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 [20] 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.
Highlights
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 explore patterns of quantum entanglement that emerge in systems containing entangled spins and ghost-spins, regarding them as toy models for subsectors with negative norm states arising in covariant formulations of theories with gauge symmetry, as mentioned earlier
In this case the state is a product state comprising spins disentangled from ghost-spins: the sign of the norm of the state enters as an overall sign in ρsA, giving SA > 0 for positive norm states, while for negative norm states, SA has a negative real part and a constant imaginary part
Summary
We review the toy model of two “ghost-spins” [20], which abstracts away from the specific technical issues of the ghost CFTs there but mimics some of the key features. A generic normalized positive/negative norm state with norm ±1 is. A simple example of a positive norm state is |ψ = ψ++| + + + ψ−−| − − , while |ψ = ψ+−| + − + ψ−+| − + has negative norm. The reduced density matrix is normalized to have trρA = trρ = ±1 depending on whether the state (2.9) is positive or negative norm. Let us for simplicity consider a simple family of states where the reduced density matrix is diagonal, by restricting to ψ−+∗ = In this case, log ρA is diagonal and can be calculated . Where the ± pertain to positive and negative norm states respectively. We obtain (log ρA)++ = log(±x) and (log ρA)−− = log(±(1−x)), the ± referring again to positive/negative norm states respectively. It is worth noting that while (2.4) mimics the norms of the bc-ghost system in [20], there is no obvious analog of the background charge here: in particular tracing over spinA instead of spinB is equivalent, so that entanglement entropy for the subsystem is the same as that for the complement
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