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 Jamiolkowski 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.

arXiv comment34 pages in Latex including 3 figures in eps, ver 2: minor revisions

References in corpus (21)

• Separability of Mixed States: Necessary and Sufficient Conditions
• On the volume of the set of mixed entangled states
• Separability of very noisy mixed states and implications for NMR quantum computing
• Geometric Quantum Mechanics
• Largest separable balls around the maximally mixed bipartite quantum state
• On the volume of the set of mixed entangled states II
• Quantum inseparability as local pseudomixture
• Geometry of entangled states
• Separability and entanglement of composite quantum systems
• Hilbert--Schmidt volume of the set of mixed quantum states
• A generalized Pancharatnam geometric phase formula for three level systems
• Geometry of quantum systems: density states and entanglement
• Tensor products of convex sets and the volume of separable states on N qudits
• The volume of separable states is super-doubly-exponentially small
• Bipartite Mixed States of Infinite-Dimensional Systems are Generically Nonseparable
• Silver mean conjectures for 15-d volumes and 14-d hyperareas of the separable two-qubit systems
• On the structure of the body of states with positive partial transpose
• Geometry of mixed states and degeneracy structure of geometric phases for multi-level quantum systems. A unitary group approach
• Entangled states close to the maximally mixed state
• Cones of ball-ball separable elements
• Subnormalized states and trace-nonincreasing maps

Cited by in corpus (25)

• Cones of positive maps and their duality relations
• Generating random quantum channels
• Entanglement thresholds for random induced states
• Almost all quantum channels are equidistant
• Approximating the Set of Separable States Using the Positive Partial Transpose Test
• How often is a random quantum state k-entangled?
• The Inverse Eigenvalue Problem for Entanglement Witnesses
• Pauli semigroups and unistochastic quantum channels
• The absolute positive partial transpose property for random induced states
• Geometry of the Pauli maps and Pauli channels
• Norms and Cones in the Theory of Quantum Entanglement
• There are many more positive maps than completely positive maps
• Bound entangled states with extremal properties
• Measure of not-completely-positive qubit maps: the general case
• Measure of positive and not completely positive single-qubit Pauli maps
• Dvoretzky's Theorem and the Complexity of Entanglement Detection
• Log-Convex set of Lindblad semigroups acting on $N$-level system
• Geometry of generalized Pauli channels
• Convex set of quantum states with positive partial transpose analysed by hit and run algorithm
• Separability Criterion for Quantum Effects
• Geometry of symmetric and non-invertible Pauli channels
• Rogers-Shephard inequality for log-concave functions
• New examples of extremal positive linear maps
• Noncommutative maximal ergodic inequality for non-tracial L1-spaces
• Spectral characterizations of entanglement witnesses