The references that inspired this chapter are all mentioned in the References section.

Quantum channels

Quantum channels, also known as quantum maps [B2] or quantum operations [B3], are a mean to describe the three different kinds of evolution that may affect an open quantum system:

  • Unitary evolution obeying the dynamics prescribed by the closed system’s Hamiltonian

  • Measurement

  • Erratic interaction with the environment (noise)


  • Density operator, the key representation of the state of open systems

  • Quantum maps, Kraus sum representation /
    Quantum operations, operator-sum representation

More to investigate:

  • Tensor networks and graphical calculus for open quantum systems [136],
    in relation with the Pauli Transfer Matrix (PTM) representation of a Quantum Channel (see PTM on qiskit), and completely positive trace preserving (CPTP) maps

Master equation

The Lindblad master equation.

A differential equation that formalizes the evolution of the system under the action of its environment. The involved \(L_\mu\) operators describe the relaxation events which affect the system, and can be, in many cases, guessed from a careful analysis of the system.

The evolution according to the master equation can be numerically evaluated by Monte Carlo simulations.


  • Quantum maps, Kraus sum representation: section 4.2 [B2], alias
    “Quantum operations” incl. operator-sum representation, section 8.2 [B3]
  • “The Lindblad master equation”, section 4.3 [B2]