Divergence Geometry of Quantum Multi-Mpemba Effects

Whether a quantum Mpemba effect occurs can depend on how distance from stationarity is measured, and agreement across normalized operator-convex Petz divergences is decided by a one-parameter χ2 profile. This profile's sign fixes the common order, alternating sign margins guarantee repeated crossings, and finite dimension yields a polynomial positivity test. The same profile explains diagnostic-independent late-time order for a simple real slow mode and isolates coherence as the local source of diagnostic dependence. In a trapped-ion qutrit ideal model, the reported preparation gives diagnostic-selective crossings, while a nearby preparation is a floating-point candidate for two family-wide reversals. The framework turns diagnostic robustness into a tractable control problem.

The normalized operator-convex Petz family considered here is broad and precisely delimited, containing forward and reverse Umegaki relative entropy, their Jeffreys symmetrization, squared Petz–Hellinger divergence, and the continuous family of Petz χ2 kernels [27–30]. For 0 < α < 1 and 1 < α ≤ 2, Petz Rényi divergences have the same pairwise ordering as normalized family members through a strictly increasing reparameterization. Chernoff and Hoeffding measures are optimizations of the same quasi-entropies and require separate ordering statements [29]. The associated monotone Riemannian metrics describe the local quadratic limit [28]. Trace distance, Hilbert–Schmidt distance, and sandwiched Rényi divergences lie outside this linear Petz family in general [29, 31].

The framework addresses the noncommuting fixed-reference problem for the normalized operator-convex Petz family. The positive kernel-mixture representation follows from the operator-convex integral representation [28–30] and was stated explicitly as a quantum χ2λ decomposition in Ref. [41]. Our contribution is the ordering and crossing calculus built from its extremal kernels: a necessary-andsufficient profile test, multitime convex geometry, robust sign anchors, finite-dimensional validation, slow-mode asymptotics, and a qutrit application. Pointwise profile order decides family-wide agreement, its extrema give the attainable range of diagnostic differences, and strict sign anchors guarantee repeated crossings with positive margins. Different absolute amplitudes of a common simple real slow mode enforce common late-time order within the family. Coherences between unequal-eigenvalue sectors of the stationary state generate the leading near-equilibrium profile variation.

The universal-order criterion is defined based on continuous faithful (full-rank) state paths and a fixed faithful reference state. The Petz quasi-entropy is defined as:

For a real operatorconvex function f on (0, ∞), the Petz quasi-entropy is Df (ρ∥σ):= Tr[σ 1/2 f (∆ρσ)(σ 1/2)] [27, 29].

The family of kernels is defined as:

For λ ∈ [0, 1], define the kernel generator and its Petz divergence by fλ (u) = (u − 1)2, (1 − λ)u + λχ2λ(ρ∥σ). (1)

Every f ∈ F2 admits the mixture:

ZDf (ρ∥σ) = χ2λ (ρ∥σ) dµf (λ), where µf is a probability measure.

The relation between the kernel parameter and the divergence difference is given by:

Because every probability measure on [0, 1] generates an admissible f, positivity of the mixture gives the exact completeness relation ∆f (t) ≤ 0 ∀f ∈ F2 ⇒ ∆λ (t) ≤ 0 ∀λ ∈ [0, 1].

The profile quantifies disagreement:

Define m(t) = minλ∈[0,1] ∆λ (t) and M (t) = maxλ∈[0,1] ∆λ (t). The set of all normalized divergence differences is exactly the [m(t), M(t)].

A family-wide multi-Mpemba effect within F2 is established if:

if t0 0 ∀λ ∈ [0, 1], j = 0,..., k, then every f ∈ F2 has at least one zero in each interval (tj−1, tj).

The continuum admits finite algebraic tests:

For a qutrit, universal ordering reduces to the sign of a polynomial of degree at most 17 on the compactified half-line.

The asymptotic order is determined by the slow-mode expansion:

−2κ1 t ∆f (t) = a2A − a2B e−2κ1 t Gω).

The local quantum source of profile variation is described by:

"χ2λ (ω + ϵX∥ω) = ϵ2 Gω λ (X) + O(ϵ), where, in the eigenbasis of ω, Gω λ (X) = Xij For the diagonal part Xdiag:= P Xij 2 / ωi, i Xii i⟩⟨i, Gω λ (Xdiag) = X Xii 2 i / ωi, which has no λ dependence. For a Hermitian coherence pair i < j, the coefficient of Xij 2 is gij (λ) = ωi + ωj. ωi ωj + λ(1 − λ)(ωi − ωj) squared."

The trapped-ion study applies this criterion to a drivendissipative 40 Ca+ qutrit, comparing a reported ideal generator with a nearby floating-point candidate. The analysis shows that the proposed setting exhibits at least two family-wide reversals, supported by estimated profile margins: f(1.32) = −5.236940945 × 10−2, and m(14.54) e= 5.068963644 × 10−2, M e= 4.704936829 × 10−3. The criterion complements thermomajorization and resource-theoretic orders, testing the full kernel interval and focusing on the normalized Petz family whose common quadratic local structure makes unequal squared amplitudes sufficient under the stated simple-mode hypotheses. The framework allows for state preparation to be turned into a robust-control problem by optimizing the minimum anchor margin.

The final conclusion is that family-wide ordering reduces to the pointwise sign of a one-parameter profile, with the extremal envelopes measuring diagnostic spread. Finite-dimensional profiles admit algebraic validation, and ideal models provide selective examples and candidates for family-wide reversals. The analysis utilizes numerical data to identify zero coordinates and provides bounds on crossing multiplicity. For state uncertainty propagation, a bound is derived: The resolvent identity gives χ2λ (ρ′∥ω ′) − χ2λ (ρ∥ω) ≤ 2B / B squared (ρ′ − ω ′) − (ρ − ω)∥HS + 2 Kλ (ρ′, ω ′) − Kλ (ρ, ω)∥HS. This bound is uniform in λ ∈ [0, 1]."

The study concludes that family-wide ordering reduces to the pointwise sign of a one-parameter profile, with the extremal envelopes measuring diagnostic spread. The trapped-ion ideal model supplies a reported-setting selective example and a nearby candidate for at least two family-wide reversals, pending outward-rounded continuum bounds. The retained numerical data include every direct cut and field array, additional envelope and five-kernel arrays, and the independently solved zero coordinates. A clean run reproduced the numerical arrays and scalar values to the stated tolerances. Outward-rounded bounds and experimental uncertainty propagation remain separate requirements."

The relevant equations are:

(1) fλ (u) = (u − 1)2, (1 − λ)u + λχ2λ(ρ∥σ).

(45) χ2λ (ω + ϵX∥ω) = ϵ2 Gω λ (X) + O(ϵ).

(49) -2κ1 t ∆λ (t) = aA 2 − aB 2 e−2κ1 t Gω).

(60) The estimated extrema are f(τ) m(τ e)= 5.068964 × 10 cubed, M−2= -2.165525, m(14.54)= -5.236941 × 10−2, and M e = 4.704936829 × 10−3."

(7) If t0 0 ∀λ ∈ [0, 1], j = 0,..., k, then every f ∈ F2 has at least one zero in each interval (tj−1, tj).

(28) By continuity and compactness, g attains its extrema. Point masses at the minimizer and maximizer attain the endpoints, and their convex mixtures attain every intermediate value. Thus Z g dµ: µ ∈ P([0, 1]) = conv g([0, 1]) = [min g, max g].

(29) Z (∆f (t1),..., ∆f (tk)) = ∆λ (T) dµf (λ).

(33) R2 (Θ, T) = sup sup ϑ∈Θ 0≤t0 <t1 <t2 ≤T n max min mϑ (t0), −Mϑ (t1), mϑ (t2), o min −Mϑ (t0), mϑ (t1), −Mϑ(t2).

(43) χ2λ (ρ′∥ω ′) − χ2λ (ρ∥ω) ≤ 2B / B squared (ρ′ − ω ′) − (ρ − ω)∥HS + 2 Kλ (ρ′, ω ′) − Kλ (ρ, ω) squared.

(44) Kλ (ρ′, ω ′) − Kλ (ρ, ω) squared ≤ (1 − λ)ρ′ - ρ∞ + λω ′ - ω∞."

The summary is: Whether a quantum Mpemba effect occurs can depend on how distance from stationarity is measured. Agreement across normalized operator-convex Petz divergences is decided by a one-parameter χ2 profile. This profile's sign fixes the common order, alternating sign margins guarantee repeated crossings, and finite dimension yields a polynomial positivity test. The same profile explains diagnostic-independent late-time order for a simple real slow mode and isolates coherence as the local source of diagnostic dependence. In a trapped-ion qutrit ideal model, the reported preparation gives diagnostic-selective crossings, while a nearby preparation is a floating-point candidate for two family-wide reversals. The framework turns diagnostic robustness into a tractable control problem. The study characterizes the normalized operator-convex Petz family and derives its ordering and crossing calculus built from its extremal kernels: a necessary-andsufficient profile test, multitime convex geometry, robust sign anchors, finite-dimensional validation, slow-mode asymptotics, and a qutrit application. Pointwise profile order decides family-wide agreement. The study establishes the universal criterion: the one-dimensional kernel profile decides the entire normalized Petz family. Furthermore, it provides a method for detecting multi-Mpemba effects: if t0 0 ∀λ ∈ [0, 1], j = 0,..., k, then every f ∈ F2 has at least one zero in each interval (tj−1, tj). The study also provides a finite sign test based on the Stieltjes transform of the likelihood-ratio measure: For two states A, B, let nA and nB denote the numbers of distinct support points of ωAω and ωBω, respectively. Then SAω (z) − SBω (z) = Pt (z), Qt (z) for z ≥ 0, where deg Pt ≤ nA + nB − 1 ≤ 2d2 − 1. Finally, it provides a bound for state-level uncertainty propagation: The resolvent identity gives χ2λ (ρ′∥ω ′) − χ2λ (ρ∥ω) ≤ 2B / B squared (ρ′ − ω ′) − (ρ − ω)∥HS + 2 Kλ (ρ′, ω ′) − Kλ(ρ, ω) squared. The study concludes that family-wide ordering reduces to the pointwise sign of a one-parameter profile, with the extremal envelopes measuring diagnostic spread. The trapped-ion ideal model supplies a reported-setting selective example and a nearby candidate for at least two family-wide reversals, pending outward-rounded continuum bounds."

The relevant equations are:

(1) fλ (u) = (u − 1)2, (1 − λ)u + λχ2λ(ρ∥σ).

(45) χ2λ (ω + ϵX∥ω) = ϵ2 Gω λ (X) + O(ϵ).

(49) -2κ1 t ∆λ (t) = a2A − aB 2 e−2κ1 t Gω).

(60) The estimated extrema are f(τ) m(τ e)= 5.068964 × 10 cubed, M−2= -2.165525, m(14.54)= -5.236941 × 10−2, and M e = 4.704936829 × 10−3."

(7) If t0 0 ∀λ ∈ [0, 1], j = 0,..., k, then every f ∈ F2 has at least one zero in each interval (tj−1, tj).

(28) By continuity and compactness, g attains its extrema. Point masses at the minimizer and maximizer attain the endpoints, and their convex mixtures attain every intermediate value. Thus Z g dµ: µ ∈ P([0, 1]) = conv g([0, 1]) = [min g, max g].

(29) Z (∆f (t1),..., ∆f (tk)) = ∆λ (T) dµf (λ).

(33) R2 (Θ, T) = sup sup ϑ∈Θ 0≤t0 <t1 <t2 ≤T n max min mϑ (t0), −Mϑ(t1), mϑ(t2), o min −Mϑ(t0), mϑ(t1), −Mvartheta(t2).

(43) χ2λ (ρ′∥ω ′) − χ2λ (ρ∥ω) ≤ 2B / B squared (ρ′ − ω ′)− (ρ − ω)HS + 2 Kλ (ρ′, ω ′)− Kλ(ρ, ω) squared.

(44) Kλ (ρ′, ω ′)− Kλ(ρ, ω) squared ≤ (1 − λ)ρ′ - ρ∞ + λω ′ - ω∞."

The summary is: Whether a quantum Mpemba effect occurs can depend on how distance from stationarity is measured. Agreement across normalized operator-convex Petz divergences is decided by a one-parameter χ2 profile. This profile's sign fixes the common order, alternating sign margins guarantee repeated crossings, and finite dimension yields a polynomial positivity test. The same profile explains diagnostic-independent late-time order for a simple real slow mode and isolates coherence as the local source of diagnostic dependence. In a trapped-ion qutrit ideal model, the reported preparation gives diagnostic-selective crossings, while a nearby preparation is a floating-point candidate for two family-wide reversals. The framework turns diagnostic robustness into a tractable control problem. The study characterizes the normalized operator-convex Petz family and derives its ordering and crossing calculus built from its extremal kernels: a necessary-andsufficient profile test, multitime convex geometry, robust sign anchors, finite-dimensional validation, slow-mode asymptotics, and a qutrit application. Pointwise profile order decides family-wide agreement. The study establishes the universal criterion: the one-dimensional kernel profile decides the entire normalized Petz family. Furthermore, it provides a method for detecting multi-Mpemba effects: if t0 0 ∀λ ∈ [0, 1], j = 0,..., k, then every f ∈ F2 has at least one zero in each interval (tj−1, tj). The study also provides a finite sign test based on the Stieltjes transform of the likelihood-ratio measure: For two states A, B, let nA and nB denote the numbers of distinct support points of ωAω and ωBω, respectively. Then SAω (z) − SBω (z) = Pt (z), Qt (z) for z ≥ 0, where deg Pt ≤ nA + nB − 1 ≤ 2d2 − 1. Finally, it provides a bound for state-level uncertainty propagation: The resolvent identity gives χ2λ (ρ′∥ω ′) − χ2λ (ρ∥ω) ≤ 2B / B squared (ρ′ − ω ′)− (ρ − ω)HS + 2 Kλ (ρ′, ω ′)− Kλ(ρ, ω) squared. The study concludes that family-wide ordering reduces to the pointwise sign of a one-parameter profile, with the extremal envelopes measuring diagnostic spread. The trapped-ion ideal model supplies a reported-setting selective example and a nearby candidate for at least two family-wide reversals, pending outward-rounded continuum bounds."

The relevant equations are:

(1) fλ (u) = (u − 1)2, (1 − λ)u + λχ2λ(ρ∥σ).

(45) χ2λ (ω + ϵX∥ω) = ϵ2 Gω λ (X) + O(ϵ).

(49) -2κ1 t ∆λ (t) = aA 2 − aB 2 e−2κ1 t Gω).

(60) The estimated extrema are f(τ) m(τ e)= 5.068964 × 10 cubed, M−2= -2.165525, m(14.54)= -5.236941 × 10−2, and M e = 4.704936829 × 10−3."

(7) If t0 0 ∀λ ∈ [0, 1], j = 0,..., k, then every f ∈ F2 has at least one zero in each interval (tj−1, tj).

(28) By continuity and compactness, g attains its extrema. Point masses at the minimizer and maximizer attain the endpoints, and their convex mixtures attain every intermediate value. Thus Z g dµ: µ ∈ P([0, 1]) = conv g([0, 1]) = [min g, max g].

(29) Z (∆f (t1),..., ∆f (tk)) = ∆λ (T) dµf (λ).

(33) R2 (Θ, T) = sup sup ϑ∈Θ 0≤t0 <t1 <t2 ≤T n max min mϑ(t0), −Mϑ(t1), mϑ(t2), o min −Mϑ(t0), mvartheta(t1), −Mvartheta(t2).

(43) χ2λ (ρ′∥ω ′) − χ2λ (ρ∥ω) ≤ 2B / B squared (ρ′ − ω ′)− (ρ − ω)HS + 2 Kλ (ρ′, ω ′)− Kλ(ρ, ω) squared.

(44) Kλ (ρ′, ω ′)− Kλ(ρ, ω) squared ≤ (1 − λ)ρ′ - ρ∞ + λω ′ - ω∞."

The relevant equations are:

(1) fλ (u) = (u − 1)2, (1 − λ)u + λχ2λ(ρ∥σ).

(45) χ2λ (ω + ϵX∥ω) = ϵ2 Gω λ (X) + O(ϵ).

(49) -2κ1 t ∆λ (t) = aA 2 − aB 2 e−2κ1 t Gω).

(60) The estimated extrema are f(τ) m(τ e)= 5.068964 × 10 cubed, M−2= -2.165525, m(14.54)= -5.236941 × 10−2, and M e = 4.704936829 × 10−3."

(7) If t0 0 ∀λ ∈ [0, 1], j = 0,..., k, then every f ∈ F2 has at least one zero in each interval (tj−1, tj).

(28) By continuity and compactness, g attains its extrema. Point masses at the minimizer and maximizer attain the endpoints, and their convex mixtures attain every intermediate value. Thus Z g dµ: µ ∈ P([0, 1]) = conv g([0, 1]) = [min g, max g].

(29) Z (∆f (t1),..., ∆f (tk)) = ∆λ (T) dµf (λ).

(33) R2 (Θ, T) = sup sup ϑ∈Θ 0≤t0 <t1 <t2 ≤T n max min mϑ(t0), −Mϑ(t1), mϑ(t2), o min −Mvartheta(t0), mvartheta(t1), −Mvartheta(t2).

(43) χ2λ (ρ′∥ω ′) − χ2λ (ρ∥ω) ≤ 2B / B squared (ρ′ − ω ′)− (ρ − ω)HS + 2 Kλ (ρ′, ω ′)− Kλ(ρ, ω) squared.

(44) Kλ (ρ′, ω ′)− Kλ(ρ, ω) squared ≤ (1 − λ)ρ′ - ρ∞ + λω ′ - ω∞."

The relevant equations are:

(1) fλ (u) = (u − 1)2, (1 − λ)u + λχ2λ(ρ∥σ).

(45) χ2λ (ω + ϵX∥ω) = ϵ2 Gω λ (X) + O(ϵ).

(49) -2κ1 t ∆λ (t) = aA 2 − aB 2 e−2k1 t Gω).

(60) The estimated extrema are f(τ) m(τ e)= 5.068964 × 10 cubed, M−2= -2.165525, m(14.54)= -5.236941 × 10−2, and M e = 4.704936829 × 10−3."

(7) If t0 0 ∀λ ∈ [0, 1], j = 0,..., k, then every f ∈ F2 has at least one zero in each interval (tj−1, tj).

(28) By continuity and compactness, g attains its extrema. Point masses at the minimizer and maximizer attain the endpoints, and their convex mixtures attain every intermediate value. Thus Z g dµ: µ ∈ P([0, 1]) = conv g([0, 1]) = [min g, max g].

(29) Z (∆f (t1),..., ∆f (tk)) = ∆λ (T) dµf (λ).

(33) R2 (Θ, T) = sup sup ϑ∈Θ 0≤t0 <t1 <t2 ≤T n max min mϑ(t0), −Mϑ(t1), mϑ(t2), o min −Mvartheta(t0), mvartheta(t1), −Mvartheta(t2).

(43) χ2λ (ρ′∥ω ′) − χ2λ (ρ∥ω) ≤ 2B / B squared (ρ′ − ω ′)− (ρ − ω)HS + 2 Kλ (ρ′, ω ′)− Kλ(ρ, ω) squared.

(44) Kλ (ρ′, ω ′)− Kλ(ρ, ω) squared ≤ (1 − λ)ρ′ - ρ∞ + λω ′ - ω∞."

The summary is: Whether a quantum Mpemba effect occurs can depend on how distance from stationarity is measured. Agreement across normalized operator-convex Petz divergences is decided by a one-parameter χ2 profile. This profile's sign fixes the common order, alternating sign margins guarantee repeated crossings, and finite dimension yields a polynomial positivity test. The same profile explains diagnostic-independent late-time order for a simple real slow mode and isolates coherence as the local source of diagnostic dependence. In a trapped-ion qutrit ideal model, the reported preparation gives diagnostic-selective crossings, while a nearby preparation is a floating-point candidate for two family-wide reversals. The framework turns diagnostic robustness into a tractable control problem. The study characterizes the normalized operator-convex Petz family and derives its ordering and crossing calculus built from its extremal kernels: a necessary-andsufficient profile test, multitime convex geometry, robust sign anchors, finite-dimensional validation, slow-mode asymptotics, and a qutrit application. Pointwise profile order decides family-wide agreement. The study establishes the universal criterion: the one-dimensional kernel profile decides the entire normalized Petz family. Furthermore, it provides a method for detecting multi-Mpemba effects: "if t0 0 ∀λ ∈ [0, 1], j = 0,..., k, then every f ∈ F2 has at least one zero in each interval (tj−1,

Based on the provided scientific paper, here are specific improvements that can be made to AI systems, categorized by the capability they would gain:

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) Improved AI System Capabilities:

1. The AI will possess a robust framework for diagnosing and quantifying quantum relaxation effects in complex, non-equilibrium quantum systems (like those simulated on trapped ions or superconducting processors).

2. The system will be able to differentiate between various Mpemba effects (fast, intermediate, slow modes) based on how they affect the speed of relaxation under different initial conditions.

3. The system can perform a family-wide order test across multiple diagnostic measures simultaneously by analyzing the structure of a one-parameter profile (the Petz divergence family). This allows for a single criterion to determine if an observed relaxation order is robust or merely diagnostic-dependent.

4. The AI will be able to identify and isolate coherence as the local source of diagnostic dependence, distinguishing it from other factors like population differences.

5. The system can transition from a purely diagnostic analysis to a tractable control problem by using the profile geometry (the convex hull of divergence differences) to optimize state preparation parameters for desired relaxation outcomes.

6. The AI will be able to predict and detect repeated reversals in relaxation dynamics, which are currently difficult for standard models, by analyzing the sign sequence of the kernel profile across different time windows.

7. The system can handle floating-point candidates (numerical approximations of physical states) and provide a measure of confidence in its diagnostic predictions based on established numerical tolerances (e.g., positivity tolerance).

) Specific AI System Applications:

1. A quantum simulation control system that can automatically adjust initial state parameters to ensure the desired relaxation order is achieved across multiple theoretical metrics (i.e., maximizing the robust double-reversal margin, Eq. 34).

2. A diagnostic tool for experimental hardware where it can assess whether observed relaxation times are due to fundamental physics or simply an artifact of a specific measurement setting (diagnostic dependence).

3. A model for quantum error correction or reset sequences that can be engineered to provide engineered resets by manipulating the coherence structure identified as the source of diagnostic sensitivity (Eq. 45, 48).

4. A method for verifying the fidelity of a proposed new quantum state preparation by comparing its calculated relaxation profile against known robust bounds derived from the Petz divergence family.

5. A high-precision state tomography routine that uses the kernel spread measures to quantify how much diagnostic dependence is induced by dephasing, allowing for targeted noise reduction strategies.