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Choi matrices revisited. III

Kyung Hoon Han, Seung-Hyeok Kye

Abstract

We look for all linear isomorphisms from the mapping spaces onto the tensor products of matrices which send $k$-superpositive maps onto unnormalized bi-partite states of Schmidt numbers less than or equal to $k$. They also send $k$-positive maps onto $k$-block-positive matrices. We also look for all the bilinear pairings between the mapping spaces and tensor products of matrices which retain the usual duality between $k$-positivity and Schmidt numbers $\le k$. They also retain the duality between $k$-superpositivity and $k$-block-positivity.

Choi matrices revisited. III

Abstract

We look for all linear isomorphisms from the mapping spaces onto the tensor products of matrices which send -superpositive maps onto unnormalized bi-partite states of Schmidt numbers less than or equal to . They also send -positive maps onto -block-positive matrices. We also look for all the bilinear pairings between the mapping spaces and tensor products of matrices which retain the usual duality between -positivity and Schmidt numbers . They also retain the duality between -superpositivity and -block-positivity.

Paper Structure

This paper contains 6 sections, 15 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Linear isomorphisms and Bilinear pairings
  3. Hermiticity preserving bilinear pairings on mapping spaces and tensor products
  4. $k$-superpositivity, Schmidt number $k$ and $k$-positivity
  5. Conclusion
  6. APPENDIX: Isomorphisms preserving separability

Key Result

Lemma 2.1

Suppose that $\Lambda$ is a linear functional on a vector space $Y$, and $\alpha$ is a scalar-valued function on $Y\setminus\{0\}$. If $y\mapsto \alpha(y)\Lambda(y)$ extends to a linear functional on $Y$, then $\alpha$ is a constant function.

Theorems & Definitions (15)

  • Lemma 2.1
  • Proposition 2.2
  • Corollary 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3: han_kye_choi_2
  • Proposition 3.4
  • ...and 5 more