Relating Entropies of Quantum Channels

Area of Science:

• Quantum Information Theory
• Quantum Channel Characterization
• Entropy in Quantum Systems

Background:

• Defining and quantifying the entropy of quantum channels is crucial for understanding information processing capabilities.
• Existing methods for quantum channel entropy lack a unified framework, necessitating comparative studies.
• The Choi-Jamiołkowski state and relative entropy are key concepts in quantum information theory.

Purpose of the Study:

• To investigate and compare two distinct approaches for defining quantum channel entropy.
• To establish the relationship between entropy defined via the Choi-Jamiołkowski state and relative entropy.
• To determine conditions under which these entropy measures are equivalent.

Main Methods:

• Utilized the von Neumann entropy of the Choi-Jamiołkowski state as one definition.
• Employed relative entropy of the extended channel output relative to a depolarizing channel as the second definition.
• Performed analytical derivations and numerical investigations for various quantum channels, including qubit channels.

Main Results:

• Demonstrated that the entropy derived from the Choi-Jamiołkowski state serves as an upper bound for the optimized relative entropy.
• Proved that this upper bound is precisely met (saturated) for all unital qubit channels.
• Provided evidence and conjectures suggesting saturation for random channels in the limit of infinite dimensions.

Conclusions:

• The Choi-Jamiołkowski state entropy is a robust upper bound for a key relative entropy measure of quantum channels.
• Unital qubit channels represent a class where these two entropy definitions coincide, simplifying analysis.
• The findings suggest a potential universality of this saturation in broader channel classes and high-dimensional regimes.