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arxiv: 2606.19649 · v2 · pith:FRESVIEGnew · submitted 2026-06-17 · 🪐 quant-ph · physics.ins-det

Optimized Quantum States for Sensing in the Presence of Loss and Phase Noise

Shruti Maliakal , Zachary Mann , Christopher Wipf , Rana X Adhikari , Su Direkci , Yanbei Chen This is my paper

Pith reviewed 2026-07-01 07:25 UTC · model grok-4.3

classification 🪐 quant-ph physics.ins-det
keywords quantum sensingnon-Gaussian statesphase noisephoton lossquantum Fisher informationsqueezed vacuumgravitational wave detection
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The pith

Non-Gaussian states outperform any Gaussian state for sensing when both loss and phase noise are present.

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

Squeezed vacuum is known to be optimal for quantum sensing in the presence of loss alone, yet phase noise removes that optimality. The authors maximize the quantum Fisher information numerically over all possible input states for varying amounts of loss and phase noise. They locate families of non-Gaussian states that return higher information than the best Gaussian state at every point in the parameter space. For an average of five photons, five percent loss, and two hundred milliradians of phase noise, the improvement reaches 2.2 dB and survives even when the final measurement is limited to homodyne detection.

Core claim

Numerical optimization of the quantum Fisher information across the loss and phase-noise landscape identifies three classes of non-Gaussian states—Fock-like, cubic-phase-like, and states with discrete rotational symmetry—that achieve strictly higher information than any Gaussian state for every combination of loss and phase noise.

What carries the argument

Numerical maximization of the quantum Fisher information for a loss-plus-phase-noise channel acting on an input state with fixed average photon number.

If this is right

  • With five photons on average, five percent loss, and two hundred milliradians of phase noise, the best non-Gaussian state yields a 2.2 dB improvement over the best Gaussian state.
  • The non-Gaussian advantage survives restriction of the measurement to homodyne detection.
  • The optimal states belong to three distinct families: Fock-like, cubic-phase-like, and rotationally symmetric states.

Where Pith is reading between the lines

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

  • The same numerical search could be repeated for other noise models such as thermal noise or detector inefficiency to map the full landscape of useful non-Gaussian resources.
  • Approximate preparation of the reported states may be feasible with existing nonlinear optics or cavity-QED methods, allowing near-term tests of the predicted advantage.
  • Current gravitational-wave detectors that rely on squeezed vacuum may gain sensitivity by replacing part of the squeezed light with one of the identified non-Gaussian states when phase noise is the dominant imperfection.

Load-bearing premise

The states located by the numerical search are physically realizable and the quantum Fisher information remains the correct figure of merit once the loss and phase-noise model is applied.

What would settle it

An explicit construction of a Gaussian state whose quantum Fisher information equals or exceeds that of the reported non-Gaussian optimum at any tested loss and phase-noise value would falsify the central claim.

Figures

Figures reproduced from arXiv: 2606.19649 by Christopher Wipf, Rana X Adhikari, Shruti Maliakal, Su Direkci, Yanbei Chen, Zachary Mann.

Figure 1
Figure 1. Figure 1: FIG. 1: Optimized non-Gaussian probe states for [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: a for η = 0.9 and σϕ = 0.1 (the box with the orange, diamond marker). This is consistent with the expectation that states with rotational symmetry are favored when loss and phase noise are both significant: their rotational structure resists the random dephasing rotation, while their phase-space localization resists loss. At η = 1, by contrast, the parity protection of squeezed vacuum suf￾fices. Low loss a… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Homodyne ( [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: (a) CFI of each measurement (PNR, parity, homodyne at optimized angle [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1: Quadrature variance as a function of homodyne angle [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Comparison of the exact quadrature variance expression with the Lindblad-based [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Evolution of annihilation operators in a Mach-Zehnder (left) and a Michelson (right) interferometer. [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Comparison of the analytical upper bounds for the QFI. We plot the convexity bound (Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Optimization schematic. The states obtained from the channel evolution are used to calculate the QFI, [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: QFI convergence with Fock basis size for the optimized states at [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Fidelity between a squeezed vacuum state and the same state with Fock basis truncation, as given by [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: QFI convergence with Fock basis size for the optimized states at [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Wigner functions of the optimized Fock-superposition states across the ( [PITH_FULL_IMAGE:figures/full_fig_p029_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Wigner functions of the optimized squeezed-vacuum superposition states across the ( [PITH_FULL_IMAGE:figures/full_fig_p030_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Wigner functions of the optimized displaced squeezed-vacuum superposition states across the ( [PITH_FULL_IMAGE:figures/full_fig_p031_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: QFI of a displaced, squeezed state [PITH_FULL_IMAGE:figures/full_fig_p032_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Wigner functions of two Gaussian states with the same squeezing level [PITH_FULL_IMAGE:figures/full_fig_p032_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Wigner functions of the cubic phase with best fidelity match to states in at each ( [PITH_FULL_IMAGE:figures/full_fig_p033_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: Optimization to maximize the overlap [PITH_FULL_IMAGE:figures/full_fig_p034_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: Sensitivity achieved with the binomial-amplitude state input for optimized photon counting [PITH_FULL_IMAGE:figures/full_fig_p035_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: The Fock distribution of the optimized state belonging to the Fock superposition ansatz class at each ( [PITH_FULL_IMAGE:figures/full_fig_p035_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18: Evolution of the optimized state with increasing [PITH_FULL_IMAGE:figures/full_fig_p036_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19: Evolution of the optimized state with increasing [PITH_FULL_IMAGE:figures/full_fig_p037_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20: Classical Fisher information (CFI) advantage, under balanced homodyne at fixed local-oscillator phase [PITH_FULL_IMAGE:figures/full_fig_p038_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21: CFI vs homodyne angle for the cat state and the best squeezed vacuum state at [PITH_FULL_IMAGE:figures/full_fig_p038_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22: Classical Fisher information (CFI) for several measurement strategies, as a fraction of the QFI, across the [PITH_FULL_IMAGE:figures/full_fig_p039_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23 [PITH_FULL_IMAGE:figures/full_fig_p040_23.png] view at source ↗
read the original abstract

Squeezed vacuum lets gravitational-wave detectors and other quantum sensors surpass the standard quantum limit, and is optimal in the loss-limited regime; phase noise breaks this optimality. Numerically optimizing the quantum Fisher information across the loss and phase-noise landscape, we identify non-Gaussian states that outperform any Gaussian state. These fall into three classes: Fock-like, cubic-phase-like, and states with discrete rotational symmetry. Limiting the average number of photons in the input state to $\bar{n}=5$, with $1-\eta = 5\%$ photon loss and 200 mrad phase noise, the non-Gaussian advantage reaches up to 2.2 dB. Furthermore, we observe that the non-Gaussian advantage can persist even when the measurement strategy is homodyne detection.

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

3 major / 2 minor

Summary. The manuscript numerically optimizes the quantum Fisher information over input quantum states subject to a composite channel of photon loss followed by phase noise. It reports that non-Gaussian states belonging to three classes (Fock-like, cubic-phase-like, and states with discrete rotational symmetry) outperform the best Gaussian states, with a maximum advantage of 2.2 dB at ar n=5, 5% loss, and 200 mrad phase noise; the advantage is further shown to survive when the final measurement is restricted to homodyne detection.

Significance. If the numerical results hold, the work supplies concrete, non-Gaussian state families that improve sensing precision when both loss and phase noise are present, extending beyond the known optimality of squeezed vacuum in the loss-only regime. The persistence of the advantage under homodyne detection is practically relevant. The direct numerical maximization of QFI against an external Gaussian benchmark, without fitted parameters or self-referential definitions, is a methodological strength.

major comments (3)
  1. [Numerical methods] Numerical methods section: the optimization procedure (state parameterization, search algorithm, convergence criteria, and stopping tolerances) is not described, so it is impossible to judge whether the reported 2.2 dB advantage at ar n=5, 5% loss, 200 mrad noise is robust or an artifact of the search.
  2. [Results] Gaussian benchmark: no independent analytical upper bound or exhaustive numerical verification of the maximum QFI attainable with Gaussian states is supplied; without this the claimed non-Gaussian advantage cannot be confirmed to exceed the true Gaussian optimum.
  3. [Model] Channel model and QFI validity: the paper assumes the composite loss-plus-phase-noise channel commutes with the parameter encoding in the manner required for the standard QFI formula; the order of the two noise processes and the explicit Kraus-operator derivative used should be stated and justified, as any mismatch would invalidate the state ranking.
minor comments (2)
  1. [Results] The three state classes are named in the abstract but lack explicit parameterizations or example density matrices in the main text; adding these would aid reproducibility.
  2. [Figures] Figure captions should explicitly state the precise values of ar n, loss, and phase-noise strength used for each panel.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which have helped us identify areas for clarification. We address each major comment below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

  1. Referee: [Numerical methods] Numerical methods section: the optimization procedure (state parameterization, search algorithm, convergence criteria, and stopping tolerances) is not described, so it is impossible to judge whether the reported 2.2 dB advantage at ar n=5, 5% loss, 200 mrad noise is robust or an artifact of the search.

    Authors: We agree that the numerical optimization details were insufficiently described. In the revised manuscript we will add a dedicated subsection specifying: (i) state parameterization via a truncated Fock-space representation (dimension chosen to ensure convergence for ar n=5) with separate ansätze for the Fock-like, cubic-phase-like, and discrete-rotational-symmetry families; (ii) the search algorithm (a combination of gradient ascent on the QFI and a derivative-free global optimizer for initial exploration); and (iii) convergence criteria together with the stopping tolerances (relative change in QFI < 10^{-6} over 100 iterations and absolute tolerance on photon-number truncation). These additions will allow independent assessment of the robustness of the reported 2.2 dB advantage. revision: yes

  2. Referee: [Results] Gaussian benchmark: no independent analytical upper bound or exhaustive numerical verification of the maximum QFI attainable with Gaussian states is supplied; without this the claimed non-Gaussian advantage cannot be confirmed to exceed the true Gaussian optimum.

    Authors: The manuscript already performs a direct numerical maximization of the QFI over the Gaussian manifold (displaced squeezed states) using the identical optimization engine and truncation as for the non-Gaussian search; this constitutes an internal benchmark rather than an external analytical bound. While a closed-form upper bound for the composite loss-plus-phase-noise channel is not known, we will augment the revised text with (a) an explicit statement that the Gaussian optimization was run to the same convergence tolerances and (b) additional cross-checks against the known loss-only optimum (squeezed vacuum) and against a dense grid of Gaussian parameters. We believe these steps suffice to substantiate the advantage, but will present them more prominently. revision: partial

  3. Referee: [Model] Channel model and QFI validity: the paper assumes the composite loss-plus-phase-noise channel commutes with the parameter encoding in the manner required for the standard QFI formula; the order of the two noise processes and the explicit Kraus-operator derivative used should be stated and justified, as any mismatch would invalidate the state ranking.

    Authors: We will expand the channel-model section to state explicitly that photon loss is applied first, followed by the phase-noise channel, reflecting the typical experimental sequence (propagation loss preceding phase fluctuations). The QFI is evaluated via the standard formula involving the derivative of the composite Kraus operators with respect to the encoded phase; we will supply the explicit ordering of the Kraus operators and a short justification that the phase-encoding unitary commutes with the loss channel in the chosen ordering, thereby validating the QFI expression used for state ranking. If the referee prefers the reverse ordering we can recompute and report the difference. revision: yes

Circularity Check

0 steps flagged

Numerical QFI maximization against Gaussian benchmark is external and non-circular

full rationale

The paper's central result is obtained by direct numerical optimization of quantum Fisher information over input states subject to explicit loss and phase-noise channels, with explicit comparison to the performance of Gaussian states. No step reduces a claimed prediction to a fitted parameter by construction, invokes a self-citation as the sole justification for a uniqueness claim, or renames an input ansatz. The derivation chain consists of standard QFI formulas applied to a composite channel followed by numerical search; the non-Gaussian advantage is an output of that search rather than an input. This is the most common honest finding for a computational optimization paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the assumption that quantum Fisher information is the right figure of merit and that the numerical search can locate globally optimal states; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Quantum Fisher information remains the appropriate bound on sensing precision after loss and phase noise are applied
    Invoked when the optimization target is chosen.

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Reference graph

Works this paper leans on

40 extracted references · 6 canonical work pages · 4 internal anchors

  1. [1]

    Noiseless, phase-sensitive sensing (squeezed vacuum states are optimal) 6

  2. [2]

    Phase-sensitive sensing with decoherence (squeezed vacuum states are optimal) 6

  3. [3]

    Lossless phase-insensitive sensing (Fock states are optimal) 8

  4. [4]

    Convexity Bound 10

    QFI of Fock states in the presence of loss 9 D. Convexity Bound 10

  5. [5]

    Convexity bound for Gaussian states 10

  6. [6]

    The Variance Bound 11 F

    Convexity bound for General States 11 E. The Variance Bound 11 F. Combined Channel Bound 13 G. Comparison of the Analytical Bounds 13 H. Iterative Algorithm for Global QFI Optimization 14 III. Optimization details 15 A. State parametrization 15 B. Quantum channel 15 C. QFI computation 16 D. Objective function 16 E. Two-phase hybrid optimization 16 F. Warm...

  7. [7]

    Noiseless, phase-sensitive sensing (squeezed vacuum states are optimal) In the absence of any loss, an input pure state is mapped to an output pure state with the encoding in Eq. 11. The QFI for a given pure state|ψ⟩can be computed from Eq. (7), withG= ˆa+ ˆa †. Here, it is implicitly assumed that the phase of the displacement is known (i.e., the displace...

  8. [8]

    loss during displacement

    Phase-sensitive sensing with decoherence (squeezed vacuum states are optimal) In the case of phase-sensitive sensing with loss only, the optimal state is known to be squeezed vacuum [8]. For completeness, we re-derive this fact below. The effect of loss can be simplified as follows. Loss modifies an annihilation operator ˆaas ˆa→√ηˆa+ p 1−η ˆb(13) where ˆ...

  9. [9]

    We will first show that the optimal input state is a Fock state

    Lossless phase-insensitive sensing (Fock states are optimal) Let us now consider the high phase-noise limit. We will first show that the optimal input state is a Fock state. We will then compute the QFI of Fock states in the presence of loss. Let us write the two-stage sensing channel in terms of the channels Φ and Λ ϵ, which will denote the preparation p...

  10. [10]

    These results provide support to our numerical results that Fock states are optimal for low losses, but not for high losses

    QFI of Fock states in the presence of loss Let us now compute the QFI of a Fock state after experiencing the photon loss channel with transmission rateη. These results provide support to our numerical results that Fock states are optimal for low losses, but not for high losses. The photon loss channel will act only on the diagonal elements. For a Fock sta...

  11. [11]

    We will consider the optimization over states ˆρsatisfying the constraint⟨ˆn⟩ ˆρ≤N target

    Convexity bound for Gaussian states Let us first consider the convexity bound for Gaussian states, which will indicate a suppression of QFI due to phase noise. We will consider the optimization over states ˆρsatisfying the constraint⟨ˆn⟩ ˆρ≤N target. We will drop explicit references to this constraint from the equations to help simplify the notation. We f...

  12. [12]

    More specifically, we relax the bound by ignoring the contribution of phase noise to once again obtain a computation using Gaussian states

    Convexity bound for General States For general states, we get a weaker bound. More specifically, we relax the bound by ignoring the contribution of phase noise to once again obtain a computation using Gaussian states. max ˆρ FQ(Dϵ ◦Λ◦Φ(ˆρ))≤max ˆρ Z dφP(φ)F Q(Dϵ ◦Λ(R(φ)ˆρR†(φ))) (47) ≤ Z dφP(φ) max ˆρ FQ(Dϵ ◦Λ(ˆρ)) (48) = max ˆρ FQ(Dϵ ◦Λ(ˆρ)).(49) We know...

  13. [13]

    The derivative∂ρ/∂θis computed by central finite differences with step sizeh= 10 −5

  14. [14]

    The density matrix without displacement is eigendecomposed:ρ= P i λi|ui⟩⟨ui|

  15. [15]

    The derivative is rotated into the eigenbasis: (∂ρ/∂θ) rot =U †(∂ρ/∂θ)U

  16. [16]

    The QFI is evaluated as FQ = X i,j λi+λj >10−8 2 (∂ρ/∂θ) rot ij 2 λi +λ j .(77) D. Objective function The optimizer minimizes f(x) =−F Q[ψ(x)] +w n max 0,⟨ ˆN⟩ −N target 2 +w b X i max(0, x i −x ub i )2 + max(0, x lb i −x i)2 ,(78) wherew n = 106 penalizes mean photon number exceeding the targetN target, andw b = 104 penalizes parameter bound violations f...

  17. [17]

    A single displaced squeezed-vacuum state|α, r⟩= ˆD(α) ˆS(r)|0⟩is parameterized by the displacementα∈Cand squeezingr∈C(i.e.,n sup = 1 in the displaced squeezed-vacuum ansatz of Sec. III). 20 FIG. 8: QFI convergence with Fock basis size for the optimized states at ¯Ntarget = 5 in the low-loss, low-phase-noise regime. Each panel shows a different (η, σ ϕ) po...

  18. [18]

    For each grid point (η, σ ϕ), the two-phase hybrid optimization (ECA global search followed by L-BFGS local refinement) is run to maximizeF Q subject to the photon number constraint⟨ ˆN⟩ ≤N target, using the penalized objective function of Sec. III. We use the same optimization procedure for consistency despite knowing that a global search is not necessar...

  19. [19]

    III, so that the optimizer can efficiently explore the trade-off between displacement and squeezing at each noise configuration

    Warm-starting from the best squeezed vacuum state is used as described in Sec. III, so that the optimizer can efficiently explore the trade-off between displacement and squeezing at each noise configuration. This procedure yields a potentially different Gaussian state at each (η, σϕ) point, reflecting how the optimal balance between displacement and squee...

    1906

  20. [20]

    M. G. Genoni, S. Olivares, and M. G. A. Paris, Optical phase estimation in the presence of phase diffusion, Phys. Rev. Lett.106, 153603 (2011)

    2011

  21. [21]

    McCuller, S

    L. McCuller, S. E. Dwyer, A. C. Green, H. Yu, L. Barsotti, C. D. Blair, D. D. Brown, A. Effler, M. Evans, A. Fernandez- Galiana, P. Fritschel, V. V. Frolov, N. Kijbunchoo, G. L. Mansell, F. Matichard, N. Mavalvala, D. E. McClelland, T. McRae, A. Mullavey, D. Sigg, B. J. J. Slagmolen, M. Tse, T. Vo, R. L. Ward, C. Whittle,et al., LIGO’s quantum response to...

    2021

  22. [22]

    J. W. Gardner, T. Gefen, S. A. Haine, J. J. Hope, J. Preskill, Y. Chen, and L. McCuller, Stochastic waveform estimation at the fundamental quantum limit, PRX Quantum6, 030311 (2025)

    2025

  23. [23]

    V. L. Gorshenin, Using Schroedinger cat quantum state for detection of a given phase shift, Laser Physics Letters21, 065201 (2024), arXiv:2403.03787 [quant-ph]

    work page arXiv 2024

  24. [24]

    V. L. Gorshenin and F. Y. Khalili, Using non-Gaussian quantum states for detection of a given phase shift, JOSA B42, 1448 (2025)

    2025

  25. [25]

    M. G. A. Paris, Displacement operator by beam splitter, Physics Letters A217, 78 (1996)

    1996

  26. [26]

    M. D. Lang and C. M. Caves, Optimal Quantum-Enhanced Interferometry Using a Laser Power Source, Physical Review Letters111, 173601 (2013)

    2013

  27. [27]

    C. L. Latune, B. M. Escher, R. L. de Matos Filho, and L. Davidovich, Quantum limit for the measurement of a classical force coupled to a noisy quantum-mechanical oscillator, Physical Review A88, 042112 (2013)

    2013

  28. [28]

    B. M. Escher, R. L. de Matos Filho, and L. Davidovich, General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology, Nature Phys.7, 406 (2011)

    2011

  29. [29]

    Demkowicz-Dobrza´ nski, J

    R. Demkowicz-Dobrza´ nski, J. Ko lody´ nski, and M. Gut ¸˘ a, The elusive Heisenberg limit in quantum-enhanced metrology, Nat. Commun.3, 1063 (2012)

    2012

  30. [30]

    Ko lody´ nski and R

    J. Ko lody´ nski and R. Demkowicz-Dobrza´ nski, Efficient tools for quantum metrology with uncorrelated noise, New Journal of Physics15, 073043 (2013)

    2013

  31. [31]

    Fadel, N

    M. Fadel, N. Roux, and M. Gessner, Quantum metrology with a continuous-variable system, Reports on Progress in Physics 88, 106001 (2025)

    2025

  32. [32]

    Weedbrook, S

    C. Weedbrook, S. Pirandola, R. Garc´ ıa-Patr´ on, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum information, Reviews of Modern Physics84, 621 (2012)

    2012

  33. [33]

    Pinel, P

    O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun, Quantum parameter estimation using general single-mode Gaussian states, Physical Review A88, 040102 (2013)

    2013

  34. [34]

    Lower bounds on the quantum Fisher information based on the variance and various types of entropies

    G. Toth, Lower bounds on the quantum Fisher information based on the variance and various types of entropies (2018), arXiv:1701.07461 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  35. [35]

    Extremal properties of the variance and the quantum Fisher information

    G. Toth and D. Petz, Extremal properties of the variance and the quantum Fisher information, Physical Review A87, 032324 (2013), arXiv:1109.2831 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  36. [36]

    Quantum Fisher Information: Variational principle and simple iterative algorithm for its efficient computation

    K. Macieszczak, Quantum fisher information: Variational principle and simple iterative algorithm for its efficient compu- tation (2013), arXiv:1312.1356 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  37. [37]

    Macieszczak, M

    K. Macieszczak, M. Fraas, and R. Demkowicz-Dobrza´ nski, Bayesian quantum frequency estimation in presence of collective dephasing, New Journal of Physics16, 113002 (2014)

    2014

  38. [38]

    Squeezed-vacuum bosonic codes

    N. Gutman, E. Blumenthal, S. Hacohen-Gourgy, A. Orda, and I. Kaminer, Squeezed-vacuum bosonic codes, Advanced Physics Research , e00007 (2026), arXiv:2511.06108 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  39. [39]

    Yanagimoto, T

    R. Yanagimoto, T. Onodera, E. Ng, L. G. Wright, P. L. McMahon, and H. Mabuchi, Engineering a kerr-based deterministic cubic phase gate via gaussian operations, Phys. Rev. Lett.124, 240503 (2020)

    2020

  40. [40]

    Brunelli and O

    M. Brunelli and O. Houhou, Linear and quadratic reservoir engineering of non-gaussian states, Physical Review A100, 10.1103/physreva.100.013831 (2019)

    work page doi:10.1103/physreva.100.013831 2019