Y. Liu, Q. Zhao, M.-H. Li, J.-Y. Guan, Y. Zhang, B. Bai, W. Zhang, W.-Z. Liu, C. Wu, X. Yuan, H. Li, W. J. Munro, Z. Wang, L. You, J. Zhang, X. Ma, J. Fan, Q. Zhang, and J.-W. Pan, Device-independent quantum random-number generation, Nature562, 548 (2018)

    2018

  17. [17]

    M.-H. Li, X. Zhang, W.-Z. Liu, S.-R. Zhao, B. Bai, Y. Liu, Q. Zhao, Y. Peng, J. Zhang, Y. Zhang, W. Munro, X. Ma, Q. Zhang, J. Fan, and J.-W. Pan, Experimental Realization of Device-Independent Quantum Random- ness Expansion, Physical Review Letters126, 050503 (2021)

    2021

  18. [18]

    Liu, M.-H

    W.-Z. Liu, M.-H. Li, S. Ragy, S.-R. Zhao, B. Bai, Y. Liu, P. J. Brown, J. Zhang, R. Colbeck, J. Fan, Q. Zhang, and J.-W. Pan, Device-independent randomness expan- sion against quantum side information, Nature Physics 17, 448 (2021)

    2021

  19. [19]

    J. B. Brask, N. Brunner, J. Pauwels, D. Rusca, and A. Tavakoli, Quantum correlations in prepare-and- measure scenarios and their semi-device-independent ap- plications (2026), arXiv:2603.23604 [quant-ph]

    arXiv 2026

  20. [20]

    Z. Cao, H. Zhou, and X. Ma, Loss-tolerant measurement- device-independent quantum random number genera- tion, New Journal of Physics17, 125011 (2015)

    2015

  21. [21]

    Avesani, D

    M. Avesani, D. G. Marangon, G. Vallone, and P. Villoresi, Source-device-independent heterodyne- based quantum random number generator at 17 gbps, Nature Communications9, 5365 (2018)

    2018

  22. [22]

    P. R. Smith, D. G. Marangon, M. Lucamarini, Z. L. Yuan, and A. J. Shields, Simple source device- independent continuous-variable quantum random num- ber generator, Phys. Rev. A99, 062326 (2019)

    2019

  23. [23]

    J. B. Brask, A. Martin, W. Esposito, R. Houlmann, J. Bowles, H. Zbinden, and N. Brunner, Megahertz- rate semi-device-independent quantum random number generators based on unambiguous state discrimination, Phys. Rev. Appl.7, 054018 (2017)

    2017

  24. [24]

    Avesani, H

    M. Avesani, H. Tebyanian, P. Villoresi, and G. Val- lone, Semi-device-independent heterodyne-based quan- tum random-number generator, Phys. Rev. Appl.15, 034034 (2021)

    2021

  25. [25]

    Li, Z.-Q

    H.-W. Li, Z.-Q. Yin, Y.-C. Wu, X.-B. Zou, S. Wang, W. Chen, G.-C. Guo, and Z.-F. Han, Semi-device- independent random-number expansion without entan- glement, Phys. Rev. A84, 034301 (2011)

    2011

  26. [26]

    H.-W. Li, M. Paw lowski, Z.-Q. Yin, G.-C. Guo, and Z.-F. Han, Semi-device-independent randomness certification usingn→1 quantum random access codes, Phys. Rev. A85, 052308 (2012)

    2012

  27. [27]

    Pauwels, S

    J. Pauwels, S. Pironio, E. Woodhead, and A. Tavakoli, Almost qudits in the prepare-and-measure scenario, Phys. Rev. Lett.129, 250504 (2022)

    2022

  28. [28]

    Tebyanian, M

    H. Tebyanian, M. Zahidy, M. Avesani, A. Stanco, P. Vil- loresi, and G. Vallone, Semi-device independent ran- domness generation based on quantum state’s indistin- guishability, Quantum Science and Technology6, 045026 (2021)

    2021

  29. [29]

    Van Himbeeck, E

    T. Van Himbeeck, E. Woodhead, N. J. Cerf, R. Garc´ ıa- Patr´ on, and S. Pironio, Semi-device-independent frame- work based on natural physical assumptions, Quantum 1, 33 (2017)

    2017

  30. [30]

    T. V. Himbeeck and S. Pironio, Correlations and ran- domness generation based on energy constraints (2019), arXiv:1905.09117 [quant-ph]

    Pith/arXiv arXiv 2019

  31. [31]

    Rusca, H

    D. Rusca, H. Tebyanian, A. Martin, and H. Zbinden, Fast self-testing quantum random number generator based on homodyne detection, Applied Physics Letters116, 264004 (2020)

    2020

  32. [32]

    Roch i Carceller, J

    C. Roch i Carceller, J. Pauwels, S. Pironio, and A. Tavakoli, Prepare-and-measure scenarios with photon- number constraints, Phys. Rev. Lett.135, 140802 (2025)

    2025

  33. [33]

    A. Peri, G. Gualandi, T. Bertapelle, M. Sabatini, G. Cor- rielli, Y. Pi´ etri, D. G. Marangon, G. Vallone, P. Villoresi, R. Osellame, and M. Avesani, High-performance het- erodyne receiver for quantum information processing in a laser-written integrated photonic platform, Advanced Photonics8, 10.1117/1.ap.8.1.016009 (2026)

    work page doi:10.1117/1.ap.8.1.016009 2026

  34. [34]

    Sabatini, T

    M. Sabatini, T. Bertapelle, P. Villoresi, G. Vallone, and M. Avesani, Hybrid encoder for discrete and continu- ous variable qkd, Advanced Quantum Technologies8, 2400522 (2025)

    2025

  35. [35]

    Metger, O

    T. Metger, O. Fawzi, D. Sutter, and R. Renner, Gen- eralised entropy accumulation, in2022 IEEE 63rd An- nual Symposium on Foundations of Computer Science (FOCS)(2022) pp. 844–850

    2022

  36. [36]

    Renner,Security of Quantum Key Distribu- tion, Ph.D

    R. Renner,Security of Quantum Key Distribu- tion, Ph.D. thesis, ETH Zurich (2005), available at http://arxiv.org/abs/quant-ph/0512258

    Pith/arXiv arXiv 2005

  37. [37]

    Tavakoli, A

    A. Tavakoli, A. Pozas-Kerstjens, P. Brown, and M. Ara´ ujo, Semidefinite programming relaxations for 9 quantum correlations, Rev. Mod. Phys.96, 045006 (2024)

    2024

  38. [38]

    Corrielli, A

    G. Corrielli, A. Crespi, and R. Osellame, Femtosecond laser micromachining for integrated quantum photonics, Nanophotonics10, 3789 (2021)

    2021

  39. [39]

    Lunghi, J

    T. Lunghi, J. B. Brask, C. C. W. Lim, Q. Lavigne, J. Bowles, A. Martin, H. Zbinden, and N. Brunner, Self- testing quantum random number generator, Phys. Rev. Lett.114, 150501 (2015)

    2015

  40. [40]

    Leone, D

    N. Leone, D. Rusca, S. Azzini, G. Fontana, F. Acerbi, A. Gola, A. Tontini, N. Massari, H. Zbinden, and L. Pavesi, An optical chip for self-testing quantum random number generation, APL Photonics5, 101301 (2020)

    2020

  41. [41]

    C. R. i. Carceller, L. N. Faria, Z.-H. Liu, N. Sguerso, U. L. Andersen, J. S. Neergaard-Nielsen, and J. B. Brask, Im- proving semi-device-independent randomness certifica- tion by entropy accumulation, Phys. Rev. A112, 022430 (2025)

    2025

  42. [42]

    Rusca, T

    D. Rusca, T. van Himbeeck, A. Martin, J. B. Brask, W. Shi, S. Pironio, N. Brunner, and H. Zbinden, Self- testing quantum random-number generator based on an energy bound, Physical Review A100, 062338 (2019), publisher: American Physical Society

    2019

  43. [43]

    Z. Lu, J. Wu, Y. Zhang, S. Ren, X. Wang, H. Zhou, and Y. Li, Semi-Device-Independent Quantum Random Number Generator Resistant to General Attacks (2026), arXiv:2602.06362 [quant-ph]

    arXiv 2026

  44. [44]

    Y. Pan, H. Wang, Y. Shao, Y. Pi, Y. Li, B. Liu, W. Huang, and B. Xu, Experimental demonstration of high-rate discrete-modulated continuous-variable quan- tum key distribution system, Optics Letters47, 3307 (2022)

    2022

  45. [45]

    A. A. Hajomer, C. Bruynsteen, I. Derkach, N. Jain, A. Bomhals, S. Bastiaens, U. L. Andersen, X. Yin, and T. Gehring, Continuous-variable quantum key distribu- tion at 10 gbaud using an integrated photonic-electronic receiver, Optica11, 1197 (2024)

    2024

  46. [46]

    Raffaelli, G

    F. Raffaelli, G. Ferranti, D. H. Mahler, P. Sibson, J. E. Kennard, A. Santamato, G. Sinclair, D. Bonneau, M. G. Thompson, and J. C. Matthews, A homodyne detector integrated onto a photonic chip for measuring quantum states and generating random numbers, Quantum Sci- ence and Technology3, 025003 (2018)

    2018

  47. [47]

    Zhang, J

    G. Zhang, J. Y. Haw, H. Cai, F. Xu, S. M. Assad, J. F. Fitzsimons, X. Zhou, Y. Zhang, S. Yu, J. Wu, W. Ser, L. C. Kwek, and A. Q. Liu, An integrated silicon pho- tonic chip platform for continuous-variable quantum key distribution, Nature Photonics13, 839 (2019)

    2019

  48. [48]

    Y. Bian, Y. Pan, X. Xu, L. Zhao, Y. Li, W. Huang, L. Zhang, S. Yu, Y. Zhang, and B. Xu, Continuous- variable quantum key distribution over 28.6 km fiber with an integrated silicon photonic receiver chip, Applied Physics Letters124, 174001 (2024)

    2024

  49. [49]

    Brown, H

    P. Brown, H. Fawzi, and O. Fawzi, Device-independent lower bounds on the conditional von Neumann entropy, Quantum8, 1445 (2024)

    2024

  50. [50]

    Bancal, L

    J.-D. Bancal, L. Sheridan, and V. Scarani, More ran- domness from the same data, New Journal of Physics 16, 033011 (2014). 10 Appendix A: Semidefinite programs to bound the Shannon entropy In this part of the supplementary material, we present the semidefinite progams (SDPs) we compute the bound the certifiable Shannon entropy in our protocol. We begin sh...

    2014

  51. [51]

    Primal SDP The Shannon entropy of Bob’s measurement outcomes (H(B|Λ E) in the main text) is lower bounded by [31, 39] H(B|Λ E)≥c m + m−1X i=1 wi ti log 2 inf {zλ i,b,x} (X x px X λ q(λ) X b pλ(b|x) 2zλ i,b,x + (1−t i)(zλ i,b,x)2 +t i(zλ i,b,x)2 ) ,(A1) forc m =P i wi ti log 2, wheret i andw i are the nodes and weights of the Gauss-Radau quadrature,mis the...

  52. [52]

    Since CVXPY internally constructs and solves both the primal and the dual problems, the dual variables associated with each constraint are directly accessible at optimality

    Dual SDP and min-tradeoff function We solve the primal SDP in (C10) using the solver MOSEK with CVXPY in Python. Since CVXPY internally constructs and solves both the primal and the dual problems, the dual variables associated with each constraint are directly accessible at optimality. Therefore, to reconstruct the dual objective function of the primal SD...