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arXiv:1501.01743 · Journal of High Energy Physics

Entanglement of spatial regions vs. entanglement of particles

Swapnamay Mondal

Published Aug 1, 2016• 1 citation
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Abstract

Consider an arbitrary local quantum field theory with a gap or an arbitrary gapless free theory. We consider states in such a theory, that describe two entangled particles localized in disjoint regions of space. We show that in such a state, to leading order, Rényi entropies of spatial regions, containing only one of the particles are same as their quantum mechanical counterparts, after subtraction of vacuum contribution. Subleading corrections depend on overlap of wave functions. These results suggest that Von Neumann entropy of a spatial region, after subtraction of vacuum contribution, can serve as a measure of entanglement of indistinguishable particles in pure states.

DOI
10.1007/jhep08(2016)121

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Consider an arbitrary local quantum field theory with a gap or an arbitrary gapless free theory. We consider states in such a theory, that describe two entangled particles localized in disjoint regions of space. We show that in such a state, to leading order, Rényi entropies of spatial regions, containing only one of the particles are same as their quantum mechanical counterparts, after subtraction of vacuum contribution. Subleading corrections depend on overlap of wave functions. These results suggest that Von Neumann entropy of a spatial region, after subtraction of vacuum contribution, can serve as a measure of entanglement of indistinguishable particles in pure states.
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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.
First, we study the entanglement entropy S AB of a d + 1 dimensional massless free scalar field on two disjoint compact spatial regions A and B whose sizes are R 1 and R 2 , respectively, and the distance between them is r.The state of the massless free scalar field is the vacuum state.We consider the mutual information S A;B ≡ S A + S B -S AB where S A and S B are the entanglement entropy on the inside region of A and B, respectively.We develop the computational method based on that of Bombelli et al and obtain the result thatWhen A and B are spheres in d = 2 and 3, we obtain explicit resultsBy using the method used in the Minkowski spacetime case with some modifications, we study the entanglement entropy, S C , of the massless free scalar field on the outside region C of two black holes A and B whose radii are R 1 and R 2 and how it depends on the distance, r( R 1 , R 2 ) , between two black holes.If we can consider the entanglement entropy as thermodynamic entropy, we can see the entropic force (we call this entanglement entropic force) acting on the two black holes from the r dependence of S C .We consider the property of the entropic force.
The entanglement between spatial regions in an interacting Bose-Einstein condensate is investigated using a quantum field theoretic formalism. Regions that are small compared to the healing length are governed by a nonrelativistic quantum field theory in the vacuum limit, and we show that the latter has vanishing entanglement. In the opposite limit of a region that is large compared to the healing length, the entanglement entropy is like in the vacuum of a relativistic theory where the velocity of light is replaced with the velocity of sound and where the inverse healing length provides a natural ultraviolet regularization scale. Besides the von Neumann entanglement entropy, we also calculate R\'enyi entanglement entropies for a one-dimensional quasicondensate.
Entanglement is defined between subsystems of a quantum system, and at fixed time two regions of space can be viewed as two subsystems of a relativistic quantum field. The entropy of entanglement between such subsystems is ill-defined unless an ultraviolet cutoff is introduced, but it still diverges in the continuum limit. This behavior is generic for arbitrary finite-energy states, hence a conceptual tension with the finite entanglement entropy typical of nonrelativistic quantum systems. We introduce a novel approach to explain the transition from infinite to finite entanglement, based on coarse graining the spatial resolution of the detectors measuring the field state. We show that states with a finite number of particles become localized, allowing an identification between a region of space and the nonrelativistic degrees of freedom of the particles therein contained, and that the renormalized entropy of finite-energy states reduces to the entanglement entropy of nonrelativistic quantum mechanics.
Operationally accessible entanglement in bipartite systems of indistinguishable particles could be reduced due to restrictions on the allowed local operations as a result of particle number conservation. In order to quantify this effect, Wiseman and Vaccaro [Phys. Rev. Lett. 91, 097902 (2003)PRLTAO0031-900710.1103/PhysRevLett.91.097902] introduced an operational measure of the von Neumann entanglement entropy. Motivated by advances in measuring Rényi entropies in quantum many-body systems subject to conservation laws, we derive a generalization of the operationally accessible entanglement that is both computationally and experimentally measurable. Using the Widom theorem, we investigate its scaling with the size of a spatial subregion for free fermions and find a logarithmically violated area law scaling, similar to the spatial entanglement entropy, with at most a double-log leading-order correction. A modification of the correlation matrix method confirms our findings in systems of up to 10^{5} particles.