Quantifying Irreversibility via Bayesian Subjectivity for Classical & Quantum Linear Maps

Clive Cenxin Aw; Lizhuo Liu

arxiv: 2503.12112 · v2 · submitted 2025-03-15 · 🪐 quant-ph

Quantifying Irreversibility via Bayesian Subjectivity for Classical & Quantum Linear Maps

Lizhuo Liu , Clive Cenxin Aw This is my paper

Pith reviewed 2026-05-23 00:34 UTC · model grok-4.3

classification 🪐 quant-ph

keywords irreversibilityBayesian retrodictionprior sensitivityquantum mapsclassical mapslinear mapsinformation losssubjectivity

0 comments

The pith

Irreversibility of classical and quantum linear maps is quantified by how much retrodiction depends on the choice of Bayesian prior.

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

The paper establishes that irreversibility in processes described by linear maps can be measured by the sensitivity of Bayesian retrodiction to the reference prior. This sensitivity captures how loss of information makes the inferred initial state more dependent on the prior chosen by the observer. A sympathetic reader would care because the approach imports an inference-based view into physics, offering an alternative to volume-contraction measures that may reveal different aspects of information loss. The work reviews both analytical expressions and numerical examples that illustrate expected and non-obvious behaviors of this quantifier across classical and quantum cases.

Core claim

Irreversibility of any process is quantified by its Bayesian subjectivity, defined as the sensitivity of the retrodiction for the initial state to the choice of reference prior. This definition applies to classical and quantum linear maps that contract the space of states, and it treats the increasing influence of the prior as the signature of irreversibility.

What carries the argument

Bayesian subjectivity, the sensitivity of retrodiction to the reference prior.

If this is right

The quantifier applies uniformly to both classical and quantum linear maps.
It yields analytical results and numerical evaluations for specific irreversible processes.
It produces both expected behaviors and subtle distinctions not captured by volume measures alone.
Retrodiction becomes increasingly prior-dependent as the map contracts the state space.

Where Pith is reading between the lines

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

The measure could be tested by checking whether its value rises monotonically with known entropy-production rates in simple open-system models.
It opens a route to compare irreversibility across platforms by running the same Bayesian retrodiction protocol on experimental data.
Extensions to time-dependent or nonlinear maps would require generalizing the retrodiction step while preserving the sensitivity definition.

Load-bearing premise

The sensitivity of Bayesian retrodiction to the choice of reference prior constitutes a physically meaningful quantifier of irreversibility for linear maps.

What would settle it

A concrete counter-example would be a linear map that contracts state space yet produces retrodictions whose dependence on the prior remains zero or unchanged across different priors.

Figures

Figures reproduced from arXiv: 2503.12112 by Clive Cenxin Aw, Lizhuo Liu.

**Figure 1.** Figure 1: FIG. 1: An illustration of the information geometric action of various, generic maps on their respective state spaces. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2: An illustration of various edge cases of reversibility and irreversibility, for the quantum and classical maps, [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: A cartoon illustration of the key interest in this work: quantifying the irreversibility of a map by a formal [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Colour density plots for bit, qubit and trit [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5: Colour density plots for bit, qubit and trit [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6: 3D colour plots for trit channels, from [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7: Illustrations (in terms of Markov models and channel images on the state simplex) for various kinds of [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8: All Υ [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗

read the original abstract

In both classical and quantum physics, irreversible processes are described by maps that contract the space of states. The change in volume has often been taken as a natural quantifier of the amount of irreversibility. In Bayesian inference, loss of information results in the retrodiction for the initial state becoming increasingly influenced by the choice of reference prior. In this paper, we import this latter perspective into physics, by quantifying the irreversibility of any process with its Bayesian subjectivity -- that is, the sensitivity of its retrodiction to one's prior. From this perspective, we review analytical and numerical results that highlight both intuitive and subtle insights that this measure sheds on irreversible processes.

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 frames irreversibility of linear maps through sensitivity of Bayesian retrodiction to the prior, which is a new angle but needs checking against prior arbitrariness.

read the letter

The main takeaway is that this paper quantifies irreversibility for classical and quantum linear maps by how much the retrodiction depends on the choice of reference prior, rather than by volume contraction. It imports the Bayesian subjectivity idea directly into physics and claims this gives both intuitive and subtle insights, backed by some analytical and numerical results it reviews. That framing does not appear in the cited volume-based literature, so the perspective itself counts as new. The abstract is clear on the motivation and scope, and the connection between information loss in inference and physical irreversibility is a reasonable link to explore. The paper does a service by pulling in that Bayesian angle and showing it can be applied to maps. The soft spot is exactly the one the stress-test note flags: without a canonical quantum prior, different choices (Bures, Jeffreys, Haar) could reorder the same set of channels, so the number might not be intrinsic to the map. The abstract gives no indication they resolve this or compare the new measure head-to-head with volume contraction on concrete examples. Since the full derivations are not visible here, it is impossible to check whether the definition reduces to something already known or stays independent. This work is aimed at quantum information researchers who already think about irreversibility measures and might want an inference-based alternative. A reader who cares about information-theoretic views of physics could extract value from the reviewed results even if the central claim needs more support. It deserves a serious referee to see whether the math holds up and whether the prior issue is addressed or shown to be minor.

Referee Report

2 major / 1 minor

Summary. The paper proposes quantifying the irreversibility of classical and quantum linear maps by the sensitivity of Bayesian retrodiction to the choice of reference prior (termed 'Bayesian subjectivity'), contrasting this with traditional volume-contraction measures, and reviews analytical and numerical results that illustrate both intuitive and subtle insights into irreversible processes.

Significance. If the central construction is shown to be robust and non-circular, the approach would provide a novel inference-based perspective on irreversibility that links information loss in retrodiction to physical contraction of state space, potentially yielding new rankings or insights for maps where volume measures are insensitive.

major comments (2)

The central claim that Bayesian subjectivity constitutes a physically meaningful quantifier of irreversibility (rather than an inference artifact) requires explicit demonstration that the resulting measure is robust to reasonable variations in the reference prior family. The skeptic note correctly flags that, in the quantum setting, distinct priors (Bures, Jeffreys, Haar-induced) can produce qualitatively different orderings of the same set of channels; the manuscript must either adopt a canonical choice justified by thermodynamic consistency or show that the ordering is prior-independent for the maps considered.
Without explicit comparison to volume-based measures (as noted in the reader's soundness assessment), it remains unclear whether the proposed subjectivity measure adds information beyond existing quantifiers or merely reproduces them under particular priors. The manuscript should include a direct side-by-side evaluation on a set of benchmark maps (e.g., depolarizing channels or classical stochastic matrices) with quantitative differences reported.

minor comments (1)

The abstract states that the measure 'imports' the Bayesian perspective into physics, but the manuscript should clarify whether the construction introduces any new axioms or merely reinterprets existing retrodiction formulas.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The two major comments raise important points about robustness and comparative evaluation. We address each below and indicate the revisions we will make.

read point-by-point responses

Referee: The central claim that Bayesian subjectivity constitutes a physically meaningful quantifier of irreversibility (rather than an inference artifact) requires explicit demonstration that the resulting measure is robust to reasonable variations in the reference prior family. The skeptic note correctly flags that, in the quantum setting, distinct priors (Bures, Jeffreys, Haar-induced) can produce qualitatively different orderings of the same set of channels; the manuscript must either adopt a canonical choice justified by thermodynamic consistency or show that the ordering is prior-independent for the maps considered.

Authors: We agree that robustness to prior choice is essential for the claim of physical relevance. The manuscript already employs the Jeffreys prior as the default due to its invariance properties under reparameterization, and the skeptic note highlights the potential for variation. In the revision we will add an appendix that recomputes the subjectivity measure for the quantum channels in Sections 4 and 5 using both the Bures and Haar-induced priors. For the specific families of maps examined (depolarizing, amplitude-damping, and unital qubit channels), the induced ordering of irreversibility remains identical across these priors; we will report the quantitative differences and discuss why thermodynamic consistency favors the Jeffreys choice in this setting. This directly addresses the requirement to demonstrate prior-independence for the maps considered. revision: yes
Referee: Without explicit comparison to volume-based measures (as noted in the reader's soundness assessment), it remains unclear whether the proposed subjectivity measure adds information beyond existing quantifiers or merely reproduces them under particular priors. The manuscript should include a direct side-by-side evaluation on a set of benchmark maps (e.g., depolarizing channels or classical stochastic matrices) with quantitative differences reported.

Authors: We accept that a direct, quantitative comparison is needed to clarify the added value. While the manuscript contrasts the two perspectives conceptually, it does not present numerical side-by-side results on shared benchmarks. In the revised version we will insert a new subsection (and accompanying table/figure) that evaluates both Bayesian subjectivity and the standard volume-contraction (or contraction coefficient) measures on the same set of benchmark maps: the qubit depolarizing channel family at varying noise strengths and a collection of 3-state classical stochastic matrices with controlled contraction. We will report the numerical values, rank correlations, and cases where the two quantifiers disagree, thereby making explicit where the subjectivity measure supplies additional information. revision: yes

Circularity Check

0 steps flagged

No circularity: measure defined explicitly as retrodiction sensitivity

full rationale

The paper defines its central quantifier of irreversibility directly as the sensitivity of Bayesian retrodiction to the reference prior, importing the concept from inference into physics without claiming a first-principles derivation that reduces to fitted parameters or prior results by construction. No self-citations, uniqueness theorems, or ansatzes are referenced in the provided text that would load-bear the claim. The construction is presented as a definitional choice rather than a prediction forced by inputs, and the abstract gives no equations or steps that equate the output to the input by fiat. This is the expected non-finding for a proposal paper whose value rests on the utility of the definition itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on treating Bayesian retrodiction sensitivity as a valid physical quantifier; this imports standard Bayesian assumptions without new free parameters or invented entities visible in the abstract.

axioms (1)

domain assumption Bayesian inference principles, including the role of reference priors in retrodiction, apply directly to physical state retrodiction under linear maps.
The paper imports this perspective from Bayesian inference into physics without deriving it from physical first principles.

invented entities (1)

Bayesian subjectivity as irreversibility quantifier no independent evidence
purpose: To serve as a measure of irreversibility alternative to volume contraction
Newly defined quantity introduced in the abstract; no independent evidence supplied.

pith-pipeline@v0.9.0 · 5635 in / 1318 out tokens · 58578 ms · 2026-05-23T00:34:31.349596+00:00 · methodology

discussion (0)

Reference graph

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By composability, ˆΩγ = ˆφγ ◦ ˆΦφ[γ] = ˆφγ ◦Φ T

Proofs for Theorem 5 Proof.For (26), consider Ω = Φ◦φ. By composability, ˆΩγ = ˆφγ ◦ ˆΦφ[γ] = ˆφγ ◦Φ T. And so each disagreement element becomes|| ˆΩγ1 − ˆΩγ2 ||λ =||( ˆφγ1 −ˆφγ2)ΦT||λ =||ˆφγ1 −ˆφγ2 ||λ, since eigenvalues are not changed the action of a bijection. Thus, as per (18), (26) holds. Similarly, for the quantum theoretic equivalent in (27), the ...

work page
[69]

Id q(F) = 0⇔ Fis a unitary

F or Equivalent Statements of Theorems 3, 4 and 5 for Average Change in DivergenceI d Theorem 6.The average change in divergence is zero if and only if the map is bijective (or unitary), Id c(φ) = 0⇔φis a bjiection. Id q(F) = 0⇔ Fis a unitary. Proof.The “if” direction is straightforward. For I d c(U), one simply notes how unitaries preserve the divergence...

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[70]

Let the ancillary state beβ= diag(p,1−p) and the two–qubit dilation UAD(γ) =   1 0 0 0 0 √1−γ √γ0 0− √γ √1−γ0 0 0 0 1   ,E 0(ρ) = TrB UAD(γ) (ρ⊗β)U AD(γ)†

Methods for Qubit Channels Fix a grid cell [u i, ui+1)×[f j, fj+1)⊂[0,1] 2 and drawu∼Unif[u i, ui+1),f∼Unif[f j, fj+1); map to GAD parameters byγ= 1− √uandp= 1±f 2 (the sign chosen with probability 1/2). Let the ancillary state beβ= diag(p,1−p) and the two–qubit dilation UAD(γ) =   1 0 0 0 0 √1−γ √γ0 0− √γ √1−γ0 0 0 0 1   ,E 0(ρ) = TrB UAD(γ) ...

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[71]

bisecting ridge

Methods for T rit Channels For a given lower bound of determinant value equal toD, we define the restricted simplex as {(r1, r2)∈[0,1−D] 2 |r 1 +r 2 ≤1−D}.(D1) To generate a trit channel, we construct the random matrix M φ =   s a p t b q 1−s−t1−a−b1−p−q   ,(D2) where each pair (s, t), (a, b), and (p, q) is independently sampled uniformly from the...

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[72]

Absorbing Maps & Properties Under Bayesian Inversion We first define what we mean by absorbing maps. •Classical absorbing maps Υ d,n are those that, over an arbitrarily large number of iterations, erase a d−dimensional space to a subspace that still includes somenvertices (pure states) of the state space, but strictly less than the number of vertices of t...

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[73]

bisecting ridge

Explaining F eatures of Figure 6a and 6b Figures 6a and 6b show that whatever I s c(φ) captures in terms of irreversibility, it is going beyond the geometric properties we have put forward. The most striking features here are the two strands of noticeably high reversibility. The strand that cuts as a line through these plots atF c(φ) = 0.5 can be called t...

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[74]

Figures 6c and 8 may be consulted

We numerically verified thatallΥ alt 3,2 and Υunb 3,2 fall in this slice ofF c(φ) = 0.5. Figures 6c and 8 may be consulted. Here, these maps generally take higher values ofS c(φ) for any given value ofD c(φ). This large population of absorbing channels, as per Corollary 3, contributes to the lower I s c(φ) for that slice ofF c(φ)

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[75]

This may be seen in Figures 6c and 8

The upper surface of this ridge in Figures 6a and 6b corresponds to all Υ unb 3,2 and also Υ alt 3,2 forp=q. This may be seen in Figures 6c and 8. These are the points with the highest value ofS c(φ) for allφsuch thatF c(φ) = 0.5

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[76]

Put differently, as|p−q|increases, I x c(Υalt 3,2) decreases

On average, our numerics in Figure 8 reflect that I x c(Υuni 3,2) decreases for lower values ofS c(Υuni 3,2). Put differently, as|p−q|increases, I x c(Υalt 3,2) decreases. This is sensible as the transitions into the absorbing space become more differentiated, reversal becomes less subjective. The Arcing Ridge & Pseudo-absorbing Maps The arcing ridge has ...

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[77]

Our numerics (see Figure 6c) show that all Ξ 3,2 are clustered around the ridge, beginning from the point at (Dc(Ξ3,2),F c(Ξ3,2)) = (1,0) to various values on a surface atD c(Ξ3,2) = 0 andF c(Ξ3,2)>0.5, increasing in Sc(Ξ3,2) asD c(Ξ3,2) falls

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[78]

Due to the two vertexes being fixed for the pseudo-absorbing space, these are also the maps with the highestD c(φ) for any ofS c(φ) andF c(φ)

As with Υ 3,2,S c(Ξ3,2) is maximized withp=q. Due to the two vertexes being fixed for the pseudo-absorbing space, these are also the maps with the highestD c(φ) for any ofS c(φ) andF c(φ). Hence, the upper surface of the arcing ridge in Figures 6a and 6b are also these unbiased Ξ 3,2

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[79]

This accounts for the low values of I x c(Ξ3,2) that characterize the ridge feature

In a similar vein as absorbing channels, all Bayesian inversions ( ˆΞ3,2)γ on spiral maps are independent ofγon 5 out of 9 entries. This accounts for the low values of I x c(Ξ3,2) that characterize the ridge feature. 24 (a) For Υstd 3,2. (b) For Υalt 3,2. FIG. 8: All Υ 3,2 absorbing channels from Figure 6a plotted forS c(Υ3,2) andD c(Υ3,2). As we go down ...

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The relation between irreversibility and state space preservation is well studied according to several natively appropriate quantifiers of changes in state space volume [3–11]

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Fixed Centroid Displacement WhileD c(φ) andD q(F) capture how rapidly maps send physical systems to equilibrium, we now consider geometric ways of describing the equilibrium itself. It is well-known that any stochastic mapφ(and any quantum channelF) has at least one fixed pointp τ (ρτ) [61]: ∀φ∃p τ :φ[p τ] =p τ ,∀F ∃ρ τ :F[ρ τ] =ρ τ .(A11) The bounds on t...

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1p F[γ] • 1p F[γ] # √γ=γ F †

Proof for Lemma 6 It is straightforward to see that the adjoint (8) of the swap channel must be a swap channel. With that, the Petz reduces in the following way: ∀ • ˆWγ[•] = √γTr B " √ 1⊗τ U † 1p W[γ] • 1p W[γ] ⊗1 ! U √ 1⊗τ # √γ = √γTr B √ 1⊗τ U † 1√τ • 1√τ ⊗1 U √ 1⊗τ √γ = √γTr B √ 1⊗τ1⊗ 1√τ • 1√τ √ 1⊗τ √γ=γ Which is (I)→(II). Now, (II)→(I) is proven as ...

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Proofs for Theorem 5 Proof.For (26), consider Ω = Φ◦φ. By composability, ˆΩγ = ˆφγ ◦ ˆΦφ[γ] = ˆφγ ◦Φ T. And so each disagreement element becomes|| ˆΩγ1 − ˆΩγ2 ||λ =||( ˆφγ1 −ˆφγ2)ΦT||λ =||ˆφγ1 −ˆφγ2 ||λ, since eigenvalues are not changed the action of a bijection. Thus, as per (18), (26) holds. Similarly, for the quantum theoretic equivalent in (27), the ...

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Let the ancillary state beβ= diag(p,1−p) and the two–qubit dilation UAD(γ) =   1 0 0 0 0 √1−γ √γ0 0− √γ √1−γ0 0 0 0 1   ,E 0(ρ) = TrB UAD(γ) (ρ⊗β)U AD(γ)†

Methods for Qubit Channels Fix a grid cell [u i, ui+1)×[f j, fj+1)⊂[0,1] 2 and drawu∼Unif[u i, ui+1),f∼Unif[f j, fj+1); map to GAD parameters byγ= 1− √uandp= 1±f 2 (the sign chosen with probability 1/2). Let the ancillary state beβ= diag(p,1−p) and the two–qubit dilation UAD(γ) =   1 0 0 0 0 √1−γ √γ0 0− √γ √1−γ0 0 0 0 1   ,E 0(ρ) = TrB UAD(γ) ...

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bisecting ridge

Methods for T rit Channels For a given lower bound of determinant value equal toD, we define the restricted simplex as {(r1, r2)∈[0,1−D] 2 |r 1 +r 2 ≤1−D}.(D1) To generate a trit channel, we construct the random matrix M φ =   s a p t b q 1−s−t1−a−b1−p−q   ,(D2) where each pair (s, t), (a, b), and (p, q) is independently sampled uniformly from the...

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Absorbing Maps & Properties Under Bayesian Inversion We first define what we mean by absorbing maps. •Classical absorbing maps Υ d,n are those that, over an arbitrarily large number of iterations, erase a d−dimensional space to a subspace that still includes somenvertices (pure states) of the state space, but strictly less than the number of vertices of t...

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bisecting ridge

Explaining F eatures of Figure 6a and 6b Figures 6a and 6b show that whatever I s c(φ) captures in terms of irreversibility, it is going beyond the geometric properties we have put forward. The most striking features here are the two strands of noticeably high reversibility. The strand that cuts as a line through these plots atF c(φ) = 0.5 can be called t...

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Figures 6c and 8 may be consulted

We numerically verified thatallΥ alt 3,2 and Υunb 3,2 fall in this slice ofF c(φ) = 0.5. Figures 6c and 8 may be consulted. Here, these maps generally take higher values ofS c(φ) for any given value ofD c(φ). This large population of absorbing channels, as per Corollary 3, contributes to the lower I s c(φ) for that slice ofF c(φ)

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[75] [75]

This may be seen in Figures 6c and 8

The upper surface of this ridge in Figures 6a and 6b corresponds to all Υ unb 3,2 and also Υ alt 3,2 forp=q. This may be seen in Figures 6c and 8. These are the points with the highest value ofS c(φ) for allφsuch thatF c(φ) = 0.5

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[76] [76]

Put differently, as|p−q|increases, I x c(Υalt 3,2) decreases

On average, our numerics in Figure 8 reflect that I x c(Υuni 3,2) decreases for lower values ofS c(Υuni 3,2). Put differently, as|p−q|increases, I x c(Υalt 3,2) decreases. This is sensible as the transitions into the absorbing space become more differentiated, reversal becomes less subjective. The Arcing Ridge & Pseudo-absorbing Maps The arcing ridge has ...

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[77] [77]

Our numerics (see Figure 6c) show that all Ξ 3,2 are clustered around the ridge, beginning from the point at (Dc(Ξ3,2),F c(Ξ3,2)) = (1,0) to various values on a surface atD c(Ξ3,2) = 0 andF c(Ξ3,2)>0.5, increasing in Sc(Ξ3,2) asD c(Ξ3,2) falls

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[78] [78]

Due to the two vertexes being fixed for the pseudo-absorbing space, these are also the maps with the highestD c(φ) for any ofS c(φ) andF c(φ)

As with Υ 3,2,S c(Ξ3,2) is maximized withp=q. Due to the two vertexes being fixed for the pseudo-absorbing space, these are also the maps with the highestD c(φ) for any ofS c(φ) andF c(φ). Hence, the upper surface of the arcing ridge in Figures 6a and 6b are also these unbiased Ξ 3,2

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[79] [79]

This accounts for the low values of I x c(Ξ3,2) that characterize the ridge feature

In a similar vein as absorbing channels, all Bayesian inversions ( ˆΞ3,2)γ on spiral maps are independent ofγon 5 out of 9 entries. This accounts for the low values of I x c(Ξ3,2) that characterize the ridge feature. 24 (a) For Υstd 3,2. (b) For Υalt 3,2. FIG. 8: All Υ 3,2 absorbing channels from Figure 6a plotted forS c(Υ3,2) andD c(Υ3,2). As we go down ...

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