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Notes on Quantum Entanglement of Local Operators

Masahiro Nozaki A

Published Aug 16, 2016• 67 citations
OpenAlex

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.

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Notes on quantum entanglement of local operators

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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.
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.
We introduce a series of quantities which characterize a given local operator in any conformal field theory from the viewpoint of quantum entanglement. It is defined by the increased amount of (Rényi) entanglement entropy at late time for an excited state defined by acting the local operator on the vacuum. We consider a conformal field theory on an infinite space and take the subsystem in the definition of the entanglement entropy to be its half. We calculate these quantities for a free massless scalar field theory in two, four and six dimensions. We find that these results are interpreted in terms of quantum entanglement of a finite number of states, including Einstein-Podolsky-Rosen states. They agree with a heuristic picture of propagations of entangled particles.
In this paper we study the time evolution of (Rényi) entanglement entropies for locally excited states in four dimensional free massless fermionic field theory. Locally excited states are defined by being acted by various local operators on the ground state. Their excesses are defined by subtracting (Rényi) entanglement entropy for the ground state from those for locally excited states. They finally approach some constant if the subsystem is given by half of the total space. They have spin dependence. They can be interpreted in terms of quasi-particles.
In this paper, we study Renyi and von Neumann entanglement entropy of excited states created by local operators in large-|NNN| (or large-central-charge) conformal field theories (CFTs). First we point out that a naive large-|NNN| expansion can break down for the von Neumann entanglement entropy, while it does not for the Renyi entanglement entropy. This happens even for the excited states in free Yang–Mills theories. Next, we analyze strongly coupled large-|NNN| CFTs from both the field-theoretic and holographic viewpoints. We find that the Renyi entanglement entropy of the excited state, produced by a local operator, grows logarithmically under its time evolution and its coefficient is proportional to the conformal dimension of the local operator.
We develop the computational method of entanglement entropy based on the idea that TrρΩn \mathrm{T}\mathrm{r}{\rho}_{{}^{\varOmega}}^n TrρΩn​ is written as the expectation value of the local operator, where ρ Ω is a density matrix of the subsystem Ω. We apply it to consider the mutual Rényi information I (n)(A, B) = S () + S () − S ∪ () of disjoint compact spatial regions A and B in the locally excited states defined by acting the local operators at A and B on the vacuum of a (d + 1)-dimensional field theory, in the limit when the separation r between A and B is much greater than their sizes R A,B . For the general QFT which has a mass gap, we compute I (n)(A, B) explicitly and find that this result is interpreted in terms of an entangled state in quantum mechanics. For a free massless scalar field, we show that for some classes of excited states, I (n)(A, B) − I (n)(A, B)| r → ∞ = C () /r α(d − 1) where α = 1 or 2 which is determined by the property of the local operators under the transformation ϕ → −ϕ and α = 2 for the vacuum state. We give a method to compute C (2) systematically.
In this paper, we investigate the second and third pseudo Rényi entanglement entropies (PREE) for a locally excited state $| ψ\rangle $ and its time evolution ∣ϕ⟩=e−iHt∣ψ⟩| ϕ\rangle = e^{- i H t} | ψ\rangle∣ϕ⟩=e−iHt∣ψ⟩ in a two-dimensional conformal field theory whose field content is a free massless scalar field. We consider excited states which are constructed by applying primary operators at time t=0t=0t=0, on the vacuum state. We study the time evolution of the PREE for an entangling region in the shape of finite and semi-infinite intervals at zero temperature. It is observed that the PREE is always a complex number for t≠0t \neq 0t=0 and is a pure real number at t=0t=0t=0. Moreover, we discuss on its dependence on the location xmx_mxm​ of the center of the entangling region.