First Law of Information Thermodynamics

• The First Law of Information Thermodynamics is defined as a framework that recasts energy conservation using information-theoretic measures like entropy and relative entropy.
• It applies across quantum, classical, equilibrium, and nonequilibrium systems by substituting traditional thermal baths with intrinsic temperature measures and Gibbs-like states.
• The law explains that energy changes during state transformations are determined by conserved information quantities, establishing limits on work extraction and dissipation.

The First Law of Information Thermodynamics expresses energy conservation in thermodynamic systems by recasting classical thermodynamic quantities in terms of information-theoretic measures, most prominently entropy and relative entropy. Diverse frameworks demonstrate that fundamental energy balances, as well as the operational distinctions between heat and work, can be formulated as direct consequences of information conservation—often making explicit the deep correspondence between physical laws and the structure of optimal inference. This approach applies seamlessly to quantum, classical, equilibrium, and driven nonequilibrium scenarios, and generalizes the classical first law without dependence on a global bath temperature or even the notion of a physical bath.

1. Foundations: Information, Entropy, and Energy Functionals

The unifying concept across all rigorous treatments is the role of entropy as a quantifier of coarse-grained information, linking the microscopic quantum or classical state to macroscopic thermodynamic constraints. For a quantum system in state $\rho$ with Hamiltonian $H$, the von Neumann entropy $S(\rho) = -\mathrm{Tr}[\rho\ln\rho]$ quantifies the information content, while the mean energy $E(\rho) = \mathrm{Tr}[H\rho]$ defines the internal energy (Bera et al., 2017, Bera et al., 2018).

The minimum energy compatible with fixed entropy, called "bound energy" $$E_{\mathrm{bound}}(\rho) = \min_{\sigma: S(\sigma)=S(\rho)} \mathrm{Tr}[H\sigma]$$, is achieved by a Gibbs-form "completely passive" state $\gamma(\rho) = e^{-\beta(\rho) H}/Z_{\beta(\rho)}$, where $\beta(\rho)$ is the intrinsic inverse temperature determined by the entropy constraint. The "free energy" $F(\rho) = E(\rho) - E_{\mathrm{bound}}(\rho)$ quantifies the energy extractable by entropy-preserving (adiabatic) operations, independent of the existence of a physical thermal bath (Bera et al., 2017, Bera et al., 2018). Classical analogues use the Shannon entropy or Kullback-Leibler divergence $D(p\|\pi)$, where $p$ is an arbitrary distribution and $H$0 is the equilibrium (Gibbs) state (Gopalkrishnan, 2013).

2. First Law as an Information Identity and Its Generalization

In its most general form, the First Law of Information Thermodynamics states that for an iso-entropic (entropy-preserving) transformation $H$1, the energy change decomposes as

$H$2

where $H$3 (change in bound energy) is identified as heat, and $H$4 (change in free energy) as work. The only assumption is conservation of information ($H$5); the result holds in quantum and classical systems, with or without baths, and without any fixed environmental temperature (Bera et al., 2017, Bera et al., 2018). In the classical limit, $H$6 when the intrinsic temperature $H$7 is well-defined.

Several alternative, but equivalent, information-theoretic formulations demonstrate that the first law—traditionally an equality—also implies universal Landauer-like inequalities bounding energy-entropy tradeoffs even in nonthermal or unknown bath scenarios (Liu et al., 2023). In these, system-intrinsic reference temperatures and relative entropy corrections replace explicit heat-bath exchange terms, further emphasizing the foundational role of information (Liu et al., 2023, Tsuruyama, 31 Dec 2025).

3. Information-Analogues of Heat, Work, and Free Energy

Different frameworks converge on the identification of thermodynamic variables with information-theoretic primitives:

• Bound energy ($H$8): minimal energy at fixed entropy, the portion locked by information constraints, interpreted as 'heat capacity' within the information-theoretic regime (Bera et al., 2017, Bera et al., 2018).
• Free energy ($H$9): energy available for work under entropy-preserving evolution.
• Heat ($S(\rho) = -\mathrm{Tr}[\rho\ln\rho]$0): generalized as the change in bound energy, which in traditional contexts corresponds to $S(\rho) = -\mathrm{Tr}[\rho\ln\rho]$1.
• Work ($S(\rho) = -\mathrm{Tr}[\rho\ln\rho]$2): change in free energy, operationally corresponding to extractable energy.

Table: Summary of Key Quantities

Thermodynamic Variable	Information-Theoretic Analogue	Reference(s)
Entropy $S(\rho) = -\mathrm{Tr}[\rho\ln\rho]$3	von Neumann/ Shannon / observational	(Bera et al., 2017, Tsuruyama, 31 Dec 2025)
Internal Energy $S(\rho) = -\mathrm{Tr}[\rho\ln\rho]$4	Mean energy $S(\rho) = -\mathrm{Tr}[\rho\ln\rho]$5	(Bera et al., 2017)
Free Energy $S(\rho) = -\mathrm{Tr}[\rho\ln\rho]$6	$S(\rho) = -\mathrm{Tr}[\rho\ln\rho]$7	(Bera et al., 2017)
Heat $S(\rho) = -\mathrm{Tr}[\rho\ln\rho]$8	$S(\rho) = -\mathrm{Tr}[\rho\ln\rho]$9	(Bera et al., 2017, Liu et al., 2023)
Work $E(\rho) = \mathrm{Tr}[H\rho]$0	$E(\rho) = \mathrm{Tr}[H\rho]$1 or $E(\rho) = \mathrm{Tr}[H\rho]$2	(Bera et al., 2017, Barzi et al., 2024)

Further, the formulation by Barzi & Fethi (Barzi et al., 2024) replaces entropy by information (number of mole-bits $E(\rho) = \mathrm{Tr}[H\rho]$3), so that heat exchanged is given by $E(\rho) = \mathrm{Tr}[H\rho]$4, and all energetic changes are directly tracked by changes in stored information and volume.

4. Dynamical Laws and Nonequilibrium Extensions

The first law extends rigorously to nonequilibrium, driven, and steady-state scenarios, where the roles of information flow and entropy production acquire dynamical significance. In stochastic systems, the Information Processing First Law (IPFL) relates the change in Shannon entropy and the Kullback-Leibler divergence from the local steady state distribution to the difference in average excess work and average excess heat:

$E(\rho) = \mathrm{Tr}[H\rho]$5

for system state vector $E(\rho) = \mathrm{Tr}[H\rho]$6 and reference steady state $E(\rho) = \mathrm{Tr}[H\rho]$7 (Semaan et al., 2022). This equality generalizes nonequilibrium second laws (Hatano–Sasa and total-entropy-production theorems), and enables explicit identification of energetic and informational contributions to system evolution, including the role of information-bearing degrees of freedom (Semaan et al., 2022).

Path-space KL divergences quantify the total entropy production for general (not necessarily reversible) processes and allow separation of dissipation into system and environment components (Tsuruyama, 31 Dec 2025). "Heat" is defined as that part of the total information-theoretic dissipation not accounted for by the system's observational entropy change, a model-independent approach not relying on the existence or characterization of a physical bath.

5. Axiomatic and Resource-Theoretic Approaches

Axiomatic derivations based on minimum-relative-entropy inference subject to constraints (mean energy, particle numbers, and log-information volume) yield a generalized Legendre structure and a differential first law:

$E(\rho) = \mathrm{Tr}[H\rho]$8

where $E(\rho) = \mathrm{Tr}[H\rho]$9 is the maximum relative entropy (observational entropy), $$E_{\mathrm{bound}}(\rho) = \min_{\sigma: S(\sigma)=S(\rho)} \mathrm{Tr}[H\sigma]$$0 is the conjugate "information-pressure," and $$E_{\mathrm{bound}}(\rho) = \min_{\sigma: S(\sigma)=S(\rho)} \mathrm{Tr}[H\sigma]$$1 is the incremental change in mean log-information-volume (Tsuruyama, 31 Dec 2025). This structure naturally leads to all intensive thermodynamic variables (temperature, chemical potential, pressure) as duals to information constraints, unifying statistical inference with thermodynamic bookkeeping. This approach also recovers standard thermodynamics when the information-volume constraint is identified with geometrical volume, but enables generalization to arbitrary measurement-coarse grainings and reference measures.

Resource-theoretic frameworks further operationalize the first law, defining allowed state transformations and the task of work extraction exclusively in terms of information conservation and redistribution between bound and free energy (Bera et al., 2017, Bera et al., 2018).

6. Operational and Conceptual Implications

The information-theoretic framing of the first law reveals several key operational and conceptual consequences:

• Energy changes in physical or informational processes are strictly bounded by, or even entirely determined by, information-theoretic variables such as entropy, relative entropy, and information flow.
• Foundational results such as Landauer's principle emerge as corollaries or special cases of energy-information identities or inequalities, with new, tighter bounds applying even in non-standard or nonthermal environments (Liu et al., 2023, Gopalkrishnan, 2013).
• No external or bath temperature is required for these formulations; intrinsic temperatures or reference Gibbs distributions suffice, enabling application to finite, correlated, or non-equilibrium systems.
• In systems designed for information processing—e.g., quantum engines, information ratchets, statistical inference protocols—the first law constrains possible work extraction, dissipation, and efficiency solely on the basis of information-theoretic quantities (Semaan et al., 2022, Vieland, 2013).

7. Representative Examples and Applications

Concrete examples clarify the information first law across different physical regimes:

• Isothermal Ideal Gas Expansion: In the information reformulation, the heat exchanged during a reversible isothermal expansion is $$E_{\mathrm{bound}}(\rho) = \min_{\sigma: S(\sigma)=S(\rho)} \mathrm{Tr}[H\sigma]$$2, where $$E_{\mathrm{bound}}(\rho) = \min_{\sigma: S(\sigma)=S(\rho)} \mathrm{Tr}[H\sigma]$$3 is the information-modified gas constant, and $$E_{\mathrm{bound}}(\rho) = \min_{\sigma: S(\sigma)=S(\rho)} \mathrm{Tr}[H\sigma]$$4 the number of moles (Barzi et al., 2024).
• Quantum Two-Level System: The decomposition of an arbitrary state into the canonical state of equal entropy (with minimal energy) and the extractable free part demonstrates direct calculation of work and heat as changes in free and bound energy, respectively (Bera et al., 2017, Bera et al., 2018).
• Information Engines: Explicit application of the IPFL to a model ratchet interacting with an information-bearing tape illuminates how entropy production, excess heat, and work depend on both information flows and the detailed violation of detailed balance in nonequilibrium steady states (Semaan et al., 2022).

The universal applicability of the First Law of Information Thermodynamics is underpinned by its independence from microspecifics: all that is required is a consistent account of information and energy, with the dynamics constrained by conservation of the former and the structure of the latter. This synthesis enables new directions in quantum thermodynamics, stochastic processes, and the theory of logical/physical information processing, and subsumes traditional energy-bookkeeping as a special case (Bera et al., 2017, Tsuruyama, 31 Dec 2025, Semaan et al., 2022, Barzi et al., 2024, Liu et al., 2023, Gopalkrishnan, 2013, Vieland, 2013).