[Paper Review] The Kraus representation for the dynamics of open quantum systems

Jonas Maziero|arXiv (Cornell University)|Oct 30, 2015

Quantum Mechanics and Applications3 citations

TL;DR

This paper presents a simplified derivation of the Kraus representation for open quantum systems using unitary evolution of the total system-environment and partial trace operations. It demonstrates the formalism through a two-level atom interacting with the electromagnetic vacuum, showing how quantum coherence decays over time, offering a pedagogically accessible route to understanding non-unitary dynamics in quantum systems.

ABSTRACT

The necessity and utility of considering the interaction of a quantum system with its environment when describing its time evolution have been recognized in several branches of physics and of other sciences. The Kraus' representation is a general and succinct approach to describe such open system dynamics in a wide range of relevant physical scenarios. In this article, by abdicating from the generality of the formalism of quantum operations and with this avoiding its associated complications, we show in a simple manner how we can obtain the Kraus' representation using basically the closed system (system plus environment) unitary dynamics and the partial trace function. The example of a two-level atom interacting with the vacuum of the electromagnetic field is regarded for the sake of instantiating this formalism, which is then applied to study the time evolution of the atom's quantum coherence.

Motivation & Objective

• To provide a clear, accessible derivation of the Kraus representation for open quantum systems without relying on advanced quantum operation formalism.
• To illustrate the formalism using a physically relevant example: a two-level atom interacting with the electromagnetic vacuum.
• To study the time evolution of quantum coherence in the atom using the derived Kraus representation.
• To support quantum information education by simplifying the understanding of non-unitary dynamics in realistic quantum systems.

Proposed method

• The total system (atom + environment) is assumed to evolve unitarily according to the Schrödinger equation.
• The Kraus operators are derived by tracing out the environmental degrees of freedom using the partial trace operation.
• The resulting dynamics are expressed in the Kraus form: ρ(t) = ∑_k A_k ρ(0) A_k^†, where A_k are the Kraus operators.
• The formalism is applied to a two-level atom coupled to the vacuum of the electromagnetic field, a standard model of spontaneous emission.
• The time evolution of quantum coherence is analyzed by computing the off-diagonal elements of the density matrix in the atomic basis.
• The derivation avoids the full generality of quantum operations, focusing instead on the physically intuitive link between unitary evolution and open system dynamics.

Experimental results

Research questions

• RQ1 How can the Kraus representation be derived in a simple and intuitive way from unitary evolution and partial trace?
• RQ2 What is the explicit form of the Kraus operators for a two-level atom interacting with the electromagnetic vacuum?
• RQ3 How does quantum coherence evolve in time under this open system dynamics?
• RQ4 What insights does this formalism provide for understanding decoherence in realistic quantum systems?

Key findings

• The Kraus representation is successfully derived using only unitary evolution of the closed system and partial trace, avoiding complex formalism.
• For the two-level atom in vacuum, the Kraus operators correspond to the standard spontaneous emission process, with one operator representing emission and another representing no emission.
• The time evolution of the atomic density matrix shows exponential decay of the off-diagonal coherence term, consistent with standard spontaneous emission theory.
• The formalism clearly illustrates how environmental entanglement leads to decoherence, with the partial trace removing environmental degrees of freedom and resulting in mixed state dynamics.