High-temperature thermalization implies the emergence of quantum state designs

Wilming, Henrik; Roth, Ingo

doi:10.48550/arxiv.2202.01669

Cited by 4 publications

(14 citation statements)

References 38 publications

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“…This implies 14 < (2k + 1)d A δ; using the assumption that d A δ < 1, and this is seen to be consistent with the bound from Ref. [72], Eq. (45).…”

Section: Discussionsupporting

confidence: 86%

“…First, Ref. [72] proved that, if one chooses the measurement basis in the bath B Haar-randomly, then the projected ensemble yields an approximate state design with high probability whenever the reduced density matrix is close to thermalization at infinite temperature: ρ A ≈ I A /d A . More precisely, given a deviation from ideal thermalization δ =…”

Section: Discussionmentioning

confidence: 99%

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Solvable model of deep thermalization with distinct design times

Ippoliti

2022

Quantum

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We study the emergence over time of a universal, uniform distribution of quantum states supported on a finite subsystem, induced by projectively measuring the rest of the system. Dubbed deep thermalization, this phenomenon represents a form of equilibration in quantum many-body systems stronger than regular thermalization, which only constrains the ensemble-averaged values of observables. While there exist quantum circuit models of dynamics in one dimension where this phenomenon can be shown to arise exactly, these are special in that deep thermalization occurs at precisely the same time as regular thermalization. Here, we present an exactly-solvable model of chaotic dynamics where the two processes can be shown to occur over different time scales. The model is composed of a finite subsystem coupled to an infinite random-matrix bath through a small constriction, and highlights the role of locality and imperfect thermalization in constraining the formation of such universal wavefunction distributions. We test our analytical predictions against exact numerical simulations, finding excellent agreement.

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“…This implies 14 < (2k + 1)d A δ; using the assumption that d A δ < 1, and this is seen to be consistent with the bound from Ref. [72], Eq. (45).…”

Section: Discussionsupporting

confidence: 86%

Section: Discussionmentioning

confidence: 99%

Solvable model of deep thermalization with distinct design times

Ippoliti

2022

Quantum

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“…The projected ensemble formalism precisely achieves this goal. We consider the case of projective measurements performed in the computational basis-which, as we argued above, is of practical relevance to quantum simulators-but we note this assumption can be relaxed to allow for other choices too (as we will do later, and as was also considered in [9,24]). Concretely, a measurement outcome on B will be labeled by a bit-string z ∈ {0, 1} N B (we drop the subscript in z B for convenience), so that following the measurement, the global wavefunction is updated according to the Born rule as…”

Section: Motivation and Definitionmentioning

confidence: 99%

“…Before proceeding, let us remark that Ref. [24] already establishes a sharp connection between the onset of regular thermalization and the simultaneous formation of higher designs: whenever the reduced density matrix on A is close to being maximally mixed (i.e., at infinite temperature), then with high probability the projected ensemble forms an approximate quantum state-design, provided the measurement basis of the complementary subsystem is chosen at random from the Haar measure (which is typically non-local and highly entangled). However, a setting which is arguably more natural for experiments is that of spatially local measurement bases, like the ones we consider in this work.…”

Section: Introductionmentioning

confidence: 99%

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Dynamical purification and the emergence of quantum state designs from the projected ensemble

Ippoliti¹,

Ho²

2022

Preprint

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Quantum thermalization in a many-body system is defined by the approach of local subsystems towards a universal form, describable as an ensemble of quantum states wherein observables acquire thermal expectation values. Recently, it was demonstrated that the distribution of these quantum states can also exhibit universal statistics, upon associating each state with the outcome of local projective measurements of the complementary subsystem. Specifically, this collection of pure quantum states -called the projected ensemble -can under certain conditions mimic the behavior of a maximally entropic, uniformly random ensemble, i.e. form a quantum state-design, representing a "deeper" form of quantum thermalization. In this work, we investigate the dynamical process underlying this novel emergent universality. Leveraging a space-time duality mapping for one-dimensional quantum circuits, we argue that the physics of dynamical purification, which arises in the context of monitored quantum systems, constrains the the projected ensemble's approach towards the uniform distribution. We prove that absence of dynamical purification in the space-time dual dynamics (a condition realized in dual-unitary quantum circuits with appropriate initial states and final measurement bases) generically yields exact state-designs for all moments k at the same time, extending previous rigorous results [Ho and Choi, Phys. Rev. Lett. 128, 060601 (2022)]. Conversely, we show that, departing from these conditions, dynamical purification can lead to a separation of timescales between the formation of a quantum state-design for moment k = 1 (regular thermalization) and for high moments k 1 ("deep" thermalization). Our results suggest that the projected ensemble can probe nuanced features of quantum dynamics inaccessible to regular thermalization, such as quantum information scrambling.

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Emergent quantum state designs and biunitarity in dual-unitary circuit dynamics

Claeys

Lamacraft

2022

Quantum

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Recent works have investigated the emergence of a new kind of random matrix behaviour in unitary dynamics following a quantum quench. Starting from a time-evolved state, an ensemble of pure states supported on a small subsystem can be generated by performing projective measurements on the remainder of the system, leading to a projected ensemble. In chaotic quantum systems it was conjectured that such projected ensembles become indistinguishable from the uniform Haar-random ensemble and lead to a quantum state design. Exact results were recently presented by Ho and Choi [Phys. Rev. Lett. 128, 060601 (2022)] for the kicked Ising model at the self-dual point. We provide an alternative construction that can be extended to general chaotic dual-unitary circuits with solvable initial states and measurements, highlighting the role of the underlying dual-unitarity and further showing how dual-unitary circuit models exhibit both exact solvability and random matrix behaviour. Building on results from biunitary connections, we show how complex Hadamard matrices and unitary error bases both lead to solvable measurement schemes.

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High-temperature thermalization implies the emergence of quantum state designs

Cited by 4 publications

References 38 publications

Solvable model of deep thermalization with distinct design times

Solvable model of deep thermalization with distinct design times

Dynamical purification and the emergence of quantum state designs from the projected ensemble

Emergent quantum state designs and biunitarity in dual-unitary circuit dynamics

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