Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the maximum purity of absolutely separable bipartite states

Hoang Phi Dung, Vu The Khoi

TL;DR

This work characterizes the maximum purity of absolutely separable and absolutely PPT bipartite states, framing the problem as the Euclidean radius of the smallest Hilbert-Schmidt ball around the maximally mixed state that contains all such states. It provides an exact analytic result for the maximum purity in the two-qubit case ($\mathrm{tr}(\rho^2)=\tfrac{3}{8}$) and uses numerical optimization to propose conjectured maxima for higher dimensions, including explicit eigenvalue configurations. For qubit-qudit and qutrit-qudit systems, the authors present conjectured formulas for the maximum purity that depend on dimension and residue classes, supported by numerical evidence. Geometrically, the results suggest these sets lie inside a ball of radius $O(1/\sqrt{n})$ around the maximally mixed state, offering insight into the boundary between classical and quantum correlations in high dimensions.

Abstract

In this study, we investigate the problem of determining the maximum purity for absolutely separable and absolutely PPT quantum states. From the geometric viewpoint, this problem is equivalent to asking for the exact Euclidean radius of the smallest ball around the maximally mixed state that encompasses the set of all absolutely separable or absolutely PPT states. Our results provide an analytic solution for two qubit states. Based on numerical computation, we propose a conjectured maximum purity for absolutely separable qubit-qudit states and absolutely PPT qutrit-qudit states.

On the maximum purity of absolutely separable bipartite states

TL;DR

This work characterizes the maximum purity of absolutely separable and absolutely PPT bipartite states, framing the problem as the Euclidean radius of the smallest Hilbert-Schmidt ball around the maximally mixed state that contains all such states. It provides an exact analytic result for the maximum purity in the two-qubit case () and uses numerical optimization to propose conjectured maxima for higher dimensions, including explicit eigenvalue configurations. For qubit-qudit and qutrit-qudit systems, the authors present conjectured formulas for the maximum purity that depend on dimension and residue classes, supported by numerical evidence. Geometrically, the results suggest these sets lie inside a ball of radius around the maximally mixed state, offering insight into the boundary between classical and quantum correlations in high dimensions.

Abstract

In this study, we investigate the problem of determining the maximum purity for absolutely separable and absolutely PPT quantum states. From the geometric viewpoint, this problem is equivalent to asking for the exact Euclidean radius of the smallest ball around the maximally mixed state that encompasses the set of all absolutely separable or absolutely PPT states. Our results provide an analytic solution for two qubit states. Based on numerical computation, we propose a conjectured maximum purity for absolutely separable qubit-qudit states and absolutely PPT qutrit-qudit states.
Paper Structure (7 sections, 1 theorem, 15 equations, 2 figures, 2 tables)

This paper contains 7 sections, 1 theorem, 15 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Maximum purity of absolutely separable and absolutely PPT states
  4. The two-qubit case
  5. The qubit-qudit case
  6. The qutrit-qudit case
  7. Conclusions

Key Result

Theorem 3.1

The maximum purity of absolutely separable two-qubit states is $\dfrac{3}{8}$ and this maximum value is attained iff the eigenvalues $\lambda_1 = \lambda_2 = \dfrac{1}{4} + \dfrac{1}{4\sqrt{2}}$ and $\lambda_3 = \lambda_4=\dfrac{1}{4} - \dfrac{1}{4\sqrt{2}}$.

Figures (2)

  • Figure 1: Plot of the maximum purity of absolutely separable states versus the dimension $n$ in a $2\otimes n$ system, where the red curve represents the numerical values and the blue curve represents the conjectured values.
  • Figure 2: Plot of the maximum purity of absolutely PPT states versus the dimension $n$ in a $3\otimes n$ system, where the red curve represents the numerical values and the blue curve represents the conjectured values.

Theorems & Definitions (7)

  • Theorem 3.1
  • proof
  • Conjecture 3.1
  • Conjecture 3.2
  • Remark 3.1
  • Conjecture 3.3
  • Conjecture 3.4