Mutual Information Energy in Thermodynamics & ML

• Mutual Information Energy is a framework that rigorously connects mutual information with energy, entropy production, and thermodynamic work in coupled systems.
• It employs methods such as thermodynamic decomposition, energy-based copula formulations, and mutual information-energy inequalities to quantify statistical and quantum correlations.
• Applications include optimizing communication channels, enhancing energy-based machine learning models, and refining physical bounds like Landauer’s limit in experimental designs.

Mutual information energy refers to a set of rigorous and physically motivated relationships connecting mutual information—a measure of statistical or quantum correlations between systems—to energy, entropy production, and thermodynamic resources. These connections underlie large areas of statistical physics, quantum information, non-equilibrium thermodynamics, and modern machine learning, and are made precise through a variety of formalisms including entropy decompositions, energy-based probabilistic models, work/information trade-offs, and inequalities directly bounding or relating mutual information to physical energy quantities.

1. Thermodynamic Decomposition: Information and Entropy Production

Mutual information enters the nonequilibrium thermodynamics of coupled stochastic (classical or quantum) systems as a quantifiable energetic resource. For a universe consisting of two subsystems $X$ and $Y$ (e.g., a system and a memory), plus heat baths $B$ at inverse temperatures $\{\beta_k\}$, the total entropy production can be decomposed as: $$\Delta S_{\rm tot} = \Delta S_{\rm th} + \Delta S_{\rm info}, \qquad \Delta S_{\rm info} = -\Delta I_{XY}$$ Here, $\Delta S_{\rm th}$ denotes the conventional thermodynamic entropy of $X$ and baths, and $\Delta S_{\rm info}$ is an information-theoretic term given by minus the change in mutual information $I_{XY}$ between $X$ and $Y$ during the process. This decomposition yields nonequilibrium equalities and fluctuation theorems: $$\langle e^{-\Delta S_{\rm tot}} \rangle = 1 \quad \Longleftrightarrow \quad \langle e^{-\Delta S_{\rm th} + \Delta I_{XY}} \rangle = 1$$ and enforces a generalized second law $$\langle \Delta S_{\rm th} \rangle \geq \langle \Delta I_{XY} \rangle$$, demonstrating that information acquisition can offset thermodynamic entropy production and vice versa. This framework underpins refined Landauer-type bounds: the minimum work to erase information is directly proportional to the acquired mutual information, $\langle W \rangle \geq k_BT \langle I_{XY} \rangle$ (Sagawa et al., 2013).

2. Energy-Based and Copula Formulations: Mutual Information as Expected Energy

Mutual information possesses an explicit “energy” representation in energy-based models and copula theory. For two random variables $X, Y$ with continuous marginals, there exists a copula function $C$ relating their joint distribution to the marginals. The copula density $c(u,v)$ defines an “energy” $E_c(u,v) = -\log c(u,v)$, such that: $$I(X;Y) = -H_c(X,Y) = -\iint c(u,v)\log c(u,v) \,du\,dv = \mathbb{E}_c\big[-E_c(u,v)\big]$$ Mutual information is thereby interpreted as the negative average copula energy. Parametric or neural energy-based copula models $c_\theta(u,v) = \exp[-E_\theta(u,v)]/Z_\theta$ can be trained to maximize mutual information, establishing a strong parallel between dependence structure and energetics (0808.0845).

3. Mutual Information-Energy Inequalities in Quantum and Statistical Systems

In quantum thermodynamics, mutual information between parts of a bipartite thermal state is bounded directly by interaction energy and partition functions: $$I(\rho_{AB}) \leq -\beta \operatorname{Tr}[\rho_{AB} H_{\rm int}] + \ln \frac{\operatorname{Tr} e^{-\beta H_A} \operatorname{Tr} e^{-\beta H_B}}{\operatorname{Tr} e^{-\beta H}}$$ where $H=H_A\otimes I_B + I_A\otimes H_B + H_{\rm int}$ is the full Hamiltonian and $\beta$ is the inverse temperature. At high temperature, this bound is nearly tight and quantifies the maximum correlations sustainable by a given interaction energy (Fedorov et al., 2014). In the two-spin $\tfrac12$ XY Heisenberg model, the bound is saturated as $T \to \infty$ and diverges as $T \to 0$ with an entangled ground state.

Quantum mutual information also appears as a constraint for energy exchanges in unitary dynamics and heat flows. The difference in mutual information between pre- and post-interaction states bounds the possible “anomalous” heat exchanges, providing a direct thermodynamic role for information (Jevtic et al., 2011).

4. Thermodynamic Representations: Mutual Information as Work and Free Energy

In communication channels (notably the Gaussian channel), mutual information can be formulated as a thermodynamic work or free energy difference. By mapping the signal-to-noise ratio (SNR) to inverse temperature and channel output statistics to canonical (Gibbs) distributions, the mutual information becomes: $$I(X;Y) = -\left[\gamma \mathbb{E}_Y U(Y;\gamma)\right]_{0}^{\beta} + \int_{0}^{\beta} \mathbb{E}_Y U(Y;\gamma) d\gamma$$ where $U(Y;\gamma)$ is the “internal energy” at inverse temperature $\gamma$ and $F(Y;\beta)$ is the corresponding free energy. This renders $I$ as the reversible work extracted by “heating” the system from zero noise (infinite temperature) to finite SNR. The I-MMSE relationship $dI/d\,\text{SNR} = \frac12 \text{MMSE}$ further connects information gain with thermodynamic susceptibilities (0806.3133).

5. Mutual Information in Holography and Quantum Field Theory

In holographic duals of quantum field theories, mutual information controls spatial correlations and is sensitive to the bulk energy scales and the number of degrees of freedom. In non-conformal backgrounds, increasing an explicit energy scale $\Lambda$ generally enhances holographic mutual information $I(A,B)$ between boundary subregions and moves the disentangling transition $x_{DT}$ to larger separations compared to the conformal (CFT) case, despite a concurrent decrease in effective degrees of freedom along the renormalization group flow. This competition produces a “Mutual Information Energy” effect where non-conformal energy scale effects dominate over degrees-of-freedom reduction, while strong subadditivity and monogamy of mutual information are preserved (Ali-Akbari et al., 2019).

Holographic Regime	Effect of $\Lambda$ (energy scale)	Effect of Reduced Degrees of Freedom
Small $l$ (UV)	$I_{NCFT} > I_{CFT}$ if $\Lambda \gg 1$	$I$ decreases as $N_{IR}/N_{UV}$ decreases
Large $l$ (IR)	$I_{NCFT} \simeq I_{CFT}$ if $\Lambda \gg 1$	$I$ decreases with $N_{IR}/N_{UV}$

6. Energy-Efficient Communication and Channel Mutual Information

In communication theory, the mutual information $I(s)$ for channels with signal energy $s$ determines the minimum energy per bit $E_b$ required for reliable transmission. In the Poisson channel, for any fixed input constellation, $I(s) = c_1 s + c_2 s^2 + o(s^2)$ at low $s$, but the channel capacity grows as $-\!s\ln s$—which is achievable only via vanishing-probability “flash signaling” strategies that maximize energetic efficiency. With additive noise, the leading term is quadratic, e.g., $I(s) = (Var(X)/2n)s^2 + o(s^2)$ for additive Poisson noise of mean $n$. The minimum $E_b$ can be zero (Poisson noise) or $n\ln2$ (geometric noise), but fixed constellations cannot attain this due to their suboptimal $I(s)$ scaling (0808.2703).

7. Learning, Estimation, and Mutual Information as an Optimization Principle

Contemporary energy-based and variational machine learning methods increasingly operationalize mutual information energy concepts. Mutual information estimation may proceed via energy-based models such as MINE (mutual information neural estimation), where the Donsker–Varadhan (DV) and annealed importance sampling (AIS) lower bounds recast MI as practical objectives incorporating energy-based critic networks and partition function estimation. Advanced estimators (GIWAE, MINE-AIS) leverage multichain AIS and MCMC to provide scalable, unbiased MI estimates in deep generative models, tightly matching ground-truth values even at high MI (Brekelmans et al., 2023). These approaches exhibit marked advantages over earlier variational methods in representing and harnessing the “energy” structure of the data.

For scientific instrument optimization (e.g., calorimeter design), mutual information is used directly as the scalar objective to optimize detector layer thicknesses for maximal energy resolution. Task-agnostic MI-based optimization recovers essentially the same detector configurations as reconstruction-based surrogates, but is invariant under invertible transformations and robust to target ambiguities, provided enough samples for MI estimation—a direct application of the mutual information energy principle in experimental design (Wozniak et al., 18 Mar 2025).

• Role of Mutual Information in Entropy Production under Information Exchanges (Sagawa et al., 2013)
• Mutual information is copula entropy (0808.0845)
• Mutual information-energy inequality for thermal states of a bipartite quantum system (Fedorov et al., 2014)
• Quantum Mutual Information Along Unitary Orbits (Jevtic et al., 2011)
• Shannon Meets Carnot: Mutual Information Via Thermodynamics (0806.3133)
• Holographic Mutual and Tripartite Information in a Non-Conformal Background (Ali-Akbari et al., 2019)
• Low-Signal-Energy Asymptotics of Capacity and Mutual Information for the Discrete-Time Poisson Channel (0808.2703)
• Improving Mutual Information Estimation with Annealed and Energy-Based Bounds (Brekelmans et al., 2023)
• End-to-End Optimal Detector Design with Mutual Information Surrogates (Wozniak et al., 18 Mar 2025)