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A triality pattern in entanglement theory

Daniel Cariello

TL;DR

Addresses separability for bipartite quantum states by studying PPT, SPC, and invariant-under-realignment classes. Develops a unified toolkit based on contraction maps and the interrelations among these classes, yielding a universal spectral-radius bound and a method to normalize SPC and invariant-under-realignment states into a structured filter normal form. Proves a lower bound on rank for all three classes and shows separability when the bound is tight, leveraging the complete reducibility property. The work reveals a triality pattern linking the three state types and suggests that insights in one class can inform the others, strengthening the entanglement theory toolkit.

Abstract

In this work, we present new connections between three types of quantum states: positive under partial transpose states, symmetric with positive coefficients states and invariant under realignment states. First, we obtain a common upper bound for their spectral radii and a result on their filter normal forms. Then we prove the existence of a lower bound for their ranks and the fact that whenever this bound is attained the states are separable. These connections add new evidence to the pattern that for every proven result for one of these types, there are counterparts for the other two, which is a potential source of information for entanglement theory.

A triality pattern in entanglement theory

TL;DR

Addresses separability for bipartite quantum states by studying PPT, SPC, and invariant-under-realignment classes. Develops a unified toolkit based on contraction maps and the interrelations among these classes, yielding a universal spectral-radius bound and a method to normalize SPC and invariant-under-realignment states into a structured filter normal form. Proves a lower bound on rank for all three classes and shows separability when the bound is tight, leveraging the complete reducibility property. The work reveals a triality pattern linking the three state types and suggests that insights in one class can inform the others, strengthening the entanglement theory toolkit.

Abstract

In this work, we present new connections between three types of quantum states: positive under partial transpose states, symmetric with positive coefficients states and invariant under realignment states. First, we obtain a common upper bound for their spectral radii and a result on their filter normal forms. Then we prove the existence of a lower bound for their ranks and the fact that whenever this bound is attained the states are separable. These connections add new evidence to the pattern that for every proven result for one of these types, there are counterparts for the other two, which is a potential source of information for entanglement theory.
Paper Structure (7 sections, 16 theorems, 36 equations)

This paper contains 7 sections, 16 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. An upper bound for the spectral radius of the special triad
  4. Filter normal form for SPC states and invariant under realignment states
  5. Lower bound for the rank of the special triad
  6. Summary and Conclusion
  7. Disclosure Statement

Key Result

Lemma 2.3

Let $\gamma,\delta \in \mathcal{M}_k\otimes \mathcal{M}_k$ and $v=\sum_{i}a_i\otimes b_i,\ w=\sum_{j} c_j\otimes d_j \in \mathbb{C}^k\otimes \mathbb{C}^k$.

Theorems & Definitions (36)

  • Definition 2.1: Generalization of the Hadamard product
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 3.1
  • proof
  • ...and 26 more