Measure of positive and not completely positive single-qubit Pauli maps

Abstract

The time evolution of an initially uncorrelated system is governed by a completely positive (CP) map. More generally, the system may contain initial (quantum) correlations with an environment, in which case the system evolves according to a not-completely positive (NCP) map. It is an interesting question what the relative measure is for these two types of maps within the set of positive maps. After indicating the scope of the full problem of computing the true volume for generic maps acting on a qubit, we study the case of Pauli channels in an abstract space whose elements represent an equivalence class of maps that are identical up to a non-Pauli unitary. In this space, we show that the volume of NCP maps is twice that of CP maps.

arXiv comment10 pages, Accepted in Phys. Rev. A for publication

References in corpus (11)

• Non-Markovian dynamics in open quantum systems
• Entanglement and non-Markovianity of quantum evolutions
• Concepts of quantum non-Markovianity: a hierarchy
• Dividing Quantum Channels
• Dynamics of initially entangled open quantum systems
• A Parametrization of Bipartite Systems Based on SU(4) Euler Angles
• Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive
• Simulating a single qubit channel using a mixed state environment
• Volume of the space of qubit-qubit channels and state transformations under random quantum channels
• The operator sum-difference representation for quantum maps: application to the two-qubit amplitude damping channel
• Amplitude Damping for single-qubit System with single-qubit mixed-state Environment

Cited by in corpus (7)

• Geometry of the Pauli maps and Pauli channels
• Singularities, mixing and non-Markovianity of Pauli dynamical maps
• Measure of not-completely-positive qubit maps: the general case
• Log-Convex set of Lindblad semigroups acting on $N$-level system
• Geometry of symmetric and non-invertible Pauli channels
• Guaranteeing Completely Positive Quantum Evolution
• No-Signaling in Steepest Entropy Ascent: A Nonlinear Non-local Non-equilibrium Quantum Dynamics of Composite Systems