Coherence transformations in single qubit systems.

Area of Science:

• Quantum Information Science
• Quantum Computing Theory

Background:

• Coherence is a key resource in quantum computation.
• Understanding coherence-free operations is crucial for robust quantum information processing.

Purpose of the Study:

• To investigate single qubit transformations under various coherence-free operations.
• To establish quantitative connections and hierarchical relationships between these operations.
• To provide a new proof for necessary and sufficient conditions of single qubit transformations via incoherent operations (IO) and strictly incoherent operations (SIO).

Main Methods:

• Analysis of single qubit transformations under four types of coherence-free operations: IO, SIO, physically incoherent operation (PIO), and coherence-preserving operation (CPO).
• Development of a quantitative connection between IO and SIO.
• Geometric illustration using the Bloch sphere to depict transformation regions.

Main Results:

• A clear hierarchy of coherence-free operations in single qubit systems was established: CPO ⊂ PIO ⊂ SIO=IO.
• A novel and explicit proof for the conditions of single qubit transformation via IO or SIO was provided.
• It was shown that two Kraus operators are sufficient for SIO to achieve these transformations.
• Transformation regions for CPO and PIO were explicitly defined.

Conclusions:

• The study clarifies the relationships and capabilities of different coherence-free operations for single qubits.
• The geometric approach offers an intuitive method for analyzing coherence transformations.
• The findings provide practical guidance for constructing specific coherence-free operations.