Quantum Mutual Entropy Overview

• Quantum mutual entropy is a measure of total correlation in quantum systems, capturing both classical correlations and entanglement using a C*-algebra framework.
• It demonstrates a thermal area law where, at finite temperatures, mutual entropy scales with the boundary size rather than the volume, ensuring spatial locality.
• The framework extends to infinite systems and disjoint regions, elucidating how temperature suppresses critical entanglement and supporting tensor network methods.

Quantum mutual entropy (also known as quantum mutual information) quantifies the total amount of correlation—both classical and quantum—between parts of a quantum system. It generalizes classical mutual information to the quantum domain, where it serves as a key tool for characterizing correlations, entanglement, and information flow in quantum systems, including infinite lattice models, field theories, and many-body systems at finite temperature. Recent advances have established rigorous definitions of quantum mutual entropy in the setting of quasi-local C*-algebras, proven thermal area laws, and elucidated the profound effect of temperature on entanglement in extended quantum lattice systems (Moriya, 7 Oct 2025).

1. Quantum Mutual Entropy in Infinite Quantum Systems

The formulation of quantum mutual entropy in infinite quantum spin and fermion lattice systems avoids the shortcomings of finite-dimensional approaches by utilizing the C*-algebraic framework of quantum statistical mechanics. In this context, the algebra of observables is constructed as the inductive limit of local algebras associated to finite regions Λ of an infinite lattice Γ, allowing a precise treatment of boundary and surface terms.

Given a global state ψ (a positive, normalized linear functional) on the total quasi-local C*-algebra, the mutual entropy between a region Λ and its complement c is defined via local entropy functionals (von Neumann entropy for finite subalgebras) and conditional entropy. Three equivalent formulations are used:

1. Direct difference of entropies:

$I_{ψ}(Λ : c) = S_{Λ ∪ c}(ψ) - S(ψ)$

where $S_{Λ}$ is the von Neumann entropy of the local restriction to Λ, and $S(ψ)$ denotes the global entropy functional.

1. In terms of conditional entropy:

$I_{ψ}(Λ : c) = S_{Λ}(ψ) - S(ψ | Λ)$

where the conditional entropy $S(ψ | Λ)$ is defined as an infimum or limit over increasing regions in the complement.

1. As a relative entropy:

$I_{ψ}(Λ : c) = S(ψ \| ψ_{Λ} \otimes ψ_{c})$

with $S(· \| ·)$ denoting the quantum relative entropy and $ψ_{Λ}$, $ψ_{c}$ the local restrictions.

This formalism applies equally to quantum spin systems, where local algebras are full matrix algebras, and to fermion systems, where the local algebra is constructed from canonical anti-commutation relations (CAR).

2. The Thermal Area Law

A principal result is the establishment of a thermal area law for mutual entropy in infinite quantum systems. For Gibbs or KMS (Kubo-Martin-Schwinger) states at finite inverse temperature β, the mutual entropy $I_{ψ}(Λ : c)$ grows at most proportionally to the size of the boundary ∂ of region Λ, not to its volume. Explicitly, for general finite-range interactions and appropriate surface Hamiltonian $H_{∂}$, one finds the bound:

$I_{ρ_{\text{Gibbs}}}(Λ : c) \le 2β \| H_{∂} \|$

or, if $\| H_{∂} \| \le c|\partial|$,

$I_{ρ_{\text{Gibbs}}}(Λ : c) \le 2β c|\partial|$

where $‖H_{∂}‖$ is the operator norm of the boundary Hamiltonian and $|\partial|$ is the “area” (or, in 1D, the number of boundary links between $\Lambda$ and $c$).

The proof leverages local thermodynamical stability (LTS), a variational principle framed in terms of conditional free energy:

$F(ψ) = ψ(H) - \frac{1}{β} S(ψ)$

and exploits the difference in free energy between the equilibrium state and product states decoupling $\Lambda$ and $c$. This ensures that the excess correlation—quantified by mutual entropy—is entirely due to interactions crossing the boundary.

3. Mutual Entropy Across Disjoint Infinite Regions

For translation-invariant finite-range Hamiltonians on one-dimensional lattices, mutual entropy between left and right infinite half-chains ($\mathbb{Z}_{-}$ and $\mathbb{Z}_{+}$) remains finite in any thermal equilibrium state at β < ∞:

$$I(\mathbb{Z}_{-} : \mathbb{Z}_{+}) \le 2β \| H_{∂} \|$$

with $H_{∂}$ the finite interaction across the cut. This result follows from the same variational principles and entropy inequalities used in the finite case and applies to both spin and fermionic chains.

This assertion is significant because in critical ground states, which display long-range quantum entanglement, the mutual entropy between infinite halves may diverge or even become infinite—signaling a qualitatively different correlation structure at T = 0 compared to T > 0.

4. Effect of Temperature: Thermal Destruction of Critical Entanglement

A key implication is the dramatic effect of temperature on quantum entanglement. Critical (gapless) ground states in one dimension often exhibit an infinite mutual entropy, corresponding to strong long-range entanglement across any bipartition—a hallmark of criticality. However, the introduction of an arbitrarily small positive temperature is sufficient to render the mutual entropy finite:

$$\lim_{\beta \to \infty} I(\mathbb{Z}_{-} : \mathbb{Z}_{+}) = \infty,\quad I(\mathbb{Z}_{-} : \mathbb{Z}_{+}) < \infty\;\text{for}\;\beta < \infty$$

Thus, thermal fluctuations act as a “cutoff,” suppressing long-range quantum correlations and restoring an effective locality even in systems that are critical at zero temperature. This identifies a sharp contrast between ground state entanglement structure and the mixed-state correlations characteristic of finite temperature.

5. Mathematical Formulations and Key Inequalities

Several mathematical formulas and inequalities are central to the analysis:

• Von Neumann entropy: $S(ρ) = -\operatorname{Tr}(D_{ρ} \log D_{ρ})$
• Mutual entropy for subsystems Λ and $c$: $I_{ρ}(Λ : c) = S(ρ_{Λ}) + S(ρ_{c}) - S(ρ_{Λ∪c})$
• Relative entropy expression: $$I_{\psi}(Λ : c) = S(\psi \| \psi_{Λ} \otimes \psi_{c})$$
• Thermal area law bound: $I_{ψ}(Λ : c) \le 2β c |\partial|$
• In 1D infinite chains: $$I(\mathbb{Z}_{-} : \mathbb{Z}_{+}) \le 2β \| H_{∂} \|$$

These expressions are rigorously justified using the algebraic machinery of quantum statistical mechanics and entropy theory, including strong subadditivity, properties of conditional entropy, and modular theory for states on C*-algebras.

6. Physical and Theoretical Implications

The rigorous definition and bounding of quantum mutual entropy in infinite quantum systems has multiple consequences:

• It provides a metric for quantifying correlations and the degree of nonlocality or entanglement in equilibrium and non-equilibrium states of infinite lattices.
• The thermal area law demonstrates that equilibrium states (except ground states at T = 0 in critical systems) obey spatial locality, with mutual entropy scaling only with the boundary.
• These findings underpin the effectiveness of tensor network approaches and justifications for area-law-driven numerical techniques in quantum many-body physics.
• The suppression of critical entanglement at finite temperature delineates a sharp distinction between quantum and classical correlations in mixed states.

7. Extensions and Open Directions

The framework extends naturally to quasi-local algebras for boson systems, general higher-dimensional lattices, and systems with more complex symmetries or interactions, including fermions with grading automorphisms. A plausible implication is that the area law applies widely to non-integrable and disordered systems as long as interactions are finite-range and thermal equilibrium is established.

Open problems include rigorous characterization of the sharp crossover from infinite to finite mutual entropy at low but nonzero temperature, quantitative refinements for systems with long-range interactions or fractal boundaries, and the extension to non-equilibrium states or dynamical scenarios such as quantum quenches.

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In summary, quantum mutual entropy in infinite systems, as formalized in the C*-algebraic approach, is a robust, operationally meaningful measure of correlation. The proof of the thermal area law for mutual entropy connects microscopic quantum interactions to macroscopic entanglement structure and uncovers the fundamental ways in which temperature alters the organization of quantum correlations across extended systems (Moriya, 7 Oct 2025).