ABB Entropy: Multiple Definitions and Applications

• ABB entropy is a family of context-dependent entropy measures arising in deformed distributions, quantum processes, and algebraic dynamics.
• In deformed binomial distributions, ABB entropy highlights non-extensive Boltzmann-Gibbs scaling versus extensive Rényi behavior, illustrating key statistical differences.
• In quantum systems and algebraic dynamics, ABB entropy underpins robust entanglement measures, distillation bounds, and links to Mahler measures and the Lehmer problem.

Searching arXiv for recent and foundational uses of “ABB entropy” to ground the article in the cited literature. ABB entropy is not a single invariant with a universally fixed definition. In current arXiv usage, the label appears in several non-equivalent senses: as an allusion to the Boltzmann-Gibbs entropy of the modified Abel-polynomial deformation of the binomial law; as a von Neumann entropy attached to a normalized transition matrix in post-selected quantum systems; and, in algebraic dynamics, as the algebraic entropy of endomorphisms of abelian groups. A separate acronymic usage occurs in the entropy accumulation literature, where “ABB” denotes the original Arnon-Friedman, Renner, and Berta formulation of the entropy accumulation theorem rather than a distinct entropy functional (Bergeron et al., 2014, Chen et al., 12 Aug 2025, Dikranjan et al., 2010, Metger et al., 2022).

1. Terminological scope and competing definitions

The term is best understood as a family of context-dependent designations rather than a canonical quantity. In the deformed-binomial literature, “ABB entropy” refers to the Abel-related Boltzmann-Gibbs entropy and is explicitly stated not to be “defined as a new quantity in the text”; in the transition-matrix literature it is a formally defined entropy functional; in algebraic dynamics it denotes a fully axiomatized invariant for endomorphisms of abelian groups; and in entropy accumulation it is an author acronym attached to the original theorem rather than the name of an entropy measure (Bergeron et al., 2014, Chen et al., 12 Aug 2025, Dikranjan et al., 2010, Metger et al., 2022).

Usage	Underlying object	Defining feature
Abel-Boltzmann-Gibbs usage	Modified Abel-polynomial deformed binomial distribution	BG entropy scales as $\sqrt{n}$, while Rényi entropy is extensive
Transition-matrix ABB entropy	State-normalized transition matrix $\tilde{\tau}\tilde{\tau}^\dagger$	Von Neumann entropy $$-\operatorname{Tr}(\tilde{\tau}\tilde{\tau}^\dagger \ln(\tilde{\tau}\tilde{\tau}^\dagger))$$
Algebraic ABB entropy	Endomorphism $\phi$ of an abelian group	Growth rate of finite trajectories, with Addition and Uniqueness Theorems
“ABB EAT” usage	Sequential quantum process	“ABB” names the original EAT authors, not a new entropy

This multiplicity is not merely terminological. Each usage isolates a different type of structure: correlated discrete probability laws, post-selected quantum channels, algebraic dynamics on groups, or entropy accumulation under sequential non-signalling constraints. A plausible implication is that any occurrence of “ABB entropy” must be interpreted locally from the surrounding formalism rather than from the acronym alone.

2. Abel-type deformed binomial distributions

In "Entropies of deformed binomial distributions" (Bergeron et al., 2014), the Abel case is one of three symmetric generalizations of the binomial distribution. Its generating function is

$$\mathcal{N}(t)=e^{-\alpha W(-t/\alpha)}, \qquad \alpha>0,$$

with $W$ the Lambert function. The associated probability law is symmetric under $k\to n-k$ and $\eta\to 1-\eta$, and the paper treats this case as exceptional among the three deformations studied.

The key entropy result is asymptotic. For the modified Abel-polynomial distribution, the Boltzmann-Gibbs entropy requires regularization because of divergences at the boundaries $x=0$ and $x=1$, and the dominant term is

$\tilde{\tau}\tilde{\tau}^\dagger$0

Accordingly, the Boltzmann-Gibbs entropy is non-extensive: it grows like $\tilde{\tau}\tilde{\tau}^\dagger$1 rather than proportionally to $\tilde{\tau}\tilde{\tau}^\dagger$2. The Tsallis entropy is likewise non-extensive for any $\tilde{\tau}\tilde{\tau}^\dagger$3. By contrast, the Rényi entropy satisfies

$\tilde{\tau}\tilde{\tau}^\dagger$4

and its leading-order term is independent of the Rényi parameter $\tilde{\tau}\tilde{\tau}^\dagger$5 (Bergeron et al., 2014).

The same paper emphasizes that this “ABB entropy” is not introduced as a distinct new entropy functional; it is an allusion to the Boltzmann-Gibbs entropy computed for the Abel-polynomial model. The Abel case stands out because only the Rényi entropy is extensive, whereas in the $\tilde{\tau}\tilde{\tau}^\dagger$6-exponential and Hermite cases both Boltzmann-Gibbs and Rényi entropies are extensive. The paper also provides a combinatorial or alphabet-word probabilistic interpretation for the Abel distribution, and states that the Abel class exhibits a non-trivial and highly correlated statistical structure. This suggests that the Abel deformation supplies a concrete example in which Rényi extensivity survives even when Boltzmann-Gibbs extensivity fails.

3. ABB entropy for transition matrices in random quantum systems

"Entropy Measures for Transition Matrices in Random Systems" (Chen et al., 12 Aug 2025) introduces a formally defined ABB entropy for a transition matrix $\tilde{\tau}\tilde{\tau}^\dagger$7 obtained by post-selection between two quantum states. After state normalization,

$\tilde{\tau}\tilde{\tau}^\dagger$8

the positive matrix $\tilde{\tau}\tilde{\tau}^\dagger$9 has eigenvalues

$$-\operatorname{Tr}(\tilde{\tau}\tilde{\tau}^\dagger \ln(\tilde{\tau}\tilde{\tau}^\dagger))$$0

where the $$-\operatorname{Tr}(\tilde{\tau}\tilde{\tau}^\dagger \ln(\tilde{\tau}\tilde{\tau}^\dagger))$$1 are singular values of $$-\operatorname{Tr}(\tilde{\tau}\tilde{\tau}^\dagger \ln(\tilde{\tau}\tilde{\tau}^\dagger))$$2. The ABB entropy is then

$$-\operatorname{Tr}(\tilde{\tau}\tilde{\tau}^\dagger \ln(\tilde{\tau}\tilde{\tau}^\dagger))$$3

with Rényi generalization

$$-\operatorname{Tr}(\tilde{\tau}\tilde{\tau}^\dagger \ln(\tilde{\tau}\tilde{\tau}^\dagger))$$4

The paper states that $$-\operatorname{Tr}(\tilde{\tau}\tilde{\tau}^\dagger \ln(\tilde{\tau}\tilde{\tau}^\dagger))$$5 is real, non-negative, bounded above by $$-\operatorname{Tr}(\tilde{\tau}\tilde{\tau}^\dagger \ln(\tilde{\tau}\tilde{\tau}^\dagger))$$6, and invariant under bi-unitary transformations $$-\operatorname{Tr}(\tilde{\tau}\tilde{\tau}^\dagger \ln(\tilde{\tau}\tilde{\tau}^\dagger))$$7 (Chen et al., 12 Aug 2025).

Operationally, the paper assigns ABB entropy a distinct probabilistic meaning. It identifies $$-\operatorname{Tr}(\tilde{\tau}\tilde{\tau}^\dagger \ln(\tilde{\tau}\tilde{\tau}^\dagger))$$8 with the entanglement entropy of the normalized Choi-Jamiołkowski state associated with $$-\operatorname{Tr}(\tilde{\tau}\tilde{\tau}^\dagger \ln(\tilde{\tau}\tilde{\tau}^\dagger))$$9, and argues that only ABB entropy among the entropies considered has a distillation interpretation: in the many-copy limit, a dominant sub-transition matrix of rank $\phi$0 with $\phi$1 can be robustly distilled with maximal joint success probability. The asymptotic relation

$\phi$2

encodes that interpretation (Chen et al., 12 Aug 2025).

The paper also contrasts ABB entropy with pseudo entropy, modified pseudo entropy, and SVD entropy. Pseudo entropy can be complex and unbounded; modified pseudo entropy can be negative or complex; SVD entropy is real, positive, and bounded but lacks the distillation interpretation and is not monotonic under operations in general. By contrast, the ABB entropy is reported to never increase under post-selected measurements and trace-preserving quantum operations:

$\phi$3

For transition matrices built from independent Haar-random states, bi-orthogonal eigenstates of non-Hermitian random systems, and bi-orthogonal states in $\phi$4-symmetric systems near exceptional points, ABB and SVD entropies track Page-curve-type behavior, while pseudo-type entropies do not; specifically,

$\phi$5

The paper further states that ABB entropy remains finite at exceptional points, unlike pseudo and modified pseudo entropies (Chen et al., 12 Aug 2025).

4. Algebraic entropy of endomorphisms of abelian groups

In "Entropy on abelian groups" (Dikranjan et al., 2010), ABB entropy refers to algebraic entropy for endomorphisms of arbitrary abelian groups. Let $\phi$6 be an abelian group, $\phi$7, and $\phi$8 a non-empty finite subset. The $\phi$9-th $$\mathcal{N}(t)=e^{-\alpha W(-t/\alpha)}, \qquad \alpha>0,$$0-trajectory is

$$\mathcal{N}(t)=e^{-\alpha W(-t/\alpha)}, \qquad \alpha>0,$$1

and the entropy with respect to $$\mathcal{N}(t)=e^{-\alpha W(-t/\alpha)}, \qquad \alpha>0,$$2 is

$$\mathcal{N}(t)=e^{-\alpha W(-t/\alpha)}, \qquad \alpha>0,$$3

The algebraic entropy is

$$\mathcal{N}(t)=e^{-\alpha W(-t/\alpha)}, \qquad \alpha>0,$$4

The existence of the limit is guaranteed by subadditivity via Fekete’s lemma (Dikranjan et al., 2010).

This entropy has a structural theory closely modeled on entropy in dynamics. The paper lists invariance under conjugation, the logarithmic law $$\mathcal{N}(t)=e^{-\alpha W(-t/\alpha)}, \qquad \alpha>0,$$5 for $$\mathcal{N}(t)=e^{-\alpha W(-t/\alpha)}, \qquad \alpha>0,$$6, continuity for direct limits, additivity on direct products, and monotonicity with respect to invariant subgroups and quotients. Its cornerstone is the Addition Theorem:

$$\mathcal{N}(t)=e^{-\alpha W(-t/\alpha)}, \qquad \alpha>0,$$7

for every $$\mathcal{N}(t)=e^{-\alpha W(-t/\alpha)}, \qquad \alpha>0,$$8-invariant subgroup $$\mathcal{N}(t)=e^{-\alpha W(-t/\alpha)}, \qquad \alpha>0,$$9 and induced endomorphism $W$0 (Dikranjan et al., 2010).

The number-theoretic content enters through the Algebraic Yuzvinski Formula. For $W$1,

$W$2

where $W$3 is the characteristic polynomial of $W$4, $W$5 makes $W$6 primitive integral, and $W$7 is the Mahler measure. The same paper proves a Uniqueness Theorem characterizing the entropy by conjugation invariance, continuity on direct systems, the Addition Theorem, Bernoulli normalization, and Yuzvinski/Mahler normalization. It also relates the infimum of positive algebraic entropies to the Lehmer problem:

$W$8

where $W$9 is the infimum of positive Mahler measures of primitive integer polynomials (Dikranjan et al., 2010).

Examples anchor the abstract theory. Multiplication by $k\to n-k$0 on $k\to n-k$1 has entropy $k\to n-k$2 for $k\to n-k$3, with value $k\to n-k$4 when $k\to n-k$5. For a Bernoulli shift $k\to n-k$6 on $k\to n-k$7, one has $k\to n-k$8 when $k\to n-k$9 is finite and $\eta\to 1-\eta$0 for infinite $\eta\to 1-\eta$1. In this sense, algebraic ABB entropy is a functorial growth invariant measuring orbit complexity of finite sets under iteration.

5. ABB in entropy accumulation theory

A different usage appears in "Generalised entropy accumulation" (Metger et al., 2022). There, “ABB” refers to the original entropy accumulation theorem of Arnon-Friedman, Renner, and Berta, rather than to a separate entropy measure. The original ABB EAT assumes that past side information is not updated and imposes a quantum Markov chain condition on newly generated side information. The generalised theorem replaces that model by a fully updatable side-information register subject only to a non-signalling condition.

For channels $\eta\to 1-\eta$2, the non-signalling condition is the existence of $\eta\to 1-\eta$3 such that

$\eta\to 1-\eta$4

Under this condition, the paper proves a generalized min-entropy lower bound of the form

$\eta\to 1-\eta$5

together with explicit second-order corrections and a testing version incorporating observed statistics (Metger et al., 2022).

The technical novelty lies in improved chain rules for sandwiched Rényi divergence and entropy and in a new variant of Uhlmann’s theorem for $\eta\to 1-\eta$6. In this literature, the ABB label therefore functions bibliographically: it identifies the prior EAT framework to be generalized. A common misconception is to read “ABB entropy” here as naming another entropy functional; the source instead concerns entropy accumulation bounds for smoothed min-entropy in sequential quantum processes.

6. Related abbreviations, neighboring entropies, and recurrent confusions

The literature also contains nearby uses of “ABB” that are not entropy definitions. In "A Parallel Privacy-Preserving Shortest Path Protocol from a Path Algebra Problem" (Anagreh et al., 2022), ABB denotes the Arithmetic Black Box model for secure multiparty computation. The model exposes primitives such as storing a value, performing arithmetic on secret handles, and declassifying only final outputs. Entropy is discussed there as the uncertainty of secret-shared values that persists until declassification, but ABB names the computational interface, not a standard entropy functional (Anagreh et al., 2022).

Other neighboring entropy notions can further obscure the label. "Rare dense solutions clusters in asymmetric binary perceptrons -- local entropy via fully lifted RDT" (Stojnic, 24 Jun 2025) studies local entropy as a geometric observable on solution clusters, with a breakdown window for $\eta\to 1-\eta$7 that matches the reported algorithmic window $\eta\to 1-\eta$8–$\eta\to 1-\eta$9. "Epistasis and Entropy" (Crona, 2015) studies shared entropy or mutual information between loci and shows that non-zero shared entropy does not imply pairwise epistasis. These works are conceptually distinct from all three principal ABB-entropy usages discussed above (Stojnic, 24 Jun 2025, Crona, 2015).

Taken together, the terminology supports a narrow conclusion. “ABB entropy” has no field-independent meaning. In one context it is an Abel-specific Boltzmann-Gibbs quantity whose non-extensivity is contrasted with extensive Rényi behavior; in another it is a post-selection-based quantum entropy with monotonicity and a distillation interpretation; in a third it is the algebraic entropy of endomorphisms of abelian groups with deep links to Mahler measure and the Lehmer problem; and in entropy accumulation theory it is merely an acronymic reference to earlier authors. Any rigorous use of the term therefore requires explicit specification of the ambient formalism and the object whose entropy is being computed.