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Universal scaling of finite-temperature quantum adiabaticity in driven many-body systems

Li-Ying Chou, Jyong-Hao Chen

TL;DR

The work develops a finite-temperature framework for quantum adiabaticity in closed many-body systems by marrying a mixed-state quantum speed limit with fidelity susceptibility in Liouville space. For protocols that drive a Gibbs state toward a quasi-Gibbs target, it yields an explicit threshold driving rate Γth that separates adiabatic from nonadiabatic dynamics, with Γth factorizing into a zero-temperature size-dependent part ΓN and a universal temperature factor f(β). In gapped local Hamiltonians, f(β) is exponentially close to unity at low temperature and grows linearly with temperature at high temperature, a structure verified exactly in TFIC, QXYC, and MFIC chains via transfer-matrix methods. The results provide model-insensitive criteria for finite-temperature adiabaticity and offer practically accessible diagnostics for experimental implementations and quantum-state engineering in thermal settings. The approach paves the way for extending adiabaticity criteria to open systems, enabling rigorous assessments of thermal-state preparation and adiabatic quantum computation under decoherence.

Abstract

Establishing quantitative adiabaticity criteria at finite temperature remains substantially less developed than in the pure-state setting, despite the fact that realistic quantum systems are never at absolute zero. Here we derive rigorous bounds on the Hilbert-Schmidt fidelity between mixed states by combining a mixed-state quantum speed limit with mixed-state fidelity susceptibility within the Liouville space formulation of quantum mechanics. Applied to protocols that drive an initial Gibbs state toward a quasi-Gibbs target, these bounds yield an explicit threshold driving rate for the onset of nonadiabaticity. For a broad class of local Hamiltonians in gapped phases, we show that, in the thermodynamic limit, the threshold factorizes into two factors: a system-size contribution that recovers the zero-temperature scaling and a universal temperature-dependent factor. The latter is exponentially close to unity at low temperature, whereas at high temperature it increases linearly with temperature. We verify the predicted scaling in several spin-1/2 chains by obtaining closed-form expressions for the threshold driving rate. Our results provide practical and largely model-independent criteria for finite-temperature adiabaticity in closed many-body systems.

Universal scaling of finite-temperature quantum adiabaticity in driven many-body systems

TL;DR

The work develops a finite-temperature framework for quantum adiabaticity in closed many-body systems by marrying a mixed-state quantum speed limit with fidelity susceptibility in Liouville space. For protocols that drive a Gibbs state toward a quasi-Gibbs target, it yields an explicit threshold driving rate Γth that separates adiabatic from nonadiabatic dynamics, with Γth factorizing into a zero-temperature size-dependent part ΓN and a universal temperature factor f(β). In gapped local Hamiltonians, f(β) is exponentially close to unity at low temperature and grows linearly with temperature at high temperature, a structure verified exactly in TFIC, QXYC, and MFIC chains via transfer-matrix methods. The results provide model-insensitive criteria for finite-temperature adiabaticity and offer practically accessible diagnostics for experimental implementations and quantum-state engineering in thermal settings. The approach paves the way for extending adiabaticity criteria to open systems, enabling rigorous assessments of thermal-state preparation and adiabatic quantum computation under decoherence.

Abstract

Establishing quantitative adiabaticity criteria at finite temperature remains substantially less developed than in the pure-state setting, despite the fact that realistic quantum systems are never at absolute zero. Here we derive rigorous bounds on the Hilbert-Schmidt fidelity between mixed states by combining a mixed-state quantum speed limit with mixed-state fidelity susceptibility within the Liouville space formulation of quantum mechanics. Applied to protocols that drive an initial Gibbs state toward a quasi-Gibbs target, these bounds yield an explicit threshold driving rate for the onset of nonadiabaticity. For a broad class of local Hamiltonians in gapped phases, we show that, in the thermodynamic limit, the threshold factorizes into two factors: a system-size contribution that recovers the zero-temperature scaling and a universal temperature-dependent factor. The latter is exponentially close to unity at low temperature, whereas at high temperature it increases linearly with temperature. We verify the predicted scaling in several spin-1/2 chains by obtaining closed-form expressions for the threshold driving rate. Our results provide practical and largely model-independent criteria for finite-temperature adiabaticity in closed many-body systems.
Paper Structure (33 sections, 1 theorem, 199 equations, 2 figures, 1 table)

This paper contains 33 sections, 1 theorem, 199 equations, 2 figures, 1 table.

Table of Contents

  1. Derivation of the mixed-state quantum speed limit [Eq. (\ref{['eq: QSL mixed states']})]
  2. Liouville space setup
  3. Hilbert--Schmidt angle and its rate of change
  4. Derivation of mixed-state fidelity bounds [Eq. (\ref{['eq: fidelity bound']})]
  5. Proof of Theorem \ref{['thm:GammaThTempScaling']}
  6. Useful expressions for $\delta V$ and $\chi^{\,}_{\mathrm{F}}$
  7. Useful expression for $\delta V$
  8. Useful expression for $\chi^{\,}_{\mathrm{F}}$
  9. Low-temperature regime
  10. $\delta V$ in the low-temperature regime
  11. $\chi^{\,}_{\mathrm F}$ in the low-temperature regime
  12. Threshold driving rate $\Gamma^{\,}_{\mathrm{th}}$ in the low-temperature regime
  13. High-temperature regime
  14. $\delta V$ in the high-temperature regime
  15. $\chi^{\,}_{\mathrm{F}}$ in the high-temperature regime
  16. ...and 18 more sections

Key Result

Theorem 1

Consider a broad class of local Hamiltonians in gapped phases, i.e., Hamiltonians that can be written as sums of local operators and possess a nonzero spectral gap above the ground state. In the thermodynamic limit $N\to\infty$, the threshold driving rate $\Gamma^{\,}_{\mathrm{th}}$ factorizes as in Here, $c^{\,}_{1}\in(0,2]$ is a model-dependent dimensionless constant, whereas $c^{\,}_{2}>0$ is a

Figures (2)

  • Figure 1: Temperature-dependent factor $f(\beta)$ in the threshold driving rate $\Gamma^{\,}_{\mathrm{th}}$ [Eqs. \ref{['eq: define threshold driving rate']}, \ref{['eq: factorization of threshold driving rate']}] for the TFIC and QXYC [Eq. \ref{['eq: V_models_def']}] in the thermodynamic limit. The solid curve shows the exact result $f(\beta)=\coth(2\beta J)$ [Eq. \ref{['eq: exact Ising temperature dependence']}], while the dotted curves show the low- and high-temperature asymptotics, $f(\beta)\simeq 1+2e^{-4\beta J}$ (low temperature) and $f(\beta)\simeq 1/(2\beta J)$ (high temperature).
  • Figure 2: Adiabatic fidelity $\mathcal{F}(\lambda)$ [Eq. \ref{['eq:adiabatic_fidelity_defined']}] (cyan curve) and thermal-state overlap $\mathcal{C}(\lambda)$ [Eq. \ref{['eq: thermal state overlap defined']}] (black curve) for the driven transverse-field Ising chain $\hat{H}^{\,}_{\lambda}=\hat{H}^{\,}_{0}+\lambda \hat{V}^{\,}_{\mathrm{TFIC}}$\ref{['eq: V_TFIC_def']} at $\beta J=5$ and $\Gamma/J=2$, plotted as a function of $\lambda=h/J$. Panels (a) and (b) correspond to $N=10^{3}$ and $N=10^{4}$, respectively. Over the range shown, $\mathcal{F}(\lambda)$ and $\mathcal{C}(\lambda)$ are visually indistinguishable. The blue (red) shaded band indicates the values of $\mathcal{F}(\lambda)$ allowed by inequality \ref{['eq: fidelity bound a']} (inequality \ref{['eq: fidelity bound b']}).

Theorems & Definitions (2)

  • Theorem 1: Temperature scaling of $\Gamma^{\,}_{\mathrm{th}}$
  • proof : Proof (sketch).