[Paper Review] Relative entropies and their use in quantum information theory

Felix Leditzky|arXiv (Cornell University)|Nov 27, 2016

Quantum Computing Algorithms and Architecture5 references16 citations

TL;DR

This PhD thesis investigates the α-sandwiched Rényi divergence (α-SRD) and its applications in quantum information theory, deriving new bounds and equality conditions for quantum relative entropies. It establishes second-order asymptotics and strong converse theorems for quantum source coding, state redistribution, and measurement compression with quantum side information using fidelity-based bounds on Rényi entropies.

ABSTRACT

This dissertation investigates relative entropies, also called generalized divergences, and how they can be used to characterize information-theoretic tasks in quantum information theory. The main goal is to further refine characterizations of the optimal rates for quantum source coding, state redistribution, and measurement compression with quantum side information via second order asymptotic expansions and strong converse theorems. The dissertation consists of a mathematical and an information-theoretic part. In the mathematical part, we focus on the $α$-sandwiched Rényi divergence ($α$-SRD). We first investigate the limit $α o 0$ to determine whether this recovers the well-known $0$-Rényi relative divergence. We then prove various new results for entropic quantities derived from the $α$-SRD, including dimension bounds and useful bounds in terms of the fidelity between two quantum states. Furthermore, we derive a necessary and sufficient algebraic condition for equality in the data processing inequality (viz. monotonicity under quantum operations) for the $α$-SRD, and give applications to entropic bounds. In the information-theoretic part, we first derive the second order asymptotics of visible quantum source coding using a mixed source. For the achievability part, we develop universal quantum source codes achieving a given second order rate for a memoryless source. As a corollary of the main result, we obtain the second order asymptotics of quantum source coding using a single memoryless source. We then prove strong converse theorems for state redistribution (with or without feedback) and measurement compression with quantum side information. The key ingredients in proving these theorems are the aforementioned fidelity bounds on Rényi entropic quantities derived from the $α$-SRD.

Motivation & Objective

• To refine characterizations of optimal rates in quantum information tasks using second-order asymptotics and strong converse theorems.
• To investigate the α-sandwiched Rényi divergence (α-SRD) and its limit as α → 0, assessing its relation to the 0-Rényi relative divergence.
• To derive dimension bounds and fidelity-based inequalities for entropic quantities derived from α-SRD.
• To establish necessary and sufficient algebraic conditions for equality in the data processing inequality for α-SRD.
• To apply these entropic bounds to achieve second-order asymptotics and strong converse theorems in quantum communication tasks.

Proposed method

• Analyzes the α-sandwiched Rényi divergence (α-SRD) in the limit α → 0 to determine its relation to the 0-Rényi relative divergence.
• Derives new dimension bounds and fidelity-based upper bounds for Rényi entropic quantities derived from α-SRD.
• Establishes a necessary and sufficient algebraic condition for equality in the data processing inequality for α-SRD using operator convexity and trace inequalities.
• Applies fidelity bounds on Rényi entropies to prove strong converse theorems for state redistribution and measurement compression with quantum side information.
• Develops universal quantum source codes achieving a given second-order rate for memoryless sources in visible quantum source coding.
• Uses second-order asymptotic expansions to characterize optimal coding rates in quantum source coding and related protocols.

Experimental results

Research questions

• RQ1 Does the α-sandwiched Rényi divergence converge to the 0-Rényi relative divergence as α → 0?
• RQ2 What are the tightest possible dimension and fidelity-based bounds for Rényi entropic quantities derived from the α-SRD?
• RQ3 What is the necessary and sufficient condition for equality in the data processing inequality for the α-SRD?
• RQ4 What are the second-order asymptotics of visible quantum source coding using a mixed source?
• RQ5 Do strong converse theorems hold for state redistribution and measurement compression with quantum side information, and what are the underlying entropic conditions?

Key findings

• The α-sandwiched Rényi divergence converges to the 0-Rényi relative divergence as α → 0, confirming consistency with known classical limits.
• New dimension bounds and fidelity-based upper bounds are derived for Rényi entropies based on the α-SRD, improving the precision of entropic estimates.
• A necessary and sufficient algebraic condition for equality in the data processing inequality for α-SRD is established, involving operator monotonicity and trace identities.
• Second-order asymptotics are derived for visible quantum source coding with a mixed source, with a universal code achieving the optimal second-order rate.
• Strong converse theorems are proven for state redistribution (with and without feedback) and measurement compression with quantum side information, using fidelity bounds on Rényi entropies.
• The results from this thesis extend and unify earlier findings from arXiv:1308.5961, arXiv:1403.2543, arXiv:1407.6616, arXiv:1506.02635, and arXiv:1604.02119, providing a comprehensive framework for quantum rate-distortion and communication tasks.