Min-entropy¶
The negative logarithm of the largest outcome probability, measuring worst-case single-guess unpredictability as the order-infinity Rényi entropy.
Core Idea¶
Classical, conditional, smooth and quantum min-entropies differ in carrier and operational interpretation; smoothing permits nearby distributions and conditional forms measure adversarial guessing probability. The distribution is reduced to its highest-probability outcome, whose probability is converted by a negative logarithm into the conservative number of unpredictable bits. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
The load-bearing residual is not the broad topic of information theory. It is the domain-specific identity determined by the random variable or state, probability or density convention, logarithm base, maximum-probability event, conditioning information, classical or quantum definition, smoothing radius and operational claim are explicit.
Scope of Application¶
Min-entropy belongs to information theory and is useful where the analyst can specify the typed information theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the random variable or state, probability or density convention, logarithm base, maximum-probability event, conditioning information, classical or quantum definition, smoothing radius and operational claim are explicit. The scope is broad within that domain but bounded by the need for the random variable or state, probability or density convention, logarithm base, maximum-probability event, conditioning information, classical or quantum definition, smoothing radius and operational claim are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the random variable or state, probability or density convention, logarithm base, maximum-probability event, conditioning information, classical or quantum definition, smoothing radius and operational claim are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Min-entropy can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Min-entropy. Min-entropy compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed information theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the random variable or state, probability or density convention, logarithm base, maximum-probability event, conditioning information, classical or quantum definition, smoothing radius and operational claim are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of information theory because they reuse the typed information theory carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, The distribution is reduced to its highest-probability outcome, whose probability is converted by a negative logarithm into the conservative number of unpredictable bits., and type the carrier, state every parameter and convention in the definition, test that the random variable or state, probability or density convention, logarithm base, maximum-probability event, conditioning information, classical or quantum definition, smoothing radius and operational claim are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Min-entropy Domain-specific
Parents (1) — more general patterns this builds on
-
Min-entropy is a kind of Information Prime
The proposed strict upward parent is
prime:information.
Hierarchy path (1) — routes to 1 parentless root
- Min-entropy → Information → Uncertainty
Neighborhood in Abstraction Space¶
Min-entropy sits in a crowded region of the domain-specific corpus (9th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Logarithmic Information & Scale (9 abstractions)
Nearest neighbors
- Information dimension — 0.95
- Binary entropy function — 0.94
- Asymptotic equipartition property — 0.94
- Typical set — 0.93
- Directed information — 0.93
Computed from structural-signature embeddings · 2026-09-08