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

Abstract

We investigate the set a) of positive, trace preserving maps acting on density matrices of size N , and a sequence of its nested subsets: the sets of maps which are b) decomposable, c) completely positive, d) extended by identity impose positive partial transpose and e) are superpositive. Working with the Hilbert-Schmidt (Euclidean) measure we derive tight explicit two-sided bounds for the volumes of all five sets. A sample consequence is the fact that, as N increases, a generic positive map becomes not decomposable and, a fortiori, not completely positive. Due to the Jamio lkowski isomorphism, the results obtained for quantum maps are closely connected to similar relations between the volume of the set of quantum states and the volumes of its subsets (such as states with positive partial transpose or separable states) or supersets. Our approach depends on systematic use of duality to derive quantitative estimates, and on various tools of classical convexity, high-dimensional probability and geometry of Banach spaces, some of which are not standard.

How often is a random quantum statek-entangled?

“…In particular, we conclude that vrad(BP (5) and then to proposition 6 of [26]. The difficulty is that, as indicated in the discussion following (5), we cannot (for example) identify P…”

“…Let C 1 := { ∈ C : tr (I C d ) = 1} be the base of C and let C TP := { ∈ C : ∀ ρ ∈ M d tr (ρ) = trρ} be the section of d C 1 (a rescaled base) defined by the trace-preserving condition. The argument from [26] (proposition 6 in [26] and the discussion in the paragraph following it) shows then that the volume radii of d C 1 and of C TP differ by a factor 1 ± O(log d/d 2 ). In our setting, this implies that vrad P…”

“…Let C ⊂ K be a closed convex cone and let e ∈ C ∩ C * be a unit vector. Put V b := {x ∈ K : x, e = 1} and let C b = C ∩ V b be the base of the cone C. Making use of lemma 1 from [26] one obtains a relation…”

“…Heuristically, proposition 6 of [26] shows that if an m-dimensional convex body K is 'reasonably balanced', then all its central sections whose codimension is 'substantially smaller' than m have volume radii close to that of K. More specifically, suppose C is any convex cone of maps on M d which contains the superpositive cone and is contained in the positive cone (all cones considered here verify this condition). Let C 1 := { ∈ C : tr (I C d ) = 1} be the base of C and let C TP := { ∈ C : ∀ ρ ∈ M d tr (ρ) = trρ} be the section of d C 1 (a rescaled base) defined by the trace-preserving condition.…”

“…These maps, also called k-entanglement breaking channels, correspond by the Choi-Jamiołkowski isomorphism to k-entangled states of a bipartite system [21,25]. Thus, investigating relations between sets of maps of different degree of positivity, one can establish the properties of the subsets of M d 2 characterized by different classes of quantum entanglement [21,26], and vice versa.…”