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Kraus is King: High-order Completely Positive and Trace Preserving (CPTP) Low Rank Method for the Lindblad Master Equation

Daniel Appelo, Yingda Cheng

Abstract

We design high order accurate methods that exploit low rank structure in the density matrix while respecting the essential structure of the Lindblad equation. Our methods preserves complete positivity and are trace preserving.

Kraus is King: High-order Completely Positive and Trace Preserving (CPTP) Low Rank Method for the Lindblad Master Equation

Abstract

We design high order accurate methods that exploit low rank structure in the density matrix while respecting the essential structure of the Lindblad equation. Our methods preserves complete positivity and are trace preserving.
Paper Structure (8 sections, 1 theorem, 21 equations, 4 figures, 1 table)

This paper contains 8 sections, 1 theorem, 21 equations, 4 figures, 1 table.

Table of Contents

  1. Integrating Factor Induced CP Scheme
  2. Truncated SVD is a CP Map
  3. CPTP Low Rank IF Scheme
  4. Alternative Truncation
  5. Numerical Examples

Key Result

Theorem 1

The truncated SVD operator $\mathcal{T}_{\epsilon,r_{\rm max}}[A],$ where $A$ is SPSD, is on Kraus form, and thus is a CP map.

Figures (4)

  • Figure 1: Evolution of the population $\rho_{33}$ in the example from riesch2019analyzing. The units for the $x$-axis is pico seconds. The number of timesteps in the left and right insets are 300 and 60 respectively. To the left we compare the classic RK4 method (black) and the integrating factor method with matrix exponentiation for the flow (red, dashed) using a timestep of 0.1 femto-seconds. To the right we use a five times larger timestep and have also included results when the flow is computed using a Taylor series method of order 6.
  • Figure 2: Computation of revival for a small $m$ case using the classic fourth order accurate Runge-Kutta method, the full-rank integrating factor method and two low-rank methods (matrix exponentiation and Taylor series). See the text for explanation of the subfigures.
  • Figure 3: Errors at the final time for the two low-rank methods (matrix exponentiation (left) and Taylor series (right)) as a function of the timestep. The dashed lines are orders, one to four. See the text for explanation of the subfigures.
  • Figure 4: Low-rank computation of revival for a large $m$ case. See the text for explanation of the subfigures.

Theorems & Definitions (2)

  • Theorem 1
  • proof