Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2606.00749

Quantum Ergodicity and Thermalization in Interval Quantum Mechanics

Abbas Edalat

quant-phmath-phSubmitted May 30, 2026• Announced Jun 2, 2026• 0 citations
arXivHTMLPDF

Abstract

We combine Reimann's spectral typicality theorem -- a modern formulation of quantum ergodicity -- with the framework of Interval Quantum Mechanics (IQM). In IQM, quantum states are represented not by points but by \emph{quantum parcels}: weak open convex sets of density matrices defined by finitely many expectation intervals. Such parcels are the exact mathematical representation of the epistemic knowledge obtained from finite-precision measurements of macroscopic observables. We prove that for a single parcel in which every state has large effective dimension (a condition that ensures thermalization), the expectation interval of any bounded observable becomes concentrated around the microcanonical value for most late times. The asymptotic bound depends only on the minimal effective dimension within the parcel, not on its detailed shape. For a double parcel \((O_1,O_2)\) with both components contained in an energy shell, separated by a conserved quantity \(Q^*\) that is supported on the range of the measurement projector, we show that the expectation intervals of both parcels become concentrated near the microcanonical values of bounded observables, the separation is preserved exactly, and the updated double parcel after a fuzzy measurement remains valid.

Related papers

Dynamical Typicality Approach to Eigenstate Thermalization

Peter Reimann

We consider the set of all initial states within a microcanonical energy shell of an isolated many-body quantum system, which exhibit an arbitrary but fixed nonequilibrium expectation value for some given observable A. On the condition that this set is not too small, it is shown by means of a dynamical typicality approach that most such initial states exhibit thermalization if and only if A satisfies the so-called weak eigenstate thermalization hypothesis (wETH). Here, thermalization means that the expectation value of A spends most of its time close to the microcanonical value after initial transients have died out. The wETH means that, within the energy shell, most eigenstates of the pertinent system Hamiltonian exhibit very similar expectation values of A.

arXiv:1806.03193•Physical Review Letters•Jun 6, 2018•31 citations

Thermalization in the two-body random ensemble

V K B Kota, A Relaño, J. Retamosa +1 more

Using the ergodicity principle for the expectation values of several types of observables, we investigate the thermalization process in isolated fermionic systems.These are described by the two-body random ensemble, which is a paradigmatic model to study quantum chaos and especially the dynamical transition from integrability to chaos.By means of exact diagonalizations we analyze the relevance of the eigenstate thermalization hypothesis as well as the influence of other factors, such as the energy and structure of the initial state, or the dimension of the Hilbert space.We also obtain analytical expressions linking the degree of thermalization for a given observable with the so-called number of principal components for transition strengths originating at a given energy, with the dimensions of the whole Hilbert space and microcanonical energy shell, and with the correlations generated by the observable.As the strength of the residual interaction is increased, an order-to-chaos transition takes place, and we show that the onset of Wigner spectral fluctuations, which is the standard signature of chaos, is not sufficient to guarantee thermalization in finite systems.When all the signatures of chaos are fulfilled, including the quasicomplete delocalization of eigenfunctions, the eigenstate thermalization hypothesis is the mechanism responsible for the thermalization of certain types of observables, such as (linear combinations of) occupancies and strength function operators.Our results also suggest that fully chaotic systems will thermalize relative to most observables in the thermodynamic limit.

arXiv:1102.0528•Journal of Statistical Mechanics Theory and Experiment•Oct 31, 2011•25 citations

Aperiodicity is sufficient for macroscopic thermalization

Amit Vikram

We identify a general mechanism for the finite-time thermalization of macroscopic observables, such as coarse-grained charge densities, in terms of elementary forms of the quantum dynamics of initial states: (1) aperiodicity, which provides a computable measure of (2) a dynamical partially ergodic exploration of the Hilbert space. Specifically, this mechanism predicts the equilibration of all (concentrated) macroscopic observables, in almost all states in an initial ensemble and almost all times within finite and longer intervals, given only the observable-independent information that the return probability of the ensemble of initial states is small over a finite time range. As a special case, it also accesses standard results on equilibration over infinitely long times in terms of (stronger versions of) the effective dimension of initial state delocalization in the energy eigenbasis. Our results incorporate macroscopic thermalization into the domain of operational quantum statistical mechanics, recently developed to provide finitely computable criteria for microscopic thermalization. We discuss an overall characterization of this approach as establishing connections between (1) the decay of a (theoretically or experimentally) computable probe indicating memorylessness, (2) a fundamental invariant mechanism in terms of the alignment of observables or states in the Hilbert space, and (3) predicting different natural forms of (classical and) quantum thermalization, most of which rigorously recover conventional eigenstate-based descriptions of infinite-time thermalization as a special case but provide stronger accessible predictions over finite observation times in the thermodynamic limit.

quant-pharXiv:2608.13462•Aug 14, 2026•0 citations

11 Thermalization in the Two-Body Random Ensemble

V. K. B. Kota, A. Relaño, J. Retamosa +1 more

Using the ergodicity principle for the expectation values of several types of observables, we investigate the thermalization process in isolated fermionic systems. These are described by the two-body random ensemble, which is a paradigmatic model to study quantum chaos and especially the dynamical transition from integrability to chaos. By means of exact diagonalizations we analyze the relevance of the eigenstate thermalization hypothesis as well as the influence of other factors, such as the energy and structure of the initial state, or the dimension of the Hilbert space. We also obtain analytical expressions linking the degree of thermalization for a given observable with the so-called number of principal components for transition strengths originating at a given energy, with the dimensions of the whole Hilbert space and microcanonical energy shell, and with the correlations generated by the observable. As the strength of the residual interaction is increased, an order-to-chaos transition takes place, and we show that the onset of Wigner spectral fluctuations, which is the standard signature of chaos, is not sufficient to guarantee thermalization in finite systems. When all the signatures of chaos are fulfilled, including the quasicomplete delocalization of eigenfunctions, the eigenstate thermalization hypothesis is the mechanism responsible for the thermalization of certain types of observables, such as (linear combinations of) occupancies and strength function operators. Our results also suggest that fully chaotic systems will thermalize relative to most observables in the thermodynamic limit.

•Jan 1, 2008•24 citations

Quantum Mechanics from Ergodic Average of Microstates

Marco Matone

We formulate quantum mechanics as an effective theory of an underlying structure characterized by microstates $|{\mathcal M}^j(t)\rangle$, each one defined by the quantum state $|Ψ(t)\rangle$ and a complete set of commutative observables $O^j$. At any time $t$, $|{\mathcal M}^j(t)\rangle$ corresponds to a state $|O^j_k\rangle$, for some $k$ depending on $t$, and jumps after time intervals whose duration, of the order of the Compton time $τ$, is proportional to the probability $|\langle O_k^j|Ψ(t)\rangle|^2$. This reproduces the Born rule and mimics the wave-particle duality. The theory is based on a partition of time whose flow is characterized by quantum probabilities. Ergodicity arises at ordinary quantum scales with the expectation values corresponding to time averaging over a period $τ$. The measurement of $O^j$ provides a new partition of time and the outcome is the state $|O_k^j\rangle$ to which $|{\mathcal M}^j(t)\rangle$ corresponds at that time. The formulation, that shares some features with the path integral, can be tested by experiments involving time intervals of order $τ$.

arXiv:1710.08419•Oct 23, 2017•0 citations

Symmetry-driven thermalization via finite de Finetti theorems

Uttam Singh, Nicolas J. Cerf

Thermal behavior in subsystems of closed quantum systems is commonly attributed to dynamical chaos, quantum ergodicity, canonical typicality, or the eigenstate thermalization hypothesis, suggesting a fundamentally statistical origin of thermalization. Here, we propose a potential alternative mechanism in which thermal structures emerge deterministically from symmetry considerations alone, without recourse to statistical arguments. We prove a finite de Finetti-type theorem for quantum states invariant under energy-preserving unitaries, establishing that the reduced marginals of any such invariant $N$-qudit state are close (both in trace distance and relative entropy) to convex mixtures of thermal product states, with explicit error bounds vanishing as $N \to \infty$. We further present an example of energy-conserving Lindblad dynamics whose long-time limit is invariant under energy-preserving unitaries, providing a dynamical realization of the desired symmetry class. These results imply that invariance under energy-preserving unitaries suffices as a sole fundamental, deterministic principle to enforce thermal structures.

quant-pharXiv:2604.09410•Physical Review Letters•Apr 13, 2026•0 citations
We combine Reimann's spectral typicality theorem -- a modern formulation of quantum ergodicity -- with the framework of Interval Quantum Mechanics (IQM). In IQM, quantum states are represented not by points but by \emph{quantum parcels}: weak open convex sets of density matrices defined by finitely many expectation intervals. Such parcels are the exact mathematical representation of the epistemic knowledge obtained from finite-precision measurements of macroscopic observables.
We prove that for a single parcel in which every state has large effective dimension (a condition that ensures thermalization), the expectation interval of any bounded observable becomes concentrated around the microcanonical value for most late times. The asymptotic bound depends only on the minimal effective dimension within the parcel, not on its detailed shape.
For a double parcel ((O_1,O_2)) with both components contained in an energy shell, separated by a conserved quantity (Q^*) that is supported on the range of the measurement projector, we show that the expectation intervals of both parcels become concentrated near the microcanonical values of bounded observables, the separation is preserved exactly, and the updated double parcel after a fuzzy measurement remains valid.
We consider the set of all initial states within a microcanonical energy shell of an isolated many-body quantum system, which exhibit an arbitrary but fixed nonequilibrium expectation value for some given observable A. On the condition that this set is not too small, it is shown by means of a dynamical typicality approach that most such initial states exhibit thermalization if and only if A satisfies the so-called weak eigenstate thermalization hypothesis (wETH). Here, thermalization means that the expectation value of A spends most of its time close to the microcanonical value after initial transients have died out. The wETH means that, within the energy shell, most eigenstates of the pertinent system Hamiltonian exhibit very similar expectation values of A.
Using the ergodicity principle for the expectation values of several types of observables, we investigate the thermalization process in isolated fermionic systems.These are described by the two-body random ensemble, which is a paradigmatic model to study quantum chaos and especially the dynamical transition from integrability to chaos.By means of exact diagonalizations we analyze the relevance of the eigenstate thermalization hypothesis as well as the influence of other factors, such as the energy and structure of the initial state, or the dimension of the Hilbert space.We also obtain analytical expressions linking the degree of thermalization for a given observable with the so-called number of principal components for transition strengths originating at a given energy, with the dimensions of the whole Hilbert space and microcanonical energy shell, and with the correlations generated by the observable.As the strength of the residual interaction is increased, an order-to-chaos transition takes place, and we show that the onset of Wigner spectral fluctuations, which is the standard signature of chaos, is not sufficient to guarantee thermalization in finite systems.When all the signatures of chaos are fulfilled, including the quasicomplete delocalization of eigenfunctions, the eigenstate thermalization hypothesis is the mechanism responsible for the thermalization of certain types of observables, such as (linear combinations of) occupancies and strength function operators.Our results also suggest that fully chaotic systems will thermalize relative to most observables in the thermodynamic limit.
We identify a general mechanism for the finite-time thermalization of macroscopic observables, such as coarse-grained charge densities, in terms of elementary forms of the quantum dynamics of initial states: (1) aperiodicity, which provides a computable measure of (2) a dynamical partially ergodic exploration of the Hilbert space. Specifically, this mechanism predicts the equilibration of all (concentrated) macroscopic observables, in almost all states in an initial ensemble and almost all times within finite and longer intervals, given only the observable-independent information that the return probability of the ensemble of initial states is small over a finite time range. As a special case, it also accesses standard results on equilibration over infinitely long times in terms of (stronger versions of) the effective dimension of initial state delocalization in the energy eigenbasis. Our results incorporate macroscopic thermalization into the domain of operational quantum statistical mechanics, recently developed to provide finitely computable criteria for microscopic thermalization. We discuss an overall characterization of this approach as establishing connections between (1) the decay of a (theoretically or experimentally) computable probe indicating memorylessness, (2) a fundamental invariant mechanism in terms of the alignment of observables or states in the Hilbert space, and (3) predicting different natural forms of (classical and) quantum thermalization, most of which rigorously recover conventional eigenstate-based descriptions of infinite-time thermalization as a special case but provide stronger accessible predictions over finite observation times in the thermodynamic limit.
Using the ergodicity principle for the expectation values of several types of observables, we investigate the thermalization process in isolated fermionic systems. These are described by the two-body random ensemble, which is a paradigmatic model to study quantum chaos and especially the dynamical transition from integrability to chaos. By means of exact diagonalizations we analyze the relevance of the eigenstate thermalization hypothesis as well as the influence of other factors, such as the energy and structure of the initial state, or the dimension of the Hilbert space. We also obtain analytical expressions linking the degree of thermalization for a given observable with the so-called number of principal components for transition strengths originating at a given energy, with the dimensions of the whole Hilbert space and microcanonical energy shell, and with the correlations generated by the observable. As the strength of the residual interaction is increased, an order-to-chaos transition takes place, and we show that the onset of Wigner spectral fluctuations, which is the standard signature of chaos, is not sufficient to guarantee thermalization in finite systems. When all the signatures of chaos are fulfilled, including the quasicomplete delocalization of eigenfunctions, the eigenstate thermalization hypothesis is the mechanism responsible for the thermalization of certain types of observables, such as (linear combinations of) occupancies and strength function operators. Our results also suggest that fully chaotic systems will thermalize relative to most observables in the thermodynamic limit.
We formulate quantum mechanics as an effective theory of an underlying structure characterized by microstates ∣Mj(t)⟩|{\mathcal M}^j(t)\rangle∣Mj(t)⟩, each one defined by the quantum state ∣Ψ(t)⟩|Ψ(t)\rangle∣Ψ(t)⟩ and a complete set of commutative observables OjO^jOj. At any time ttt, ∣Mj(t)⟩|{\mathcal M}^j(t)\rangle∣Mj(t)⟩ corresponds to a state ∣Okj⟩|O^j_k\rangle∣Okj​⟩, for some kkk depending on ttt, and jumps after time intervals whose duration, of the order of the Compton time τττ, is proportional to the probability ∣⟨Okj∣Ψ(t)⟩∣2|\langle O_k^j|Ψ(t)\rangle|^2∣⟨Okj​∣Ψ(t)⟩∣2. This reproduces the Born rule and mimics the wave-particle duality. The theory is based on a partition of time whose flow is characterized by quantum probabilities. Ergodicity arises at ordinary quantum scales with the expectation values corresponding to time averaging over a period τττ. The measurement of OjO^jOj provides a new partition of time and the outcome is the state ∣Okj⟩|O_k^j\rangle∣Okj​⟩ to which ∣Mj(t)⟩|{\mathcal M}^j(t)\rangle∣Mj(t)⟩ corresponds at that time. The formulation, that shares some features with the path integral, can be tested by experiments involving time intervals of order τττ.
Thermal behavior in subsystems of closed quantum systems is commonly attributed to dynamical chaos, quantum ergodicity, canonical typicality, or the eigenstate thermalization hypothesis, suggesting a fundamentally statistical origin of thermalization. Here, we propose a potential alternative mechanism in which thermal structures emerge deterministically from symmetry considerations alone, without recourse to statistical arguments. We prove a finite de Finetti-type theorem for quantum states invariant under energy-preserving unitaries, establishing that the reduced marginals of any such invariant NNN-qudit state are close (both in trace distance and relative entropy) to convex mixtures of thermal product states, with explicit error bounds vanishing as N→∞N \to \inftyN→∞. We further present an example of energy-conserving Lindblad dynamics whose long-time limit is invariant under energy-preserving unitaries, providing a dynamical realization of the desired symmetry class. These results imply that invariance under energy-preserving unitaries suffices as a sole fundamental, deterministic principle to enforce thermal structures.