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Extreme points of absolutely PPT states with exactly three distinct eigenvalues

Nalan Wang, Lin Chen, Zhiwei Song

Abstract

Whether the sets of absolutely separable (AS) and absolutely two-qutrit positive-partial-transpose (AP) states are the same has been an open problem in entanglement theory for decades. Since they are both convex sets, we investigate the boundary and extreme points of full-rank two-qutrit AP states with exactly three distinct eigenvalues. We show that every boundary point is an extreme point, with exactly one exception. We explicitly characterize the expressions of such points, each of which turns out to contain at most one parameter in some intervals. When the parameter approaches the ends of intervals, most points become the known extreme points of exactly two distinct eigenvalues. We present our results by tables and figures.

Extreme points of absolutely PPT states with exactly three distinct eigenvalues

Abstract

Whether the sets of absolutely separable (AS) and absolutely two-qutrit positive-partial-transpose (AP) states are the same has been an open problem in entanglement theory for decades. Since they are both convex sets, we investigate the boundary and extreme points of full-rank two-qutrit AP states with exactly three distinct eigenvalues. We show that every boundary point is an extreme point, with exactly one exception. We explicitly characterize the expressions of such points, each of which turns out to contain at most one parameter in some intervals. When the parameter approaches the ends of intervals, most points become the known extreme points of exactly two distinct eigenvalues. We present our results by tables and figures.
Paper Structure (17 sections, 9 theorems, 249 equations, 8 figures, 7 tables)

This paper contains 17 sections, 9 theorems, 249 equations, 8 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. An inequality
  4. Existence of extreme points with three distinct eigenvalues
  5. Extreme points with maximum eigenvalue of multiplicity one
  6. Extreme points with maximum eigenvalue of multiplicity two
  7. Extreme points with maximum eigenvalue of multiplicity more than two
  8. multiplicity three
  9. multiplicity four
  10. multiplicity five
  11. multiplicity six
  12. multiplicity seven
  13. Conclusions
  14. The proof of Lemma \ref{['le:3cbbbbbccc']}
  15. The proof of Theorem \ref{['all_3eigen']}
  16. ...and 2 more sections

Key Result

Lemma 2

A state $\rho \in {\cal AS}_{m,n}$ (resp. ${\cal AP}_{m,n}$) is an interior point if and only if for any state $v \in {\cal AS}_{m,n}$ (resp. ${\cal AP}_{m,n}$), there exists $\epsilon > 0$ s.t. for any $0 < \delta < \epsilon$, we have

Figures (8)

  • Figure 1: Classification of extreme points of $\nu_{1,k,8-k}$ where $k\in[1,7]$ with Remark \ref{['rem:nu1,4,4']}
  • Figure 2: Classification of extreme points of $\nu_{2,k,7-k}$ where $k\in[1,6]$ with Remark \ref{['rem:table2']}
  • Figure 3: Classification of extreme points of $\nu_{3,k,6-k}$ where $k\in[1,5]$ with Remark \ref{['rem:table3']}.
  • Figure 4: Classification of extreme points of $\nu_{4,k,5-k}$ where $k\in[1,4]$ with Remark \ref{['rem:table4']}
  • Figure 5: Classification of extreme points of $\nu_{5,k,4-k}$ where $k\in[1,3]$ with Remark \ref{['rem:table5']}
  • ...and 3 more figures

Theorems & Definitions (16)

  • Definition 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Theorem 10
  • ...and 6 more