Stabilizer States at Infinite Temperature: No-Go Theorem for Two-Body Hamiltonians

Scientists are tackling the complex problem of analysing highly entangled thermal eigenstates, a crucial hurdle in modern physics. Akihiro Hokkyo from The University of Tokyo, alongside colleagues, demonstrate a novel stabilizer-based method for constructing analytically solvable energy eigenstates within nonintegrable Hamiltonians, offering a fresh perspective on thermal behaviour. Their research, detailed in a new Letter, establishes a definitive limit , a ‘no-go’ theorem , proving that stabilizer eigenstates of two-body Hamiltonians cannot fully replicate microscopic thermal equilibrium. Significantly, the team also exhibits this bound’s precision by designing two nonintegrable Hamiltonians where stabilizer eigenstates accurately predict thermal values for both two- and three-body observables, revealing a fundamental constraint linked to the limited range of interactions.

Stabilizer States and Thermalisation in Quantum Systems

Scientists have demonstrated a novel approach to understanding thermalisation in isolated quantum many-body systems, a central question in quantum statistical mechanics. Researchers introduced a stabilizer-based framework to construct analytically tractable energy eigenstates of nonintegrable Hamiltonians, focusing on zero-energy eigenstates at infinite temperature. This breakthrough reveals a sharp no-go theorem: stabilizer eigenstates of two-body Hamiltonians cannot satisfy k-body microscopic thermal equilibrium for any k greater than or equal to 4. The team proved this limitation is not absolute, explicitly constructing two-body nonintegrable Hamiltonians whose stabilizer eigenstates reproduce thermal expectation values for all two-body and all three-body observables.
This work establishes a fundamental constraint imposed by the few-body nature of interactions, identifying the structural origin of this limitation by characterizing the conditions under which a stabilizer state can appear as a zero-energy eigenstate of a Hamiltonian. Experiments show that traditional approaches, like random-matrix theory, struggle to incorporate essential physical constraints such as locality and few-body interactions, hindering fully analytic verification of the eigenstate thermalization hypothesis (ETH) for realistic Hamiltonians. Rigorous analytic results are often restricted to limited settings, including non-thermal counterexamples like quantum many-body scars and Hilbert-space fragmentation, or valid only in constrained regimes such as free systems or at low temperatures. The research addresses a significant gap between the entanglement structure required for thermalisation and that accessible within controlled analytic frameworks, as thermal pure states exhibit volume-law entanglement while many tractable states obey area-law entanglement scaling.

Scientists achieved explicit families of energy eigenstates that are analytically tractable yet display genuine thermal features, building upon recent constructions like entangled antipodal pair (EAP) states which reproduce thermal expectation values at infinite temperature for local observables. However, the EAP states are distinguishable from thermal states when probed by two-body observables, failing to realise thermal equilibrium at the level of few-body correlations. Consequently, the team investigated whether explicit energy eigenstates could exhibit thermal structure beyond local observables, utilising the stabilizer formalism and graph states to explore thermal typicality beyond dimer constructions. Their main results establish the aforementioned no-go theorem alongside its tightness, demonstrating that stabilizer eigenstates of two-body Hamiltonians cannot exhibit microscopic thermal equilibrium for arbitrary k-body observables with k greater than or equal to 4. The study further clarifies how the few-body nature of interactions fundamentally limits the achievable degree of microscopic thermal equilibrium, providing a controlled family of examples relevant to analytic studies of ETH-like behaviour.

Stabilizer States and Thermalisation Limits Demonstrated

Scientists pioneered a stabilizer-based approach to construct analytically tractable energy eigenstates for nonintegrable Hamiltonians, addressing a central challenge in understanding highly entangled thermal eigenstates. The research focused on zero-energy eigenstates at infinite temperature, leading to a sharp no-go theorem demonstrating that stabilizer eigenstates of two-body Hamiltonians cannot satisfy microscopic thermal equilibrium for any k ≥ 4. This rigorous result establishes a fundamental limit on the thermalization achievable with stabilizer states. To prove this, the study employed a detailed analysis of stabilizer states and their properties within the context of many-body quantum systems.

Researchers meticulously constructed two-body nonintegrable Hamiltonians, then explicitly demonstrated that their stabilizer eigenstates reproduce thermal expectation values for all two- and three-body observables, confirming the tightness of the established bound. This construction involved leveraging the well-studied properties of graph states, a subclass of stabilizer states, to facilitate exact evaluation of reduced density matrices and provide a classical description of entanglement. Experiments utilized N-qubit systems, where each qubit resides in a Hilbert space Hi isomorphic to C2, and subsystems A are defined as subsets of the N sites labelled by i ∈ [N] := {1, ., N}. The team engineered these systems to explore the conditions under which a stabilizer state can function as a zero-energy eigenstate of a Hamiltonian, revealing a fundamental constraint imposed by the few-body nature of interactions.

This innovative approach enabled the identification of the structural origin of the limitation, clarifying why achieving full microscopic thermal equilibrium is challenging within the stabilizer and graph-state paradigm. Furthermore, the work built upon previous constructions of entangled antipodal pair (EAP) states, recognizing their limitations in capturing few-body correlations. The study extended this framework by developing a more general stabilizer-based approach, allowing for the construction of explicit examples relevant to analytic studies of eigenstate thermalization hypothesis (ETH)-like behaviour. This method achieves a controlled family of eigenstates, clarifying both the possibilities and limitations of realizing thermal eigenstates within the chosen framework, and offering a significant advancement in the field of quantum statistical mechanics.

Stabilizer States and Thermalisation Limits Demonstrated

Scientists have developed a stabilizer-based approach to construct analytically tractable energy eigenstates for nonintegrable Hamiltonians, addressing a central challenge in understanding highly entangled thermal eigenstates. The research focuses on zero-energy eigenstates at infinite temperature, leading to a sharp no-go theorem: stabilizer eigenstates of two-body Hamiltonians cannot satisfy microscopic thermal equilibrium for any k ≥ 4. This fundamentally limits the thermalization achievable with these states, even with complex entanglement structures. Experiments revealed that while stabilizer states offer a controlled framework for studying thermal behaviour, they are constrained by the few-body nature of interactions.

The team proved that stabilizer eigenstates of two-body Hamiltonians cannot exhibit microscopic thermal equilibrium for arbitrary k-body observables with k greater than or equal to 4, establishing a precise boundary on their thermal properties. Data shows this bound is tight, as researchers explicitly constructed two nonintegrable Hamiltonians whose stabilizer eigenstates reproduce thermal expectation values for all two- and three-body observables, demonstrating the limit is not merely theoretical. Measurements. The work rigorously analysed N-qubit systems, defining microscopic thermal equilibrium (MITE) as indistinguishability of reduced states from a Gibbs state at infinite temperature.

Scientists established that a state is in k-body MITE if it is in MITE on every subsystem A with |A| = k, meaning all k-body observables are thermal. Tests prove that the EAP states, previously introduced as entangled states reproducing thermal expectation values for local observables, can be represented as graph states and thus explored using the stabilizer formalism. This breakthrough delivers a deeper understanding of entanglement structure required for thermalization and its accessibility within controlled analytic frameworks.

MITE Bound Fails for Stabilizer States

Scientists have demonstrated a sharp and tight bound on the few-body microscopic thermal equilibrium (MITE) property for stabilizer eigenstates of two-body Hamiltonians. This research introduces a stabilizer-based approach to construct analytically tractable energy eigenstates of nonintegrable Hamiltonians, focusing on zero-energy eigenstates at infinite temperature. The authors proved a no-go theorem establishing that stabilizer eigenstates of two-body Hamiltonians cannot satisfy microscopic thermal equilibrium for any system size. Furthermore, the study confirms this bound is tight by explicitly constructing two nonintegrable Hamiltonians where stabilizer eigenstates accurately reproduce thermal expectation values for both two- and three-body observables.

This work clarifies fundamental limitations on realizing thermal eigenstates within stabilizer and graph-state frameworks, offering controlled analytic examples relevant to eigenstate thermalization (ETH)-like behaviour beyond simpler dimer constructions. The authors acknowledge a limitation in focusing solely on the infinite-temperature regime and suggest future research could explore extending the analysis to stabilizer states of higher-spin systems. Future investigations might also consider whether this construction can be used to simulate thermal pure states at finite temperature, mirroring the capabilities of EAP states. The results indicate a fundamental interplay between the few-body nature of interactions, the achievable degree of few-body MITE, and the amount of nonstabilizerness, or ‘magic’, present in the system. Quantifying this trade-off could illuminate the structural differences between stabilizer states and thermal many-body quantum states found in realistic Hamiltonians, representing a crucial direction for ongoing research.