Minimal and maximal operator spaces and operator systems in entanglement theory

Abstract

We examine k-minimal and k-maximal operator spaces and operator systems, and investigate their relationships with the separability problem in quantum information theory. We show that the matrix norms that define the k-minimal operator spaces are equal to a family of norms that have been studied independently as a tool for detecting k-positive linear maps and bound entanglement. Similarly, we investigate the k-super minimal and k-super maximal operator systems that were recently introduced and show that their cones of positive elements are exactly the cones of k-block positive operators and (unnormalized) states with Schmidt number no greater than k, respectively. We characterize a class of norms on the k-super minimal operator systems and show that the completely bounded versions of these norms provide a criterion for testing the Schmidt number of a quantum state that generalizes the recently-developed separability criterion based on trace-contractive maps.

Role Of Partial Transpose And Generalized Choi Maps In Quantum Dynamical Semigroups Involving Separable And Entangled States

“…The density matrix has the Schmidt number at least r + 1 if and only if there exists an r-positive linear map Λ : M n → M n such that (I ⊗ Λ)(ρ) ≥ 0. Remarkable progress on the topic has been made in recent years as can be gauged from ( [17], [19], [20], [21], [22], [23], [24], [44], [52], [55], [58], [60], [62], [63], [76], [85]) The name "partially entanglement breaking " has been associated with maps having Schmidt number < n by some authors. We shall not go into details in this paper.…”

On k-Minimal and k-Maximal Operator Space Structures

“…These spaces were first noticed by Junge [10] and more generally studied by Lehner [11]. Recently, the relationship of kminimal and k-maximal operator space structures to norms that have been used in quantum information theory [7,8] have been investigated by Johnston et al [9].…”

“…is any operator space, the matrix norms in Min k (X ) and Max k (X ) are explicitly given ( [9,13]) as follows:…”

Mapping cones are operator systems

“…It was shown in [5] that if OM IN k (M n ) and OM AX k (M n ) denote the super k-minimal and super k-maximal operator systems on M n [4], respectively, then we have that…”

“…By using [5,Theorem 5] and the fact that P k (M n ) is a semigroup, it is not difficult to see that CP(OM IN k (M n )) = P k (M n ). By using [1, Theorem 3.8] we can similarly see that CP(OM AX k (M n )) = P k (M n ), so we can't possibly hope for a uniqueness result as strong as that of Theorem 6 or Corollary 8 in this setting.…”