Complete Positivity of Subsystems in Quantum Dynamics

TL;DR

This paper explores the conditions under which subsystems in quantum dynamics maintain complete positivity, revealing that joint unitary evolution requires both system and environment channels to be either CP or NCP, with implications for quantum memory effects.

Contribution

It establishes that for a joint system-environment evolution to be unitary, both subsystems must be either CP or NCP, providing new insights into quantum channel divisibility and memory effects.

Findings

Subsystem channels must both be CP or NCP for joint unitarity

Memory effects relate to CP-divisibility of subsystems

Illustrated with Bell, GHZ, W states, and partial transpose operations

Abstract

Although many quantum channels satisfy Completely Positive Trace Preserving (CPTP) condition, there are valid quantum channels that can be non-completely positive (NCP). As memory effects can provide advantages in the dynamics of noisy quantum systems, we investigate the relative CP condition and the CP-divisibility condition of the system and environment subsystems of a joint system-environment quantum state evolving noiselessly. We show that the system and environment channels must be both CP (NCP) or CP-divisible (CP-indivisible) for the evolution in the joint system-environment space to be unitary. We illustrate our results with examples of Bell state created from $|00\rangle$, GHZ state created from $|000\rangle$, W state created from $|100\rangle$, and the partial transpose (PT) operation acting on the Bell state