Topics in quantum information and the theory of open quantum systems

TL;DR

This thesis explores fundamental topics in open quantum systems, quantum measurements, and error correction, providing new insights into measurement universality, entanglement, non-Markovian dynamics, and fault-tolerant quantum computation.

Contribution

It introduces new conditions for quantum error correction under continuous and non-Markovian dynamics, and demonstrates the scalability of holonomic quantum computation.

Findings

Every generalized measurement can be realized through weak measurements.

A new entanglement monotone for three-qubit states is derived.

Continuous QEC shows quadratic improvement in non-Markovian regimes.

Abstract

This thesis examines seven topics in the areas of deterministic open-quantum-system dynamics, quantum measurements, and quantum error correction (QEC). The first topic concerns weak measurements and their universality as a means of generating quantum operations. It is shown that every generalized measurement can be implemented as a sequence of weak (infinitesimal) measurements. The second topic is an application of this result to the theory of entanglement. Necessary and sufficient differential conditions for entanglement monotones are derived and are used to find a new entanglement monotone for three-qubit states. The third topic is a study of the performance of different master equations for the description of non-Markovian dynamics. The system studied is a qubit coupled to a spin bath via the Ising interaction. The fourth topic investigates continuous QEC in the presence of non-Markovian noise. It is shown that due to the existence of a Zeno regime in non-Markovian dynamics, the performance of continuous QEC exhibits a quadratic improvement for a sufficiently high time resolution of the error-correcting operations. The fifth topic studies the conditions for correctability of subsystem codes under continuous dynamics. Necessary and sufficient conditions on the Lindbladian and the system-environment Hamiltonian are derived. The sixth topic examines the robustness of operator QEC codes against initialization errors. A new measure of fidelity for encoded information is introduced and is used to show that operator codes are robust against imperfect initialization without the need for restriction of the standard operator QEC conditions. The last topic concerns holonomic quantum computation (HQC) and stabilizer codes. A fault-tolerant scheme for HQC is presented, proving the scalability of the holonomic approach.