Excitation Flow, Positivity, and Fisher Information for Open Subsystems of an $N$-Qubit Network

TL;DR

This paper derives explicit formulas for propagators of open subsystems in an N-qubit network with conserved excitation, linking excitation flow to positivity, entanglement, and Fisher information, revealing fundamental constraints and behaviors.

Contribution

It provides a unified analytical framework connecting excitation flow, positivity, entanglement, and Fisher information in N-qubit networks with conserved excitation.

Findings

Single transition amplitude controls excitation flow and positivity.

Positivity depends solely on excitation flow direction, independent of subsystem size.

Fisher information decomposes into state and process parts, with bounds on each.

Abstract

We derive closed-form propagators for any $K$-qubit subsystem of a closed $N$-qubit network with a single conserved excitation. A single transition amplitude simultaneously controls excitation flow between subsystems, the positivity and complete positivity of every propagator, the entanglement entropy of every subsystem, and the quantum Fisher information for global parameters. Positivity and complete positivity coincide, determined solely by the direction of excitation flow, independently of subsystem size, coherence, or entanglement structure. A propagator is positive and completely positive if and only if it contracts the subsystem state toward its fixed point. The ensemble of propagators collectively constrains global properties inaccessible to any single subsystem. For single-qubit subsystems, we characterize the ensemble's fixed-point distribution and domain of positivity, finding a band of states that lies inside the positivity domain of every propagator yet is never visited by the physical dynamics. The quantum Fisher information decomposes into state and process contributions over any observation window $[t_1,t_2]$, with the state contribution bounded while the process contribution grows secularly. The total Fisher information is minimal when all future propagators are nonpositive and not completely positive, and near its maximum when they are positive and completely positive.