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Lecture Notes on the Theory of Open Quantum Systems

Daniel A. Lidar

TL;DR

This work develops a comprehensive framework for open quantum systems centered on completely positive (CP) quantum maps and master equations. It builds from density-operator formalism and Kraus representations to expose how system–bath interactions yield CP dynamical maps, including detailed Lindblad-type generators. Through both first-principles models (spin–bath systems, phase damping, amplitude damping) and coarse-graining analyses, it derives and analyzes Markovian and non-Markovian behaviors, culminating in exact spin-boson solutions and their coherence dynamics. The results provide a rigorous toolkit for modeling decoherence, dissipation, and information flow in quantum technologies, with clear connections to Bloch-sphere intuition, quantum discord, and entanglement criteria such as PPT.

Abstract

This is a self-contained set of lecture notes covering various aspects of the theory of open quantum system, at a level appropriate for a one-semester graduate course. The main emphasis is on completely positive maps and master equations, both Markovian and non-Markovian.

Lecture Notes on the Theory of Open Quantum Systems

TL;DR

This work develops a comprehensive framework for open quantum systems centered on completely positive (CP) quantum maps and master equations. It builds from density-operator formalism and Kraus representations to expose how system–bath interactions yield CP dynamical maps, including detailed Lindblad-type generators. Through both first-principles models (spin–bath systems, phase damping, amplitude damping) and coarse-graining analyses, it derives and analyzes Markovian and non-Markovian behaviors, culminating in exact spin-boson solutions and their coherence dynamics. The results provide a rigorous toolkit for modeling decoherence, dissipation, and information flow in quantum technologies, with clear connections to Bloch-sphere intuition, quantum discord, and entanglement criteria such as PPT.

Abstract

This is a self-contained set of lecture notes covering various aspects of the theory of open quantum system, at a level appropriate for a one-semester graduate course. The main emphasis is on completely positive maps and master equations, both Markovian and non-Markovian.

Paper Structure

This paper contains 192 sections, 8 theorems, 878 equations, 11 figures.

Table of Contents

  1. Preface and Acknowledgments
  2. Review of Quantum Mechanics
  3. Postulate 1
  4. Postulate 2
  5. Postulate 3
  6. Postulate 4
  7. Projective (von Neumann) measurements
  8. Examples of measuring observables
  9. Expectation value of an observable
  10. Heisenberg Uncertainty Principle
  11. Positive Operator Valued Measures (POVMs)
  12. Density Operators
  13. Properties of the density operator
  14. Dynamics of the density operator
  15. Restatement of the postulates of quantum mechanics
  16. ...and 177 more sections

Key Result

Theorem 1

A linear operator $A:V \rightarrow V$ obeys $A^\dagger A = A A^\dagger$ (i.e., it is a normal operator) if and only if $A=\sum_{a} \lambda_a \ket{a}\bra{a}$ for a set of orthonormal basis vectors $\{\ket{a}\}$ for $V$, which are also the eigenvectors of $A$ with respective eigenvalues $\{\lambda_a\}

Figures (11)

  • Figure 1: The Bloch sphere is a geometric representation of the collection of all Bloch vectors $\vec{v}$ which describe valid qubit density operators. Thus, the sphere is of radius $1$, its surface represents all pure states, and its interior represents all mixed states. In this diagram the blue vector lies on the surface of the sphere indicating a pure state, whereas the red vector lies in its interior indicating a mixed state.
  • Figure 2: A commutative diagram showing that the quantum map $\Phi$ can be viewed as a composition of three maps.
  • Figure 3: The Bloch sphere become an ellipsoid after transformation by the phase damping channel. The invariant states are those on the $\sigma_z$ axis. The major axis has length $2$, the minor axis has length $2(2p-1)$.
  • Figure 4: The Bloch sphere transformed by the depolarizing channel. As $p\rightarrow1$, all states converge to the fully mixed state at the origin.
  • Figure 5: Comparison of the exact solution of the spin-boson model for single-qubit phase damping to the result obtained from the coarse-grained Markovian master equation. Plotted are the arguments $\Gamma(t)$ of the exponentials in Eq. \ref{['eq:324']}. Straight lines correspond to the Markovian solution, which intersects the exact solution (thick line) at $t=\tau$, as seen from Eqs. \ref{['eq:SME-Debye']} and \ref{['eq:exact-Debye']}. The bosonic bath density of states is represented by the Debye model [Eq. \ref{['eq:Debye']}]. The results shown correspond to $C=0.05$ and $\omega_c=1$. Reproduced from Ref. Lidar200135.
  • ...and 6 more figures

Theorems & Definitions (19)

  • Theorem 1: Spectral Theorem
  • Theorem 1
  • proof
  • Theorem 2
  • Claim 1
  • proof
  • Theorem 2
  • proof
  • Definition 1
  • Lemma 1
  • ...and 9 more