[Paper Review] Second Order Asymptotics of Optimal Approximate Conversion for Probability Distributions and Entangled States and Its Application to LOCC Cloning.

Wataru Kumagai, Masahito Hayashi|arXiv (Cornell University)|Jun 18, 2013

Quantum Information and Cryptography3 citations

TL;DR

This paper derives the second-order asymptotic rates for optimal approximate conversion of i.i.d. probability distributions and entangled pure states under LOCC constraints, generalizing resolvability, intrinsic randomness, entanglement dilution, and concentration. It introduces LOCC cloning for known entangled states and establishes optimal second-order performance using the majorization method in both classical and quantum settings.

ABSTRACT

We consider approximate conversion problems for probability distributions and derive the asymptotically optimal second-order conversion rate for independent and identical distributions on finite sets. Then, we apply those results to approximate conversion problems of entangled pure states in quantum systems when only local operations and classical communiactions (LOCC) are allowed. Our results can be regarded as a generalization of those for the resolvability and the intrinsic randomness, and the quantum application of our results can be regarded as a generalization of those for the entanglement dilution and concentration. Moreover, we will introduce the notion of LOCC cloning for a known pure entangled state and derive its optimal asymptotic performance. To derive the optimal second-order rates, the majorization method is used, which is a basic tool in the conversion theory of quantum entangled pure states by LOCC. In this paper, we show the efficiency of the majorization method not only in quantum settings but also in classical settings.

Motivation & Objective

• To determine the optimal second-order asymptotic rate for approximate conversion of i.i.d. probability distributions on finite sets.
• To extend these results to approximate conversion of entangled pure states under local operations and classical communication (LOCC).
• To generalize prior work on resolvability, intrinsic randomness, entanglement dilution, and concentration in the second-order asymptotic regime.
• To introduce and analyze the concept of LOCC cloning for a known pure entangled state in the asymptotic setting.
• To demonstrate the effectiveness of the majorization method in both classical and quantum settings for second-order analysis.

Proposed method

• The majorization method is employed as the foundational tool for analyzing convertibility in both classical and quantum settings.
• Second-order asymptotic expansions are derived by analyzing the variance of the distribution of outcomes in the limit of many i.i.d. copies.
• The analysis leverages the structure of majorization to characterize the optimal conversion rates beyond the first-order asymptotic regime.
• The framework is applied to entangled pure states to determine the optimal rate of approximate conversion under LOCC constraints.
• LOCC cloning is formalized as a specific approximate conversion task, and its second-order optimal performance is derived.
• The results unify and generalize previous results in quantum information theory, particularly in entanglement theory and classical information theory.

Experimental results

Research questions

• RQ1 What is the second-order asymptotic rate for approximate conversion of i.i.d. probability distributions on finite sets?
• RQ2 How does the second-order performance of LOCC conversion for entangled pure states compare to first-order rates?
• RQ3 Can the majorization method be effectively applied to derive second-order results in classical information theory as well as quantum theory?
• RQ4 What is the optimal second-order performance of LOCC cloning for a known pure entangled state?
• RQ5 How do the results generalize prior work on resolvability, intrinsic randomness, entanglement dilution, and concentration?

Key findings

• The second-order asymptotic rate for approximate conversion of i.i.d. probability distributions is derived using the majorization method, extending beyond first-order asymptotics.
• The results generalize the theory of resolvability and intrinsic randomness to the second-order regime in classical settings.
• For entangled pure states, the second-order optimal rate under LOCC is characterized, generalizing entanglement dilution and concentration results.
• LOCC cloning for a known pure entangled state is introduced, and its optimal second-order performance is derived.
• The majorization method proves effective not only in quantum settings but also in classical settings for second-order analysis.
• The framework unifies classical and quantum approximate conversion problems under a common second-order asymptotic theory.