Notes on quantum entanglement of local operators

Abstract

This is an expanded version of the short report arXiv:1401.0539, where we studied the time evolution of (Renyi) entanglement entropies for the excited state defined by acting a given local operator on the ground state. In the present paper, we introduce (Renyi) entanglement entropies of given local operators which are defined by late time values of excesses of (Renyi) entanglement entropies. They measure the degrees of freedom of local operators and characterize them in conformal field theories from the viewpoint of quantum entanglement. We explain how to compute them in free massless scalar field theories and we also investigate their time evolution. Our results can be interpreted in terms of the relativistic propagation of entangled pairs. The main new results which we acquire in the present paper are as follows. Firstly, we provide an explanation which shows that (Renyi) entanglement entropies of a specific operator are given by (Renyi) entanglement entropies whose reduced density matrices are given by the binomial distribution. That operator is constructed of only the scalar field. Secondly, we found the sum rule which (Renyi) entanglement entropies of those local operators obey. Those local operators are located separately. Moreover we argue that (Renyi) entanglement entropies of specific operators in conformal field theories are given by (Renyi) entanglement entropies whose reduced density matrices are given by the binomial distribution. These specific operators are constructed of single-species operators. We also argue that general operators obey the sum rule which we mentioned above.

Entanglement between two interacting CFTs and generalized holographic entanglement entropy

“…Note that this instantaneous increasing of the entanglement entropy occurs because the interaction introduces the entanglement between the two scalar field theories homogeneously at the same time. This is in contrast with evolutions of entanglement entropy under quantum quenches [47,48] or local operator excitations [44][45][46], where the causal propagation of entangled pairs play an important role.…”

“…This is naturally understood as in [44][45][46] because the state we are looking at is 25) in the original (unreplicated) theory and this is an maximally entangled state (EPR state).…”

Left-right entanglement entropy of boundary states

“…There have also been some recent work discussing excited states in CFT's in various dimensions [11][12][13][14].…”

Entanglement constant for conformal families

“…In the computation of the late time value (semi-inifinite interval) of the second Rényi entanglement entropy we use the (3.7). If we first take the limit of large central charge, the contribution to the second Rényi entropy becomes ∆S (2) 8 log 2t − 2 log c 2 (E.1) This is the logarithmic growth with time obtained in [6] (the negative part comes form the normalisation) and it should be possible to perform a similar computation for the entanglement entropy and match it with gravity using [34] as in [7]. On the other hand, as we saw in the text, if we know the answer for the full correlator (all quantum gravity effects taken into account) we can first take → 0 and obtain the entanglement constant.…”

“…In fact, at late times, the Rényi entanglement entropies saturate to constants that can be used to characterize local operators by the way they change quantum entanglement in a given state. Detailed reviews can be found in [6,7] and further results and applications to finite temperature and holography in [8,9,10,11,12,13].…”