Horodeckis criterion of separability of mixed states in von Neumann and C*-algebras

Abstract

Communicated by M. OhyaThe Horodeckis necessary and sufficient condition of separability of mixed states is generalized to arbitrary composite quantum systems.

Stable Subspaces of Positive Maps of Matrix Algebras

“…This is a consequence of the celebrated Krein-Milman theorem [15], as for every positive number r > 0, the set of positive maps such that their operator norm: ||S|| ≤ r, is compact. This is certainly true for maps on finite-dimensional matrix algebras, but applies also to the general setting of von Neumann algebras [18]. Despite considerable effort, examples of extremal positive maps, even in the low-dimensional case, are scarce [6,20,10,7].…”

“…It turns out that there is a one-to-one correspondence between positive maps and entanglement witnesses [8], and the Peres-Horodecki criterion, whether a quantum state is separable, is computationally feasible as long as a structure theorem similar to that proven by Størmer and Woronowicz holds true. Unfortunately, whereas that correspondence does exists even for the most general infinitedimensional quantum systems [24,18], higher dimensional situation lacks the complete description of positive maps and one needs a deeper understanding of their highly nontrivial structure. Because the set of positive maps forms a convex cone, its elements can be characterised as convex combinations of extremal ones.…”

Extremal Positive Maps on M3(ℂ) and Idempotent Matrices

“…The well established criterion of separability, proposed in the mentioned papers, reveals a one-to-one correspondence between positive maps and entanglement witnesses [5]. This Peres-Horodecki criterion, originally proposed for maps on algebras M n of square complex matrices of size n, which holds true even in the most general setting of injective von Neumann algebras [9], is computationally feasible as long as the structure of general positive maps on operator algebras representing a composite quantum system in question is known. To this day, the complete characterisation of positive maps have been obtained only for the algebra M 2 and the maps between M 2 and M 3 [14,16].…”