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arXiv:2304.03120 · Journal of Physics A Mathematical and Theoretical

On separable states in relativistic quantum field theory

Ko Sanders

Published Nov 10, 2023• 3 citations

Abstract

Abstract We initiate an investigation into separable, but physically reasonable, states in relativistic quantum field theory. In particular we will consider the minimum amount of energy density needed to ensure the existence of separable states between given spacelike separated regions. This is a first step towards improving our understanding of the balance between entanglement entropy and energy (density), which is of great physical interest in its own right and also in the context of black hole thermodynamics. We will focus concretely on a linear scalar quantum field in a topologically trivial, four-dimensional globally hyperbolic spacetime. For rather general spacelike separated regions A and B we prove the existence of a separable quasi-free Hadamard state. In Minkowski spacetime we provide a tighter construction for massive free scalar fields: given any R > 0 we construct a quasi-free Hadamard state which is stationary, homogeneous, spatially isotropic and separable between any two regions in an inertial time slice <?CDATA $t = \mathrm{const.}$?> t = c o n s t . all of whose points have a distance <?CDATA ${\gt} R$?> > R . We also show that the normal ordered energy density of these states can be made <?CDATA ${\unicode{x2A7D}} 10^{31}\frac{m^4}{(mR)^8}e^{-\frac14mR}$?> ⩽ 10 31 m 4 ( m R ) 8 e − 1 4 m R (in Planck units). To achieve these results we use a rather explicit construction of test-functions f of positive type for which we can get sufficient control on lower bounds on <?CDATA $\hat{f}$?> f ˆ .

DOI: 10.1088/1751-8121/ad0bca

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Abstract We initiate an investigation into separable, but physically reasonable, states in relativistic quantum field theory. In particular we will consider the minimum amount of energy density needed to ensure the existence of separable states between given spacelike separated regions. This is a first step towards improving our understanding of the balance between entanglement entropy and energy (density), which is of great physical interest in its own right and also in the context of black hole thermodynamics. We will focus concretely on a linear scalar quantum field in a topologically trivial, four-dimensional globally hyperbolic spacetime. For rather general spacelike separated regions A and B we prove the existence of a separable quasi-free Hadamard state. In Minkowski spacetime we provide a tighter construction for massive free scalar fields: given any R > 0 we construct a quasi-free Hadamard state which is stationary, homogeneous, spatially isotropic and separable between any two regions in an inertial time slice <?CDATA $t = \mathrm{const.}$?> t = c o n s t . all of whose points have a distance <?CDATA ${\gt} R$?> > R . We also show that the normal ordered energy density of these states can be made <?CDATA ${\unicode{x2A7D}} 10^{31}\frac{m^4}{(mR)^8}e^{-\frac14mR}$?> ⩽ 10 31 m 4 ( m R ) 8 e − 1 4 m R (in Planck units). To achieve these results we use a rather explicit construction of test-functions f of positive type for which we can get sufficient control on lower bounds on <?CDATA $\hat{f}$?> f ˆ .

This paper is the second and final part of a two-part study. We construct positive-energy relativistic spatial localization observables in Minkowski spacetime within standard quantum field theory, using the stress--energy--momentum tensor smeared with suitable test functions. For each fixed timelike direction, the construction gives positive operator-valued measures (POVMs) on spacelike hypersurfaces, well defined on every

n

-particle sector and satisfying a relativistic causality condition excluding superluminal propagation of detection probabilities. The observables are built from local or quasi-local field-theoretic quantities, thus providing a rigorous version of earlier heuristic proposals. In the one-particle sector, the construction reduces to the observable previously introduced by the author, and its first moment gives the Newton--Wigner position operator under appropriate normalization and centering assumptions. Because the Reeh--Schlieder theorem prevents the normally ordered stress--energy--momentum tensor from being positive on the full Fock space, we use quantum energy inequalities to obtain lower bounds controlling deviations from positivity. This leads to regularized operator families, bounded from below, which approximate the localization effects. Finally, we define conditional localization observables for finite laboratories through modified local energy operators. By Haag duality, the corresponding conditional POVMs belong to local von Neumann algebras and commute for causally separated regions, in accordance with the Araki--Haag--Kastler framework. The results show how commutativity of localization observables is recovered for conditional measurements in finite spacetime regions.

It is well known that entanglement is widespread in quantum field theory, in the following sense: every Reeh-Schlieder state contains entanglement between any two spatially separated regions. This applies, in particular, to the vacuum of a noninteracting scalar theory in Minkowski spacetime. Discussions on entanglement in field theory have focused mainly on subsystems containing infinitely many degrees of freedom---typically, the field modes that are supported within a compact region of space. In this article, we study entanglement in subsystems made of finitely many field degrees of freedom, in a free scalar theory in

D+1

-dimensional Minkowski spacetime. The focus on finitely many modes of the field is motivated by the finite capabilities of real experiments. We find that entanglement between finite-dimensional subsystems is not common at all, and that one needs to carefully select the support of modes for entanglement to show up. We also find that entanglement is increasingly sparser in higher dimensions. We conclude that entanglement in Minkowski spacetime is significantly less ubiquitous than normally thought.

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.

Recently, several notions of entanglement in time have emerged as a novel frontier in quantum many-body physics, quantum field theory, and gravity. We propose a systematic prescription to characterize temporal entanglement in relativistic quantum field theory in a general state for an arbitrary subregion on a flat, constant time slice in a flat spacetime. Our prescription starts with the standard entanglement entropy of a spatial subregion and amounts to transporting the unchanged subregion to boosted time slices all the way across the light cone when it becomes, in general, a complex characterization of the corresponding temporal subregion. For holographic quantum field theories, our prescription amounts to an analytic continuation of all codimension-two bulk extremal surfaces satisfying the homology constraint and picking the one with the smallest real value of the area as the leading saddle point. We implement this prescription for holographic conformal field theories in thermal states on both a two-dimensional Lorentzian cylinder and three-dimensional Minkowski space, and we show that it leads to results with self-consistent physical properties of temporal entanglement.