Complete Positivity of Subsystems in Quantum Dynamics

TL;DR

This work investigates how the CP and CP-divisibility properties of system and environment channels relate under joint unitary evolution, revealing that these properties must be mirrored across subsystems in memory-affected quantum dynamics. By formalizing the Kraus representation and the CP criterion, the authors show that a CP (or CP-divisible) channel on the system enforces the same status on the environment, and vice versa. Through explicit constructions of Bell, GHZ, and W states and a partial transpose example, they illustrate conditions under which subsystem channels are CP or CP-divisible and when non-CP behavior can arise in a joint noiseless evolution. The results provide a conceptual and technical basis for leveraging memory effects and engineered dissipation in quantum information processing, clarifying when non-CP maps can be embedded in unitary SE dynamics.

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

Theorems & Definitions (7)

• Theorem 1
• Corollary 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof