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The Hidden Nature of Non-Markovianity

Jihong Cai, Advith Govindarajan, Marius Junge

TL;DR

This work shows that non-Markovianity, often diagnosed via memory effects in quantum dynamics, is invisible at the level of a single trajectory: under mild regularity any differentiable density-trajectory $\rho_t$ can be realized as the outcome of a time-dependent Lindbladian, i.e., a Lindbladian lift exists for $\dot\rho_t = L_t(\rho_t)$. By linking the tangent cone of the quantum state space to admissible Lindblad velocities, the authors prove that NM is a global property of the dynamical map rather than detectable from individual trajectories or finite trajectory ensembles. They construct explicit lifting strategies, including parametric and geometric lifts, and prove existence results for continuous and differentiable lifts under spectral-regularity assumptions. The results imply that NM, while useful as a control resource, cannot be unambiguously identified from trajectory data alone and highlight the importance of global process information or process-tensor style descriptions for NM detection.

Abstract

The theory of open quantum systems served as a tool to prepare entanglement at the beginning stage of quantum technology and more recently provides an important tool for state preparation. Dynamics given by time dependent Lindbladians are Markovian and lead to decoherence, decay of correlation and convergence to equilibrium. In contrast Non-Markovian evolutions can outperform their Markovian counterparts by enhancing memory. In this letter we compare the trajectories of Markovian and Non-Markovian evolutions starting from a fixed initial value. It turns out that under mild assumptions every trajectory can be obtained from a family of time dependent Lindbladians. Hence Non-Markovianity is invisible if single trajectories are concerned.

The Hidden Nature of Non-Markovianity

TL;DR

This work shows that non-Markovianity, often diagnosed via memory effects in quantum dynamics, is invisible at the level of a single trajectory: under mild regularity any differentiable density-trajectory can be realized as the outcome of a time-dependent Lindbladian, i.e., a Lindbladian lift exists for . By linking the tangent cone of the quantum state space to admissible Lindblad velocities, the authors prove that NM is a global property of the dynamical map rather than detectable from individual trajectories or finite trajectory ensembles. They construct explicit lifting strategies, including parametric and geometric lifts, and prove existence results for continuous and differentiable lifts under spectral-regularity assumptions. The results imply that NM, while useful as a control resource, cannot be unambiguously identified from trajectory data alone and highlight the importance of global process information or process-tensor style descriptions for NM detection.

Abstract

The theory of open quantum systems served as a tool to prepare entanglement at the beginning stage of quantum technology and more recently provides an important tool for state preparation. Dynamics given by time dependent Lindbladians are Markovian and lead to decoherence, decay of correlation and convergence to equilibrium. In contrast Non-Markovian evolutions can outperform their Markovian counterparts by enhancing memory. In this letter we compare the trajectories of Markovian and Non-Markovian evolutions starting from a fixed initial value. It turns out that under mild assumptions every trajectory can be obtained from a family of time dependent Lindbladians. Hence Non-Markovianity is invisible if single trajectories are concerned.
Paper Structure (5 sections, 10 theorems, 37 equations, 1 figure)

This paper contains 5 sections, 10 theorems, 37 equations, 1 figure.

Table of Contents

  1. Tangent Cone and Lifting Problem
  2. Existence of Reguarlized Lifts
  3. Parametric and Geometric Lifts
  4. Measurable and Integrable Lifts
  5. Continuous and Differentiable Lifts

Key Result

Theorem 1

For any $C^2$ path $\rho_t$ satisfying properties (1) and (2), there exists a continuous Lindbladian lift on the interval $[t_0,t_1)$.

Figures (1)

  • Figure 1: The blue line shows a path of densities with tangent given by the green arrow. On the left we see that a jump operator with negative sign can lead out of the space of densities if the path hits the boundary. On the right we see that the derivative stays inside the space of densities if path does not hit the boundary and can be lifted.

Theorems & Definitions (20)

  • Theorem 1: Lindbladian Lifting
  • Corollary 2
  • Corollary 3
  • Definition 4
  • Example 1
  • Definition 5
  • Example 2
  • Proposition 6
  • proof
  • Proposition 7
  • ...and 10 more