Catalytic Relative Majorization

• Catalytic relative majorization is a comparison framework where an auxiliary catalyst pair enables transformations that are impossible under standard majorization.
• It uses α–z relative entropies and tropical limits to establish both exact one-shot and asymptotic conversion criteria for flat states and cq-pure pairs.
• The framework integrates quantum resource theories, thermo-majorization, and classical matrix majorization, offering a unified view of catalytic transformations.

Searching arXiv for papers on catalytic relative majorization and closely related majorization frameworks. Searching for the 2025 paper on $\alpha$-$z$ relative entropies and catalytic relative majorization of flat states. Catalytic relative majorization is a convertibility relation for pairs of states, distributions, or statistical experiments in which an otherwise impossible transformation may become possible after adjoining an auxiliary pair that is returned unchanged. In the quantum formulation, one asks whether there exists a single quantum channel $\mathcal C$ such that

$$\mathcal C(\rho)=\rho',\qquad \mathcal C(\sigma)=\sigma',$$

and in the catalytic variant whether there exist catalyst states $(\tau,\omega)$ and a quantum channel $\mathcal C$ such that

$$\mathcal C(\rho\otimes\tau)=\rho'\otimes\tau,\qquad \mathcal C(\sigma\otimes\omega)=\sigma'\otimes\omega,$$

with $F(\tau,\omega)>0$ in order to exclude orthogonal catalysts that would trivialize the problem. The subject lies at the interface of relative majorization, catalytic majorization or trumping, Blackwell comparison of experiments, thermo-majorization, and quantum resource theories, and recent work has produced exact, asymptotic, large-sample, and symmetry-restricted formulations with distinct monotone families (Verhagen et al., 10 Jul 2025).

1. Basic order structure and catalytic variants

Relative majorization is an order on pairs rather than on single states. In the quantum setting, $(\rho,\sigma)$ majorizes $(\rho',\sigma')$ if there exists a single channel mapping both components exactly. The large-sample version asks whether for all sufficiently large $z$0 there exists a channel $z$1 such that

$z$2

The catalytic version instead asks for a single-copy transformation made possible by an auxiliary pair $z$3 returned exactly. The same work also considers asymptotic variants in which the target $z$4 is reached only approximately, with arbitrary precision, while $z$5 remains exact (Verhagen et al., 10 Jul 2025).

A closely related one-sided relaxation is relative submajorization. For nonnegative vectors $z$6 and $z$7, relative majorization requires a stochastic matrix $z$8 with

$z$9

whereas relative submajorization requires only a substochastic $\mathcal C$0 such that

$\mathcal C$1

This relaxation is exact for several finite-shot tasks: probabilistic conversion, approximation error, and assistance by standard resources such as work batteries, maximally entangled states, or coherence batteries. When $\mathcal C$2 and $\mathcal C$3, submajorization collapses to strict relative majorization (Renes, 2015).

The order also admits a statistical interpretation. In the classical case, relative majorization is the comparison of pairs of distributions by a common stochastic post-processing map. In thermodynamic language this becomes comparison of $\mathcal C$4, where $\mathcal C$5 is a Gibbs state. A plausible implication is that catalytic relative majorization should be viewed not as a separate order on single resources, but as a higher-order comparison of resource-reference pairs under tensor-product extension.

2. Flat states, cq-pure pairs, and structural assumptions

The sharpest quantum results currently available concern a restricted state class. The relevant pairs belong to a class $\mathcal C$6 of cq-states with pure components,

$\mathcal C$7

with some overlap in the sense that there exists at least one $\mathcal C$8 such that $\mathcal C$9. The paper also distinguishes “non-parallel” pairs, meaning

$$\mathcal C(\rho)=\rho',\qquad \mathcal C(\sigma)=\sigma',$$0

A central subclass is formed by flat states,

$$\mathcal C(\rho)=\rho',\qquad \mathcal C(\sigma)=\sigma',$$1

with $$\mathcal C(\rho)=\rho',\qquad \mathcal C(\sigma)=\sigma',$$2 projections. By Jordan’s lemma, such pairs decompose blockwise into the cq-pure form above, so flat states are contained in $$\mathcal C(\rho)=\rho',\qquad \mathcal C(\sigma)=\sigma',$$3. Weighted direct sums of flat states are also included because they admit the same type of decomposition. The main catalytic results for flat states therefore extend to a larger cq-pure class rather than remaining confined to projection-normalized states (Verhagen et al., 10 Jul 2025).

Two structural conditions recur. The first,

$$\mathcal C(\rho)=\rho',\qquad \mathcal C(\sigma)=\sigma',$$4

excludes perfectly parallel components. The second,

$$\mathcal C(\rho)=\rho',\qquad \mathcal C(\sigma)=\sigma',$$5

requires at least one genuinely orthogonal block. These arise from the semiring proof method through the notion of a power universal element. In the “everywhere overlapping” semiring, power universality is equivalent to

$$\mathcal C(\rho)=\rho',\qquad \mathcal C(\sigma)=\sigma',$$6

whereas in the larger “minimal restrictions” semiring, $$\mathcal C(\rho)=\rho',\qquad \mathcal C(\sigma)=\sigma',$$7 is sufficient. This explains why $$\mathcal C(\rho)=\rho',\qquad \mathcal C(\sigma)=\sigma',$$8 is needed in exact and asymptotic large-sample theorems but not in the full catalytic theorem.

The same theme reappears in classical matrix majorization with unequal supports. When support restrictions vary, the admissible monotone family changes substantially. In particular, the nested-support dichotomy

$$\mathcal C(\rho)=\rho',\qquad \mathcal C(\sigma)=\sigma',$$9

behaves differently from the equal-support case, and this distinction is decisive in asymptotic catalytic relative majorization of pairs (Verhagen et al., 2024).

3. Entropic monotones and exact or asymptotic criteria

The most detailed current characterization of catalytic relative majorization for flat-state-type quantum pairs is expressed through $(\tau,\omega)$0-$(\tau,\omega)$1 relative entropies. For $(\tau,\omega)$2, the paper uses

$(\tau,\omega)$3

for

$(\tau,\omega)$4

together with the tropical limit

$(\tau,\omega)$5

For $(\tau,\omega)$6, $(\tau,\omega)$7 coincides with the standard $(\tau,\omega)$8 on $(\tau,\omega)$9, up to a prefactor. The admissible region

$\mathcal C$0

is not ad hoc: it is forced by the classification of nondegenerate monotone homomorphisms. The resulting family includes the sandwiched Rényi line $\mathcal C$1 and the Petz line $\mathcal C$2, but the transformation criteria genuinely require the full two-parameter region. This is presented as the first operational interpretation of $\mathcal C$3-$\mathcal C$4 relative entropies in which $\mathcal C$5 and $\mathcal C$6 are truly independent (Verhagen et al., 10 Jul 2025).

In exact form, the main theorem states that if $\mathcal C$7, $\mathcal C$8 satisfies $\mathcal C$9, and

$$\mathcal C(\rho\otimes\tau)=\rho'\otimes\tau,\qquad \mathcal C(\sigma\otimes\omega)=\sigma'\otimes\omega,$$0

together with

$$\mathcal C(\rho\otimes\tau)=\rho'\otimes\tau,\qquad \mathcal C(\sigma\otimes\omega)=\sigma'\otimes\omega,$$1

then $$\mathcal C(\rho\otimes\tau)=\rho'\otimes\tau,\qquad \mathcal C(\sigma\otimes\omega)=\sigma'\otimes\omega,$$2 majorizes $$\mathcal C(\rho\otimes\tau)=\rho'\otimes\tau,\qquad \mathcal C(\sigma\otimes\omega)=\sigma'\otimes\omega,$$3 both in the large-sample setting and in the catalytic setting, with a catalyst pair in $$\mathcal C(\rho\otimes\tau)=\rho'\otimes\tau,\qquad \mathcal C(\sigma\otimes\omega)=\sigma'\otimes\omega,$$4. Conversely, large-sample or catalytic convertibility implies the same inequalities non-strictly. For the catalytic conclusion, $$\mathcal C(\rho\otimes\tau)=\rho'\otimes\tau,\qquad \mathcal C(\sigma\otimes\omega)=\sigma'\otimes\omega,$$5 need not satisfy $$\mathcal C(\rho\otimes\tau)=\rho'\otimes\tau,\qquad \mathcal C(\sigma\otimes\omega)=\sigma'\otimes\omega,$$6. Exact catalytic relative majorization is therefore “almost necessary and sufficient”: strict inequalities suffice, while actual convertibility yields only non-strict inequalities.

The asymptotic catalytic theorem is cleaner. If $$\mathcal C(\rho\otimes\tau)=\rho'\otimes\tau,\qquad \mathcal C(\sigma\otimes\omega)=\sigma'\otimes\omega,$$7, the source is non-parallel, and the target pair does not commute, then the following are equivalent: $$\mathcal C(\rho\otimes\tau)=\rho'\otimes\tau,\qquad \mathcal C(\sigma\otimes\omega)=\sigma'\otimes\omega,$$8 and the existence, for every $$\mathcal C(\rho\otimes\tau)=\rho'\otimes\tau,\qquad \mathcal C(\sigma\otimes\omega)=\sigma'\otimes\omega,$$9, of a state $F(\tau,\omega)>0$0 with

$F(\tau,\omega)>0$1

a quantum channel $F(\tau,\omega)>0$2, and a catalyst pair $F(\tau,\omega)>0$3 such that

$F(\tau,\omega)>0$4

Thus asymptotic catalytic relative majorization is characterized exactly by the family of $F(\tau,\omega)>0$5-$F(\tau,\omega)>0$6 relative entropy inequalities. The same paper also gives the optimal conversion rate: $F(\tau,\omega)>0$7 or, equivalently, by the minimum over the endpoint-extended $F(\tau,\omega)>0$8 family together with $F(\tau,\omega)>0$9.

4. Classical, matrix, and statistical-experiment formulations

The classical and finite-alphabet theory places catalytic relative majorization inside matrix majorization. A $(\rho,\sigma)$0-tuple $(\rho,\sigma)$1 majorizes $(\rho,\sigma)$2 if there exists a single stochastic matrix $(\rho,\sigma)$3 such that $(\rho,\sigma)$4, equivalently $(\rho,\sigma)$5 for all $(\rho,\sigma)$6. The case $(\rho,\sigma)$7 is exactly relative majorization. Catalytic matrix majorization asks whether there exists a tuple $(\rho,\sigma)$8 such that

$(\rho,\sigma)$9

For full-support finite alphabets, approximate catalytic matrix majorization is characterized exactly by the multivariate Rényi-type monotones $(\rho',\sigma')$0: for all relevant $(\rho',\sigma')$1,

$(\rho',\sigma')$2

if and only if $(\rho',\sigma')$3 catalytically majorizes arbitrarily good approximants of $(\rho',\sigma')$4. Specializing to $(\rho',\sigma')$5, this yields catalytic relative majorization with vanishing error, governed by Rényi divergences in both directions (Farooq et al., 2023).

Support conditions substantially alter the monotone family. In the nested-support dichotomy

$(\rho',\sigma')$6

exact catalytic relative majorization is guaranteed by the strict one-sided family

$(\rho',\sigma')$7

and exact catalytic conversion implies the same inequalities non-strictly. Under strict support inclusion, asymptotic catalytic relative majorization is characterized exactly by the non-strict inequalities for $(\rho',\sigma')$8, and one may keep the second target component exact. This completes the asymptotic catalytic picture for the $(\rho',\sigma')$9 support patterns treated in that work (Verhagen et al., 2024).

The theory also extends from finite sample spaces to regular finite statistical experiments over standard Borel spaces. If $z$00 and $z$01 are regular finite statistical experiments, then $z$02 means that there exists a Markov kernel $z$03 with $z$04 for all $z$05. Here $z$06 is again relative majorization. The multivariate continuous-space theorem gives sufficient and almost necessary conditions for both large-sample and catalytic majorization in terms of three families: temperate multivariate Rényi divergences $z$07, tropical multivariate Rényi divergences $z$08, and derivations given by convex combinations of pairwise Kullback–Leibler divergences. If all three families are strictly ordered, then there exists $z$09 such that

$z$10

and exact convertibility implies the same inequalities non-strictly. The catalyst can be chosen explicitly as

$z$11

This places catalytic relative majorization of binary experiments inside a broader real-algebraic asymptotic-spectrum theory (Haapasalo, 30 Jun 2026).

A different but foundational viewpoint is provided by quantum relative Lorenz curves. For a pair $z$12, the testing region

$z$13

defines a preorder

$z$14

This is equivalent to the family of trace-norm inequalities

$z$15

equivalently all hypothesis-testing relative entropies, and equivalently Hilbert $z$16-divergences in both directions. Although this work does not develop catalysts, it provides the noncommutative benchmark that later catalytic theories extend (Buscemi et al., 2016).

5. Thermodynamic, correlated, and symmetry-restricted forms

In thermodynamics, relative majorization specializes to thermo-majorization by fixing the second component to a Gibbs state. In the quasi-classical setting, catalytic relative majorization with correlations can be formulated as follows: for distributions $z$17, there exist a distribution $z$18, a joint distribution $z$19, and a classical channel $z$20 such that

$z$21

where $z$22 is uniform on the support of $z$23, the marginals of $z$24 are $z$25 and $z$26, $z$27 is arbitrarily close to $z$28, and the residual correlation can be made arbitrarily small in relative entropy. In this correlated-catalytic model, approximate pair conversion is equivalent to the single inequality

$z$29

Under additional rationality and support assumptions, an exact-marginal version requires

$z$30

This gives ordinary relative entropy a one-shot operational role in quasi-classical asymmetric distinguishability (Rethinasamy et al., 2019).

A complementary formulation replaces additive Rényi divergences by non-additive Tsallis divergences. In the uncorrelated catalytic thermal regime, this yields generalized free energies with an explicit catalyst-dependent correction term because

$z$31

The same catalytic relative-majorization preorder can therefore be represented equivalently by Tsallis-type monotones. In particular, a corollary states

$z$32

The same paper also shows, through explicit thermo-majorization examples, that in correlated catalysis reduced-state data are generally insufficient: the thermo-majorization behavior of the joint transformation can change while the system and catalyst marginals remain fixed, and even states with identical marginals and the same mutual information can differ in thermo-majorization accessibility (Günhan et al., 23 Apr 2026).

Symmetry adds another layer. Equivariant relative submajorization studies triples $z$33 under completely positive trace-nonincreasing maps $z$34 satisfying

$z$35

for all $z$36. Catalysis is defined semiring-theoretically by tensoring with an exact-return auxiliary equivariant object. Strict inequality of all spectral monotones implies existence of a catalyst, and the asymptotic relaxation is characterized by the full asymptotic spectrum. In the classical and commuting-$z$37 regimes, the explicit monotones are sandwiched Rényi-type quantities

$z$38

and these generate second laws for time-translation symmetric Gibbs-preserving maps and thermal processes even in catalytic and multicopy settings (Bunth et al., 2021).

6. Scope, monotone complexity, and open directions

The current theory is structurally rich but incomplete. The flat-state $z$39-$z$40 theorem does not solve the fully general noncommutative problem; its exact and asymptotic large-sample statements require additional source assumptions, and the paper explicitly notes as an open problem the possible removal of $z$41 by analyzing a different semiring. The continuous-space multivariate theory likewise depends on a regularity condition of mutual absolute continuity plus essential boundedness of likelihood ratios, and the authors note that removing or weakening this assumption—especially for catalytic majorization—remains open (Verhagen et al., 10 Jul 2025, Haapasalo, 30 Jun 2026).

A second issue is monotone complexity. For ordinary majorization, thermo-majorization, and trumping, exact scalar-monotone characterizations exhibit an infinite-second-laws phenomenon. In particular, if $z$42, the smallest family of second laws for majorization is countably infinite; for $z$43-majorization with nonuniform reference, even $z$44 already forces countably infinite second laws; and any characterization of strict trumping of the form

$z$45

cannot use finite $z$46. This does not directly prove the same statement for full catalytic relative majorization, but it suggests that finite exact families of real-valued monotones are unlikely except in specially collapsed regimes (Hack et al., 2022).

At the same time, finite sufficient criteria do exist in related special cases. For ordinary catalytic majorization, a finite family of symmetric-polynomial inequalities $z$47, together with entropy conditions, suffices to certify trumping; for catalytic thermal transitions with a fixed Gibbs reference $z$48, the same approach gives finite sufficient criteria after an embedding that maps $z$49 to a uniform distribution. These are certificates of sufficiency rather than complete criteria, and they do not solve the full pair-to-pair catalytic relative-majorization problem. A plausible implication is that future work may continue to bifurcate into exact but infinite complete families, and finite but one-sided certification schemes (Elkouss et al., 27 Feb 2025).

Across the subject, catalytic relative majorization now has several distinct faces: exact one-shot catalysis, large-sample comparison, asymptotic catalytic conversion with vanishing error, correlated catalysis, relative submajorization, and symmetry-restricted variants. The existing literature establishes complete criteria in some of these regimes, especially for classical pairs, matrix majorization, binary experiments, and flat-state-type quantum pairs, while leaving the fully general noncommutative, support-irregular, and correlated-joint-state problems only partially resolved.