Geometric phase of arbitrary Mueller evolutions and its two-level quantum analogue

Jos\'e J Gil

arxiv: 2602.14245 · v2 · submitted 2026-02-15 · 🪐 quant-ph

Geometric phase of arbitrary Mueller evolutions and its two-level quantum analogue

Jos\'e J Gil This is my paper

Pith reviewed 2026-05-15 21:35 UTC · model grok-4.3

classification 🪐 quant-ph

keywords Mueller matrixgeometric phasecharacteristic decompositionholonomic contentPancharatnam phasepolarization opticsChoi representationtwo-level quantum dynamics

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The pith

Any physically realizable Mueller matrix carries a single invariant geometric phase fixed by the retarding part of its characteristic pure component.

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

The paper establishes that Mueller transformations, which describe how polarized light changes under linear optical systems, possess one intrinsic geometric phase that remains the same no matter how the transformation is physically built or measured. This phase is isolated through the characteristic decomposition, which extracts a pure component, and specifically its retarding (phase-shifting) portion defines the canonical holonomic content. Observed phases in actual interferometers can vary because other components in the decomposition affect visibility and can average the apparent phase, but they add no unique holonomy of their own. The same structure maps directly to the Choi representation of open two-level quantum dynamics, giving a quantum analogue. If correct, this supplies a well-defined geometric invariant for arbitrary Mueller evolutions where none was previously available.

Core claim

For a general physically realizable Mueller transformation, the only intrinsic geometric phase structure that can be assigned to it in an invariant manner is the retarding part of the characteristic pure component selected by the characteristic decomposition, which defines a canonical holonomic content. A Mueller matrix does not determine a unique observed interferometric (Pancharatnam) geometric phase, since the latter depends on the specific physical realization of the transformation and on the interferometric readout. The remaining characteristic layers may modify the measured complex visibility, and even its observed argument through convex averaging, but they do not define a unique holy

What carries the argument

The characteristic decomposition, which extracts from any Mueller matrix a unique pure component whose retarding (phase) part alone supplies the invariant geometric phase.

If this is right

Every Mueller matrix acquires a canonical geometric phase independent of its physical implementation.
Interferometric measurements can yield phases that differ from the invariant value due to averaging over the non-pure layers.
The same invariant applies to the Choi operator of any open two-level quantum channel.
Polarization optics gains a well-defined holonomic quantity for systems previously treated as having only setup-dependent phases.

Where Pith is reading between the lines

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

This invariant could serve as a conserved quantity when composing sequences of Mueller matrices, allowing phase tracking without reference to specific realizations.
In quantum information, the retarding part might label equivalence classes of channels that share the same intrinsic holonomy.
Experimental tests could compare the predicted invariant phase against visibility-weighted averages in multi-path interferometers.

Load-bearing premise

The characteristic decomposition is assumed to uniquely isolate a pure component whose retarding part carries the sole invariant geometric phase for every physically realizable Mueller matrix.

What would settle it

Construct a Mueller matrix whose characteristic pure component has a known retarding phase, then realize it in two physically distinct ways that produce different observed Pancharatnam phases not equal to that retarding value.

read the original abstract

We identify, for a general physically realizable Mueller transformation, the only intrinsic geometricphase structure that can be assigned to it in an invariant manner: the retarding part of the characteristic pure component selected by the characteristic decomposition, which defines a canonical holonomic content. A Mueller matrix does not, in general, determine a unique observed interferometric (Pancharatnam) geometric phase, since the latter depends on the specific physical realization of the transformation and on the interferometric readout. The remaining characteristic layers may modify the measured complex visibility, and even its observed argument through convex averaging, but they do not define a unique geometric holonomy of their own. We further establish the quantum analogue for open two-level dynamics within the Choi representation.

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.

Desk Editor's Note private letter to a colleague

The paper isolates the retarding part of the characteristic pure component as the invariant geometric phase for Mueller matrices and sketches its two-level quantum version, but the uniqueness of that decomposition is not obviously guaranteed for all physical cases.

read the letter

The core contribution is the claim that only the retarding sub-matrix of the pure component picked out by the characteristic decomposition carries an intrinsic, realization-independent geometric phase for any physically realizable Mueller matrix. Everything else in the decomposition can affect measured visibilities through averaging but does not define its own holonomy. The two-level quantum analogue is presented via the Choi representation, which at least makes the mapping concrete rather than hand-wavy. That distinction between invariant structure and readout-dependent phase is the part that could be useful for people working on polarization devices or open-system quantum optics. The abstract is clear on this point and the authors cite the standard decomposition literature without obvious circularity. The main soft spot is the stress-test concern: characteristic decomposition is not guaranteed to yield a unique pure component when the Mueller matrix sits on the boundary of the physical domain or has zero eigenvalues. If multiple convex decompositions exist, the extracted retarding part becomes decomposition-dependent and the asserted invariance weakens. The paper would need to show explicitly that the retarding phase is the same across all admissible decompositions or restrict the claim to the interior of the domain. Without that, the central invariance result rests on an assumption that may not hold universally. This is a targeted optics-plus-quantum-information note rather than a broad advance, so it is worth a referee's time for the subfield but not urgent for a general journal. I would bring it to a reading group for the decomposition discussion and the Choi link, though I would not cite it in my own work unless the uniqueness issue is closed. Send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript identifies the retarding part of the characteristic pure component (selected by the characteristic decomposition of a physically realizable Mueller matrix) as the only intrinsic, invariant geometric-phase structure assignable to the transformation. It contrasts this canonical holonomic content with the non-unique, realization-dependent Pancharatnam phase observed in interferometry, noting that the remaining layers affect visibility but do not define unique holonomy. A quantum analogue is derived for open two-level dynamics in the Choi representation.

Significance. If the uniqueness and invariance claims hold, the result supplies a canonical, decomposition-based definition of geometric phase for the full class of Mueller transformations, bridging classical polarization optics with quantum geometric phases. This could enable invariant characterizations in quantum information and optical metrology independent of specific realizations or readouts.

major comments (2)

[Section defining the characteristic decomposition and its uniqueness] The central claim that the characteristic decomposition canonically isolates a unique pure component (whose retarding sub-matrix supplies the invariant geometric phase) is not demonstrated for all physically realizable Mueller matrices. Matrices on the boundary of the physical domain (singular or rank-deficient covariance) admit multiple convex decompositions into pure and depolarizing parts, rendering the extracted retarding part decomposition-dependent and therefore not intrinsically fixed by the Mueller matrix alone.
[Section establishing invariance of the holonomic content] The invariance argument relies on the retarding part being the sole holonomic carrier, yet the manuscript does not provide an explicit proof that alternative decompositions (when they exist) cannot yield a different retarding phase while still reproducing the same Mueller matrix. This is load-bearing for the assertion that the structure is 'the only intrinsic' one.

minor comments (2)

[Notation section] Notation for the retarding sub-matrix and its extraction from the pure component should be introduced with an explicit equation reference at first use to improve readability.
[Quantum analogue section] The quantum analogue section would benefit from a brief comparison table contrasting the Mueller and Choi representations for the geometric phase extraction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below. We agree that the uniqueness and invariance properties require more explicit demonstration for boundary cases and will revise the manuscript to include the necessary proofs and clarifications.

read point-by-point responses

Referee: [Section defining the characteristic decomposition and its uniqueness] The central claim that the characteristic decomposition canonically isolates a unique pure component (whose retarding sub-matrix supplies the invariant geometric phase) is not demonstrated for all physically realizable Mueller matrices. Matrices on the boundary of the physical domain (singular or rank-deficient covariance) admit multiple convex decompositions into pure and depolarizing parts, rendering the extracted retarding part decomposition-dependent and therefore not intrinsically fixed by the Mueller matrix alone.

Authors: We acknowledge the referee's point that uniqueness must be shown explicitly, including for singular or rank-deficient cases. The characteristic decomposition is canonically defined by selecting the pure component associated with the dominant eigenvalue of the covariance matrix constructed from the Mueller matrix; this selection is unique by construction whenever the dominant eigenvalue is non-degenerate, which holds throughout the physical domain. To fully address the concern, we will add a dedicated paragraph (or short subsection) in the revised manuscript that proves the extracted pure component—and therefore its retarding sub-matrix—remains fixed for every physically realizable Mueller matrix, even when alternative convex decompositions exist. revision: yes
Referee: [Section establishing invariance of the holonomic content] The invariance argument relies on the retarding part being the sole holonomic carrier, yet the manuscript does not provide an explicit proof that alternative decompositions (when they exist) cannot yield a different retarding phase while still reproducing the same Mueller matrix. This is load-bearing for the assertion that the structure is 'the only intrinsic' one.

Authors: We agree that an explicit invariance proof is needed to confirm that no alternative decomposition can produce a different retarding phase for the identical Mueller matrix. In the revised version we will insert a concise proof showing that the retarding sub-matrix of the characteristic pure component is invariant: any convex combination of other pure Mueller matrices that reproduces the original transformation cannot alter the holonomic content carried by the characteristic retarding part without violating the defining eigenvalue selection of the decomposition. This addition will directly support the claim that the structure is the only intrinsic geometric-phase feature. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's derivation identifies the retarding part of the characteristic pure component (via the characteristic decomposition) as the invariant geometric phase for physically realizable Mueller matrices, then extends the construction to the Choi representation for two-level open quantum dynamics. This identification is presented as a consequence of the decomposition's properties rather than a self-referential definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain. No equations or steps in the provided text reduce the claimed holonomic content to the input Mueller matrix by construction, and the uniqueness assertion is not shown to collapse into an ansatz or prior author result without independent content. The overall chain remains self-contained against external benchmarks in polarimetry.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard characteristic decomposition of Mueller matrices being well-defined and unique for physically realizable cases, with no new free parameters or invented entities introduced in the abstract.

axioms (1)

domain assumption The characteristic decomposition uniquely selects the pure component whose retarding part defines the invariant geometric phase for any physically realizable Mueller matrix.
This is invoked directly in the identification of the canonical holonomic content.

pith-pipeline@v0.9.0 · 5411 in / 1295 out tokens · 20399 ms · 2026-05-15T21:35:15.621810+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArrowOfTime.lean forward_accumulates / arrow_from_z echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

the retarding part of the characteristic pure component selected by the characteristic decomposition, which defines a canonical holonomic content
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

characteristic decomposition of ˆM = P1 ˆMJ + (1-P1) ˆMnp

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

[1]

Antisymmetric Mueller generator as the uni- versal origin of geometric phase in classical polarization and quantum two-level systems,

J. J. Gil, “Antisymmetric Mueller generator as the uni- versal origin of geometric phase in classical polarization and quantum two-level systems,”J. Opt. Soc. Am. A43, 507–515 (2026)

work page 2026
[2]

Generalized theory of interference, and its applications. part I. coherent pencils,

S. Pancharatnam, “Generalized theory of interference, and its applications. part I. coherent pencils,”Proc. Indian Acad. Sci. A44, 247–262 (1956)

work page 1956
[3]

Quantal phase factors accompanying adia- batic changes,

M. V. Berry, “Quantal phase factors accompanying adia- batic changes,”Proc. R. Soc. Lond. A392, 45–57 (1984)

work page 1984
[4]

General setting for Berry’s phase,

J. Samuel and R. Bhandari, “General setting for Berry’s phase,”Phys. Rev. Lett.60, 2339–2342 (1988)

work page 1988
[5]

Geometric phases for mixed states in interferometry,

E. Sjöqvist, A. K. Pati, A. Ekert,et al., “Geometric phases for mixed states in interferometry,”Phys. Rev. Lett.85, 2845–2849 (2000)

work page 2000
[6]

Polarimetric characterization of light and media – physical quantities involved in polarimetric phenomena,

J. J. Gil, “Polarimetric characterization of light and media – physical quantities involved in polarimetric phenomena,” Eur. Phys. J. Appl. Phys.40, 1–47 (2007)

work page 2007
[7]

Structure of polarimetric purity of a Mueller matrix and sources of depolarization,

J. J. Gil, “Structure of polarimetric purity of a Mueller matrix and sources of depolarization,”Opt. Commun. 368, 165–173 (2016)

work page 2016
[8]

Group theory and polarisation algebra,

S. R. Cloude, “Group theory and polarisation algebra,” Optik75, 26–36 (1986)

work page 1986
[9]

Modelo matricial para el estudio de fenó- menos de polarización de la luz,

P. M. Arnal, “Modelo matricial para el estudio de fenó- menos de polarización de la luz,” Ph.D. thesis, Universi- dad de Zaragoza, Zaragoza, Spain (1990)

work page 1990
[10]

Invariant indices of polarimetric purity. generalized indices of purity forn×n covariance matrices,

I. San José and J. J. Gil, “Invariant indices of polarimetric purity. generalized indices of purity forn×n covariance matrices,”Opt. Commun.284, 38–47 (2011)

work page 2011
[11]

On optimal filtering of measured Mueller ma- trices,

J. J. Gil, “On optimal filtering of measured Mueller ma- trices,”Appl. Opt.55, 5449–5455 (2016)

work page 2016
[12]

Eigenvalue-based depolar- ization metric spaces for Mueller matrices,

R. Ossikovski and J. Vizet, “Eigenvalue-based depolar- ization metric spaces for Mueller matrices,”J. Opt. Soc. Am. A36, 1173–1186 (2019)

work page 2019
[13]

Singular Mueller matrices,

J. J. Gil, R. Ossikovski, and I. San José, “Singular Mueller matrices,”J. Opt. Soc. Am. A33, 600–609 (2016)

work page 2016
[14]

Obtainment of the polarizing and retardation parameters of a non-depolarizing opti- cal system from the polar decomposition of its Mueller matrix,

J. J. Gil and E. Bernabéu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing opti- cal system from the polar decomposition of its Mueller matrix,”Optik76, 67–71 (1987)

work page 1987
[15]

Interpretation of Mueller 5 matrices based on polar decomposition,

S.-Y. Lu and R. A. Chipman, “Interpretation of Mueller 5 matrices based on polar decomposition,”J. Opt. Soc. Am. A13, 1106–1113 (1996)

work page 1996
[16]

Sources of asymmetry and the concept of non- regularity of n-dimensional density matrices,

J. J. Gil, “Sources of asymmetry and the concept of non- regularity of n-dimensional density matrices,”Symmetry 12, 1002 (2020)

work page 2020
[17]

Linear transformations which preserve trace and positive semidefiniteness of operators,

A. Jamiołkowski, “Linear transformations which preserve trace and positive semidefiniteness of operators,”Rep. Math. Phys.3, 275–278 (1972)

work page 1972
[18]

Completely positive linear maps on complex matrices,

M.-D. Choi, “Completely positive linear maps on complex matrices,”Linear Algebra Appl.10, 285–290 (1975)

work page 1975
[19]

Analysis of depolarizing Mueller matrices through a symmetric decomposition,

R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,”J. Opt. Soc. Am. A26, 1109–1118 (2009)

work page 2009
[20]

Characterization of retardance of nondepolarizing and depolarizing media,

J. J. Gil, I. San José, and R. Ossikovski, “Characterization of retardance of nondepolarizing and depolarizing media,” J. Opt. Soc. Am. A41, 1544–1553 (2024)

work page 2024

[1] [1]

Antisymmetric Mueller generator as the uni- versal origin of geometric phase in classical polarization and quantum two-level systems,

J. J. Gil, “Antisymmetric Mueller generator as the uni- versal origin of geometric phase in classical polarization and quantum two-level systems,”J. Opt. Soc. Am. A43, 507–515 (2026)

work page 2026

[2] [2]

Generalized theory of interference, and its applications. part I. coherent pencils,

S. Pancharatnam, “Generalized theory of interference, and its applications. part I. coherent pencils,”Proc. Indian Acad. Sci. A44, 247–262 (1956)

work page 1956

[3] [3]

Quantal phase factors accompanying adia- batic changes,

M. V. Berry, “Quantal phase factors accompanying adia- batic changes,”Proc. R. Soc. Lond. A392, 45–57 (1984)

work page 1984

[4] [4]

General setting for Berry’s phase,

J. Samuel and R. Bhandari, “General setting for Berry’s phase,”Phys. Rev. Lett.60, 2339–2342 (1988)

work page 1988

[5] [5]

Geometric phases for mixed states in interferometry,

E. Sjöqvist, A. K. Pati, A. Ekert,et al., “Geometric phases for mixed states in interferometry,”Phys. Rev. Lett.85, 2845–2849 (2000)

work page 2000

[6] [6]

Polarimetric characterization of light and media – physical quantities involved in polarimetric phenomena,

J. J. Gil, “Polarimetric characterization of light and media – physical quantities involved in polarimetric phenomena,” Eur. Phys. J. Appl. Phys.40, 1–47 (2007)

work page 2007

[7] [7]

Structure of polarimetric purity of a Mueller matrix and sources of depolarization,

J. J. Gil, “Structure of polarimetric purity of a Mueller matrix and sources of depolarization,”Opt. Commun. 368, 165–173 (2016)

work page 2016

[8] [8]

Group theory and polarisation algebra,

S. R. Cloude, “Group theory and polarisation algebra,” Optik75, 26–36 (1986)

work page 1986

[9] [9]

Modelo matricial para el estudio de fenó- menos de polarización de la luz,

P. M. Arnal, “Modelo matricial para el estudio de fenó- menos de polarización de la luz,” Ph.D. thesis, Universi- dad de Zaragoza, Zaragoza, Spain (1990)

work page 1990

[10] [10]

Invariant indices of polarimetric purity. generalized indices of purity forn×n covariance matrices,

I. San José and J. J. Gil, “Invariant indices of polarimetric purity. generalized indices of purity forn×n covariance matrices,”Opt. Commun.284, 38–47 (2011)

work page 2011

[11] [11]

On optimal filtering of measured Mueller ma- trices,

J. J. Gil, “On optimal filtering of measured Mueller ma- trices,”Appl. Opt.55, 5449–5455 (2016)

work page 2016

[12] [12]

Eigenvalue-based depolar- ization metric spaces for Mueller matrices,

R. Ossikovski and J. Vizet, “Eigenvalue-based depolar- ization metric spaces for Mueller matrices,”J. Opt. Soc. Am. A36, 1173–1186 (2019)

work page 2019

[13] [13]

Singular Mueller matrices,

J. J. Gil, R. Ossikovski, and I. San José, “Singular Mueller matrices,”J. Opt. Soc. Am. A33, 600–609 (2016)

work page 2016

[14] [14]

Obtainment of the polarizing and retardation parameters of a non-depolarizing opti- cal system from the polar decomposition of its Mueller matrix,

J. J. Gil and E. Bernabéu, “Obtainment of the polarizing and retardation parameters of a non-depolarizing opti- cal system from the polar decomposition of its Mueller matrix,”Optik76, 67–71 (1987)

work page 1987

[15] [15]

Interpretation of Mueller 5 matrices based on polar decomposition,

S.-Y. Lu and R. A. Chipman, “Interpretation of Mueller 5 matrices based on polar decomposition,”J. Opt. Soc. Am. A13, 1106–1113 (1996)

work page 1996

[16] [16]

Sources of asymmetry and the concept of non- regularity of n-dimensional density matrices,

J. J. Gil, “Sources of asymmetry and the concept of non- regularity of n-dimensional density matrices,”Symmetry 12, 1002 (2020)

work page 2020

[17] [17]

Linear transformations which preserve trace and positive semidefiniteness of operators,

A. Jamiołkowski, “Linear transformations which preserve trace and positive semidefiniteness of operators,”Rep. Math. Phys.3, 275–278 (1972)

work page 1972

[18] [18]

Completely positive linear maps on complex matrices,

M.-D. Choi, “Completely positive linear maps on complex matrices,”Linear Algebra Appl.10, 285–290 (1975)

work page 1975

[19] [19]

Analysis of depolarizing Mueller matrices through a symmetric decomposition,

R. Ossikovski, “Analysis of depolarizing Mueller matrices through a symmetric decomposition,”J. Opt. Soc. Am. A26, 1109–1118 (2009)

work page 2009

[20] [20]

Characterization of retardance of nondepolarizing and depolarizing media,

J. J. Gil, I. San José, and R. Ossikovski, “Characterization of retardance of nondepolarizing and depolarizing media,” J. Opt. Soc. Am. A41, 1544–1553 (2024)

work page 2024