Entanglement Entropy from String Field Theory (and a Higher-Spin Example)

We study the new class of solutions in linearized open string field theory (OSFT) involving higher-spin modes. Unlike the elementary OSFT solutions (on-shell vertex operators) that, acting on a vacuum, define wavefunctions of pure states (e.g. a tachyon), the solutions that we describe correspond to the reduced density matrices which eigenvalues describe the entanglement between higher-spin modes with different spin values. We compute the entanglement entropy on these OSFT solutions, and the answer is expressed in terms of converging series in inverse weighted partition numbers. In the case of $D$-dimensional bosonic string theory, the entanglement entropy of spin $1$ subsystem and the system of all the spin values is given by $D{\log{\lambda_0}}+{D\over{\lambda_0}}\sum_{N=3}^\infty{{|\beta(N)|}\over{\lambda(N)}} {\log{({{\lambda(N)}\over{|\beta(N)|}})}}$, where $\lambda(N)$ is the weighted number of partitions of $N$, $\beta(N)={{(N-1)\zeta(3)-\zeta(2)}\over{(N-1)^4}}$ and $\lambda_0=\sum_{N=1}^{\infty}{{\beta(N)}\over{\lambda(N)}}$ ($\zeta$ is Riemann's zeta-function). The first term, $D{\log{\lambda_0}}$, represents the entanglement swapping between string vacuum and string excitations. We generalize this result to obtain the entanglement for a subsystem of a given spin $s$ in a given space-time dimension. We also discuss how open string field theory may be used to study the entanglement of systems other than higher spin excitations in string theory.