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:2606.08756

Quantum resource localizability transitions in deep thermalization

Xiaozhou Feng, Chang Liu, Zihan Cheng, Wen Wei Ho, Matteo Ippoliti

quant-phcond-mat.dis-nncond-mat.stat-mechSubmitted Jun 7, 2026• Announced Jun 9, 2026• 0 citations
arXivHTMLPDF

Abstract

We investigate how quantum resource constraints affect deep thermalization, the emergence of universal local wavefunction distributions from partial measurements of a quantum many-body state. Quantum resources, such as non-stabilizerness (magic), coherence, asymmetry, imaginarity, and non-Gaussianity, are essential for quantum information processing, and constraints on their global abundance can reshape these emergent distributions. To address this question, we develop a unified framework for deep thermalization within general quantum resource theories (QRTs). Our central result is that QRTs fall into two classes: ``smoothly localizable'' (SL) QRTs, where the resource content of local post-measurement states changes continuously with the global resource density, set by the initial state and measurement basis, yielding continuously tunable wavefunction distributions; and ``threshold localizable'' (TL) QRTs, where the local resource content jumps discontinuously from minimal to near-maximal past a critical global resource threshold, producing a sharp transition between a resourceless, ``deep-ergodicity breaking'' distribution and a resourceful, maximally random one. We trace this SL-TL dichotomy to an information-theoretic mechanism, block sharpening: by viewing each QRT as coherence between blocks in Hilbert space, we show that the local resource content depends on the measurement's ability to collapse an initial superposition into a single resourceless block. Our theory is analytically tractable and quantitatively predicts the phase boundaries across all studied QRTs, which we validate with extensive numerical simulations. Finally, we highlight two consequences: a novel magic transition in zero-rate quantum error-correcting codes--previously believed to occur only at finite rates--and new implications for quantum resource certification protocols based on post-measurement state ensembles.

Comments
29+6 pages,10 figures

Related papers

Theory of Magic Phase Transitions in Encoding-Decoding Circuits

Piotr Sierant, Xhek Turkeshi

Quantum magic resources, or nonstabilizerness, are a central ingredient for universal quantum computation. In noisy many-body systems, the interplay between these resources and errors leads to sharp magic phase transitions. However, the microscopic mechanism behind these critical phenomena is still an open question, especially since early empirical evidence showed conflicting results regarding their universality classes. In this work, we provide a comprehensive picture of magic phase transitions for the class of encoding-decoding quantum circuits to resolve these ambiguities. We analytically show that the behavior of magic resources is fundamentally dictated by the chosen measurement protocol. When we fix, or post-select, the class of measurement syndromes, the magic transition inherits the universal features of the error-resilience phase transition in the circuits. Interestingly, this clean transition survives even for fully random Haar encoders showing that it is a consequence of initial's state retrieval, and not an artifact of the Clifford encoders. On the other hand, if we consider realistic Born-rule sampling, the intrinsic statistical fluctuations of a given syndrome measurement act as a relevant perturbation. This brings in strong finite-size drifts and an apparent multifractality, which end up altering the critical behavior of the system. We reveal that magic phase transitions are actually direct manifestations of error-resilience thresholds, rather than independent critical phenomena, reconciling conflicting observations from the earlier literature. Ultimately, our framework clarifies how the quantum computational power can survive, or be irreversibly destroyed, due to the competition between scrambling, measurements, and errors.

quant-pharXiv:2603.00235•Mar 3, 2026•0 citations

Coherence-Induced Deep Thermalization Transition in Random Permutation Quantum Dynamics

Chang Liu, Matteo Ippoliti, Wen Wei Ho

We report a phase transition in the projected ensemble-the collection of postmeasurement wave functions of a local subsystem obtained by measuring its complement. The transition emerges in systems undergoing random permutation dynamics, a type of quantum time evolution wherein computational basis states are shuffled without creating superpositions. It separates a phase exhibiting deep thermalization, where the projected ensemble is distributed over Hilbert space in a maximally entropic fashion (Haar random), from a phase where it is minimally entropic ("classical bit-string ensemble"). Crucially, this deep thermalization transition is invisible to the subsystem's density matrix, which always exhibits thermalization to infinite temperature across the phase diagram. Through a combination of analytical arguments and numerical simulations, we show that the transition is tuned by the total amount of coherence injected by the input state and the measurement basis, and is exhibited robustly across different microscopic models. Our findings represent a novel form of ergodicity-breaking universality in quantum many-body dynamics, characterized not by a failure of regular thermalization, but rather by a failure of deep thermalization.

arXiv:2510.18369•Physical Review Letters•Feb 18, 2026•3 citations

Deep thermalization and Hilbert space ergodicity

Daniel K. Mark, Manuel Endres, Matteo Ippoliti +2 more

Recent advances in quantum simulation have enabled the discovery of novel forms of universality in quantum many-body dynamics. In this Review, we discuss deep thermalization and Hilbert space ergodicity--two recently discovered phenomena characterized by the emergence of distributions of quantum states that are "maximally random" in a precise sense. They offer a new perspective on how irreversible statistical mechanics arises from reversible unitary quantum dynamics, going beyond conventional theories of quantum thermalization and equilibration. We review a unifying framework, rooted in quantum information theory and principles of maximum entropy, that explains the different forms of ergodicity which arise under various physical constraints. We discuss open questions and active research directions, including generalizations beyond ensembles of pure quantum states, phase transitions in deep thermalization tied to deeper forms of ergodicity-breaking, connections to broader topics in quantum thermalization and ergodicity, and applications in quantum information science, such as for benchmarking or tomography.

quant-pharXiv:2609.30248•Sep 25, 2026•0 citations

Nonlocality of deep thermalization

Harshank Shrotriya, Wen Wei Ho

We study the role of global system topology in governing deep thermalization, the relaxation of a local subsystem towards a maximally-entropic, uniform distribution of post-measurement states, upon observing the complementary subsystem in a local basis. Concretely, we focus on a class of (1+1)d systems exhibiting ‘maximally-chaotic’ dynamics, and consider how the rate of the formation of such a universal wavefunction distribution depends on boundary conditions of the system. We find that deep thermalization is achieved exponentially quickly in the presence of either periodic or open boundary conditions; however, the rate at which this occurs is twice as fast for the former than for the latter. These results are attained analytically using the calculus of integration over unitary groups, and supported by extensive numerical simulations. Our findings highlight the nonlocal nature of deep thermalization, and clearly illustrates that the physics underlying this phenomenon goes beyond that of standard quantum thermalization, which only depends on the net build-up of entanglement between a subsystem and its complement.

arXiv:2305.08437•SciPost Physics•Mar 21, 2025•9 citations

Maximum Entropy Principle in Deep Thermalization and in Hilbert-Space Ergodicity

Daniel K. Mark, Federica Maria Surace, Andreas Elben +5 more

We report universal statistical properties displayed by ensembles of pure states that naturally emerge in quantum many-body systems. Specifically, two classes of state ensembles are considered: those formed by (i) the temporal trajectory of a quantum state under unitary evolution or (ii) the quantum states of small subsystems obtained by partial, local projective measurements performed on their complements. These cases, respectively, exemplify the phenomena of “Hilbert-space ergodicity” and “deep thermalization.” In both cases, the resultant ensembles are defined by a simple principle: The distributions of pure states have maximum entropy, subject to constraints such as energy conservation, and effective constraints imposed by thermalization. We present and numerically verify quantifiable signatures of this principle by deriving explicit formulas for all statistical moments of the ensembles, proving the necessary and sufficient conditions for such universality under widely accepted assumptions, and describing their measurable consequences in experiments. We further discuss information-theoretic implications of the universality: Our ensembles have maximal information content while being maximally difficult to interrogate, establishing that generic quantum state ensembles that occur in nature hide (scramble) information as strongly as possible. Our results generalize the notions of Hilbert-space ergodicity to time-independent Hamiltonian dynamics and deep thermalization from infinite to finite effective temperature. Our work presents new perspectives to characterize and understand universal behaviors of quantum dynamics using statistical and information-theoretic tools. Published by the American Physical Society 2024

arXiv:2403.11970•Physical Review X•Nov 25, 2024•50 citations

Deep Thermalization and Measurements of Quantum Resources

Naga Dileep Varikuti, Soumik Bandyopadhyay, Philipp Hauke

Quantum resource theories provide a unified framework for characterizing useful quantum phenomena subject to physical constraints, but are notoriously hard to assess in experimental systems. In this letter, we introduce a unified protocol for quantifying the resource-generating power of arbitrary quantum evolutions applicable to multiple resource theories. It is based on deep thermalization, which has recently gained attention for its role in the emergence of quantum state designs from partial projective measurements. Central to our approach is the use of projected ensembles, recently employed to probe deep thermalization, together with new twirling identities that allow us to directly infer the resource-generating powers of the underlying dynamics. These identities further reveal how resources build up and thermalize in generic quantum circuits. Finally, we show that quantum resources themselves undergo deep thermalization at the subsystem level, offering a complementary and another experimentally accessible route to infer the resource-generating powers. Our work connects deep thermalization to resource quantification, offering a new perspective on the essential role of various resources in generating state designs and on efficient estimation of quantum resources in experiment.

quant-pharXiv:2512.09999•Physical Review Letters•Dec 12, 2025•0 citations
We investigate how quantum resource constraints affect deep thermalization, the emergence of universal local wavefunction distributions from partial measurements of a quantum many-body state. Quantum resources, such as non-stabilizerness (magic), coherence, asymmetry, imaginarity, and non-Gaussianity, are essential for quantum information processing, and constraints on their global abundance can reshape these emergent distributions. To address this question, we develop a unified framework for deep thermalization within general quantum resource theories (QRTs). Our central result is that QRTs fall into two classes: smoothly localizable'' (SL) QRTs, where the resource content of local post-measurement states changes continuously with the global resource density, set by the initial state and measurement basis, yielding continuously tunable wavefunction distributions; and threshold localizable'' (TL) QRTs, where the local resource content jumps discontinuously from minimal to near-maximal past a critical global resource threshold, producing a sharp transition between a resourceless, ``deep-ergodicity breaking'' distribution and a resourceful, maximally random one. We trace this SL-TL dichotomy to an information-theoretic mechanism, block sharpening: by viewing each QRT as coherence between blocks in Hilbert space, we show that the local resource content depends on the measurement's ability to collapse an initial superposition into a single resourceless block. Our theory is analytically tractable and quantitatively predicts the phase boundaries across all studied QRTs, which we validate with extensive numerical simulations. Finally, we highlight two consequences: a novel magic transition in zero-rate quantum error-correcting codes--previously believed to occur only at finite rates--and new implications for quantum resource certification protocols based on post-measurement state ensembles.
Quantum magic resources, or nonstabilizerness, are a central ingredient for universal quantum computation. In noisy many-body systems, the interplay between these resources and errors leads to sharp magic phase transitions. However, the microscopic mechanism behind these critical phenomena is still an open question, especially since early empirical evidence showed conflicting results regarding their universality classes. In this work, we provide a comprehensive picture of magic phase transitions for the class of encoding-decoding quantum circuits to resolve these ambiguities. We analytically show that the behavior of magic resources is fundamentally dictated by the chosen measurement protocol. When we fix, or post-select, the class of measurement syndromes, the magic transition inherits the universal features of the error-resilience phase transition in the circuits. Interestingly, this clean transition survives even for fully random Haar encoders showing that it is a consequence of initial's state retrieval, and not an artifact of the Clifford encoders. On the other hand, if we consider realistic Born-rule sampling, the intrinsic statistical fluctuations of a given syndrome measurement act as a relevant perturbation. This brings in strong finite-size drifts and an apparent multifractality, which end up altering the critical behavior of the system. We reveal that magic phase transitions are actually direct manifestations of error-resilience thresholds, rather than independent critical phenomena, reconciling conflicting observations from the earlier literature. Ultimately, our framework clarifies how the quantum computational power can survive, or be irreversibly destroyed, due to the competition between scrambling, measurements, and errors.
We report a phase transition in the projected ensemble-the collection of postmeasurement wave functions of a local subsystem obtained by measuring its complement. The transition emerges in systems undergoing random permutation dynamics, a type of quantum time evolution wherein computational basis states are shuffled without creating superpositions. It separates a phase exhibiting deep thermalization, where the projected ensemble is distributed over Hilbert space in a maximally entropic fashion (Haar random), from a phase where it is minimally entropic ("classical bit-string ensemble"). Crucially, this deep thermalization transition is invisible to the subsystem's density matrix, which always exhibits thermalization to infinite temperature across the phase diagram. Through a combination of analytical arguments and numerical simulations, we show that the transition is tuned by the total amount of coherence injected by the input state and the measurement basis, and is exhibited robustly across different microscopic models. Our findings represent a novel form of ergodicity-breaking universality in quantum many-body dynamics, characterized not by a failure of regular thermalization, but rather by a failure of deep thermalization.
Recent advances in quantum simulation have enabled the discovery of novel forms of universality in quantum many-body dynamics. In this Review, we discuss deep thermalization and Hilbert space ergodicity--two recently discovered phenomena characterized by the emergence of distributions of quantum states that are "maximally random" in a precise sense. They offer a new perspective on how irreversible statistical mechanics arises from reversible unitary quantum dynamics, going beyond conventional theories of quantum thermalization and equilibration. We review a unifying framework, rooted in quantum information theory and principles of maximum entropy, that explains the different forms of ergodicity which arise under various physical constraints. We discuss open questions and active research directions, including generalizations beyond ensembles of pure quantum states, phase transitions in deep thermalization tied to deeper forms of ergodicity-breaking, connections to broader topics in quantum thermalization and ergodicity, and applications in quantum information science, such as for benchmarking or tomography.
We study the role of global system topology in governing deep thermalization, the relaxation of a local subsystem towards a maximally-entropic, uniform distribution of post-measurement states, upon observing the complementary subsystem in a local basis. Concretely, we focus on a class of (1+1)d systems exhibiting ‘maximally-chaotic’ dynamics, and consider how the rate of the formation of such a universal wavefunction distribution depends on boundary conditions of the system. We find that deep thermalization is achieved exponentially quickly in the presence of either periodic or open boundary conditions; however, the rate at which this occurs is twice as fast for the former than for the latter. These results are attained analytically using the calculus of integration over unitary groups, and supported by extensive numerical simulations. Our findings highlight the nonlocal nature of deep thermalization, and clearly illustrates that the physics underlying this phenomenon goes beyond that of standard quantum thermalization, which only depends on the net build-up of entanglement between a subsystem and its complement.
We report universal statistical properties displayed by ensembles of pure states that naturally emerge in quantum many-body systems. Specifically, two classes of state ensembles are considered: those formed by (i) the temporal trajectory of a quantum state under unitary evolution or (ii) the quantum states of small subsystems obtained by partial, local projective measurements performed on their complements. These cases, respectively, exemplify the phenomena of “Hilbert-space ergodicity” and “deep thermalization.” In both cases, the resultant ensembles are defined by a simple principle: The distributions of pure states have maximum entropy, subject to constraints such as energy conservation, and effective constraints imposed by thermalization. We present and numerically verify quantifiable signatures of this principle by deriving explicit formulas for all statistical moments of the ensembles, proving the necessary and sufficient conditions for such universality under widely accepted assumptions, and describing their measurable consequences in experiments. We further discuss information-theoretic implications of the universality: Our ensembles have maximal information content while being maximally difficult to interrogate, establishing that generic quantum state ensembles that occur in nature hide (scramble) information as strongly as possible. Our results generalize the notions of Hilbert-space ergodicity to time-independent Hamiltonian dynamics and deep thermalization from infinite to finite effective temperature. Our work presents new perspectives to characterize and understand universal behaviors of quantum dynamics using statistical and information-theoretic tools. Published by the American Physical Society 2024
Quantum resource theories provide a unified framework for characterizing useful quantum phenomena subject to physical constraints, but are notoriously hard to assess in experimental systems. In this letter, we introduce a unified protocol for quantifying the resource-generating power of arbitrary quantum evolutions applicable to multiple resource theories. It is based on deep thermalization, which has recently gained attention for its role in the emergence of quantum state designs from partial projective measurements. Central to our approach is the use of projected ensembles, recently employed to probe deep thermalization, together with new twirling identities that allow us to directly infer the resource-generating powers of the underlying dynamics. These identities further reveal how resources build up and thermalize in generic quantum circuits. Finally, we show that quantum resources themselves undergo deep thermalization at the subsystem level, offering a complementary and another experimentally accessible route to infer the resource-generating powers. Our work connects deep thermalization to resource quantification, offering a new perspective on the essential role of various resources in generating state designs and on efficient estimation of quantum resources in experiment.