iMATE: Quantum Macroscopic Thermal Equilibrium

• Infinite-observable macroscopic thermal equilibrium (iMATE) is defined as the indistinguishability of quantum states from the Gibbs ensemble via all additive macroscopic observables.
• It unifies approaches from quantum statistical mechanics, information theory, and ETH, using adiabatic operations and Lieb–Robinson bounds to rigorously extend the second law to quantum systems.
• The framework guarantees macroscopic passivity and non-decreasing entropy density under finite-time operations, establishing operational limits for thermalization in many-body experiments.

Infinite-observable macroscopic thermal equilibrium (iMATE) is a rigorous framework for defining quantum thermal equilibrium in closed many-body systems via the indistinguishability of a quantum state from the canonical Gibbs ensemble with respect to all additive macroscopic observables. The iMATE paradigm unifies approaches from quantum statistical mechanics, information theory, and the eigenstate thermalisation hypothesis (ETH) to yield precise versions of the second law of thermodynamics for quantum systems under physically realistic adiabatic operations (Chiba et al., 6 Feb 2026, Anza et al., 2015).

1. Definition and Mathematical Formulation

The iMATE condition requires that a sequence of quantum states, represented as density matrices $(\rho_L)_{L\to\infty}$ on a lattice of size $L^d$, is indistinguishable from the canonical Gibbs state at inverse temperature $\beta$ by the expectation values of any additive observable in the thermodynamic limit. For a translation-invariant Hamiltonian $H_L$ and any additive observable $$A_\alpha = \sum_{\mathbf{r}\in \Lambda_L} a_\alpha(\mathbf{r})$$, where $a_\alpha(\mathbf{r})$ is supported on a finite block around site $\mathbf{r}$, iMATE is defined by

$\lim_{L\to\infty}\Tr[\rho_L A_\alpha] = \lim_{L\to\infty}\Tr[\rho^{\mathrm{can}}_L(\beta) A_\alpha],$

where

$\rho^{\mathrm{can}}_L(\beta) = \frac{e^{-\beta H_L}}{\Tr[e^{-\beta H_L}]}.$

No macroscopic, even infinite, set of additive observables can distinguish $\rho_L$ from the canonical ensemble in this limit (Chiba et al., 6 Feb 2026).

2. Macroscopic Operations and Adiabatic Evolution

Macroscopic (adiabatic) operations in the iMATE framework are defined by unitary evolution generated by Hamiltonians subject to time-dependent, finite-parameter macroscopic external controls. Explicitly, the evolution operator is governed by

$$i\frac{d}{dt}U_L(t,0) = \left[ H_L - \sum_{\mu=1}^m f^\mu(t) B^\mu_L \right] U_L(t,0), \qquad U_L(0,0)=I,$$

where each control parameter $f^\mu(t)$ couples to an additive observable $B^\mu_L$ and $m=O(1)$ is independent of system size. These operations capture the physically relevant manipulations—such as varying magnetic fields or trap potentials—accessible in experiment or computation, always for timescales $t^*=O(1)$ independent of $L$ (Chiba et al., 6 Feb 2026).

3. Passivity and Work Extraction in iMATE

A central theorem in the iMATE formalism is macroscopic passivity: for any initial iMATE state, no macroscopic operation of finite duration can reduce the system’s energy density. Concretely,

$\lim_{L\to\infty}\Tr\left[\rho_L(t^*) \frac{H_L}{N}\right] \geq \lim_{L\to\infty}\Tr\left[\rho_L \frac{H_L}{N}\right],$

where $N=L^d$ and $\rho_L(t^*)$ denotes the state after a macroscopic operation lasting time $t^*$. This generalizes standard passivity, which fails for generic pure states, by restricting attention to macroscopic observables and operations relevant on thermodynamic timescales. The proof employs Lieb–Robinson bounds to show that the Heisenberg evolution of any additive observable remains approximately additive, so iMATE expectation values persist throughout the macroscopic operation (Chiba et al., 6 Feb 2026).

4. Entropy Density and the Quantum Second Law

The iMATE formalism defines a quantum macroscopic entropy density using coarse-grained local reduced states: $$s_\ell^{\mathrm{mac}}(\rho) = \frac{1}{K^d}\sum_{k=1}^{K^d}\frac{S_{\mathrm{vN}}(\rho_{\infty|\ell}^{(k)})}{\ell^d},$$ with $$\rho_{\infty|\ell}^{(k)} = \lim_{L\to\infty}\rho_{L|\ell}^{(k)}$$ the limit of local blocks in a macroscopic cell. In the limit $\ell\to\infty$, this density recovers the standard thermodynamic entropy: $$\lim_{\ell\to\infty} s_\ell^{\mathrm{mac}}(\rho) = s^{\mathrm{TD}}(\beta|H) = \lim_{L\to\infty} \frac{S_{\mathrm{vN}}(\rho_L^{\mathrm{can}}(\beta|H))}{N}.$$ A second key theorem states that, under a generic adiabatic process consisting of (i) preparation in iMATE, (ii) finite-time macroscopic operation, and (iii) relaxation under a new time-independent Hamiltonian, the macroscopic entropy density never decreases: $$s^{\mathrm{TD}}(\beta_0|H_0) \le s^{\mathrm{TD}}(\beta_1|H_1),$$ implementing the quantum second law for all macroscopic operations and their subsequent relaxation (Chiba et al., 6 Feb 2026).

5. Information-Theoretic and ETH Perspectives

The iMATE condition can be equivalently formulated as the imposition that the outcome probabilities of all additive observables coincide with their Gibbs ensemble values. Building on the information-theoretic framework of Anzà & Vedral, each observable’s equilibrium is found by maximizing its Shannon entropy under energy and normalization constraints. For a set of observables $\{A^{(\ell)}\}$, the maximization of each $H_{A^{(\ell)}}$ forces the global state to approach the Gibbs ensemble: $\rho_{\rm eq} = \underset{\Tr\rho=1,\,\Tr\rho H=E_0}{\mathrm{argmax}}\, S_{\mathrm{vN}}(\rho) = \frac{e^{-\beta H}}{Z}.$ Extending this variational principle to infinitely many observables ensures that the only consistent equilibrium state is the canonical ensemble (Anza et al., 2015).

Hamiltonian-unbiased observables—constructed by choosing projectors mutually unbiased with respect to the energy eigenbasis—demonstrate explicitly that an infinite family of macroscopic observables can thermalize in closed quantum dynamics. Their matrix elements exhibit ETH-type structure: diagonal expectations are constant in the microcanonical shell, and off-diagonals are exponentially small, ensuring macroscopic indistinguishability from the equilibrium ensemble (Anza et al., 2015).

6. Timescale Constraints and Physical Regimes

All iMATE results, including macroscopic passivity and entropy non-decrease, are rigorously established for adiabatic operations of finite duration, $t^*=O(1)$, independent of system size. Attempts to extend the results to longer times, such as $t^* \sim L$, fail: explicit constructions reveal that with sufficiently prolonged macroscopic or even local control, both passivity and entropy increase laws can be violated by an $O(N)$ margin. Thus, the finite timescale constraint is essential for the emergence of the macroscopic second law in quantum systems (Chiba et al., 6 Feb 2026).

7. Relation to Observable-Based Thermalisation and Limitations

Single-observable “information-theoretic equilibrium” (Anza et al., 2015) generalizes naturally to iMATE by enforcing Shannon entropy maximization across an infinite family of macroscopic observables. However, rigorous extension to infinitely many observables requires careful control of separability of the observable algebra, the existence of common unbiased bases, and convergence of corresponding entropic functionals. A practical approach restricts attention to a separable, dense set of macroscopic observables and leverages Schur-concavity and compactness to ensure that the limiting iMATE state remains the Gibbs state. The infinite-observable constraint also underpins the connection with ETH: imposing thermalization for all macroscopic observables ensures the Gibbs state structure and, asymptotically, the diagonal and off-diagonal ETH scaling for their matrix elements (Anza et al., 2015).

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Table: Core Aspects of the iMATE Formalism

Aspect	Mathematical Formulation	Significance
iMATE Condition	$$\lim_{L\to\infty} \mathrm{Tr}[\rho_L A_\alpha] = \lim_{L\to\infty}\mathrm{Tr}[\rho^{\mathrm{can}}_L(\beta)A_\alpha]$$	Macroscopic indistinguishability from Gibbs state
Macroscopic Passivity	$$\lim_{L\to\infty}\mathrm{Tr}[\rho_L(t^*) H_L/N] \geq \lim_{L\to\infty}\mathrm{Tr}[\rho_L H_L/N]$$	No extraction of $O(N)$ work in finite time
Entropy Density Consistency	$$\lim_{\ell\to\infty} s_\ell^{\mathrm{mac}}(\rho) = s^{\mathrm{TD}}(\beta|H)$$	Quantum macroscopic entropy equals thermodynamic entropy

The iMATE framework thus provides a comprehensive quantum-mechanical foundation for macroscopic thermodynamics, bridging the gap between observable-based equilibrium, quantum information theoretic variational principles, and the phenomenology of ETH, under operationally meaningful restrictions on observables and control protocols (Chiba et al., 6 Feb 2026, Anza et al., 2015).