InfoExchangeUnits: Formalizing Information Measurement

• InfoExchangeUnits are formally defined units that quantify information exchange using metrics like entropy, mutual information, and related measures.
• They extend classical SI/ISQ paradigms by integrating theoretical, thermodynamic, and operational aspects to enable consistent unit conversions and safe software representations.
• Their framework facilitates precise evaluation of information flow in communication protocols, Bayesian inference, and simulation environments, ensuring rigorous interoperability.

InfoExchangeUnits encompass the rigorous definition, representation, quantification, and operationalization of units describing information exchange. Building from the classical SI/ISQ paradigm for physical units, InfoExchangeUnits formalize the measurement and manipulation of information in both abstract (entropy, mutual information) and operational (data, energy, communication) terms. The concept spans theoretical information theory, thermodynamics, Bayesian inference, interactive communication, and practical frameworks for safe, interoperable quantity exchange in software and simulation environments.

1. Theoretical Foundations and Unified Notation

InfoExchangeUnits are defined primarily via information-theoretic quantities such as entropy, mutual information, and related specific measures, all expressible in logarithmic units—nats (base e logarithm) or bits (base 2). The unified notation (Kirsch et al., 2021) expresses:

• Shannon surprise: $h(p(x)) = -\log p(x)$, the information content of an outcome.
• Entropy: $H(X) = -\sum_x p(x)\log p(x)$, the expected surprise.
• Mutual information:

$$I(X;Y) = H(X) - H(X|Y) = D_{KL}(p(x,y)\,\|\,p(x)p(y)) = \sum_{x,y} p(x,y)\log\frac{p(x,y)}{p(x)p(y)}$$

• Pointwise mutual information (PMI): $i(x;y)=\log \frac{p(x|y)}{p(x)}$, quantifying information in the observation of $y$ about $x$.

All such quantities—when computed with natural logarithms—are measured in nats; if base-2, in bits. Each unit (nat or bit) directly quantifies the amount of information exchanged, and these are the standard InfoExchangeUnits across domains.

2. InfoExchangeUnits in Thermodynamics and Physical Systems

The link between information exchange and thermodynamics is formalized by Sagawa & Ueda (Sagawa et al., 2013), who show that increments of mutual information $\Delta I$—measured in nats—function as InfoExchangeUnits in entropy production and energetic cost:

• Total entropy production: $\Delta S_{\rm tot} = \Delta s_{XB} - \Delta I_{XY}$, where $\Delta I_{XY}$ is the change in mutual information between subsystems $X$ and $H(X) = -\sum_x p(x)\log p(x)$0.
• Energetic cost per InfoExchangeUnit: For a system at inverse temperature $H(X) = -\sum_x p(x)\log p(x)$1,

$H(X) = -\sum_x p(x)\log p(x)$2

i.e., at most $H(X) = -\sum_x p(x)\log p(x)$3 of work is extractable per nat of exchanged information, and $H(X) = -\sum_x p(x)\log p(x)$4 per bit, matching the Landauer bound.

In classical mechanics, Miller-Dickson and Rose (Miller-Dickson et al., 30 Jan 2025) define information-exchange rates in interacting particle systems by computing the mutual information $H(X) = -\sum_x p(x)\log p(x)$5 as the core InfoExchangeUnit. They show that the maximal information-exchange rate (in nats/s) is proportional to the ratio of environmental power input $H(X) = -\sum_x p(x)\log p(x)$6 to initial system energy $H(X) = -\sum_x p(x)\log p(x)$7:

$H(X) = -\sum_x p(x)\log p(x)$8

and confirm that the actual units are nats/s.

3. InfoExchangeUnits in Data Communication and Interactive Protocols

Interactive communication protocols quantify the minimal number of exchanged “exchange units”—bits—to achieve reliable data exchange between parties observing correlated variables (Tyagi et al., 2016). For random variables $H(X) = -\sum_x p(x)\log p(x)$9, $$I(X;Y) = H(X) - H(X|Y) = D_{KL}(p(x,y)\,\|\,p(x)p(y)) = \sum_{x,y} p(x,y)\log\frac{p(x,y)}{p(x)p(y)}$$0:

• One-shot exchange units: For data exchange with error $$I(X;Y) = H(X) - H(X|Y) = D_{KL}(p(x,y)\,\|\,p(x)p(y)) = \sum_{x,y} p(x,y)\log\frac{p(x,y)}{p(x)p(y)}$$1, the minimum number of bits is determined by the conditional information densities:

$$I(X;Y) = H(X) - H(X|Y) = D_{KL}(p(x,y)\,\|\,p(x)p(y)) = \sum_{x,y} p(x,y)\log\frac{p(x,y)}{p(x)p(y)}$$2

The sum $$I(X;Y) = H(X) - H(X|Y) = D_{KL}(p(x,y)\,\|\,p(x)p(y)) = \sum_{x,y} p(x,y)\log\frac{p(x,y)}{p(x)p(y)}$$3 serves as the fundamental InfoExchangeUnit required for successful exchange in each realization.

• Second-order refinements: Asymptotic results exhibit a strict reduction in the second-order term for interactive (exchange-unit aware) versus non-interactive protocols, operationalizing InfoExchangeUnits as precisely the quantal slices of the information spectrum.

4. Information Mechanics: Conservation and Local-Global Exchange

The infomechanics framework (Isomura, 21 Jan 2026) articulates two additive "conserved charges" of information exchange measured in InfoExchangeUnits:

• Global (Shannon entropy): $$I(X;Y) = H(X) - H(X|Y) = D_{KL}(p(x,y)\,\|\,p(x)p(y)) = \sum_{x,y} p(x,y)\log\frac{p(x,y)}{p(x)p(y)}$$4 in nats or bits, representing the total uncertainty “volume.”
• Local (Fisher information): $$I(X;Y) = H(X) - H(X|Y) = D_{KL}(p(x,y)\,\|\,p(x)p(y)) = \sum_{x,y} p(x,y)\log\frac{p(x,y)}{p(x)p(y)}$$5 in $$I(X;Y) = H(X) - H(X|Y) = D_{KL}(p(x,y)\,\|\,p(x)p(y)) = \sum_{x,y} p(x,y)\log\frac{p(x,y)}{p(x)p(y)}$$6(length)$$I(X;Y) = H(X) - H(X|Y) = D_{KL}(p(x,y)\,\|\,p(x)p(y)) = \sum_{x,y} p(x,y)\log\frac{p(x,y)}{p(x)p(y)}$$7 or analogous units, representing curvature/sharpness.
• State function (information potential) $$I(X;Y) = H(X) - H(X|Y) = D_{KL}(p(x,y)\,\|\,p(x)p(y)) = \sum_{x,y} p(x,y)\log\frac{p(x,y)}{p(x)p(y)}$$8: A non-additive, dimensionless function:

$$I(X;Y) = H(X) - H(X|Y) = D_{KL}(p(x,y)\,\|\,p(x)p(y)) = \sum_{x,y} p(x,y)\log\frac{p(x,y)}{p(x)p(y)}$$9

captures structural/combinatorial information beyond entropy alone.

Exchange of information in Bayesian updating is strictly balanced: any pointwise reduction in posterior surprisal is exactly offset by the information gain from data, enforcing all transformations in InfoExchangeUnits (nats or bits).

5. Representation and Safe Exchange of InfoExchangeUnits in Software Systems

Formal libraries such as the SAFE-ISQ VDM library (Freitas, 2023) introduce type-safe, interoperable representations for InfoExchangeUnits in computational environments. The scheme is grounded in the International System of Quantities (ISQ), with seven base dimensions and supports robust mapping for both physical and information-theoretic units.

• Core constructs:
  • Dimension vectors: Maps from base dimensions to integer exponents, e.g., for "nat" or "bit" units, all exponents are zero except the information dimension (if represented separately).
  • Quantities: Bundles of magnitude and dimension vector; InfoExchangeUnits are realized as $i(x;y)=\log \frac{p(x|y)}{p(x)}$0 with $i(x;y)=\log \frac{p(x|y)}{p(x)}$1 reflecting information (entropy, mutual information).
  • Measurement systems: Triples of quantity, conversion schema, and tag; allow transformations (e.g., nats $i(x;y)=\log \frac{p(x|y)}{p(x)}$2 bits) via conversion functions correctly parameterized for logarithm base.
• Serialization: JSON-like structures encapsulate both numerical magnitude and explicit dimension, ensuring precise, safe info-unit exchange.
• Algebraic safety: Type invariants and conversion preconditions catch dimension/range errors at runtime and in proofs.

Component	Example (InfoExchangeUnits context)	Notes
Quantity	$i(x;y)=\log \frac{p(x|y)}{p(x)}$3	5 nats
ConversionSchema	“Nat to Bit”: $i(x;y)=\log \frac{p(x|y)}{p(x)}$4	Enables safe nat/bit conversion
Export Schema	JSON, with “magnitude”, “dim”, “schema”	Enables interoperability

This formalization allows InfoExchangeUnits to become first-class entities in scientific modeling and simulation environments, robust against unit mismatch, and compatible with proof platforms like Isabelle/HOL.

6. Unit-System Specification and Exchange

A foundational requirement for InfoExchangeUnits is the ability to unambiguously define, convert, and exchange unit-system specifications in machine-readable formats (Quincey et al., 2017). The proposed schema comprises:

• Base unit vector: Each base quantity (e.g., information) has a name, symbol, unit symbol, SI size, dimension symbol.
• Conventions: Natural-unity conventions (e.g., $i(x;y)=\log \frac{p(x|y)}{p(x)}$5), and link-unit conventions (e.g., defining logarithm base for nats vs. bits).
• Defining constants: For SI-like systems, all reference values and conversion factors are explicated.

This approach extends ISQ/SI style frameworks to support InfoExchangeUnits, ensuring that, for any domain—physical, informational, or computational—complete and lossless exchange of unit systems is possible.

7. Applications and Operational Implications

InfoExchangeUnits provide operational tools across domains:

• Active learning and Bayesian experimental design: Quantities like information gain, BALD acquisition, and core-set selection are immediately interpretable as expectations or realizations of InfoExchangeUnits (Kirsch et al., 2021).
• Physical limits of information transfer: The $i(x;y)=\log \frac{p(x|y)}{p(x)}$6 bound sets universal unit rates for information flow in classical systems (Miller-Dickson et al., 30 Jan 2025).
• Thermodynamics of computation: The link between mutual information and extractable work quantifies the fundamental cost of information exchange in nats or bits (Sagawa et al., 2013).
• Interactive protocol design: Accurate estimation and minimization of required InfoExchangeUnits allow optimal data-exchange protocol design (Tyagi et al., 2016).
• Software and simulation safety: Rigorously typed InfoExchangeUnits in SAFE-ISQ VDM library architectures prevent trivial (but historically costly) errors in engineering and computational pipelines (Freitas, 2023).

The net effect is the unification of disparate approaches to uncertainty, communication, and measurement under a single, extensible, and machine-verifiable regime of InfoExchangeUnits, directly enhancing rigor and clarity across research and application domains.