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On the extreme points of sets of absolulely separable and PPT states

Zhiwei Song, Lin Chen

TL;DR

This paper investigates the geometry of absolutely separable $\mathcal{AS}_{m,n}$ and absolutely PPT $\mathcal{AP}_{m,n}$ states, focusing on extreme points and a robustness measure for nonabsolute separability. It provides complete characterizations of extreme points for $\mathcal{AS}_{2,n}$ and $\mathcal{AP}_{2,n}$ (with at most three distinct eigenvalues) and a solvable linear-equation criterion for extreme points of $\mathcal{AP}_{3,n}$ (extremes have at most seven eigenvalues). It also introduces the robustness of nonabsolute separability $\mathcal{AR}(\rho)$, proving its key properties and giving exact formulas for pure states and rank-two two-qubit states, thereby supplying computable NAS diagnostics. Collectively, the results illuminate the spectral geometry of AS and AP sets and provide practical criteria for identifying extreme states, while outlining open questions in higher dimensions and for general mixed states.

Abstract

The absolutely separable (resp. PPT) states remain separable (resp. positive partial transpose) under any global unitary operation. We present a compact form of the extreme points in the sets of absolutely separable states and PPT states in two-qubit and qubit-qudit systems. The results imply that each extreme point has at most three distinct eigenvalues. We establish a necessary and sufficient condition for determining extreme points of the set of absolutely PPT states in two-qutrit and qutrit-qudit systems, expressed as solvable linear equations. We also demonstrate that any extreme point in qutrit-qudit system has at most seven distinct eigenvalues. We introduce the concept of robustness of nonabsolute separability. It quantifies the minimal amount by which a state needs to mix with other states such that the overall state is absolutely separable. We show that the robustness satisfies positivity, invariance under unitary transformation, monotonicity and convexity, so it is a good measure within the resource theory of nonabsolute separability. Analytical expressions for this measure are given for pure states in arbitrary system and rank-two mixed states in two-qubit system.

On the extreme points of sets of absolulely separable and PPT states

TL;DR

This paper investigates the geometry of absolutely separable and absolutely PPT states, focusing on extreme points and a robustness measure for nonabsolute separability. It provides complete characterizations of extreme points for and (with at most three distinct eigenvalues) and a solvable linear-equation criterion for extreme points of (extremes have at most seven eigenvalues). It also introduces the robustness of nonabsolute separability , proving its key properties and giving exact formulas for pure states and rank-two two-qubit states, thereby supplying computable NAS diagnostics. Collectively, the results illuminate the spectral geometry of AS and AP sets and provide practical criteria for identifying extreme states, while outlining open questions in higher dimensions and for general mixed states.

Abstract

The absolutely separable (resp. PPT) states remain separable (resp. positive partial transpose) under any global unitary operation. We present a compact form of the extreme points in the sets of absolutely separable states and PPT states in two-qubit and qubit-qudit systems. The results imply that each extreme point has at most three distinct eigenvalues. We establish a necessary and sufficient condition for determining extreme points of the set of absolutely PPT states in two-qutrit and qutrit-qudit systems, expressed as solvable linear equations. We also demonstrate that any extreme point in qutrit-qudit system has at most seven distinct eigenvalues. We introduce the concept of robustness of nonabsolute separability. It quantifies the minimal amount by which a state needs to mix with other states such that the overall state is absolutely separable. We show that the robustness satisfies positivity, invariance under unitary transformation, monotonicity and convexity, so it is a good measure within the resource theory of nonabsolute separability. Analytical expressions for this measure are given for pure states in arbitrary system and rank-two mixed states in two-qubit system.
Paper Structure (10 sections, 23 theorems, 77 equations, 1 figure, 1 table)

This paper contains 10 sections, 23 theorems, 77 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Extreme Points of ${\cal AS}_{2,n}$ and ${\cal AP}_{2,n}$
  4. Extreme points of ${\cal AP}_{3,n}$
  5. Robustness of nonabsolute separability
  6. Conclusion
  7. Proof of Theorem \ref{['thm:main']}
  8. Proof of Theorems \ref{['ef22']}, \ref{['mmain']} and Corollary \ref{['mbn']}
  9. Proof of results in Section \ref{['sec:3x3']}
  10. Proof of Theorem \ref{['battle']}

Key Result

Lemma 1

(i) Let $x,y\in \mathbb{R}^n$. Then $x+y\succ x^{\downarrow}+y^{\uparrow}.$ (ii) Let $x,y\in \mathbb{R}^m$, $u,v\in \mathbb{R}^n$ satisfy that $x\succ y$ and $u\succ v$. Then $(x,u)\succ (y,v)$. (iii) Let $x\in \mathbb{R}^n$ with $x_1^{\downarrow}=\cdots=x_k^{\downarrow}\ge x_{k+1}^{\downarrow}=\cdo

Figures (1)

  • Figure 1: A schematic diagram of robustness of nonabsolute separability for the pure state $\rho$ in two-qubit system. Here, $M$ represents the maximal ball, and ${\cal D}$ denotes the entire set of states. The line segments at the boundary of ${\cal AS}_{2,2}$ represent those non-extreme points. The optimal state $\sigma$ is the deficient-rank extreme point of ${\cal AS}_{2,2}$, residing on the boundary but not being an extreme point of ${\cal D}$. The resulting state $\gamma$ with eigenvalues $(\frac{1}{2},\frac{1}{6},\frac{1}{6},\frac{1}{6})$ lies at the middle of $\rho$ and $\sigma$. Moreover, $\sigma$ and $\gamma$ represent all the extreme points of ${\cal AS}_{2,2}$ that reside on the maximal ball, as stated in Corollary \ref{['mbn']} (i).

Theorems & Definitions (24)

  • Lemma 1
  • Lemma 2
  • Lemma 3: gurvits2002largest
  • Lemma 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 14 more