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Modes, states, and symmetries jointly fix which resources of quantum light can be used for information processing and precision measurement.

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2026-07-30 21:51 UTC pith:IDIUEPJW

load-bearing objection Solid unifying PhD monograph built on multiple PRL-level pieces; the modes–states–symmetries story is real and usable, with SSR as an explicit modeling choice rather than a hidden crack. the 2 major comments →

arxiv 2607.26761 v1 pith:IDIUEPJW submitted 2026-07-29 quant-ph

PhD thesis: Modes, States, and Symmetries in quantum Optics for quantum Information and Metrology

\'Eloi Descamps This is my paper
classification quant-ph PACS 03.67.-a42.50.-p03.65.Ta06.20.Dk
keywords quantum opticsquantum metrologytime-frequency variablescollective entanglementHong-Ou-Mandel interferometrysuperselection rulesbosonic quantum informationGKP encoding
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This thesis argues that quantum light’s usefulness is not settled by photons or field quadratures alone, but by how modal structure, state statistics, and symmetries work together. It treats time and frequency as continuous quantum variables for encoding and metrology, then shows that entanglement along collective observables—not only local ones—sets achievable precision and supports collective encodings such as time-frequency GKP-type codes. Hong–Ou–Mandel-type interference is reframed as a symmetry property of multimode inputs, yielding metrological bounds and multiphoton/multimode generalizations, including imperfect visibility. Optical superselection rules from global phase symmetry are used to organize bosonic states and operations, bridge discrete- and continuous-variable pictures, and clarify what counts as a computational or metrological resource. A sympathetic reader cares because the same three notions reorganize interference, error encoding, and ultimate precision limits under one operational language.

Core claim

The physical resources of photonic quantum information and metrology are determined by the joint structure of modes, states, and symmetries: time-frequency continuous variables, entanglement along collective operators (with metrological inequalities and k-entanglement), symmetry-based generalized Hong–Ou–Mandel interference with quantifiable precision, and optical superselection rules that fix accessible bosonic states/operations and unify discrete- and continuous-variable resource accounts.

What carries the argument

A modes–states–symmetries framework whose load-bearing pieces are collective-variable entanglement (spectral-space inequalities and k-entanglement), symmetry-centered generalized HOM interferometry, and superselection-rule-compliant (SSR) bosonic descriptions (Schwinger/spin tools and the CV limit of fixed-photon-number sectors).

Load-bearing premise

The claim that missing a shared global phase reference really imposes a photon-number superselection rule that correctly splits discrete and continuous pictures and decides which bosonic resources count.

What would settle it

Find an optical protocol where a genuine global phase reference is absent, yet modal entanglement from passive linear optics or CV encodings outside fixed-N sectors yields computational universality or metrological scaling that the SSR resource account forbids; or show the formal CV-as-limit-of-SSR construction fails for a standard Gaussian/non-Gaussian task.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Time-frequency correlations can be treated as genuine metrological resources for collective time-parameter estimation, not only technical spectrum shape.
  • HOM-type precision limits and optimal inputs follow from input symmetry under the interferometer, extending beyond two single photons and ideal visibility.
  • Collective time-frequency variables support GKP-like encodings and error correction tied to multimode structure.
  • SSR organizes when Gaussianity, nonclassicality, modal entanglement, and particle entanglement count toward bosonic universality and metrology.
  • Discrete-variable and continuous-variable optical protocols can be compared inside one fixed-photon-number-compatible resource language.

Where Pith is reading between the lines

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

  • Labs reporting HOM metrology should quote symmetry/visibility models alongside dip depth, because the thesis ties Fisher information degradation to those quantities.
  • Resource theories for linear-optical computing may need explicit phase-reference bookkeeping whenever passive elements appear to ‘create’ modal entanglement.
  • Collective-variable codes suggest hybrid architectures that encode logically in sum/difference time-frequency quadratures while sensing with the same collective observables.
  • If SSR limits are accepted, claims of CV advantage should state whether they survive restriction to phase-reference-free, fixed-N sectors.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. This PhD thesis develops a modes–states–symmetries framework for quantum optics in quantum information and metrology. It treats time-frequency degrees of freedom as continuous variables (Ch. 2), links multimode/time-frequency entanglement along collective operators to metrological bounds and k-entanglement, and proposes collective time-frequency GKP-type encodings (Ch. 3). It reinterprets Hong-Ou-Mandel interference via input-state symmetry, quantifies Fisher information including imperfect visibility (with experiment), and generalizes to multiphoton and multimode interferometers (Ch. 4). Finally it uses optical photon-number superselection rules (Schwinger representation, spherical phase space, controlled CV limits) to organize bosonic computational and metrological resources and to relate DV and CV encodings (Ch. 5). Core technical threads are supported by published works [1–9], appendices, and standard QFI/CRB and Fock/mode formalism.

Significance. If the synthesis holds, the thesis supplies a coherent language for when modal structure, photon statistics, and exchange/phase symmetries—not only ‘entanglement’ in the abstract—set precision and computational power in photonic platforms. Strengths that should be credited include: peer-reviewed derivations of time-frequency metrology and phase-space tools [1,2]; collective-operator entanglement measures and inequalities with explicit metrological reading [6]; symmetry-based HOM generalizations and visibility-aware precision, including experimental contact [4,7]; SSR-based resource partitions and universality discussion for bosonic encodings [5,8,9]; and extensive appendices (B.*) that make many bounds checkable. The collective-variable and HOM pillars are useful even if one rejects the strongest SSR↔CV identification. The work is significant as a unifying thesis rather than a single new theorem.

major comments (2)
  1. [Chapter 5, Sections 5.2–5.3] Ch. 5 / Sec. 5.2–5.3: The load-bearing unification of DV and CV resources and the universality claims rest on treating continuous-variable systems as controlled limits of fixed-N, SSR-compliant sectors (Schwinger/spin coherent states, spherical Wigner → planar phase space) and on absence of a shared global phase reference imposing photon-number SSR. This is a standard modeling choice, but the manuscript should state operational failure modes more sharply—when a local oscillator or relative-phase reference is available, which resource counts (modal vs particle entanglement, SG/SNG vs QG/QNG) and which universality statements survive unchanged. Without an explicit ‘with vs without phase reference’ map tied to the encoding-independent conditions in Sec. 5.3.5, the strongest DV/CV resource story remains partly interpretive even though the formal limit constructions are carefully set up.
  2. [Section 3.2] Sec. 3.2 (esp. inequalities around collective variance / Eq. (3.38) and the k-entanglement and thickness ζ discussion): The metrological inequalities and partial entanglement quantifier are central to Ch. 3’s claim that entanglement along collective operators is the relevant resource. The pure-state and spectral-support arguments are clear; the mixed-state extension and the trade-off plots (e.g. k vs ζ for fixed I) need a short statement of which measurement class saturates the bound (collective vs local) and whether k-entanglement can vanish while QFI along the collective generator remains large under experimentally natural noise. A single clarifying proposition or remark would lock the resource interpretation to the QFI expressions already used in Chs. 1–2.
minor comments (5)
  1. [How to use this thesis; Chapters 3–5] Front matter and Ch. 1 are appropriately pedagogical, but the icon system (established / published / unpublished) is easy to miss; a one-page ‘original vs published’ map for Chs. 3–5 would help examiners and journal readers separate [1–9] from lightbulb material.
  2. [Chapter 2; Section 3.1.5] Notation list is thorough; still, chrono-cyclic Wigner vs standard Wigner and collective vs local generators could be cross-referenced at first use in Ch. 2–3 to avoid dual meanings of ‘phase space’.
  3. [Section 4.2] HOM visibility model and experimental figures (Ch. 4, ~Figs. 4.10–4.12, Tables 4.1–4.4) are strong; ensure all plotted Fisher curves state the estimator (e.g. MLE on coincidences) and number of shots so the approach to QFI is unambiguous.
  4. [Throughout; List of publications] Minor language/typo cleanup: e.g. ‘fondamental’, ‘acronymes’, ‘chronocyclique/chrono-cyclique’, ‘Unied framework’ in [9], and consistent capitalization of chapter titles in the Contents.
  5. [Section 1.3.5; Chapter 5] Sec. 1.3.5 and Ch. 5 overlap on phase reference/SSR; a forward/back pointer stating what is standard review versus thesis contribution would reduce apparent repetition.

Circularity Check

0 steps flagged

No significant circularity: standard definitions and constructed states, with expected thesis self-citation of compiled papers [1–9] that is not load-bearing for the derivations.

full rationale

The thesis organizes modes, states, and symmetries as an analytical frame and evaluates standard resource quantities (QFI/Fisher information, collective-operator variances, HOM coincidence statistics, SSR-compliant encodings) on explicitly constructed states and interferometers. Metrological bounds follow the usual pipeline—unitary encoding → QFI = 4Δ²Ĥ (pure) or SLD form (mixed) → classical FI from POVMs/coincidences—without fitting a target observable and relabeling it as a prediction. Collective-variable inequalities, k-entanglement, symmetry-based HOM generalizations, and the SSR/CV-limit constructions are derived from stated definitions and representation theory (with technical support in the appendices), not forced by uniqueness theorems imported only from overlapping authors or by ansatz smuggled as external fact. Self-citation of [1–9] is structural for a compilation thesis and does not close a circular loop: the manuscript restates and unifies those results rather than treating an unverified self-cite as the sole warrant for the central claims. No self-definitional reduction (X defined as Y then “predicted” as Y) or fitted-input-as-prediction pattern is evidenced in the derivation chain. Score 1 reflects only the normal presence of author-overlap citations, not load-bearing circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 6 axioms · 4 invented entities

The load-bearing content sits on standard quantum mechanics, quantum estimation (CRB/QFI/SLD), quantum optics mode quantization and Fock space, and conventional treatments of bosonic exchange and optical phase references/SSR. The thesis invents organizing notions (collective-variable entanglement/k-entanglement, symmetry-generalized HOM family, SSR-compliant resource partitions and CV-as-SSR-limit dictionary) but not new physical forces or particles. No empirical free parameters drive the central theoretical claims; experimental visibility V is a measured nuisance parameter in the HOM precision study, not a universal fitted constant of the framework.

free parameters (2)
  • HOM interference visibility V (and related peak-separation / mode-overlap parameters in examples) = State- and setup-dependent; figures show F_max(V)/F_ideal scaling versus V
    Used as an experimental/model parameter when quantifying degradation of Fisher information relative to ideal HOM metrology; characterized or swept rather than a fundamental constant of the unified theory.
  • GKP lattice scale α (and finite-squeezing / finite-peak approximations)
    Standard code-design choice trading position vs momentum correctability and physical approximability for time-frequency/collective GKP encodings; not fitted to claim existence of the encoding framework.
axioms (6)
  • standard math Quantum states on Hilbert/Fock space with unitary encodings and POVM measurements; classical and quantum Cramér–Rao bounds with Fisher and quantum Fisher information (incl. pure-state QFI = 4 Var(H)).
    Ch. 1 metrology and optics sections; used throughout precision claims.
  • domain assumption Electromagnetic field as bosonic modes: canonical quantization, mode transformations as passive linear optics, photons as indistinguishable bosons under exchange.
    Sec. 1.3 and Ch. 4 HOM analysis; underpins interference and multimode resource discussion.
  • domain assumption Time and frequency for single-photon wavepackets behave as conjugate continuous variables admitting Wigner/chrono-cyclic phase-space representation and shear/rotation optics.
    Ch. 2 framework used for metrology and GKP-style encodings in Ch. 3–4.
  • domain assumption Absent a shared optical phase reference, global U(1) phase symmetry induces a photon-number superselection rule constraining accessible states/operations (SSR-compliant optics).
    Ch. 5 foundational premise linking DV/CV and resource theories; standard in SSR literature but interpretive for ‘resources’.
  • ad hoc to paper Continuous-variable bosonic systems can be obtained as controlled limits of finite-photon-number SSR-compliant systems (Schwinger/spin coherent, spherical Wigner → planar phase space).
    Sec. 5.2 formal limit and geometric picture; central to unification narrative though built from known contractions/representations.
  • ad hoc to paper Metrological ‘resources’ may be counted as photon number, mode number, energy, modal entanglement, and/or particle entanglement depending on encoding and SSR sector.
    Stated motivation in Introduction and Ch. 5 metrology; organizes comparisons across DV/CV but is a modeling stance.
invented entities (4)
  • Entanglement along collective operators and k-entanglement (partial collective entanglement quantifier) independent evidence
    purpose: Quantify multimode correlations relevant to collective metrology and relate correlation structure to QFI-type bounds beyond simple bipartite entanglement.
    Ch. 3 general theory; operational via inequalities and spectral-support pictures, not a new particle.
  • Time-frequency / collective GKP-type encodings for single photons independent evidence
    purpose: Encode logical information in continuous modal variables and collective variables for error correction oriented photonic schemes.
    Ch. 3.3 and [3]; extends known GKP idea to time-frequency modal degrees of freedom.
  • Symmetry-centered generalized HOM interferometer family (multiphoton and multimode) independent evidence
    purpose: Replace pure indistinguishability lore with input symmetry w.r.t. interferometer modes to derive interference patterns and metrological bounds more generally.
    Ch. 4 and [7]; falsifiable via coincidence statistics vs designed symmetries.
  • SSR-compliant Gaussian / non-Gaussian resource partition for bosonic QI and metrology no independent evidence
    purpose: Classify computational and metrological resources under phase superselection and relate universality and precision to entanglement and non-Gaussianity in a representation-aware way.
    Ch. 5 and [5,8,9]; dictionary-level construct tied to SSR assumption.

pith-pipeline@v1.2.0-daily-grok45 · 58647 in / 4079 out tokens · 83492 ms · 2026-07-30T21:51:54.677459+00:00 · methodology

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read the original abstract

This thesis explores the role of modes, states, and symmetries in quantum optics, within the context of quantum information and quantum metrology. It proposes a unified framework to analyze how the modal structure of photonic fields, the statistical nature of states, and their symmetry properties determine the physical resources that can be exploited for quantum information processing and quantum parameter estimation. A first line of investigation develops a description of time-frequency degrees of freedom as continuous quantum variables, highlighting their richness for encoding and manipulating information. A second axis studies entanglement and collective variables, clarifying the link between physical resources and metrological gains, particularly in reaching ultimate precision limits. Interferometric scenarios of the Hong-Ou-Mandel type are then analyzed, and a general formalism centered on the notion of symmetry is developed. This framework enables the analysis of a broad range of situations and leads to several generalizations. Finally, the thesis examines the symmetries imposed by optical superselection rules and their consequences for the structure of quantum states and their operational performance, with the aim of providing a deeper understanding of the foundations of quantum optics.

Figures

Figures reproduced from arXiv: 2607.26761 by \'Eloi Descamps.

Figure 1.1
Figure 1.1. Figure 1.1: Geometric representation of a pure qubit state on the Bloch sphere. [PITH_FULL_IMAGE:figures/full_fig_p048_1_1.png] view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: Wigner functions of typical continuous-variable quantum states. [PITH_FULL_IMAGE:figures/full_fig_p055_1_2.png] view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: Top: basic quantum gates and their circuit representations. Bottom: example [PITH_FULL_IMAGE:figures/full_fig_p057_1_3.png] view at source ↗
Figure 1.4
Figure 1.4. Figure 1.4: Top: position-space wavefunctions of the logical states [PITH_FULL_IMAGE:figures/full_fig_p066_1_4.png] view at source ↗
Figure 1.5
Figure 1.5. Figure 1.5: Schematic representation of a quantum metrology protocol: probe prepara [PITH_FULL_IMAGE:figures/full_fig_p075_1_5.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Schematics of the non linear conversion process in a [PITH_FULL_IMAGE:figures/full_fig_p116_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Density plots of the JSA for several typical biphoton states encountered in [PITH_FULL_IMAGE:figures/full_fig_p120_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: Representation of the Wigner function of a GKP state. Each cross corresponds [PITH_FULL_IMAGE:figures/full_fig_p122_2_3.png] view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: Effect of time and frequency translations on the Wigner function of a single [PITH_FULL_IMAGE:figures/full_fig_p127_2_4.png] view at source ↗
Figure 2.5
Figure 2.5. Figure 2.5: Effect of time and frequency shears on the Wigner function of a single-photon [PITH_FULL_IMAGE:figures/full_fig_p129_2_5.png] view at source ↗
Figure 2.6
Figure 2.6. Figure 2.6: Effect of a rotation on the Wigner function of a single-photon cat-like state. [PITH_FULL_IMAGE:figures/full_fig_p131_2_6.png] view at source ↗
Figure 2.7
Figure 2.7. Figure 2.7: Effect of three shears implementing a rotation on the Wigner function of a [PITH_FULL_IMAGE:figures/full_fig_p133_2_7.png] view at source ↗
Figure 2.8
Figure 2.8. Figure 2.8: Three successively applied shears implementing a [PITH_FULL_IMAGE:figures/full_fig_p134_2_8.png] view at source ↗
Figure 2.9
Figure 2.9. Figure 2.9: Schematics of a general optical interferometric protocol. An initial probe [PITH_FULL_IMAGE:figures/full_fig_p138_2_9.png] view at source ↗
Figure 2.10
Figure 2.10. Figure 2.10: Schematic representation of the Wigner functions of various states under [PITH_FULL_IMAGE:figures/full_fig_p144_2_10.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Joint Spectral Intensity (JSI) of different separable quantum states ( [PITH_FULL_IMAGE:figures/full_fig_p151_3_1.png] view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Joint spectral intensity (JSI) of different entangled quantum states ( [PITH_FULL_IMAGE:figures/full_fig_p154_3_2.png] view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: ∆2Ωˆ/∆2ω as a function of the number of photons. The different scaling behaviors (linear and quadratic) and the transition point are displayed. Adapted from Ref. [1]. ©2023 American Physical Society. The existence of a transition from Heisenberg to shot-noise scaling is reminiscent of 124 Contents | Chapter [PITH_FULL_IMAGE:figures/full_fig_p154_3_3.png] view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: Wigner function of a cat-like state distributed in (a) the local variable [PITH_FULL_IMAGE:figures/full_fig_p157_3_4.png] view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: Examples of the representation of the support of the spectrum in the bipartite [PITH_FULL_IMAGE:figures/full_fig_p163_3_5.png] view at source ↗
Figure 3.6
Figure 3.6. Figure 3.6: Two ways to increase the collective variance [PITH_FULL_IMAGE:figures/full_fig_p167_3_6.png] view at source ↗
Figure 3.7
Figure 3.7. Figure 3.7: Spectral space of H⊗2 for evenly spaced eigenvalues. Crosses represent all possible pairs (λ1, λ2) of eigenvalues of Hˆ . The colored lines illustrate examples of supports of states saturating the inequality (3.38). Adapted from Ref. [6]. ©2025 American Physical Society. ▶ Denoting by hmax and hmin the maximal and minimal eigenvalues of Hˆ , the states saturating the simplified bound (3.45) are given by … view at source ↗
Figure 3.8
Figure 3.8. Figure 3.8: Spectral space of H⊗2 for unevenly spaced eigenvalues. Black solid lines indicate directions parallel to ⃗u that intersect the grid at a single point only, leading to zero-variance states with I = 0. Red dashed lines correspond to diagonals intersecting at least two points and thus to nontrivial states saturating the inequality (3.38). Adapted from Ref. [6]. ©2025 American Physical Society. thus be used … view at source ↗
Figure 3.9
Figure 3.9. Figure 3.9: Spectral space for two time-frequency single-photons states. Colored lines indi [PITH_FULL_IMAGE:figures/full_fig_p172_3_9.png] view at source ↗
Figure 3.10
Figure 3.10. Figure 3.10: Examples of states with non-zero width along secondary variables in two [PITH_FULL_IMAGE:figures/full_fig_p174_3_10.png] view at source ↗
Figure 3.11
Figure 3.11. Figure 3.11: Trade-off between k-entanglement and thickness ζ obtained from Eq. (3.83). The left panel shows k as a function of ζ, while the right panel shows ζ as a function of k, for different values of n. The two plots are related by inversion and can be viewed as reflections across the line y = x. All curves are monotonically decreasing, illustrating that larger thickness reduces the amount of multipartite entan… view at source ↗
Figure 3.12
Figure 3.12. Figure 3.12: Values of k and ζ required to achieve a fixed value I = f for n = 10. The left (right) panel shows the minimal value of k (maximal value of ζ) as a function of ζ (k). The two plots are related by inversion. The special case f = n corresponds to either k = 1 or ζ = 1, for which I is independent of the remaining parameter. Larger target values of f require simultaneously smaller thickness and larger multi… view at source ↗
Figure 3.13
Figure 3.13. Figure 3.13: Joint spectral amplitude of a two-mode GKP code. Red points represent [PITH_FULL_IMAGE:figures/full_fig_p186_3_13.png] view at source ↗
Figure 3.14
Figure 3.14. Figure 3.14: Joint temporal amplitude of a two-mode GKP code. Red points represent [PITH_FULL_IMAGE:figures/full_fig_p187_3_14.png] view at source ↗
Figure 3.15
Figure 3.15. Figure 3.15: Left: Ideal time-frequency GKP state exhibiting a comb structure with peak [PITH_FULL_IMAGE:figures/full_fig_p189_3_15.png] view at source ↗
Figure 3.16
Figure 3.16. Figure 3.16: Effect of imperfect state preparation or measurement in rotated variables [PITH_FULL_IMAGE:figures/full_fig_p195_3_16.png] view at source ↗
Figure 3.17
Figure 3.17. Figure 3.17: Schematic representation of a CNOT gate Dˆ 1,2 between two time-frequency GKP states (blue and red), each comprising n photons distributed over n spatial modes. Photons, represented by blue and red circles, interact pairwise, with each photon pair interacting only once via a time-frequency CNOT gate Cˆ j,l (green box). Different photon￾pairing configurations across spatial modes are possible. The config… view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Schematic representation of the Hong-Ou-Mandel interferometer. Two pho [PITH_FULL_IMAGE:figures/full_fig_p201_4_1.png] view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Interference mechanisms in a HOM interferometer. (a) For distinguishable [PITH_FULL_IMAGE:figures/full_fig_p202_4_2.png] view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Coincidence probability Pc as a function of the time delay τ between two photons occupying identical Gaussian temporal modes. The HOM dip is centered at τ = 0, where the temporal mode overlap is maximal and Pc reaches its minimum value of zero. As τ increases, the overlap between the temporal modes decreases and Pc approaches the classical value of 1/2. The width of the dip is set by the inverse spectral… view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: Coincidence probability Pc as a function of the time delay τ for Gaussian temporal modes with reduced visibility V < 1. The minimum of the HOM dip no longer reaches zero, reflecting imperfect overlap between the temporal modes of the two photons. 4.1.4 Experimental considerations Throughout this chapter, and as already done above, we mostly consider idealized as￾sumptions on the experimental setup. These… view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: (a) General HOM interferometer fed by two photons prepared in an arbitrary [PITH_FULL_IMAGE:figures/full_fig_p214_4_5.png] view at source ↗
Figure 4.6
Figure 4.6. Figure 4.6: Coincidence probability Pc as a function of the relative delay. The solid black curve shows a typical HOM dip for a symmetric state, while the dashed blue curve illustrates a HOM peak for an anti-symmetric state. 4.2.3 HOM and chrono-cyclic Wigner function Let us consider an initial two-photon state whose joint spectral amplitude (JSA) is fac￾torizable in the variables ω±, namely F(ω1, ω2) = f+(ω+)f−(ω−)… view at source ↗
Figure 4.7
Figure 4.7. Figure 4.7: HOM interferometer used to probe the anti-diagonal chrono-cyclic Wigner [PITH_FULL_IMAGE:figures/full_fig_p219_4_7.png] view at source ↗
Figure 4.8
Figure 4.8. Figure 4.8: Typical behavior of the Fisher information [PITH_FULL_IMAGE:figures/full_fig_p227_4_8.png] view at source ↗
Figure 4.9
Figure 4.9. Figure 4.9: Fisher information F as a function of the delay τ for different values of the visibility V and peak separation ∆, with σ = 1. Blue curves: exact expression. Orange line: ideal limit Fideal = ∆2 (ˆω1 −ωˆ2). Green line: approximate Gaussian envelope V 2∆2 e −2σ 2τ 2 /2. The reduction of the maximal value and the displacement of the optimal estimation point away from τ = 0 become more pronounced as V decrea… view at source ↗
Figure 4.10
Figure 4.10. Figure 4.10: Experimental setup used to investigate the metrological performance of the [PITH_FULL_IMAGE:figures/full_fig_p231_4_10.png] view at source ↗
Figure 4.11
Figure 4.11. Figure 4.11: Left column: Joint spectral amplitude of the four states analyzed in this [PITH_FULL_IMAGE:figures/full_fig_p233_4_11.png] view at source ↗
Figure 4.12
Figure 4.12. Figure 4.12: Scaling of the ratio Fmax(V )/Fideal as a function of the HOM visibil￾ity V for the four biphoton states. Points correspond to experimental data and solid lines to theoretical predictions. For the time-frequency cat-like state, a maximal value Fmax(V )/Fideal = 0.97 is obtained at V = 99.4%. Note that the absolute values of Fideal differ for each state. Adapted from Ref. [4]. ©2024 American Physical Soc… view at source ↗
Figure 4.13
Figure 4.13. Figure 4.13: Mach-Zehnder interferometer composed of two balanced beam splitters sep [PITH_FULL_IMAGE:figures/full_fig_p236_4_13.png] view at source ↗
Figure 4.14
Figure 4.14. Figure 4.14: Generalized HOM interference with an arbitrary two-mode input state. The [PITH_FULL_IMAGE:figures/full_fig_p239_4_14.png] view at source ↗
Figure 4.15
Figure 4.15. Figure 4.15: Generalized HOM interference with parameter encoding. A two-mode state [PITH_FULL_IMAGE:figures/full_fig_p244_4_15.png] view at source ↗
Figure 4.16
Figure 4.16. Figure 4.16: Single-photons Mach-Zehnder interferometer. A two-mode single-photons [PITH_FULL_IMAGE:figures/full_fig_p247_4_16.png] view at source ↗
Figure 4.17
Figure 4.17. Figure 4.17: n-mode extension of the HOM interferometer. An arbitrary input state |ψ⟩, with arbitrary photon-number statistics and internal degree-of-freedom distribution, is injected into a DFT interferometer Uˆ that symmetrically mixes all modes. Photon￾number-resolved detection is performed at the output, and the numbers mj of detected photons in each mode are recorded. This setup acts as a symmetry analyzer with… view at source ↗
Figure 4.18
Figure 4.18. Figure 4.18: n-mode interferometer with parameter-dependent evolution. The input state |ψ⟩ undergoes Vˆ (θ) = e −iHθ ˆ followed by the DFT interferometer Uˆ. Photon-number￾resolved detection yields outcomes mj , from which the quantity Pn−1 k=0 kmk modulo n is computed. Estimating the associated probabilities enables parameter estimation governed by the symmetry of Hˆ . Adapted from Ref. [7]. ©2026 American Physical… view at source ↗
Figure 4.19
Figure 4.19. Figure 4.19: An initial state |ψ⟩ with photon-number distribution {m′ k } is injected into a DFT interferometer Uˆ, which enforces a well-defined symmetry according to the modular condition on Pn−1 k=0 km′ k . The resulting symmetric or anti-symmetric state then under￾goes a parameter-dependent evolution e −iHθ ˆ , followed by a second DFT interferometer and photon-number-resolved detection. This setup generalizes t… view at source ↗
Figure 4.20
Figure 4.20. Figure 4.20: Interferometer adapted to a general permutation symmetry [PITH_FULL_IMAGE:figures/full_fig_p261_4_20.png] view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Stereographic projection of the unit sphere onto the equatorial plane based [PITH_FULL_IMAGE:figures/full_fig_p270_5_1.png] view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Photon number distribution in the reflected mode of a weakly reflecting beam [PITH_FULL_IMAGE:figures/full_fig_p298_5_2.png] view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Geometric picture of the transition from the SSRC spherical phase space to the [PITH_FULL_IMAGE:figures/full_fig_p304_5_3.png] view at source ↗

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