On Duality between Quantum Maps and Quantum States

Abstract

We investigate the space of quantum operations, as well as the larger space of maps which are positive, but not completely positive. A constructive criterion for decomposability is presented. A certain class of unistochastic operations, determined by unitary matrices of extended dimensionality, is defined and analyzed. Using the concept of the dynamical matrix and the Jamio lkowski isomorphism we explore the relation between the set of quantum operations (dynamics) and the set of density matrices acting on an extended Hilbert space (kinematics). An analogous relation is established between the classical maps and an extended space of the discrete probability distributions.

Geometric representation of two-qubit entanglement witnesses

“…In the Choi-isomorphic setting, the question is equivalent to determining whether a single-qubit map is positive, which is again known to be a hard problem (see, for example, Refs. [24,25]). However, often it will be plainly apparent whether E lies inside the Bloch sphere from a visualisation, and hence it will be immediately possible to determine block positivity of R from the ellipsoid representation.…”

Geometry of sets of quantum maps: A generic positive map acting on a high-dimensional system is not completely positive

“…We point out that the order of indices of the matrix D in (2) is different than in the previous work [24,26]. (The reason for this change will be elucidated in the next paragraph.)…”

“…where c is the (universal) constant from (24). Combining the above inequality with (iii) we obtain c [Of course, any other normalizationmost notably Tr D Φ = 1 leading to sets of states -would work just as well for comparing volumes provided we were consistent.]…”

“…In particular, the rescaled dynamical matrix D associated with a (non-zero) CP map represents a state of a bipartite system, σ := D/Tr D ∈ M tot N 2 -see e.g. [23,24], an element of the base of the positive semi-definite cone obtained by intersecting that cone with the hyperplane of trace one matrices.…”

“…Due to Jamio lkowski isomorphism [23,24] the set N −1 CP TP N can be considered as a subset of the (N 4 − 1)-dimensional set M tot N 2 of density operators acting on an extended Hilbert space, H N ⊗ H N . This useful fact contributes to our understanding of properties the set of quantum operations, but its geometry is nontrivial even in the simplest case of N = 2 [25,26].…”