Entanglement Entropy in Timelike Slices: a Free Fermion Study

We define the entanglement entropy of a quantum state in a discrete set of points in an arbitrary spacetime slice, and give the explicit formula for free fermions. We investigate timelike (causal) slices specifically. For 1D lattice free fermions, we calculated the time-direction entanglement entropy in a set of times t_n=nτ (1≤n≤K) on the same site, and identified a stabilizing transition at τ=τ₀=2π/E₀, where E₀ is the energy range of single-fermion spectrum. At zero temperature, the time-like entanglement entropy of the lattice fermion with τ<τ₀ resembles the Cardy formula for one flavor of chiral fermion. For generic slices, the entanglement entropy shows a clear transition between spacelike and timelike slices. We conjecture τ₀ is the upper bound of time period for consecutive local observations to retrieve information from a quantum state, and conjecture a similar τ₀ exists in interacting models.