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Stability of PPT in equilibrium states

Marco Merkli, Mitch Zagrodnik

TL;DR

This work proves that the positive partial transpose (PPT) property of bipartite equilibrium states in infinite-dimensional systems is stable under bounded perturbations of the Hamiltonian at high temperatures or for sufficiently small perturbations. Using a Dyson expansion and a strategic factorization of the Boltzmann operator, the authors control the perturbed state through Hilbert-Schmidt bounds on the interaction in imaginary time, deriving an explicit temperature threshold via $\beta_* = \frac{2}{b}\ln\big[1+\frac{b}{a}\frac{\ln 2}{2}\big]$. The main result, Theorem 1, states that if $0<\beta \le \max\{s_*,\beta_*\ ight\\}$, then the perturbed Gibbs state $\rho_\beta$ remains PPT under the stated bound $\| e^{-sH_0}Ve^{sH_0}\|_2 \le a e^{bs}$. The approach circumvents the infinite-dimensional obstacle of eigenvalue accumulation by preconditioning with $e^{-eta H_0/2}$ and demonstrates PPT robustness for high temperatures, while noting that it does not fully resolve bound entanglement in this setting.

Abstract

We use simple spectral perturbation theory to show that the positive partial transpose property is stable under bounded perturbations of the Hamiltonian, for equilibrium states in infinite dimensions. The result holds provided the temperature is high enough, or equivalently, provided the perturbation is small enough.

Stability of PPT in equilibrium states

TL;DR

This work proves that the positive partial transpose (PPT) property of bipartite equilibrium states in infinite-dimensional systems is stable under bounded perturbations of the Hamiltonian at high temperatures or for sufficiently small perturbations. Using a Dyson expansion and a strategic factorization of the Boltzmann operator, the authors control the perturbed state through Hilbert-Schmidt bounds on the interaction in imaginary time, deriving an explicit temperature threshold via . The main result, Theorem 1, states that if , then the perturbed Gibbs state remains PPT under the stated bound . The approach circumvents the infinite-dimensional obstacle of eigenvalue accumulation by preconditioning with and demonstrates PPT robustness for high temperatures, while noting that it does not fully resolve bound entanglement in this setting.

Abstract

We use simple spectral perturbation theory to show that the positive partial transpose property is stable under bounded perturbations of the Hamiltonian, for equilibrium states in infinite dimensions. The result holds provided the temperature is high enough, or equivalently, provided the perturbation is small enough.
Paper Structure (5 sections, 2 theorems, 41 equations)

This paper contains 5 sections, 2 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. PPT criterion & main result
  3. Proofs
  4. Proof of Theorem \ref{['thm1']}
  5. Proof of \ref{['1.12-2']}

Key Result

Theorem 2.1

Let $\rho$ be a density matrix on the bipartite Hilbert space ${\cal H}={\cal H}_A\otimes{\cal H}_B$. If $\rho$ is separable then $\rho$ is PPT.

Theorems & Definitions (2)

  • Theorem 2.1: PPT criterion
  • Theorem 2.2