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The Infinite-Dimensional Quantum Entropy: the Unified Entropy Case

Roman Gielerak, Joanna Wiśniewska, Marek Sawerwain

Abstract

By a use of the Fredholm determinant theory, the unified quantum entropy notion has been extended to a case of infinite-dimensional systems. Some of the known (in the finite-dimensional case) basic properties of the introduced unified entropies have been extended to the case study. Certain numerical approaches for computing the proposed finite and infinite-dimensional entropies are being outlined as well.

The Infinite-Dimensional Quantum Entropy: the Unified Entropy Case

Abstract

By a use of the Fredholm determinant theory, the unified quantum entropy notion has been extended to a case of infinite-dimensional systems. Some of the known (in the finite-dimensional case) basic properties of the introduced unified entropies have been extended to the case study. Certain numerical approaches for computing the proposed finite and infinite-dimensional entropies are being outlined as well.

Paper Structure

This paper contains 4 sections, 1 theorem, 7 equations.

Table of Contents

  1. Introduction
  2. On the standard notation used
  3. Entropy based entanglement measures in bipartite systems
  4. Quantum von Neumann Entropy and Fredholm Determinants

Key Result

proposition thmcounterproposition

Let $\mathcal{H}$ be a separable Hilbert space with $\dim(\mathcal{H}) = \infty$ and let $r \in (0, 1)$. Then the set $I^{\infty}_{r}(\mathcal{H}) = \{ Q \in E( \mathcal{H} ) : I_r(Q) = \infty \}$ is $L_1$ -- dense subset of $E(Q)$.

Theorems & Definitions (2)

  • proposition thmcounterproposition
  • proof