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The generalized strong subadditivity of the von Neumann entropy for bosonic quantum systems

Giacomo De Palma, Dario Trevisan

Abstract

We prove a generalization of the strong subadditivity of the von Neumann entropy for bosonic quantum Gaussian systems. Such generalization determines the minimum values of linear combinations of the entropies of subsystems associated to arbitrary linear functions of the quadratures, and holds for arbitrary quantum states including the scenario where the entropies are conditioned on a memory quantum system. We apply our result to prove new entropic uncertainty relations with quantum memory, a generalization of the quantum Entropy Power Inequality, and the linear time scaling of the entanglement entropy produced by quadratic Hamiltonians.

The generalized strong subadditivity of the von Neumann entropy for bosonic quantum systems

Abstract

We prove a generalization of the strong subadditivity of the von Neumann entropy for bosonic quantum Gaussian systems. Such generalization determines the minimum values of linear combinations of the entropies of subsystems associated to arbitrary linear functions of the quadratures, and holds for arbitrary quantum states including the scenario where the entropies are conditioned on a memory quantum system. We apply our result to prove new entropic uncertainty relations with quantum memory, a generalization of the quantum Entropy Power Inequality, and the linear time scaling of the entanglement entropy produced by quadratic Hamiltonians.

Paper Structure

This paper contains 15 sections, 31 theorems, 159 equations.

Table of Contents

  1. Introduction and main result
  2. The classical Brascamp--Lieb constant
  3. Preliminaries
  4. Stam inequality
  5. Asymptotic scaling of the conditional entropy
  6. Proof of Theorem 1
  7. Achieved supremum in (\ref{['eq:fdet']})
  8. Generic case
  9. Applications
  10. Entropic uncertainty relations
  11. Generalized quantum Entropy Power Inequality
  12. Entanglement production
  13. Classical Brascamp--Lieb inequalities
  14. Entropy of the joint measurement of commuting quadratures
  15. Entropy of Gaussian states

Key Result

Theorem 1

Let $\mathbf{B}$ be a quantum Brascamp--Lieb datum as in defn:QBL with dimension $(2m,\mathbf{n})$, and let $\mathbf{p}\in\mathbb{R}_{\ge0}^K$ satisfy $2\,m=\mathbf{p}\cdot \mathbf{n}$. Then, regardless of the dimension of the conditioning quantum system $M$, which can also be chosen trivial, we hav

Theorems & Definitions (70)

  • Definition 1: Brascamp--Lieb datum
  • Definition 2: Quantum Brascamp--Lieb datum
  • Theorem 1
  • Definition 3: Critical subspace
  • Remark 1
  • Theorem 2
  • proof
  • Theorem 3
  • Remark 2
  • Theorem 4
  • ...and 60 more