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Decomposable dynamics on matrix algebras

Katarzyna Siudzińska, Krzysztof Szczygielski

Abstract

We explore a notion of decomposably divisible (D-divisible) quantum evolution families, recently introduced in J. Phys. A: Math. Theor. 56, 485202 (2023). Both necessary and sufficient conditions are presented for highly-symmetric qubit and qudit dynamical maps. Through a restructurization of the evolution generators, we encode the decomposable divisibility into the positivity of time-dependent coefficients that multiply generators of D-divisible dynamical maps. This provides an analogy to the CP-divisibility property, which is equivalent to the positivity of decoherence rates that multiply Markovian semigroup generators.

Decomposable dynamics on matrix algebras

Abstract

We explore a notion of decomposably divisible (D-divisible) quantum evolution families, recently introduced in J. Phys. A: Math. Theor. 56, 485202 (2023). Both necessary and sufficient conditions are presented for highly-symmetric qubit and qudit dynamical maps. Through a restructurization of the evolution generators, we encode the decomposable divisibility into the positivity of time-dependent coefficients that multiply generators of D-divisible dynamical maps. This provides an analogy to the CP-divisibility property, which is equivalent to the positivity of decoherence rates that multiply Markovian semigroup generators.

Paper Structure

This paper contains 9 sections, 4 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. D-divisible quantum dynamical maps
  3. Decomposable maps on matrix algebras
  4. Quantum dynamical maps and D-divisibility
  5. Qubit evolution -- Pauli channels
  6. Generalized Pauli channels
  7. Phase-covariant channels
  8. Final remarks
  9. Acknowledgments

Key Result

Theorem 1

A differentiable dynamical map $\Lambda_t$ satisfying Master Equation eq:ODEME with a time-local $L_t$ is D-divisible iff there exists a Hermitian matrix $H_t$ and a decomposable map $\varphi_t$ s.t.

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Theorem 1: cf. Szczygielski2023
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Example 1
  • Example 2
  • Example 3
  • ...and 1 more