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arXiv:quant-ph/0609212 · Physical Review A

Long-range entanglement in the Dirac vacuum

Jonathan Silman, Benni Reznik

Published May 4, 2007• 37 citations
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Abstract

Recently, there have been a number of works investigating the entanglement properties of distinct noncomplementary parts of discrete and continuous bosonic systems in ground and thermal states. The relativistic fermionic case, however, has yet to be expressly addressed. In this paper we investigate the entanglement between a pair of far-apart regions of the $(3+1)$-dimensional massless Dirac vacuum via a previously introduced distillation protocol [Reznik et al., Phys. Rev. A 71, 042104 (2005)]. We show that entanglement persists over arbitrary distances, and that as a function of $L∕R$, where $L$ is the distance between the regions and $R$ is their typical scale, it decays no faster than $\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}(L∕R{)}^{2}]$. We discuss the similarities with and differences from analogous results obtained for the massless Klein-Gordon vacuum.

DOI
10.1103/physreva.75.052307

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Recently, there have been a number of works investigating the entanglement properties of distinct noncomplementary parts of discrete and continuous bosonic systems in ground and thermal states. The relativistic fermionic case, however, has yet to be expressly addressed. In this paper we investigate the entanglement between a pair of far-apart regions of the (3+1)(3+1)(3+1)-dimensional massless Dirac vacuum via a previously introduced distillation protocol [Reznik et al., Phys. Rev. A 71, 042104 (2005)]. We show that entanglement persists over arbitrary distances, and that as a function of L∕RL∕RL∕R, where LLL is the distance between the regions and RRR is their typical scale, it decays no faster than \ensuremath∼exp[\ensuremath−(L∕R)2]\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}(L∕R{)}^{2}]\ensuremath∼exp[\ensuremath−(L∕R)2]. We discuss the similarities with and differences from analogous results obtained for the massless Klein-Gordon vacuum.
We investigate the correlations between any number of arbitrarily far-apart regions of the vacuum of the free Klein-Gordon field by means of its finite duration coupling to an equal number of localized detectors. We show that the correlations between any NNN such regions enable us to distill an NNN-partite WWW state, and therefore exhibit true NNN-fold entanglement. Furthermore, we show that for N=3N=3N=3, the correlations cannot be reproduced by a hybrid local-nonlocal hidden-variable model. For $N\ensuremath{\geqslant}4$ the issue remains open.
We prove an upper bound on long-range distillable entanglement in D spatial dimensions. Namely, it must decay faster than 1/r, where r is the distance between entangled regions. For states that are asymptotically rotationally invariant, the bound is strengthened to 1/r^{D}. We then find explicit examples of quantum states with decay arbitrarily close to the bound. In one dimension, we construct free fermion Hamiltonians with nearest neighbor couplings that have these states as ground states. Curiously, states in conformal field theory are far from saturation, with distillable entanglement decaying faster than any polynomial.
An upper bound of the relative entanglement entropy of thermal states at an inverse temperature βββ of linear, massive Klein-Gordon and Dirac quantum field theories across two regions, separated by a nonzero distance ddd in a Cauchy hypersurface of an ultrastatic (spin-)spacetime has been computed. This entanglement measure is bounded by a negative constant times ln⁡∣tanh⁡(πd/2β)∣\ln | \tanh (πd/ 2 β) |ln∣tanh(πd/2β)∣ which signifies power law decay for asymptotic ddd where the exponent depends on β<∞β< \inftyβ<∞.
We employ an approach wherein the ground state entanglement of a relativistic free scalar field is directly probed in a controlled manner. The approach consists of having a pair of initially nonentangled detectors locally interact with the vacuum for a finite duration TTT, such that the two detectors remain causally disconnected, and then analyzing the resulting detector mixed state. We show that the correlations between arbitrarily far-apart regions of the vacuum cannot be reproduced by a local hidden-variable model, and that as a function of the distance LLL between the regions, the entanglement decreases at a slower rate than $\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}{(L∕cT)}^{3}]$.
We study the entanglement between disjoint subregions in quantum critical systems through the lens of the logarithmic negativity. We work with systems in arbitrary dimensions, including conformal field theories and their corresponding lattice Hamiltonians, as well as resonating valence-bond states. At small separations, the logarithmic negativity is big and displays universal behavior, but we show nonperturbatively that it decays faster than any power at large separations. This can already be seen in the minimal setting of single-spin subregions. The corresponding absence of distillable entanglement at large separations generalizes the one-dimensional result, and indicates that quantum critical ground states do not possess long-range bipartite entanglement, at least for bosons. For systems with fermions, a more suitable definition of the logarithmic negativity exists that takes into account fermion parity, and we show that it decays algebraically. Along the way we obtain general results for the moments of the partially transposed density matrix. Published by the American Physical Society 2024
A bstract We study the static entanglement structure in (1+1)-dimensional free Dirac-fermion theory with Lifshitz symmetry and arbitrary integer dynamical critical exponent. This model is different from the one introduced in [Hartmann et al., SciPost Phys. 11 (2021) 031] due to a proper treatment of the square Laplace operator. Dirac fermion Lifshitz theory is local as opposed to its scalar counterpart which strongly affects its entanglement structure. We show that there is quantum entanglement across arbitrary subregions in various pure (including the vacuum) and mixed states of this theory for arbitrary integer values of the dynamical critical exponent. Our numerical investigations show that quantum entanglement in this theory is tightly bounded from above. Such a bound and other physical properties of quantum entanglement are carefully explained from the correlation structure in these theories. A generalization to (2+1)-dimensions where the entanglement structure is seriously different is addressed.