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arXiv:quant-ph/0508085 · Physical Review A

Many-region vacuum entanglement: Distilling a W state

Jonathan Silman, Benni Reznik

Published May 4, 2005• 15 citations
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Abstract

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 $N$ such regions enable us to distill an $N$-partite $W$ state, and therefore exhibit true $N$-fold entanglement. Furthermore, we show that for $N=3$, the correlations cannot be reproduced by a hybrid local-nonlocal hidden-variable model. For $N\ensuremath{\geqslant}4$ the issue remains open.

DOI
10.1103/physreva.71.054301

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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⩾4N\ensuremath{\geqslant}4N\ensuremath⩾4 the issue remains open.
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 $\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}{(L∕cT)}^{3}]$.
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 $\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}(L∕R{)}^{2}]$. We discuss the similarities with and differences from analogous results obtained for the massless Klein-Gordon vacuum.
Quantum field theory is the application of quantum physics to fields. It provides a theoretical framework widely used in particle physics and condensed matter physics. One of the most distinct features of quantum physics with respect to classical physics is entanglement or the existence of strong correlations between subsystems that can even be spacelike separated. In quantum fields, observables restricted to a region of space define a subsystem. While there are proofs on the existence of local observables that would allow a violation of Bell's inequalities in the vacuum states of quantum fields as well as some explicit but technically demanding schemes requiring an extreme fine-tuning of the interaction between the fields and detectors, an experimentally accessible entanglement witness for quantum fields is still missing. Here we introduce smeared field operators which allow reducing the vacuum to a system of two effective bosonic modes. The introduction of such collective observables is motivated by the fact that no physical probe has access to fields in single spatial (mathematical) points but rather smeared over finite volumes. We first give explicit collective observables whose correlations reveal vacuum entanglement in the Klein-Gordon field. We then show that the critical distance between the two regions of space above which two effective bosonic modes become separable is of the order of the Compton wavelength of the particle corresponding to the massive Klein-Gordon field.
We analyze the correlations of the vacuum state of a charged scalar field by constructing localized observables in two disjoint regions. We consider a time-dependent homogeneous electric field and examine the time evolution of correlations by relating it with the particle-antiparticle creation through the Schwinger effect. Our results show that the electric field increases the mixedness of the states and enhances total correlations, while the distillable entanglement among the regions decreases, vanishing at a finite time for sufficiently intense pulses within the non-perturbative pair-production regime. We also discuss the effect of anisotropy induced by the external background, showing that the distribution of correlations acquires a nontrivial directional dependence.
When observing a quantum field via detectors with access to only the mixed states of spatially separated, local regions---a ubiquitous experimental design---the capacity to access the full extent of distributed entanglement can be limited, shrouded by classical correlations. By performing projective measurements of the field external to two detection patches and classically communicating the results, underlying pure states may be identified for which entanglement quantification is clear. In the Gaussian continuous-variable states of the free scalar field vacuum, this protocol uncovers a disparity between the spacelike entanglement established within the field and that which is locally detectable. This discrepancy is found to grow exponentially with the separation between observation regions. The protocol developed herein offers insight and practical guidance for clarifying the unavoidable distortion of quantum field correlations when viewed from the vantage of a pair of local observers.