Entangled spins and ghost-spins

TL;DR

This paper investigates quantum entanglement patterns in spin and ghost-spin systems, revealing how negative norm states affect entanglement entropy and identifying conditions for positive entropy in models with ghost-spins.

Contribution

It introduces a detailed analysis of entanglement in ghost-spin models, showing how entanglement entropy behaves for positive and negative norm states, and explores the impact of ghost-spin number parity.

Findings

Positive norm states yield positive entanglement entropy when spins are disentangled from ghost-spins.

Negative norm states can have negative real parts and constant imaginary parts of entanglement entropy.

Even ghost-spin numbers allow positive entropy subsectors, odd numbers do not.

Abstract

We study patterns of quantum entanglement in systems of spins and ghost-spins regarding them as simple quantum mechanical toy models for theories containing negative norm states. We define a single ghost-spin as in arXiv:1602.06505 [hep-th] as a 2-state spin variable with an indefinite inner product in the state space. We find that whenever the spin sector is disentangled from the ghost-spin sector (both of which could be entangled within themselves), the reduced density matrix obtained by tracing over all the ghost-spins gives rise to positive entanglement entropy for positive norm states, while negative norm states have an entanglement entropy with a negative real part and a constant imaginary part. However when the spins are entangled with the ghost-spins, there are new entanglement patterns in general. For systems where the number of ghost-spins is even, it is possible to find subsectors of the Hilbert space where positive norm states always lead to positive entanglement entropy after tracing over the ghost-spins. With an odd number of ghost-spins however, we find that there always exist positive norm states with negative real part for entanglement entropy after tracing over the ghost-spins.