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6CCVD Research

Noncommuting conserved charges in quantum many-body thermalization

At a Glance

Section titled “At a Glance”
MetadataDetails
Publication Date2020-04-15
JournalPhysical review. E
AuthorsNicole Yunger Halpern, Michael E. Beverland, Amir Kalev
InstitutionsMicrosoft (United States), Harvard University
Citations47
AnalysisFull AI Review Included

Executive Summary

Section titled “Executive Summary”
  • Non-Abelian Thermal State (NATS) Realization: The research introduces and numerically simulates the first experimental protocol designed to observe the Non-Abelian Thermal State (NATS), a nonclassical thermal ensemble arising from conserved quantities (charges) that do not commute.
  • System and Interaction: The protocol uses a closed, isolated spin chain (qubits) governed by nonintegrable Heisenberg interactions, which conserves total spin components (Qtotα) but allows noncommuting charges to be exchanged between the system (S) and the bath (B).
  • Superior Accuracy: Numerical simulations confirm that the long-time state of the system (ρS) thermalizes accurately to the NATS prediction, showing greater accuracy than predictions based on the standard Canonical or Grand Canonical ensembles.
  • Finite-Size Performance: For the simulated finite systems (6 to 14 qubits), the distance from the predicted state (relative entropy) decreases rapidly, scaling approximately as N-2.448, suggesting strong convergence even outside the thermodynamic limit.
  • Platform Versatility: The proposed protocol is immediately implementable across leading quantum platforms, including ultracold atoms, trapped ions, nitrogen-vacancy (NV) centers, and quantum dots.
  • Bridging Theory and Practice: This work establishes a crucial link between abstract quantum-information-theoretic thermodynamics and practical quantum many-body physics (AMO and condensed matter).
  • Robustness: The protocol demonstrates robustness against realistic experimental errors, specifically maintaining accuracy even with 1% anisotropy in the Heisenberg coupling.

Technical Specifications

Section titled “Technical Specifications”
ParameterValueUnitContext
Total System Size (Simulated)6 to 14Qubits (2N)Range of total qubits used in numerical simulations.
System of Interest (S) Sizen = 2QubitsFixed size of the subsystem whose thermalization is measured.
Interaction TypeHeisenberg (J)Energy ScaleNearest-neighbor and next-nearest-neighbor couplings (Htot).
Thermalization Time (Simulated)t = 2NnJ-1Time scale used for evolution to distinguish NATS from Canonical predictions.
Relative Entropy Scaling (D)N-2.448natsBest polynomial fit for the distance D(ρS
Analytical Scaling Bound (D)N-1/2natsPredicted bound for the distance in the thermodynamic limit.
Anisotropy Tolerance (Simulated)Δ = 0.99UnitlessRobustness check against 1% anisotropy in the Heisenberg Hamiltonian.
Inverse Temperature (β)-Etot / [3(2Nn - 3)J2]J-2First-order approximation used for high temperature calculations.
Small Parameter Constraint< 1UnitlessRequired for Taylor approximations of β and effective chemical potentials (µα).

Key Methodologies

Section titled “Key Methodologies”

  1. System Definition and Hamiltonian Construction:

    • A closed, isolated chain of Nn qubits is defined, where n=2 qubits form the system (S) and the remaining Nn-2 form the effective bath (B).
    • The total Hamiltonian (Htot) is constructed using nearest-neighbor and next-nearest-neighbor Heisenberg interactions (J), ensuring the system is nonintegrable to promote thermalization.
    • Htot is designed to conserve each total spin component (Qtotα for α = x, y, z), which are the noncommuting charges.

  2. Initial State Preparation (Approximate Microcanonical Subspace):

    • The global system is prepared in a state (ρtot) that occupies an approximate microcanonical (a.m.c.) subspace (M).
    • Soft Measurement Protocol: This protocol uses a sequence of generalized measurements (Positive Operator-Valued Measures, POVMs) to define the total charges (Sα) without causing significant disturbance to the noncommuting components.
    • The POVM envelope is modeled using a binomial distribution, which approaches a Gaussian for large Nn, ensuring the total charge standard deviation scales subextensively (O([Nn]α), where α < 1/2).

  3. Evolution and Thermalization:

    • The initial state ρtot evolves under Htot for a time t ~ 2Nn/J (in simulations) to ensure sufficient thermalization.
    • The system S is expected to thermalize internally to the NATS (ρNATS), characterized by the inverse temperature (β) and generalized chemical potentials (µα).

  4. Parameter Calculation and Readout:

    • The parameters β and µα are calculated analytically using perturbation theory (first order) based on the conserved total energy (Etot) and total charges (Sα).
    • The final system state (ρS) is inferred using quantum state tomography, which involves measuring expectation values of Pauli operator products.
    • Performance Metric: The accuracy of the NATS prediction is quantified by the relative entropy distance D(ρS||ρNATS), which is compared against the distances to the Canonical and Grand Canonical predictions.

Commercial Applications

Section titled “Commercial Applications”
Industry / PlatformRelevance to NATS Technology
Quantum Simulation (AMO Physics)Direct implementation of the spin-chain protocol using Ultracold Atoms (e.g., optical lattices) and Trapped Ions. Allows for the study of complex, nonclassical thermalization dynamics and non-Abelian symmetries in controlled environments.
Solid-State Quantum DevicesUtilizing Nitrogen-Vacancy (NV) Centers in Diamond and Quantum Dots to realize the NATS protocol. This introduces noncommuting charge conservation into solid-state systems, potentially enhancing quantum memory and storage capabilities.
Quantum Computing HardwareProvides a theoretical and experimental framework for understanding thermalization in systems with non-Abelian symmetries. This is critical for managing thermal noise and designing robust Superconducting Qubit architectures.
Quantum Information TheoryNATS physics generalizes the Eigenstate Thermalization Hypothesis (ETH) to systems with noncommuting conserved quantities, which is fundamental for advancing Quantum Error Correction and Quantum Cryptography based on non-Abelian group structures.
Nuclear Magnetic Resonance (NMR)NMR platforms can implement the spin-chain dynamics and soft measurement protocols, offering an accessible route for observing NATS and studying the effects of noncommuting charges on thermal ensembles.
View Original Abstract

In statistical mechanics, a small system exchanges conserved quantities-heat, particles, electric charge, etc.-with a bath. The small system thermalizes to the canonical ensemble or the grand canonical ensemble, etc., depending on the quantities. The conserved quantities are represented by operators usually assumed to commute with each other. This assumption was removed within quantum-information-theoretic (QI-theoretic) thermodynamics recently. The small system’s long-time state was dubbed “the non-Abelian thermal state (NATS).” We propose an experimental protocol for observing a system thermalize to the NATS. We illustrate with a chain of spins, a subset of which forms the system of interest. The conserved quantities manifest as spin components. Heisenberg interactions push the conserved quantities between the system and the effective bath, the rest of the chain. We predict long-time expectation values, extending the NATS theory from abstract idealization to finite systems that thermalize with finite couplings for finite times. Numerical simulations support the analytics: The system thermalizes to near the NATS, rather than to the canonical prediction. Our proposal can be implemented with ultracold atoms, nitrogen-vacancy centers, trapped ions, quantum dots, and perhaps nuclear magnetic resonance. This work introduces noncommuting conserved quantities from QI-theoretic thermodynamics into quantum many-body physics: atomic, molecular, and optical physics and condensed matter.

Tech Support

Section titled “Tech Support”

Original Source

Section titled “Original Source”

References

Section titled “References”
  1. 1980 - Statistical Physics: Part 1

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