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arXiv:1412.7469 · Open Systems & Information Dynamics

Stable Subspaces of Positive Maps of Matrix Algebras

Marek Miller, Robert Olkiewicz

Published Jun 1, 2015• 9 citations
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Abstract

We study stable subspaces of positive extremal maps of finite dimensional matrix algebras that preserve trace and matrix identity (so-called bistochastic maps). We have established the existence of the isometric-sweeping decomposition for such maps. As the main result of the paper, we have shown that all extremal bistochastic maps acting on the algebra of matrices of size 3×3 fall into one of the three possible categories, depending on the form of the stable subspace of the isometric-sweeping decomposition. Our example of an extremal atomic positive map seems to be the first one that handles the case of that subspace being non-trivial. Lastly, we compute the entanglement witness associated with the extremal map and specify a large family of entangled states detected by it.

DOI
10.1142/s1230161215500110

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A new method of analysing positive bistochastic maps on the algebra of complex matrices [Formula: see text] has been proposed. By identifying the set of such maps with a convex set of linear operators on [Formula: see text], one can employ techniques from the theory of compact semigroups to obtain results concerning asymptotic properties of positive maps. It turns out that the idempotent elements play a crucial role in classifying the convex set into subsets, in which representations of extremal positive maps are to be found. It has been shown that all positive bistochastic maps, extremal in the set of all positive maps of [Formula: see text], that are not Jordan isomorphisms of [Formula: see text] are represented by matrices that fall into two possible categories, determined by the simplest idempotent matrices: one by the zero matrix, and the other by a one-dimensional orthogonal projection. Some norm conditions for matrices representing possible extremal maps have been specified and examples of maps from both categories have been brought up, based on the results published previously.
We define an entanglement witness in a composite quantum system as an observable having a nonnegative expectation value in every separable state. Then a state is entangled if and only if it has a negative expectation value of some entanglement witness. Equivalent representations of entanglement witnesses are nonnegative biquadratic forms or positive linear maps of Hermitian matrices. As reported elsewhere, we have studied extremal entanglement witnesses in dimension $3\ifmmode\times\else\texttimes\fi{}3$ by constructing numerical examples of generic extremal nonnegative forms. These are so complicated that we do not know how to handle them other than by numerical methods. However, the corresponding extremal positive maps can be presented graphically, as we attempt to do in the present paper. We understand that a positive map is extremal when the image of D\mathcal{D}D, the set of density matrices, fills out D\mathcal{D}D maximally, in a certain sense. For the graphical presentation of a map we transform it to a standard form where it is unital and trace preserving. We present an iterative algorithm for the transformation, which converges rapidly in all our numerical examples and presumably works for any positive map. This standard form of an entanglement witness is unique up to unitary product transformations.
We consider entanglement witnesses arising from positive linear maps which generate exposed extremal rays. We show that every entangled state can be detected by one of these witnesses, and this witness detects a unique set of entangled states among those. Therefore, they provide a minimal set of witnesses to detect any kind of entanglement in a sense. Furthermore, if those maps are indecomposable then they detect large classes of entangled states with positive partial transposes which have nonempty relative interiors in the cone generated by all PPT states. We also provide a one-parameter family of indecomposable positive linear maps which generate exposed extremal rays. This gives the first examples of such maps in three-dimensional matrix algebra.
We investigate linear maps between matrix algebras that remain positive under tensor powers, i.e., under tensoring with n copies of themselves. Completely positive and completely co-positive maps are trivial examples of this kind. We show that for every n ∈ ℕ, there exist non-trivial maps with this property and that for two-dimensional Hilbert spaces there is no non-trivial map for which this holds for all n. For higher dimensions, we reduce the existence question of such non-trivial “tensor-stable positive maps” to a one-parameter family of maps and show that an affirmative answer would imply the existence of non-positive partial transpose bound entanglement. As an application, we show that any tensor-stable positive map that is not completely positive yields an upper bound on the quantum channel capacity, which for the transposition map gives the well-known cb-norm bound. We, furthermore, show that the latter is an upper bound even for the local operations and classical communications-assisted quantum capacity, and that moreover it is a strong converse rate for this task.