Entanglement structures in quantum field theories. II. Distortions of vacuum correlations through the lens of local observers

Natalie Klco and D. H. Beck

Phys. Rev. A 108, 012429 – Published 24 July 2023

Abstract

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.

Received 4 May 2023
Accepted 7 July 2023

DOI:https://doi.org/10.1103/PhysRevA.108.012429

Physics Subject Headings (PhySH)

Entanglement in field theory Measures of continuous variables entanglement Quantum correlations in quantum information Quantum entanglement Quantum simulation

Particles & FieldsNuclear PhysicsQuantum Information, Science & Technology

Authors & Affiliations

Natalie Klco ^1,* and D. H. Beck ^2,†

¹Duke Quantum Center and Department of Physics, Duke University, Durham, North Carolina 27708, USA
²Department of Physics and Illinois Quantum Information Science and Technology (IQUIST) Center, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA

^*natalie.klco@duke.edu
^†dhbeck@illinois.edu

Entanglement structures in quantum field theories: Negativity cores and bound entanglement in the vacuum

Natalie Klco, D. H. Beck, and Martin J. Savage

Phys. Rev. A 107, 012415 (2023)

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Vol. 108, Iss. 1 — July 2023

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Images

Figure 1
Schematic representation of the thought experiment guiding the present calculations for a noninteracting scalar field vacuum in the infinite volume of one spatial dimension, $| ψ 〉$ . When latticized, the free scalar field vacuum is a Gaussian continuous-variable (CV) system with one CV quantum degree of freedom per lattice site, each represented by a horizontal line in the quantum circuit. For local observation, two patches of the field vacuum, each with $d$ lattice sites, are extracted and compose the Hilbert space of $ϕ_{p}$ , while the remaining external volume is represented by $ϕ_{v}$ . The locally observed mixed state of the patch pair results from tracing the external volume. However, a measurement of the external volume can be classically communicated to inform local unitary displacement operators capable of unifying the mixed convex decomposition into a single measurement-basis-dependent underlying pure state, $| σ^{(m, U)} 〉$ . In particular, measurement of the infinite volume in the field and conjugate momentum bases, producing $| σ^{(m, ϕ)} 〉$ and $| σ^{(m, π)} 〉$ , respectively, are at the center of this work.
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Figure 2
Underlying and locally accessible entanglement, quantified by the logarithmic negativities $N_{A | B} (σ^{(m)})$ and $N_{A | B} (σ^{(t)})$ , respectively, between patches ( $d = 16$ sites each) of the one-dimensional free lattice scalar field vacuum in the massless limit ( $m = 10^{- 10}$ ) as a function of dimensionless lattice separation, $\tilde{r} / d$ , in the thermodynamic limit of infinite volume. The classical noise generated upon tracing of the patch-external field volume produces a parametric suppression of the entanglement from polynomial or logarithmic to exponentially decaying, culminating in compatibility with a separable decomposition at finite dimensionless separation, ${\tilde{r}}_{N /} = 264$ . The vertical dashed lines are labeled in correspondence with the wave functions in the upper row of Fig. 3. Inset: Sum of two-body, $(1_{A} \times 1_{B})$ , contributions to the logarithmic negativity between volume-measured (black) and volume-traced (gray) field patches calculated in the local Williamson [60, 70] and negativity-inspired [36] bases, as discussed in Sec. 3d. Local transformations to these bases (by $S_{W}$ and $S_{N}$ , respectively) are designed to maintain the parity symmetry of the pair of field patches, leading to the diagrammatically indicated two-body structure.
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Figure 3
Spatial dependence of $G H^{Γ}$ ground-state wave functions associated with the dominant contribution to the logarithmic negativity between field patches in the massless (top) regime with $d = 16, m = 10^{- 10}$ ( ${\tilde{r}}_{N /} = 264$ ) and in the massive (bottom) regime with $d = 16, m = 0.3$ ( ${\tilde{r}}_{N /} = 59$ ). In all cases, the wave function upon volume measurement (associated with $σ^{(m, ϕ)}$ , black) remains predominantly in the IR, while the wave function upon volume tracing (associated with $σ^{(t)}$ , gray) is driven into the UV with increasing separation $\tilde{r}$ shown in lattice units.
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