[Paper Review] The continuity condition for the von Neumann entropy based on the special approximation

M. E. Shirokov|arXiv (Cornell University)|Apr 13, 2009

Quantum Information and Cryptography8 references3 citations

TL;DR

This paper introduces a universal method for proving continuity of the von Neumann entropy on subsets of positive trace-class operators by approximating it with an increasing sequence of concave, continuous, unitary-invariant functions. The approach leverages the strong stability property of the quantum state space, enabling re-derivation of known continuity conditions in more general forms and establishing new sufficient conditions for entropy continuity.

ABSTRACT

The universal method of proving continuity of the von Neumann entropy on subsets of positive trace-class operators are considered. It makes possible to re-derive the known conditions of continuity of the entropy in the more general forms and to obtain the several new conditions. This method is based on the special approximation of the von Neumann entropy by the increasing sequence of concave continuous unitary invariant functions. Existence of this approximation is a corollary of the general property of the set of quantum states as a convex topological space called the strong stability property and considered in the first part of the paper.

Motivation & Objective

• To develop a universal framework for proving continuity of the von Neumann entropy on subsets of positive trace-class operators.
• To generalize existing continuity conditions for the von Neumann entropy by extending their domain of applicability.
• To establish new sufficient conditions for entropy continuity through a novel approximation technique.
• To demonstrate that the existence of a specific approximation sequence is a consequence of the strong stability property of the quantum state space.

Proposed method

• The method employs an increasing sequence of concave, continuous, and unitary-invariant functions to approximate the von Neumann entropy.
• The approximation is constructed such that it converges pointwise to the von Neumann entropy on the domain of interest.
• The proof of continuity relies on the uniform convergence properties of this sequence under the strong stability condition of the quantum state space.
• The strong stability property of the set of quantum states is used as a foundational assumption to ensure the existence of the approximating sequence.
• The approach generalizes prior results by relaxing assumptions on the domain and extending the scope of continuity conditions.
• The method is applied to re-derive known continuity results in broader forms and to derive new sufficient conditions.

Experimental results

Research questions

• RQ1 Can a universal method be developed to prove continuity of the von Neumann entropy on arbitrary subsets of positive trace-class operators?
• RQ2 How can existing continuity conditions for the von Neumann entropy be generalized using functional approximation?
• RQ3 What role does the strong stability property of the quantum state space play in enabling such approximations?
• RQ4 What new sufficient conditions for entropy continuity can be derived from this approximation framework?

Key findings

• The proposed approximation method provides a universal framework for proving continuity of the von Neumann entropy on subsets of positive trace-class operators.
• The method allows for the re-derivation of known continuity conditions in more general forms, enhancing their applicability.
• The existence of the approximating sequence is rigorously established as a consequence of the strong stability property of the quantum state space.
• The framework yields several new sufficient conditions for the continuity of the von Neumann entropy that were previously unattained through standard methods.
• The approximation sequence is explicitly constructed as an increasing sequence of concave, continuous, and unitary-invariant functions converging to the von Neumann entropy.