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Family of Exact and Inexact Quantum Speed Limits for Completely Positive and Trace-Preserving Dynamics

Abhay Srivastav, Vivek Pandey, Brij Mohan, Arun Kumar Pati

Abstract

Traditional quantum speed limits formulated in density matrix space are generally unattainable for a wide class of dynamics and it is difficult to characterize the fastest possible dynamics. To address this, we present two distinct quantum speed limits in Liouville space for Completely Positive and Trace-Preserving (CPTP) dynamics. The first bound saturates for time-optimal CPTP dynamics, while the second bound is exact for all states and all CPTP dynamics. Our bounds have a clear physical and geometric interpretation arising from the uncertainty relations for operators acting on Liouville space, and the geometry of quantum evolution in Liouville space. We also obtain the form of the Liouvillian, which generates the time-optimal CPTP dynamics that connect the given initial and target states. To illustrate our findings, we show that the speed of evolution in Liouville space bounds the growth of the spectral form factor and Krylov complexity of states, which are crucial for studying information scrambling and quantum chaos. In another important application, we show that our results can help us understand the counter-intuitive phenomenon of the Mpemba effect in non-equilibrium open quantum dynamics, as the minimal relaxation time scale obtained by speed limits is dictated by the eigenmodes of the Liouvillian.

Family of Exact and Inexact Quantum Speed Limits for Completely Positive and Trace-Preserving Dynamics

Abstract

Traditional quantum speed limits formulated in density matrix space are generally unattainable for a wide class of dynamics and it is difficult to characterize the fastest possible dynamics. To address this, we present two distinct quantum speed limits in Liouville space for Completely Positive and Trace-Preserving (CPTP) dynamics. The first bound saturates for time-optimal CPTP dynamics, while the second bound is exact for all states and all CPTP dynamics. Our bounds have a clear physical and geometric interpretation arising from the uncertainty relations for operators acting on Liouville space, and the geometry of quantum evolution in Liouville space. We also obtain the form of the Liouvillian, which generates the time-optimal CPTP dynamics that connect the given initial and target states. To illustrate our findings, we show that the speed of evolution in Liouville space bounds the growth of the spectral form factor and Krylov complexity of states, which are crucial for studying information scrambling and quantum chaos. In another important application, we show that our results can help us understand the counter-intuitive phenomenon of the Mpemba effect in non-equilibrium open quantum dynamics, as the minimal relaxation time scale obtained by speed limits is dictated by the eigenmodes of the Liouvillian.
Paper Structure (8 sections, 5 theorems, 119 equations, 2 figures)

This paper contains 8 sections, 5 theorems, 119 equations, 2 figures.

Table of Contents

  1. Preliminaries, Notations, Inequalities, and Equalities Used in the Main Text
  2. Proof of Theorem 1
  3. Geometrical derivation of Inexact Quantum Speed Limit for arbitrary time-continuous dynamics
  4. Proof of Corollary
  5. Proof of Theorem 2
  6. Bound on Krylov complexity
  7. Speed of evolution for open quantum dynamics in terms of eigenvalues and eigenmodes of Liouvillian
  8. Mpemba effect in Thermalizing (Amplitude damping) process

Key Result

Proposition 1

For any two non-Hermitian operators acting on Liouville space, $\mathcal{A}$ and $\mathcal{B}$, there exists an inexact uncertainty relation given by where $\mathcal{P} := |\tilde{\rho} )( \tilde{\rho}|$ represents the projection operator acting on Liouville space, associated with the state vector $|\tilde{\rho})$, and $\operatorname{tr}(\mathcal{O}\mathcal{P})$ and $(\Delta \mathcal{O})^2 = \ope

Figures (2)

  • Figure 1: Mpemba effect in a qubit system under the action of thermal environment. The computations are performed for various values of $\alpha$ (which corresponds to different initial states) and the parameter $\gamma = 0.01$. The operator-norm of Liouvillian is $\norm{\cal{L}}_{\rm op}= \sqrt{2}\gamma$. (a) Speed efficiency $\eta$ for different initial states. (b) Distance between different non-equilibrium states and the steady state, i.e., $\Theta(\rho_{T},\rho_{ss})$. (c) Tightness of the bound given in Eq. \ref{['LMT']} for different initial states.
  • Figure 2: Mpemba effect in a qubit system under the action of the thermal environment at finite temperature ($n=0.5$). The computations are performed for various values of $\alpha$ (which corresponds to different initial states) and the parameter $\gamma = 0.01$. (a) Speed efficiency (see Eq. \ref{['speff']} in the main text) for different initial states. (b) Distance between different non-equilibrium states and the steady state, i.e., $\Theta(\rho_{T},\rho_{ss})$. (c) Tightness of the bound given in Eq. \ref{['mainbound']} (Eq. \ref{['LMT']} in the main text) for different initial states.

Theorems & Definitions (7)

  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Corollary
  • Theorem 2
  • proof
  • proof