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Jensen's inequality for partial traces in von Neumann algebras

Mizanur Rahaman, Lyudmila Turowska

Abstract

Motivated by a recent result on finite-dimensional Hilbert spaces, we prove a Jensen's inequality for partial traces in semifinite von Neumann algebras. We also prove a similar inequality in the framework of general (non-tracial) von Neumann algebras.

Jensen's inequality for partial traces in von Neumann algebras

Abstract

Motivated by a recent result on finite-dimensional Hilbert spaces, we prove a Jensen's inequality for partial traces in semifinite von Neumann algebras. We also prove a similar inequality in the framework of general (non-tracial) von Neumann algebras.

Paper Structure

This paper contains 7 sections, 8 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Preliminary on semifinite von Neumann algebras
  3. Trace Jensen's inequality for positive linear maps
  4. The proof of the main theorem
  5. Jensen's inequality for states (non-tracial case)
  6. Outlook
  7. Acknowledgment

Key Result

Theorem 1

For two finite-dimensional Hilbert spaces $\mathcal{H}_1, \mathcal{H}_2$ and a self-adjoint operator $H$ acting on $\mathcal{H}_1\otimes \mathcal{H}_2$ whose spectrum lies within an interval $I$, we have for every convex function $f$ defined on $I$ and every density matrix (positive semi-definite matrix with trace $1$) $\rho$ on $\mathcal{H}_1$.

Theorems & Definitions (16)

  • Theorem 1: Carlen-Frank-Larson, 2025
  • Theorem 2: Jensen's inequality for partial traces in von Neumann algebras
  • Theorem 3: Petz. 1987
  • Definition 1
  • Theorem 4
  • Definition 2
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 6 more