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Quantum speed limit for observables from quantum asymmetry

Agung Budiyono, Michael Moody, Hadyan L. Prihadi, Rafika Rahmawati, Sebastian Deffner

TL;DR

The paper derives a quantum speed limit for observables by bounding the instantaneous speed of an observable’s expectation value with the trace-norm asymmetry ||[rho, K]||_1/2 relative to the observable K, under unitary dynamics with bounded generators. It connects this speed limit to weak measurements, quantum Fisher information, and coherence (via unspeakable coherence and Kirkwood-Dirac nonrealities), and extends the bound to a time-optimization framework and a quantum thermodynamic speed limit via entropy production. Key results include exact expressions and saturations for qubits, complementarity relations for mutually unbiased bases, and a thermodynamic bound involving the Gibbs state and inverse temperature beta. The framework provides a resource-based perspective on how noncommutativity between states and observables governs dynamical speed, with implications for quantum metrology, control, and nonequilibrium thermodynamics.

Abstract

Quantum asymmetry and coherence are genuinely quantum resources that are essential to realize quantum advantage in information technologies. However, all quantum processes are fundamentally constrained by quantum speed limits, which raises the question on the corresponding bounds on the rate of consumption of asymmetry and coherence. In the present work, we derive a formulation of the quantum speed limit for observables in terms of the trace-norm asymmetry of the time-dependent quantum state relative to the observable. This version of the quantum speed limit can be shown to be directly relevant in weak measurements and quantum metrology. It can be further related to quantum coherence relative to the observable, and we obtain a complementary relation for the speed of three mutually unbiased observables for a single qubit. As an application, we derive a notion of a quantum thermodynamic speed limit.

Quantum speed limit for observables from quantum asymmetry

TL;DR

The paper derives a quantum speed limit for observables by bounding the instantaneous speed of an observable’s expectation value with the trace-norm asymmetry ||[rho, K]||_1/2 relative to the observable K, under unitary dynamics with bounded generators. It connects this speed limit to weak measurements, quantum Fisher information, and coherence (via unspeakable coherence and Kirkwood-Dirac nonrealities), and extends the bound to a time-optimization framework and a quantum thermodynamic speed limit via entropy production. Key results include exact expressions and saturations for qubits, complementarity relations for mutually unbiased bases, and a thermodynamic bound involving the Gibbs state and inverse temperature beta. The framework provides a resource-based perspective on how noncommutativity between states and observables governs dynamical speed, with implications for quantum metrology, control, and nonequilibrium thermodynamics.

Abstract

Quantum asymmetry and coherence are genuinely quantum resources that are essential to realize quantum advantage in information technologies. However, all quantum processes are fundamentally constrained by quantum speed limits, which raises the question on the corresponding bounds on the rate of consumption of asymmetry and coherence. In the present work, we derive a formulation of the quantum speed limit for observables in terms of the trace-norm asymmetry of the time-dependent quantum state relative to the observable. This version of the quantum speed limit can be shown to be directly relevant in weak measurements and quantum metrology. It can be further related to quantum coherence relative to the observable, and we obtain a complementary relation for the speed of three mutually unbiased observables for a single qubit. As an application, we derive a notion of a quantum thermodynamic speed limit.

Paper Structure

This paper contains 12 sections, 31 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Speed and quantum asymmetry
  3. Quantum speed from weak measurements
  4. Quantum metrological consequences
  5. Quantum speed limit and quantum fluctuations
  6. Quantum speed limit time
  7. Coherence and complementarity
  8. Example: quantum speed of single qubits
  9. Complementarity bounds for qubits
  10. Relative entropy of athermality
  11. Concluding remarks
  12. Quantum speed limit for qubits

Figures (1)

  • Figure 1: Realizations of the a quantum process (blue markers), with random initial states and unitary maps. The size of the marker represents the purity of the initial states. Green markers depict realizations that are optimal.