Capacity of entanglement in local operators

Abstract

We study the time evolution of the excess value of capacity of entanglement between a locally excited state and ground state in free, massless fermionic theory and free Yang-Mills theory in four spacetime dimensions. Capacity has non-trivial time evolution and is sensitive to the partial entanglement structure, and shows a universal peak at early times. We define a quantity, the normalized “Page time”, which measures the timescale when capacity reaches its peak. This quantity turns out to be a characteristic property of the inserted operator. This firmly establishes capacity as a valuable measure of entanglement structure of an operator, especially at early times similar in spirit to the Rényi entropies at late times. Interestingly, the time evolution of capacity closely resembles its evolution in microcanonical and canonical ensemble of the replica wormhole model in the context of the black hole information paradox.

Eigenstate capacity and Page curve in fermionic Gaussian states

“…Instead, it reaches a maximal value where the entropies are not minimal. This reflects the fact that capacity is a good probe for quantifying partial entanglement structure [37] which was previously suggested in the context of replica wormholes [41][42][43][44].…”

“…We follow the approach outlined in [20,30]. First, we write the capacity for a single many-body eigenstate |m of the form [41]…”

“…We consider a different entanglement measure, the capacity of entanglement (CoE) [36][37][38], first considered in the Kitaev model to describe topologically ordered states [39] and to understand the thermodynamic properties of the entanglement spectrum [40]. Very recently, it has received considerable attention, for example, in local operator excitations [41], random pure states [42], and in the context of holography and black hole information problem, where it is shown to capture partially entangled states in replica wormhole geometry [43,44]. While the EE is the expectation value of the modular Hamiltonian K A = − log ρ A , the CoE is defined as the variance of K A , namely [37,41]…”

“…Very recently, it has received considerable attention, for example, in local operator excitations [41], random pure states [42], and in the context of holography and black hole information problem, where it is shown to capture partially entangled states in replica wormhole geometry [43,44]. While the EE is the expectation value of the modular Hamiltonian K A = − log ρ A , the CoE is defined as the variance of K A , namely [37,41]…”

“…The second equality comes from the replica method and is useful for holographic computations [36]. It has been shown that in the case of operator excitations, the departure from the maximally entangled state can be captured by its CoE, which can be interpreted as entanglement between quasi-particles [41].…”