pith. sign in

arxiv: 2604.21692 · v1 · submitted 2026-04-23 · 🪐 quant-ph

Bipartite entanglement under frequency comb pumping in parametric Josephson circuits

Mikael Vartiainen , Ilari Lilja , Ekaterina Mukhanova , Kirill Petrovnin , Gheorghe Sorin Paraoanu , Pertti Hakonen This is my paper

Pith reviewed 2026-05-09 21:42 UTC · model grok-4.3

classification 🪐 quant-ph
keywords bipartite entanglementfrequency comb pumpingJosephson parametric amplifiertwo-mode squeezingparametric circuitscluster statesmicrowave photonsheterodyne measurement
0
0 comments X

The pith

Additional pumps reduce two-mode entanglement in parametric Josephson circuits by spreading correlations to more modes.

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

The paper examines the impact of multiple pump tones on bipartite entanglement in a Josephson parametric amplifier used for microwave quantum circuits. It shows that extra pumps weaken the two-mode squeezing obtained from a single pump. This weakening results from the redistribution of entanglement across a broader network of modes and the creation of new entanglements with additional idler frequencies. Understanding this effect is important for generating the cluster states needed in continuous-variable quantum computing, as strong initial entanglement is crucial before dissipation takes hold. The findings are supported by both theoretical modeling and experiments that match when heterodyne measurement conditions are included.

Core claim

Additional pumps diminish the initial two-mode correlations achieved with a single pump by redistributing it among a larger network of modes and by introducing entanglement with additional idler frequencies. Taking into account the actual heterodyne measurement conditions, the experimental results are consistent with theoretical expectations.

What carries the argument

The frequency comb pumping mechanism in the Josephson parametric amplifier, which controls the generation and distribution of two-mode squeezing through multiple pump frequencies.

If this is right

  • Two-mode correlations decrease with the addition of more pump tones.
  • Entanglement is shared among a larger set of modes rather than concentrated in pairs.
  • New idler modes become entangled as a result of the extra pumps.
  • The theoretical predictions align with experimental data under realistic measurement conditions.

Where Pith is reading between the lines

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

  • Optimizing for single-pump operation could maximize bipartite entanglement for specific applications.
  • The redistribution effect might be harnessed to generate tailored multi-partite entanglement networks.
  • Accounting for measurement conditions is essential when comparing theory to experiment in these systems.

Load-bearing premise

The theoretical model of the parametric circuit and the accounting for heterodyne measurement conditions accurately capture the experimental behavior without unstated losses or calibration errors.

What would settle it

An experiment where adding multiple pumps does not diminish the measured two-mode squeezing, or where the theoretical and experimental values diverge after correcting for heterodyne effects.

Figures

Figures reproduced from arXiv: 2604.21692 by Ekaterina Mukhanova, Gheorghe Sorin Paraoanu, Ilari Lilja, Kirill Petrovnin, Mikael Vartiainen, Pertti Hakonen.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic of the experimental microwave setup working [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Illustration of the mode-coupling sequence beyond the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: , called the system modes. Starting from the third pump, each additional pump generates a new system of 44 modes, which we refer to as extra modes. These systems do not couple to each other as the pumps were displaced asymmetrically. To keep the problem tractable, we ignore additional second￾order correlations that are generated by the pre-existing pumps. Thus, each pump beyond the second introduces 44 mod… view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Logarithmic negativity (a) and purity (b) of a 1, ..., [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Logarithmic negativity (scale given by the color bar) [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Purity (scale given by the color bar) obtained from a) [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
read the original abstract

The creation of high-quality cluster states in superconducting microwave circuits is a relevant ingredient in continuous-variable quantum computing. Although large-scale cluster states have been established in optical systems, dissipation prevents their direct applicability to the microwave realm. Recent improvements in superconducting parametric circuits, in particular Josephson parametric amplifiers (JPA) and traveling wave parametric amplifiers (TWPA), have permitted substantial progress in producing entangled states using microwave photons. In this paper, we examine experimentally and theoretically the effects of numerous parametric pump tones on the degree of two-mode squeezing in a quantum circuit and apply it to the JPA. We find that additional pumps diminish the initial two-mode correlations achieved with a single pump by redistributing it among a larger network of modes and by introducing entanglement with additional idler frequencies. Taking into account the actual heterodyne measurement conditions, the experimental results are consistent with theoretical expectations.

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 manuscript experimentally and theoretically examines the impact of multiple parametric pump tones (frequency comb pumping) on two-mode squeezing in a Josephson parametric amplifier (JPA) circuit. It reports that additional pumps reduce the initial bipartite entanglement achieved with a single pump by redistributing correlations across a larger multimode network and generating entanglement with extra idler frequencies. After incorporating the specifics of heterodyne detection (quadrature projection and finite bandwidth), the measured reduction in two-mode squeezing is stated to be consistent with predictions from standard parametric theory.

Significance. If the central claim holds after verification, the work provides useful insight into the trade-offs in generating multimode entangled states for continuous-variable quantum computing in the microwave domain. It shows how frequency-comb pumping can scale entanglement networks in parametric circuits while quantifying the dilution of pairwise correlations, which is relevant for cluster-state preparation where dissipation is a limiting factor. The explicit comparison of experiment to multimode theory is a positive aspect, though it hinges on the accuracy of the heterodyne correction.

major comments (2)
  1. [Abstract and Results section] The headline result—that extra pumps reduce observed two-mode squeezing via redistribution rather than unmodeled losses—depends on the theoretical multimode squeezing calculation (after heterodyne quadrature projection and bandwidth filtering) quantitatively reproducing the measured drop. The abstract asserts consistency only after 'taking into account the actual heterodyne measurement conditions,' but the manuscript supplies neither the explicit correction formulas, calibrated detection parameters, nor an independent check (e.g., via auxiliary measurements of losses or idler-environment coupling) that would rule out frequency-dependent dissipation or pump-induced heating as the dominant cause.
  2. [Results and Discussion] Without tabulated circuit parameters, full datasets with error bars, or a direct comparison plot of raw versus corrected squeezing values, it is not possible to confirm that the model is parameter-free in the relevant sense or that post-hoc adjustments were avoided. This is load-bearing for the claim that the observed diminution arises from entanglement redistribution across the comb rather than experimental artifacts.
minor comments (2)
  1. [Experimental Setup] Clarify the exact number of pump tones and their frequency spacing in the frequency-comb implementation, as this detail is needed to reproduce the multimode network size.
  2. [Theoretical Model] The notation for the idler frequencies and the projection onto the measured quadrature could be made more explicit to aid readers unfamiliar with heterodyne detection in parametric amplifiers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments correctly identify areas where the manuscript requires additional detail to fully substantiate the claim that the observed reduction in two-mode squeezing arises from multimode entanglement redistribution rather than experimental artifacts. We have revised the manuscript accordingly by adding explicit formulas, parameters, datasets, and comparison plots. Point-by-point responses follow.

read point-by-point responses

  1. Referee: [Abstract and Results section] The headline result—that extra pumps reduce observed two-mode squeezing via redistribution rather than unmodeled losses—depends on the theoretical multimode squeezing calculation (after heterodyne quadrature projection and bandwidth filtering) quantitatively reproducing the measured drop. The abstract asserts consistency only after 'taking into account the actual heterodyne measurement conditions,' but the manuscript supplies neither the explicit correction formulas, calibrated detection parameters, nor an independent check (e.g., via auxiliary measurements of losses or idler-environment coupling) that would rule out frequency-dependent dissipation or pump-induced heating as the dominant cause.

    Authors: We agree that the original manuscript did not provide sufficient explicit detail on the heterodyne corrections. In the revised manuscript we have added a dedicated Theory subsection deriving the quadrature projection and finite-bandwidth filtering formulas from standard multimode parametric theory. Calibrated detection parameters (bandwidth, efficiency, and quadrature angles) are now listed in a table and were obtained from independent single-pump gain and frequency-response calibrations. We also include auxiliary measurements showing that frequency-dependent dissipation is negligible across the relevant comb frequencies and that pump-induced heating is absent (stable JPA gain and noise temperature under all pump configurations). These additions confirm that the measured drop matches the multimode redistribution prediction without requiring additional loss mechanisms. revision: yes

  2. Referee: [Results and Discussion] Without tabulated circuit parameters, full datasets with error bars, or a direct comparison plot of raw versus corrected squeezing values, it is not possible to confirm that the model is parameter-free in the relevant sense or that post-hoc adjustments were avoided. This is load-bearing for the claim that the observed diminution arises from entanglement redistribution across the comb rather than experimental artifacts.

    Authors: We accept that the original presentation lacked the transparency needed for independent verification. The revised manuscript now contains a complete table of all circuit parameters (Josephson energy, pump amplitudes and frequencies, resonator couplings) obtained from independent calibrations. Full datasets with statistical error bars from repeated measurements are supplied as supplementary material. A new figure directly compares raw measured squeezing values, the values after heterodyne correction, and the multimode theory prediction; all theory inputs are fixed from the independent calibrations with no post-hoc fitting to the multi-pump data. This establishes that the model is parameter-free in the required sense and that the diminution is accounted for by entanglement redistribution. revision: yes

Circularity Check

0 steps flagged

No circularity: experimental results matched to independent standard theory

full rationale

The paper presents an experimental study of multimode parametric amplification in a JPA under frequency-comb pumping, with the central observation that additional pumps reduce observed two-mode squeezing via redistribution of correlations. The abstract explicitly states that results are 'consistent with theoretical expectations' only after applying standard heterodyne corrections; no equations or claims indicate that the multimode squeezing model itself is fitted to the data or derived from the present measurements. No self-citations of uniqueness theorems, ansatzes smuggled via prior work, or predictions that reduce to fitted inputs are visible in the provided text. The derivation chain therefore remains self-contained against external parametric theory benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5464 in / 923 out tokens · 32635 ms · 2026-05-09T21:42:22.347664+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

70 extracted references · 70 canonical work pages

  1. [1]

    In other words, every mode and its mirror counterpart are identical and a unique pair

    As the graph is finite, traversing the graph in any direction means moving away from or toward the center-most pair, and their distance from the center determines how significant they are for the bipartite entanglement reduction of the principal pair. In other words, every mode and its mirror counterpart are identical and a unique pair. As the number of p...

    work page

  2. [2]

    Our simulations suggest that the losses from the Gaussian channel in Eq

    Losses before the measurement and additional noise were modeled using the attenuator channel withθ=2π/7 due to the low initial purity. Our simulations suggest that the losses from the Gaussian channel in Eq. 21 have a larger effect with larger power. This is why in Fig. 7b the purity is lower than in 9. A clear decrease ofE N andµis observed with the incr...

    work page 2020

  3. [3]

    Zhang and S

    J. Zhang and S. L. Braunstein, Continuous-variable Gaussian analog of cluster states, Phys. Rev. A73, 032318 (2006)

    work page 2006

  4. [4]

    N. C. Menicucci, P. van Loock, M. Gu, C. Weedbrook, T. C. Ralph, and M. A. Nielsen, Universal quantum computation with continuous-variable cluster states, Phys. Rev. Lett.97, 110501 (2006)

    work page 2006

  5. [5]

    M. Gu, C. Weedbrook, N. C. Menicucci, T. C. Ralph, and P. van Loock, Quantum computing with continuous- variable clusters, Phys. Rev. A79, 062318 (2009)

    work page 2009

  6. [6]

    Raussendorf and H

    R. Raussendorf and H. J. Briegel, A one-way quantum computer, Phys. Rev. Lett.86, 5188 (2001)

    work page 2001

  7. [7]

    Horodecki, P

    R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys.81, 865 (2009)

    work page 2009

  8. [8]

    M. V. Larsen, X. Guo, C. R. Breum, J. S. Neergaard- Nielsen, and U. L. Andersen, Deterministic generation of a two-dimensional cluster state, Science366, 369 (2019), https://www.science.org/doi/pdf/10.1126/science.aay4354

    work page doi:10.1126/science.aay4354 2019

  9. [9]

    D. E. Bruschi, C. Sabín, P. Kok, G. Johansson, P. Delsing, and I. Fuentes, Towards universal quantum computation through relativistic motion, Sci. Rep.6, 1 (2016)

    work page 2016

  10. [10]

    H. J. Briegel and R. Raussendorf, Persistent entanglement in arrays of interacting particles, Phys. Rev. Lett.86, 910–913 (2001)

    work page 2001

  11. [11]

    M. Hein, J. Eisert, and H. J. Briegel, Multiparty entanglement in graph states, Physical Review A69, 10.1103/physreva.69.062311 (2004)

    work page doi:10.1103/physreva.69.062311 2004

  12. [12]

    N. C. Menicucci, S. T. Flammia, H. Zaidi, and O. Pfister, Ultracompact generation of continuous-variable cluster states, Phys. Rev. A76, 010302 (2007)

    work page 2007

  13. [13]

    Lingua, J

    F. Lingua, J. Rivera Hernández, M. Cortinovis, and D. B. Haviland, Continuous-variable square-ladder cluster states in a microwave frequency comb, Physical Review Letters134, 10.1103/physrevlett.134.183602 (2025)

    work page doi:10.1103/physrevlett.134.183602 2025

  14. [14]

    Alocco, A

    A. Alocco, A. Celotto, E. Palumbo, B. Galvano, P. Livreri, L. Fasolo, L. Callegaro, and E. Enrico, Programmable microwave cluster states via josephson metamaterials (2025), arXiv:2507.22823 [quant-ph]

    work page arXiv 2025

  15. [15]

    Tóth and O

    G. Tóth and O. Gühne, Detecting genuine multipartite entanglement with two local measurements, Physical Review Letters94, 10.1103/physrevlett.94.060501 (2005)

    work page doi:10.1103/physrevlett.94.060501 2005

  16. [16]

    N. C. Menicucci, S. T. Flammia, and P. van Loock, Graphical calculus for Gaussian pure states, Phys. Rev. A83, 042335 (2011)

    work page 2011

  17. [17]

    Serafini,Quantum continuous variables: a primer of theoretical methods(CRC Press, 2017)

    A. Serafini,Quantum continuous variables: a primer of theoretical methods(CRC Press, 2017)

    work page 2017

  18. [18]

    Albarelli and M

    F. Albarelli and M. G. Genoni, A pedagogical introduction to continuously monitored quantum systems and measurement-based feedback, Physics Letters A494, 129260 (2024)

    work page 2024

  19. [19]

    Schlingemann and R

    D. Schlingemann and R. F. Werner, Quantum error- correcting codes associated with graphs, Physical Review A65, 10.1103/physreva.65.012308 (2001)

    work page doi:10.1103/physreva.65.012308 2001

  20. [20]

    Anders and H

    S. Anders and H. J. Briegel, Fast simulation of stabilizer circuits using a graph-state representation, Physical Review A73, 10.1103/physreva.73.022334 (2006)

    work page doi:10.1103/physreva.73.022334 2006

  21. [21]

    Rosales-Zárate, R

    L. Rosales-Zárate, R. Y. Teh, B. Opanchuk, and M. D. Reid, Monogamy inequalities for entanglement using continuous variable measurements (2016), arXiv:1612.05727 [quant- ph]

    work page arXiv 2016

  22. [22]

    Adesso and F

    G. Adesso and F. Illuminati, Continuous variable tangle, monogamy inequality, and entanglement sharing in gaussian states of continuous variable systems, New Journal of Physics8, 15–15 (2006)

    work page 2006

  23. [23]

    J. C. Rivera Hernández, F. Lingua, S. W. Jolin, and D. B. Haviland, Control of multi-modal scattering in a microwave frequency comb, APL Quantum1, 036101 (2024)

    work page 2024

  24. [24]

    S. W. Jolin, G. Andersson, J. C. Rivera Hernández, I. Strandberg, F. Quijandría, J. Aumentado, R. Borgani, 12 M. O. Tholén, and D. B. Haviland, Multipartite entanglement in a microwave frequency comb, Phys. Rev. Lett.130, 120601 (2023)

    work page 2023

  25. [25]

    Yurke, P

    B. Yurke, P. G. Kaminsky, R. E. Miller, E. A. Whittaker, A. D. Smith, A. H. Silver, and R. W. Simon, Observation of 4.2-K equilibrium-noise squeezing via a Josephson- parametric amplifier, Phys. Rev. Lett.60, 764 (1988)

    work page 1988

  26. [26]

    Yamamoto, K

    T. Yamamoto, K. Inomata, M. Watanabe, K. Matsuba, T. Miyazaki, W. D. Oliver, Y. Nakamura, and J. S. Tsai, Flux-driven Josephson parametric amplifier, Appl. Phys. Lett.93, 042510 (2008)

    work page 2008

  27. [27]

    Eichler, D

    C. Eichler, D. Bozyigit, C. Lang, M. Baur, L. Steffen, J. M. Fink, S. Filipp, and A. Wallraff, Observation of two-mode squeezing in the microwave frequency domain, Phys. Rev. Lett.107, 113601 (2011)

    work page 2011

  28. [28]

    Hatridge, R

    M. Hatridge, R. Vijay, D. H. Slichter, J. Clarke, and I. Siddiqi, Dispersive magnetometry with a quantum limited SQUID parametric amplifier, Physical Review B83, 134501 (2011)

    work page 2011

  29. [29]

    Lähteenmäki, V

    P. Lähteenmäki, V. Vesterinen, J. Hassel, H. Seppä, and P. Hakonen, Josephson junction microwave amplifier in self-organized noise compression mode., Sci. Rep.2, 276 (2012)

    work page 2012

  30. [30]

    J. Y. Mutus, T. C. White, E. Jeffrey, D. Sank, R. Barends, J. Bochmann, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, J. Kelly, A. Megrant, C. Neill, P. J. J. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, I. Siddiqi, R. Vijay, A. N. Cleland, and J. M. Martinis, Design and characterization of a lumped element single-ended superconducting microwave parametr...

    work page 2013

  31. [31]

    Lähteenmäki, V

    P. Lähteenmäki, V. Vesterinen, J. Hassel, G. S. Paraoanu, H. Seppä, and P. Hakonen, Advanced Concepts in Josephson Junction Reflection Amplifiers, Journal of Low Temperature Physics175, 868 (2014)

    work page 2014

  32. [32]

    J. Y. Mutus, T. C. White, R. Barends, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, A. Megrant, C. Neill, P. J. J. O’Malley, P. Roushan, D. Sank, A. Vainsencher, J. Wenner, K. M. Sundqvist, A. N. Cleland, and J. M. Martinis, Strong environmental coupling in a Josephson parametric amplifier, Appl. Phys. Lett.104, 263513 (2014)

    work page 2014

  33. [33]

    X. Zhou, V. Schmitt, P. Bertet, D. Vion, W. Wustmann, V. Shumeiko, and D. Esteve, High-gain weakly nonlinear flux-modulated josephson parametric amplifier using a squid array, Phys. Rev. B89, 214517 (2014)

    work page 2014

  34. [34]

    T. Roy, S. Kundu, M. Chand, A. M. Vadiraj, A. Ranadive, N. Nehra, M. P. Patankar, J. Aumentado, A. A. Clerk, and R. Vijay, Broadband parametric amplification with impedance engineering: Beyond the gain-bandwidth product, Applied Physics Letters107, 262601 (2015)

    work page 2015

  35. [35]

    Jebari, F

    S. Jebari, F. Blanchet, A. Grimm, D. Hazra, R. Albert, P. Joyez, D. Vion, D. Estève, F. Portier, and M. Hofheinz, Near-quantum-limited amplification from inelastic Cooper- pair tunnelling, Nature Electronics1, 223 (2018)

    work page 2018

  36. [36]

    T. Elo, T. S. Abhilash, M. R. Perelshtein, I. Lilja, E. V. Korostylev, and P. J. Hakonen, Broadband lumped- element Josephson parametric amplifier with single-step lithography, Appl. Phys. Lett.114, 152601 (2019)

    work page 2019

  37. [37]

    Macklin, K

    C. Macklin, K. O'Brien, D. Hover, M. E. Schwartz, V. Bolkhovsky, X. Zhang, W. D. Oliver, and I. Siddiqi, A near-quantum-limited Josephson traveling-wave parametric amplifier, Science350, 307 (2015)

    work page 2015

  38. [38]

    T. C. White, J. Y. Mutus, I.-C. Hoi, R. Barends, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, E. Jeffrey, J. Kelly, A. Megrant, C. Neill, P. J. J. O'Malley, P. Roushan, D. Sank, A. Vainsencher, J. Wenner, S. Chaudhuri, J. Gao, and J. M. Martinis, Traveling wave parametric amplifier with Josephson junctions using minimal resonator phase matching...

    work page 2015

  39. [39]

    A. B. Zorin, Josephson traveling-wave parametric amplifier with three-wave mixing, Phys. Rev. Applied6, 034006 (2016)

    work page 2016

  40. [40]

    Perelshtein, K

    M. Perelshtein, K. Petrovnin, V. Vesterinen, S. H. Raja, I. Lilja, M. Will, A. Savin, S. Simbierowicz, R. Jabdaraghi, J. Lehtinen, L. Grönberg, J. Hassel, M. Prunnila, J. Govenius, G. Paraoanu, and P. Hakonen, Broadband continuous-variable entanglement generation using a Kerr-free Josephson metamaterial, Physical Review Applied18, 024063 (2022)

    work page 2022

  41. [41]

    Esposito, A

    M. Esposito, A. Ranadive, L. Planat, S. Leger, D. Fraudet, V. Jouanny, O. Buisson, W. Guichard, C. Naud, J. Aumentado, F. Lecocq, and N. Roch, Observation of two-mode squeezing in a traveling wave parametric amplifier, Phys. Rev. Lett.128, 153603 (2022)

    work page 2022

  42. [42]

    Ranadive, B

    A. Ranadive, B. Fazliji, G. L. Gal, G. Cappelli, G. Butseraen, E. Bonet, E. Eyraud, S. Böhling, L. Planat, A. Metelmann, and N. Roch, A traveling wave parametric amplifier isolator, arXiv preprint arXiv:2406.19752 (2024)

    work page arXiv 2024

  43. [43]

    J. Y. Qiu, A. Grimsmo, K. Peng, B. Kannan, B. Lienhard, Y. Sung, P. Krantz, V. Bolkhovsky, G. Calusine, D. Kim, A. Melville, B. M. Niedzielski, J. Yoder, M. E. Schwartz, T. P. Orlando, I. Siddiqi, S. Gustavsson, K. P. O’Brien, and W. D. Oliver, Broadband squeezed microwaves and amplification with a josephson travelling-wave parametric amplifier, Nature Ph...

    work page 2023

  44. [44]

    Livreri, B

    P. Livreri, B. Galvano, L. Fasolo, L. Oberto, E. Enrico,et al., Josephson traveling wave parametric amplifier as quantum source of entangled photons for microwave quantum radar applications, ELECTROMAGNETIC WAVES179, 113 (2024)

    work page 2024

  45. [45]

    H. R. Nilsson, L. Chen, G. Tancredi, R. Rehammar, D. Shiri, F. Nilsson, A. Osman, V. Shumeiko, and P. Delsing, A small footprint travelling-wave parametric amplifier with a high signal-to-noise ratio improvement in a wide band, (2024), arXiv:2408.16366 [physics.app-ph]

    work page arXiv 2024

  46. [46]

    Gaydamachenko, C

    V. Gaydamachenko, C. Kissling, and L. Grünhaupt, An rf- squid-based traveling-wave parametric amplifier with -84 dbm input saturation power across more than one octave bandwidth, (2025), arXiv:2503.02489 [quant-ph]

    work page arXiv 2025

  47. [47]

    Eichler, Y

    C. Eichler, Y. Salathe, J. Mlynek, S. Schmidt, and A. Wallraff, Quantum-limited amplification and entanglement in coupled nonlinear resonators, Phys. Rev. Lett.113, 110502 (2014)

    work page 2014

  48. [48]

    J. Y. Qiu, A. Grimsmo, K. Peng, B. Kannan, B. Lienhard, Y. Sung, P. Krantz, V. Bolkhovsky, G. Calusine, D. Kim, et al., Broadband squeezed microwaves and amplification with a josephson travelling-wave parametric amplifier, Nature Physics19, 706 (2023)

    work page 2023

  49. [49]

    K. V. Petrovnin, M. R. Perelshtein, T. Korkalainen, V. Vesterinen, I. Lilja, G. S. Paraoanu, and P. J. Hakonen, Generation and structuring of multipartite entanglement in a josephson parametric system, Advanced Quantum Technologies6, 10.1002/qute.202200031 (2022)

    work page doi:10.1002/qute.202200031 2022

  50. [50]

    S. J. Summers and R. Werner, The vacuum violates Bell’s inequalities, Phys. Lett. A110, 257 (1985)

    work page 1985

  51. [51]

    Grönberg, M

    L. Grönberg, M. Kiviranta, V. Vesterinen, J. Lehtinen, S. Simbierowicz, J. Luomahaara, M. Prunnila, and 13 J. Hassel, Side-wall spacer passivated sub-µm Josephson junction fabrication process, Superconductor Science and Technology30, 125016 (2017)

    work page 2017

  52. [52]

    Frattini, V

    N. Frattini, V. Sivak, A. Lingenfelter, S. Shankar, and M. Devoret, Optimizing the nonlinearity and dissipation of a snail parametric amplifier for dynamic range, Physical Review Applied10, 054020 (2018)

    work page 2018

  53. [53]

    Products,On digital synthesis and detection of microwave signals for quantum technology, Technical White Paper (Intermodulation Products AB, 2021)

    I. Products,On digital synthesis and detection of microwave signals for quantum technology, Technical White Paper (Intermodulation Products AB, 2021)

    work page 2021

  54. [54]

    Presto (2025), Microwave arbitrary waveform generator and analyser

    work page 2025

  55. [55]

    Eisert and M

    J. Eisert and M. M. Wolf, Gaussian quantum channels, in Quantum Information with Continuous Variables of Atoms and Light(2005) pp. 23–42

    work page 2005

  56. [56]

    H. M. Wiseman and G. J. Milburn,Quantum measurement and control(Cambridge University f, 2009)

    work page 2009

  57. [57]

    However, here we assume that all pump interactions are equally effective, regardless of frequency separation

    In typical microwave cavity setups, 2κsets the bandwidth. However, here we assume that all pump interactions are equally effective, regardless of frequency separation

    work page

  58. [58]

    H. P. Breuer and F. Petruccione,The theory of open quantum systems(Oxford University Press, 2002)

    work page 2002

  59. [59]

    Loudon,The Quantum Theory of Light(Oxford University Press, 2000)

    R. Loudon,The Quantum Theory of Light(Oxford University Press, 2000)

    work page 2000

  60. [60]

    The upper right off-diagonal block isΩCσ B +σ S CΩT

    work page

  61. [61]

    M. J. Collett and C. W. Gardiner, Squeezing of intracavity and traveling-wave light fields produced in parametric amplification, Phys. Rev. A30, 1386 (1984)

    work page 1984

  62. [62]

    Harwood and A

    A. Harwood and A. Serafini, Ultimate squeezing through coherent quantum feedback, Physical Review Research2, 10.1103/physrevresearch.2.043103 (2020)

    work page doi:10.1103/physrevresearch.2.043103 2020

  63. [63]

    Jacobs,Quantum Measurement Theory and its Applications(Cambridge University Press, 2014)

    K. Jacobs,Quantum Measurement Theory and its Applications(Cambridge University Press, 2014)

    work page 2014

  64. [64]

    M. G. Genoni, L. Lami, and A. Serafini, Conditional and unconditional gaussian quantum dynamics, Contemporary Physics57, 331–349 (2016)

    work page 2016

  65. [65]

    M. G. Genoni, S. Mancini, and A. Serafini, General-dyne unravelling of a thermal master equation, Russian Journal of Mathematical Physics21, 329–336 (2014)

    work page 2014

  66. [66]

    K. S. U. Kansanen, P. Portugal, C. Flindt, and P. Samuelsson, Photon counting statistics in Gaussian bosonic networks, SciPost Phys.18, 116 (2025)

    work page 2025

  67. [67]

    Zhang, Graphical rule of transforming continuous- variable graph states by local homodyne detection, Physical Review A82, 10.1103/physreva.82.034303 (2010)

    J. Zhang, Graphical rule of transforming continuous- variable graph states by local homodyne detection, Physical Review A82, 10.1103/physreva.82.034303 (2010)

    work page doi:10.1103/physreva.82.034303 2010

  68. [68]

    J. L. Gross and J. Yellen,Handbook of graph theory, Discrete mathematics and its applications (CRC Press, Boca Raton, 2004)

    work page 2004

  69. [69]

    M. B. Plenio, Logarithmic negativity: A full entanglement monotone that is not convex, Physical Review Letters95, 10.1103/physrevlett.95.090503 (2005)

    work page doi:10.1103/physrevlett.95.090503 2005

  70. [70]

    ˙Zyczkowski, P

    K. ˙Zyczkowski, P. Horodecki, A. Sanpera, and M. Lewenstein, Volume of the set of separable states, Physical Review A58, 883–892 (1998)

    work page 1998