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$k$-Positive Maps: New Characterizations and a Generation Method

Frederik vom Ende, Sumeet Khatri, Sergey Denisov

TL;DR

The paper tackles the problem of understanding $k$-positivity in quantum maps by deriving optimization-based characterizations that connect to the spectral norm of order-$3$ tensors and to the separability problem, thereby providing computationally tractable criteria and a conceptual link to NP-hardness for $k<d$. A novel Lie-semigroup–driven method is developed to generate families of non-decomposable $k$-positive maps from a seed map, with demonstrations in dimensions $d=3$ and $d=4$, and an SDP framework to test an equivalent form of the PPT-square conjecture. The work yields practical tools for certifying $k$-positivity and a systematic approach to sampling $k$-positive non-decomposable maps, advancing both the theory and numerical practice of quantum positivity and separability. It also clarifies the constraints and limitations of the proposed sampling method, highlighting its Markovian scope and the ongoing challenge of deciding $k$-positivity in finite time.

Abstract

We study $k$-positive linear maps on matrix algebras and address two problems, (i) characterizations of $k$-positivity and (ii) generation of non-decomposable $k$-positive maps. On the characterization side, we derive optimization-based conditions equivalent to $k$-positivity that (a) reduce to a simple check when $k=d$, (b) reveal a direct link to the spectral norm of certain order-3 tensors (aligning with known NP-hardness barriers for $k<d$), and (c) recast $k$-positivity as a novel optimization problem over separable states, thereby connecting it explicitly to separability testing. On the generation side, we introduce a Lie-semigroup-based method that, starting from a single $k$-positive map, produces one-parameter families that remain $k$-positive and non-decomposable for small enough times. We illustrate this by generating such families for $d=3$ and $d=4$. We also formulate a semi-definite program (SDP) to test an equivalent form of the positive partial transpose (PPT) square conjecture (and do not find any violation of the latter). Our results provide practical computational tools for certifying $k$-positivity and a systematic way to sample $k$-positive non-decomposable maps.

$k$-Positive Maps: New Characterizations and a Generation Method

TL;DR

The paper tackles the problem of understanding -positivity in quantum maps by deriving optimization-based characterizations that connect to the spectral norm of order- tensors and to the separability problem, thereby providing computationally tractable criteria and a conceptual link to NP-hardness for . A novel Lie-semigroup–driven method is developed to generate families of non-decomposable -positive maps from a seed map, with demonstrations in dimensions and , and an SDP framework to test an equivalent form of the PPT-square conjecture. The work yields practical tools for certifying -positivity and a systematic approach to sampling -positive non-decomposable maps, advancing both the theory and numerical practice of quantum positivity and separability. It also clarifies the constraints and limitations of the proposed sampling method, highlighting its Markovian scope and the ongoing challenge of deciding -positivity in finite time.

Abstract

We study -positive linear maps on matrix algebras and address two problems, (i) characterizations of -positivity and (ii) generation of non-decomposable -positive maps. On the characterization side, we derive optimization-based conditions equivalent to -positivity that (a) reduce to a simple check when , (b) reveal a direct link to the spectral norm of certain order-3 tensors (aligning with known NP-hardness barriers for ), and (c) recast -positivity as a novel optimization problem over separable states, thereby connecting it explicitly to separability testing. On the generation side, we introduce a Lie-semigroup-based method that, starting from a single -positive map, produces one-parameter families that remain -positive and non-decomposable for small enough times. We illustrate this by generating such families for and . We also formulate a semi-definite program (SDP) to test an equivalent form of the positive partial transpose (PPT) square conjecture (and do not find any violation of the latter). Our results provide practical computational tools for certifying -positivity and a systematic way to sample -positive non-decomposable maps.

Paper Structure

This paper contains 11 sections, 8 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Recap: $k$-positivity and its characterizations
  3. Novel characterizations of $k$-positivity
  4. First condition: Collapse for $k=d$
  5. Second condition: Connection to 3-tensors
  6. Third condition: Connection to the separability problem
  7. A new way to sample non-decomposable $k$-positive maps
  8. The theory
  9. The workflow
  10. The PPT-square conjecture
  11. Outlook

Key Result

Proposition 1

Given any $\Phi\in\mathcal{L}(\mathbb C^{m\times m},\mathbb C^{d\times d})$, $k,m,d\in\mathbb N$ the following statements are equivalent.

Figures (1)

  • Figure 1: The set of conditionally completely positive maps (CCP) is the tangent cone of the set of completely positive (CP) maps at the identity ${\rm id}$DHKS08. Moreover, the CP cone sits strictly inside the cone of positive (P) maps. Therefore, a non-trivial portion of P is in CCP$\setminus$CP, meaning that if any element of P$\,\cap\,$CCP is taken as the generator in Prop. \ref{['prop_exp_semigroup']}, then (and only then) will the corresponding semigroup be trivial (i.e., CP) for all positive times.

Theorems & Definitions (18)

  • Proposition 1
  • Remark 1
  • Proposition 2
  • Remark 2
  • Lemma 1
  • Theorem 1
  • proof
  • Theorem 2
  • Corollary 1
  • proof
  • ...and 8 more