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REVIEW 3 major objections 4 minor 290 references

Structured coherence: A modern perspective on optical coherence as a resource

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper argues that partially coherent light, captured by a coherence matrix, can outperform coherent light in optical information processing.

desk verdict A clear, useful tutorial whose 'coherence advantage' is a real but narrowly-scoped effect, not the general resource the title implies. read the letter →

arxiv 2608.06356 v1 pith:VGL42G3O submitted 2026-08-06 physics.optics

Ayman F. Abouraddy , Bahaa E. A. Saleh This is my paper
classification physics.optics PACS 42.25.Kb42.25.Hz42.82.-m
keywords opticalcoherencestructuredmatrixadvantagepartialinformationprocessingrankentropyswapping
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This tutorial sets out to recast optical coherence as a resource rather than a deficiency. Its central claim is the 'coherence advantage': in optical information processing there are settings where partially coherent light, described by an $N\times N$ coherence matrix, can outperform fully coherent light, and recent experiments on optical computing, cryptography, and communications already point that way. The paper builds the mathematical toolbox for exploiting that advantage, covering $2\times 2$ matrices for one binary degree of freedom, $4\times 4$ matrices for two binary degrees of freedom, and the concepts of coherence rank, entropy swapping, and optical cross-purity. A reader should care because if the claim is right, information can be encoded and processed in the statistical structure of partially coherent light, which is far richer than the amplitudes of coherent modes.

What carries the argument

The load-bearing object is the $N\times N$ coherence matrix $G = \langle E_j E_k^*\rangle$, a Hermitian, positive-semidefinite, unity-trace matrix obtained by averaging outer products of the random modal amplitudes over the ensemble. Everything else hangs on how $G$ transforms: unitary operations $G \rightarrow U G U^\dagger$ (which are reversible and preserve entropy and degree of coherence), non-unitary filtering and decohering operators (which tune them), the partial trace that produces reduced spatial and polarization coherence matrices from a $4\times 4$ $G$, and the scalar diagnostics — degree of coherence $D = \lambda_1 - \lambda_2$, entropy $S = -\operatorname{Tr}(G \log_2 G)$, coherence rank (the number of nonzero eigenvalues), and optical cross-purity. These objects turn 'how much the field fluctuates' into quantitative, manipulable resources.

What would settle it

Take a rapidly varying, strongly scattering polarization channel. Encode bits as $0 \rightarrow |H\rangle$ (degree of polarization 1) and $1 \rightarrow$ a maximally unpolarized field (degree of polarization 0), and decode by thresholding the degree of polarization at 0.5. If the measured cross-talk matrix is not diagonal, meaning the bit-error rate is no better than encoding 0 and 1 in orthogonal polarized states, then the claimed coherence advantage for this channel is falsified.

Watch

Extended reading notes

Core claim

On the paper's own terms, the claim it is trying to secure is that partial coherence is not noise to be eliminated but a structured resource: in a fixed, orthonormal, closed modal basis, the coherence matrix $G = \langle E_j E_k^*\rangle$ completely characterizes the fluctuating field, and its properties — degree of coherence, von Neumann entropy, coherence rank (the number of nonzero eigenvalues), and cross-purity between degrees of freedom — determine what can and cannot be done with the field. Because a coherent $N$-mode field needs only $2N-2$ real parameters while a partially coherent one needs $N^2-1$, partially coherent fields carry more information per field; because the degree of coherence or rank can be made immune to scattering that destroys polarization states, encoding in these quantities enables communication through channels where coherent encoding fails. The tutorial systematizes the unitary, filtering, and decohering operations that manipulate the coherence matrix, showing how entropy can be swapped between degrees of freedom, concentrated into modes, or spread across them.

Load-bearing premise

The load-bearing premise is that the field is fully captured by a fixed, finite, closed set of orthonormal modes, so that randomness enters only through the complex modal amplitudes and no modes outside the basis ever contribute.

Editorial extensions

If this is right

  • A partially coherent field in $N$ modes occupies $N^2-1$ real parameters, so partial coherence enlarges the information-carrying capacity of an optical field relative to coherent encoding.
  • Encoding bits in the degree of coherence (polarized versus unpolarized) survives a channel that randomly rotates polarization, where encoding bits in orthogonal polarization states fails.
  • Coherence rank becomes a communication resource: the number of nonzero eigenvalues of the coherence matrix can carry information through strongly scattering channels.
  • Entropy can be moved between degrees of freedom, concentrated into a selected mode, or spread out, which means partial coherence can be deliberately redistributed rather than merely suppressed.
  • The $4\times 4$ coherence matrix of two binary degrees of freedom is isomorphic to a two-qubit density matrix, so partial trace, entropy inequalities, and non-separability (classical entanglement) transfer directly to classical optical fields.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the resource view suggests a quantitative definition of 'coherence advantage' as the ratio of achievable information rates under a fixed channel, with the testable prediction that the advantage grows with modal dimension $N$.
  • Editorial inference: optical cross-purity and entropy swapping, developed for two binary degrees of freedom, are natural candidates for on-chip demonstrations with programmable interferometers, which would test the claim on integrated platforms without free-space alignment.
  • Editorial inference: if the coherence advantage holds, the practical bottleneck moves to mode-selective measurement; the paper's modal-projector and modal-analyzer toolbox is precisely that missing hardware, and the next step is extending those devices from coherent to partially coherent inputs.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This tutorial proposes 'structured coherence' as the regime in which an optical field is carried by a fixed, finite, deterministic modal basis and all randomness resides in the complex modal coefficients. It develops the associated coherence-matrix formalism: 2x2 matrices for one binary degree of freedom, Stokes parameters and Pauli decompositions, unitary and non-unitary transformations, degree of coherence and entropy, and explicit measurement strategies. It then extends the formalism to two binary degrees of freedom via 4x4 coherence matrices, partial trace, coherence rank, entropy swapping, optical cross-purity, tomographic reconstruction, and a communications protocol across a strongly scattering channel. The paper's motivating claim is a 'coherence advantage': settings in which partially coherent light, described by coherence matrices, can outperform coherent light in optical information processing.

Significance. If the coherence-advantage claim is sustainable, this matrix-based resource perspective could provide a valuable unifying framework for partial coherence in on-chip photonics, multimode fibers, and information processing. The tutorial's linear-algebra core is standard, internally consistent, and clearly presented; it gives explicit reconstruction procedures (Section IV.F), concrete measurement strategies (Section II.C and Section III.D), and an unusually candid statement of channel assumptions (Section III.I). The mathematical formulation itself is a useful pedagogical contribution. However, the central 'coherence advantage' is not derived in the tutorial; it is supported by idealized channel models and by references to the authors' own experimental work. The paper would be strengthened by a clear separation between the sound tutorial mathematics and the status of the advantage claim.

major comments (3)
  1. [II.A and III.I] The central 'coherence advantage' claim is load-bearing, yet every supporting scenario assumes a closed finite modal basis and, for the communications protocol, a unitary Channel-1. Section II.A defines a closed basis as one in which 'no new modes outside the basis can contribute to the field' and immediately concedes that an infinite basis is typically required and truncation is a practical expedient. Section III.I's assumptions (single-DoF channel, unitarity, no memory) are exactly what makes the maximally incoherent state invariant and what makes the degree of coherence and coherence rank unitarily invariant. If the channel couples the polarization DoF to an unused spatial mode, the reduced 2x2 coherence matrix no longer evolves unitarily: its degree of coherence and rank can change, and the decision threshold in Fig. 33 fails. The tutorial provides no error bound, leakage model, or demonstration of robustness to small violations of the closure assumption. Please add a quantitative robustness discussion or explicitly restrict the coherence-advantage claim to the idealized closed-basis unitary-channel setting.
  2. [I.D] The parameter-counting argument compares the 2N-2 real parameters needed for a coherent N-mode field with the N^2-1 real parameters needed for a partially coherent field and suggests 'richer information-carrying capacity.' Counting parameters of a state space is not by itself a communication-theoretic advantage: a larger state space can also imply greater noise sensitivity and more demanding measurements, and no coding theorem is given. The text itself concedes that no experiment has exploited this parameter-counting opportunity to date. I recommend reframing this point as an open question rather than as evidence for the coherence advantage.
  3. [IV.K and IV.M] Entropy swapping and coherence-rank communications are introduced as key applications and as part of the coherence-advantage narrative, but their derivations are not present in the tutorial; the text defers to earlier work (e.g., Refs. [134, 138-142]). The reader therefore cannot inspect the conditions under which coherence rank, entropy, or the reduced coherence matrices are preserved. At a minimum, the tutorial should state the relevant theorems and their explicit assumptions, and it should identify which steps rely on the closed-basis and unitary-channel idealizations. Without that, the claimed advantage remains an assertion about the authors' own protocols rather than a demonstrated result of the presented formalism.
minor comments (4)
  1. [IV.F Eq. (122)] The (3,3) entry of the reconstructed matrix reads "s00 + s01 - s01 - s11", which is inconsistent with the Kronecker-Pauli expansion; it should be "s00 + s01 - s10 - s11".
  2. [III.I] The acronym for the cross-talk matrix is given as "CTM" in the text and figures, but later appears as "CMT" in the sentence reporting the flat matrix; please unify the usage.
  3. [II.D] The claim that the described modal projectors and analyzers are 'equally applicable to partially coherent light' is asserted without a supporting argument or reference; a short explanation of why the linearity of the modal projectors suffices for coherence matrices would be useful.
  4. [III.C.6] The statement that any two 2x2 coherence matrices with the same entropy can be interconverted by a unitary is correct for the qubit case, but the converse statement for higher-dimensional coherence matrices is only hinted at; a one-sentence caveat would avoid overgeneralization.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the tutorial's derivations are self-contained linear algebra; self-citations are pointers, not load-bearing premises.

full rationale

The paper's central derivation chain starts from explicit definitions: the coherence matrix G = integral of P(xi)|E(xi)><E(xi)| over the ensemble, unitary evolution G -> U G U†, degree of coherence D = lambda1 - lambda2, and coherence rank as the number of nonzero eigenvalues. The claimed 'coherence advantage' examples, including the parameter-counting argument, the Channel-1 communications protocol, and coherence-rank transmission, are mathematical consequences of these definitions together with the explicitly stated channel assumptions (e.g., 2x2 unitary scattering). No parameter is fitted to data and then presented as a prediction; the experimental confirmation in Fig. 34 implements the same unitary-channel model and verifies the derived invariance of the degree of polarization. Self-citations such as [134] and [138-142] point to the authors' prior demonstrations, but the necessary mathematics (unitary invariants, partial trace, Stokes reconstruction) is re-derived in the text, so the central claim does not reduce to those citations. The closed-modal-basis and unitarity assumptions are stated idealizations; their failure in realistic systems is a scope and robustness concern, not a circularity in the derivation. No step was found in which an output quantity is identical by construction to an input quantity or in which a fitted parameter is renamed a prediction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces definitions such as structured coherence, coherence rank, entropy swapping, and optical cross-purity, but these are mathematical descriptors, not new physical entities. No free parameters are fitted to data; the numerical choices in examples are illustrative. The axioms listed are the load-bearing background assumptions of the matrix formulation.

assumptions (5)
  • domain assumption The optical field is represented in a finite, closed, orthonormal modal basis; no modes outside the basis contribute.
    Section II.A defines modes and the modal basis as stable, fixed, deterministic, and closed, and the rest of the tutorial assumes this completeness.
  • domain assumption All statistical information about the partially coherent field is captured by ensemble-averaged second-order correlations of modal coefficients.
    Section III.C.1 defines G = integral dξ P(ξ)|E(ξ)><E(ξ)| for a binary DoF and generalizes to 4x4; this is standard coherence theory, not derived in the paper.
  • domain assumption The field is paraxial, so polarization is a two-dimensional transverse degree of freedom.
    Section II.B.1 invokes the paraxial regime to reduce polarization to the transverse plane with modes |H> and |V>.
  • standard math Hermitian matrices admit spectral decomposition, Pauli matrices span 2x2 Hermitian matrices, and the partial trace is trace-preserving.
    These standard linear-algebra facts are used throughout Sections III and IV without proof.
  • domain assumption Relevant optical transformations are unitary, filtering, or decohering maps on the coherence matrix.
    Section III.F defines filtering and decohering operators, and the resource interpretation assumes these transformations are experimentally realizable.

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Pith. "Pith review of Structured coherence: A modern perspective on optical coherence as a resource." pith.science (2026). https://pith.science/paper/VGL42G3O

@misc{pith2026260806356,
  author       = {Pith},
  title        = {Pith review of: Structured coherence: A modern perspective on optical coherence as a resource},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VGL42G3O}},
  note         = {Machine review of arXiv:2608.06356}
}
read the original abstract

Optical coherence is a well-established branch of physical optics in which the statistical properties of fluctuating optical fields are described in terms of correlation functions over continuous spatial and temporal degrees of freedom (DoFs). Nevertheless, in any practical setting, only discrete DoFs are ever accessible. In these scenarios, the modes are fixed, stable, and deterministic, and partial coherence arises from random relative complex amplitudes, a configuration we refer to as 'structured coherence'. Advances in structured coherence have recently unveiled new conceptual ground in optical communications and information processing in which partial coherence may be preferable to full coherence, which we call a 'coherence advantage'. We formulate structured coherence in terms of coherence matrices to investigate these recent theoretical and experimental breakthroughs. We first review optical fields characterized by a binary DoF via 2x2 Hermitian coherence matrices and introduce key concepts that take on new significance for larger-dimensional modal sets. Next, we examine the structured coherence of two binary DoFs, which can be described by 4x4 coherence matrices. We introduce the concept of coherence rank, entropy swapping, and optical cross-purity. In the perspective outlined here, coherence is viewed as a 'resource', which can be exchanged between DoFs, concentrated into a DoF or into particular modes, or spread over the DoFs. We then examine larger-dimensional modal sets, which allow for more versatile applications in optical information processing. The formulation presented here lends itself particularly to the manipulation of partial optical coherence in integrated photonic platforms, thereby opening myriad avenues for novel fundamental investigations of structured coherence and exploiting the coherence advantage in optical communications and information processing.

Figures

Figures reproduced from arXiv: 2608.06356 by the authors.

Figure 1
Figure 1. FIG. 1. (a) In conventional optical coherence, a continuous correlation function [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Modal bases for the spatial DoF with [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Modal bases for the spatial DoF with spatially [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (61 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Projections onto a modal bases for non-overlapping spatial DoFs. (a) A two-point field. (b) A 1D array of waveguides. [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a) A parity projector formed of a balanced MZI in which a spatial flip (here a dove prism) is inserted in one arm. [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (a) A modal projector separates spatially overlapping modes into spatially non-overlapping modes, so that a detector [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Modal projector in the OAM basis via a log-polar coordinate transformation. The device comprises two phase plates [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. (a) Conceptual scheme for an MPLC formed of a sequence of phase plates with phase distributions [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. (a) Conventional optical delay [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Concept of a Hilbert-space modal analyzer. The field is incident from the left, and two copies are produced at the [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. The field vector [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. (a) A unitary transformation for a binary DoF with a modal basis [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. One of the modes, [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. (a) The unitary [PITH_FULL_IMAGE:figures/full_fig_p018_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. (a) Conventional double-slit interference. (b) The field in two paths interfere after introducing a relative phase [PITH_FULL_IMAGE:figures/full_fig_p020_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. The mode [PITH_FULL_IMAGE:figures/full_fig_p022_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. (a) General two-mode interference, whereby the input field traverses a unitary [PITH_FULL_IMAGE:figures/full_fig_p023_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. (a) Plot of the entropy [PITH_FULL_IMAGE:figures/full_fig_p024_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19. (a) Procedure I, diagonalization. A unitary [PITH_FULL_IMAGE:figures/full_fig_p025_19.png]
Figure 20
Figure 20. Figure 20: FIG. 20. Representation of a coherent field with a binary DoF as a point on the surface of the Poincaré sphere (PS). (a) [PITH_FULL_IMAGE:figures/full_fig_p026_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21. Representation of a field with a binary DoF as a point on the PS surface in terms of the Stokes parameters. The [PITH_FULL_IMAGE:figures/full_fig_p027_21.png]
Figure 22
Figure 22. Figure 22: FIG. 22. Configurations for measuring the Stokes parameters: (a) [PITH_FULL_IMAGE:figures/full_fig_p028_22.png]
Figure 23
Figure 23. Figure 23: FIG. 23. (a) The tunability of the degree of coherence [PITH_FULL_IMAGE:figures/full_fig_p030_23.png]
Figure 24
Figure 24. Figure 24: FIG. 24. (a) The polarization DoF is spanned by the modes [PITH_FULL_IMAGE:figures/full_fig_p032_24.png]
Figure 25
Figure 25. Figure 25: FIG. 25. (a) Determining the degree of polarization [PITH_FULL_IMAGE:figures/full_fig_p034_25.png]
Figure 26
Figure 26. Figure 26: FIG. 26. Polarization Stokes parameters. (a) Measuring [PITH_FULL_IMAGE:figures/full_fig_p034_26.png]
Figure 27
Figure 27. Figure 27: FIG. 27. (a) Traditional double-slit interference. The fields from [PITH_FULL_IMAGE:figures/full_fig_p035_27.png]
Figure 28
Figure 28. Figure 28: FIG. 28. (a) A phase operator [PITH_FULL_IMAGE:figures/full_fig_p036_28.png]
Figure 29
Figure 29. Figure 29: FIG. 29. Optical devices operating on the spatial DoF obtained from [PITH_FULL_IMAGE:figures/full_fig_p037_29.png]
Figure 30
Figure 30. Figure 30: FIG. 30. (a) Diagonalizing [PITH_FULL_IMAGE:figures/full_fig_p038_30.png]
Figure 31
Figure 31. Figure 31: FIG. 31. Reconstruction of the spatial coherence matrix [PITH_FULL_IMAGE:figures/full_fig_p039_31.png]
Figure 32
Figure 32. Figure 32: FIG. 32. Reconstruction of the spatial coherence matrix [PITH_FULL_IMAGE:figures/full_fig_p039_32.png]
Figure 33
Figure 33. Figure 33: FIG. 33. Optical communications across a scattering channel (from Ref. [134]). (a) Encoding scheme-1 ( [PITH_FULL_IMAGE:figures/full_fig_p041_33.png]
Figure 34
Figure 34. Figure 34: FIG. 34. Realization of optical communications through a polarization scattering channel (channel-1). (a) Schematic of the [PITH_FULL_IMAGE:figures/full_fig_p042_34.png]
Figure 35
Figure 35. Figure 35: FIG. 35. Correspondence between a qubit in quantum mechanics and a classical optical field with a binary DoF. [PITH_FULL_IMAGE:figures/full_fig_p043_35.png]
Figure 36
Figure 36. Figure 36: FIG. 36. The direct-product formulation of the composite or joint modal basis for a field characterized by two binary DoFs: a [PITH_FULL_IMAGE:figures/full_fig_p044_36.png]
Figure 37
Figure 37. Figure 37: FIG. 37. (a) Measurements of the modal weights via four detectors. The polarization components at [PITH_FULL_IMAGE:figures/full_fig_p045_37.png]
Figure 38
Figure 38. Figure 38: FIG. 38. Interpretation of the off-diagonal elements of the [PITH_FULL_IMAGE:figures/full_fig_p047_38.png]
Figure 39
Figure 39. Figure 39: FIG. 39. Interpretation of the off-diagonal terms of the coherence matrix in terms of appropriately designed interference [PITH_FULL_IMAGE:figures/full_fig_p048_39.png]
Figure 40
Figure 40. Figure 40: FIG. 40. (a) A purely polarization unitary [PITH_FULL_IMAGE:figures/full_fig_p049_40.png]
Figure 41
Figure 41. Figure 41: FIG. 41. (a) A spatially dependent polarization unitary. Polarization unitary [PITH_FULL_IMAGE:figures/full_fig_p051_41.png]
Figure 42
Figure 42. Figure 42: the most general form of this construction, where we have made use of the decomposition of 4 × 4 unitaries to be explained in Section V A. It is sufficient here to state that a sequence of 6 2 × 2 unitaries (operating on a single DoF at a time) are sufficient to accom…
Figure 43
Figure 43. Figure 43: FIG. 43. (a,b) Reduced and (c,d) restricted coherence matrices. (a) Reduced [PITH_FULL_IMAGE:figures/full_fig_p053_43.png]
Figure 44
Figure 44. Figure 44: FIG. 44. Definition of Kronecker-Pauli matrices. (a) The [PITH_FULL_IMAGE:figures/full_fig_p058_44.png]
Figure 45
Figure 45. Figure 45: FIG. 45. (a) The modal Stokes parameters [PITH_FULL_IMAGE:figures/full_fig_p059_45.png]
Figure 46
Figure 46. Figure 46: FIG. 46. Measurements for acquiring the modal Stokes parameters. (a) Measuring the 4 spatial Stokes parameters [PITH_FULL_IMAGE:figures/full_fig_p060_46.png]
Figure 47
Figure 47. Figure 47: FIG. 47. (a) A spatially incoherent field at [PITH_FULL_IMAGE:figures/full_fig_p062_47.png]
Figure 48
Figure 48. Figure 48: FIG. 48. Concept of coherence conversion. (a) Starting with a polarized (scalar) field that is spatially incoherent (the field is [PITH_FULL_IMAGE:figures/full_fig_p063_48.png]
Figure 49
Figure 49. Figure 49: FIG. 49. Reversible coherence conversion or entropy swapping between two DoFs of a rank-2 field. The maximum-entropy [PITH_FULL_IMAGE:figures/full_fig_p066_49.png]
Figure 50
Figure 50. Figure 50: FIG. 50. Geometric representation of the coherence rank of a [PITH_FULL_IMAGE:figures/full_fig_p067_50.png]
Figure 51
Figure 51. Figure 51: FIG. 51. (a) Geometric representation of the coherence rank of a [PITH_FULL_IMAGE:figures/full_fig_p068_51.png]
Figure 52
Figure 52. Figure 52: FIG. 52. Iso-entropy surfaces. (a) [PITH_FULL_IMAGE:figures/full_fig_p069_52.png]
Figure 53
Figure 53. Figure 53: FIG. 53. Conceptual scheme for entropy swapping between the spatial and polarization DoFs. [PITH_FULL_IMAGE:figures/full_fig_p070_53.png]
Figure 54
Figure 54. Figure 54: FIG. 54. The fields [PITH_FULL_IMAGE:figures/full_fig_p071_54.png]
Figure 55
Figure 55. Figure 55: FIG. 55. (a) A vector field at [PITH_FULL_IMAGE:figures/full_fig_p072_55.png]
Figure 56
Figure 56. Figure 56: FIG. 56. (a) The rank-2 coherence matrix [PITH_FULL_IMAGE:figures/full_fig_p073_56.png]
Figure 57
Figure 57. Figure 57: FIG. 57. (a) Representation of [PITH_FULL_IMAGE:figures/full_fig_p075_57.png]
Figure 58
Figure 58. Figure 58: FIG. 58. (a) Schematic of the setup for channel-2. (b) Portion of the data stream corresponding to the image on the right. (c,d) [PITH_FULL_IMAGE:figures/full_fig_p076_58.png]
Figure 59
Figure 59. Figure 59: FIG. 59. Correspondence between quantum mechanics and classical optics with respect to two-qubit systems in the former [PITH_FULL_IMAGE:figures/full_fig_p078_59.png]
Figure 60
Figure 60. Figure 60: FIG. 60. (a) Measurement of the modal weights for an optical field comprising an [PITH_FULL_IMAGE:figures/full_fig_p079_60.png]
Figure 61
Figure 61. Figure 61: FIG. 61. (a) An arbitrary [PITH_FULL_IMAGE:figures/full_fig_p081_61.png]
Figure 62
Figure 62. Figure 62: FIG. 62. Modal bases for the temporal DoF. (a) Binary time-bins. (b) [PITH_FULL_IMAGE:figures/full_fig_p081_62.png]
Figure 63
Figure 63. Figure 63: FIG. 63. Modal bases for the spectral DoF. (a) Spectral bins and (b) laser frequency combs. [PITH_FULL_IMAGE:figures/full_fig_p082_63.png]
Figure 64
Figure 64. Figure 64: FIG. 64. Vision for exploiting structured coherence in optical communications and information processing. Generic multimoded [PITH_FULL_IMAGE:figures/full_fig_p083_64.png]

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Works this paper leans on

290 extracted references · 69 canonical work pages

  1. [1]

    Mandel in 1961 using the conventional description of optical coherence in terms of continuous functions in space and time

    Basic definition Spectral cross-purity is a phenomenon first described by L. Mandel in 1961 using the conventional description of optical coherence in terms of continuous functions in space and time. Consider superposing the spectra from points |a⟩ and |b⟩, Sa(ω) and Sb(ω), respectively, in a scalar, partially coherent field. If the spectra Sa(ω) and Sb(ω...

    1961

  2. [2]

    Symmetry: the normalized spectra at |a⟩ and |b⟩ are identical

  3. [3]

    Separability: The coherence function at these two points is independent of the spatial coordinate; i.e., the coherence function is separable with respect to the spatial and spectral DoFs at these two points. The fundamental concept of cross-spectral purity has been recently generalized in two aspects: (1) it can be applied to any pair of DoFs; and (2) it ...

  4. [4]

    Polarization cross-purity We apply generalized optical cross-purity to a scenario comprising the spatial and polarization DoFs. We thus consider the following question: in a partially coherent vector field described by a 4 × 4 coherence matrix G, in which the polarizations at |a⟩ and |b⟩ are identical, would superposing the fields from |a⟩ and |b⟩ yield t...

  5. [5]

    55(a)], we instead implement a unitary operator ˆUs that impacts the spatial DoF alone and is independent of polarization DoF [Fig

    Conditions for polarization cross-purity Rather than superposing the fields from |a⟩ and |b⟩ in the double-slit experiment [Fig. 55(a)], we instead implement a unitary operator ˆUs that impacts the spatial DoF alone and is independent of polarization DoF [Fig. 55(b)]. We start with the block-matrix form for the coherence matrix G = |α|2Ga Gab Gba |β|2Ga ,...

  6. [6]

    Optical cross-purity and the coherence rank Evaluating cross-purity requires first symmetrizing the field so that Ga = Gb. Once the field is symmetrized, is it guaranteed to be separable? Does symmetry ( Ga = Gb) imply separability (and thus polarization cross-purity)? We have recently shown that the coherence rank [134, 141, 142] is crucial in this regar...

  7. [7]

    Consequently, even if G3 = ˆU GD 3 ˆU † is symmetrized ( Ga = Gb), G3 remains non-separable

    Rank-3 fields are intrinsically non-separable [142]; no unitary transformation can undo this non-separability. Consequently, even if G3 = ˆU GD 3 ˆU † is symmetrized ( Ga = Gb), G3 remains non-separable. That is, all rank-3 fields are polarization cross-impure ; in this case, symmetry does not imply separability. 73 We consider an example to clarify the d...

  8. [8]

    The channel impacts both the polarization and spatial DoFs but not any further DoFs

Show all 290 references
  1. [9]

    The channel can be represented for any bit during data transmission by a 4 × 4 unitary ˆU that encompasses both the polarization and spatial DoFs

  2. [10]

    Rapidly varying channel: ˆU changes from bit to bit

  3. [11]

    Strong scattering:family of 4 × 4 unitaries over both polarization and spatial DoFs

  4. [12]

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