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General Entangled Partner in Quantum Field Theory

Jose Trevison, Koji Yamaguchi, Masahiro Hotta

Published Jul 10, 2018• 0 citations
OpenAlex

Abstract

We extend the definition of the entangled partner mode of an arbitrary mode of a scalar quantum field in a general state. Even in the case of a free field in a Gaussian state, this provides a new class of partner named spatially overlapped partner. A general formula of spatially overlapped partners is given for a free field in a $(d+1)$-dimensional curved spacetime. We analyze memory effects of the spatially entangled partner systems of a quantum field as quantum information storage for an expanding universe.

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We extend the definition of the entangled partner mode of an arbitrary mode of a scalar quantum field in a general state. Even in the case of a free field in a Gaussian state, this provides a new class of partner named spatially overlapped partner. A general formula of spatially overlapped partners is given for a free field in a (d+1)(d+1)(d+1)-dimensional curved spacetime. We analyze memory effects of the spatially entangled partner systems of a quantum field as quantum information storage for an expanding universe.
We introduce a locally symplectic-invariant quantifier of correlations between arbitrary bosonic Gaussian modes, with particular emphasis on quantum field theory. The quantity, denoted by~Dsym\mathcal{D}^{\mathrm{sym}}Dsym, admits a simple geometric interpretation as the symmetric overlap between each mode and the purification partner of the other, providing a direct geometric characterization of how correlations are distributed between modes. We derive a necessary and sufficient criterion for two-mode Gaussian entanglement in terms of Dsym\mathcal{D}^{\mathrm{sym}}Dsym, placing on firm quantitative footing the intuition that the entanglement of a localized mode is encoded in the spatial support of its purification partner. We demonstrate the framework for wavepacket modes of a scalar quantum field in Minkowski spacetime (illustrating how the geometry of partner modes reveals the spatial structure of quantum correlations) and discuss extensions to multimode systems and mixed Gaussian states.
We investigate entanglement of local spatial modes defined by a quantum field in a de Sitter universe. The introduced modes show disentanglement behavior when the separation between two regions where local modes are assigned becomes larger than the cosmological horizon. To understand the emergence of separability between these local modes, we apply the monogamy inequality proposed by S. Camalet. We embed the focusing bipartite mode defined by the quantum field in a pure four-mode Gaussian state, and identify its partner modes. Then applying a Gaussian version of the monogamy relation, we show that the external entanglement between the bipartite mode and its partner modes constrains the entanglement of the bipartite mode. Thus the emergence of separability of local modes in the de Sitter universe can be understood from the perspective of entanglement monogamy.
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+1D+1D+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.
We demonstrate the presence of multimode entanglement in the vacuum state of a free, massless scalar quantum field in four-dimensional flat spacetime between two sets of field modes, each contained within a spacetime region that is causally disconnected from the other. This is true despite the fact that entanglement between pairs of individual field modes is sparse and appears only when the two individual modes are carefully selected. Our results reveal that, while entanglement between individual modes is limited, bipartite multimode entanglement in quantum field theory is ubiquitous. We further argue that such multimode entanglement is operationally extractable, and it forms the basis of the entanglement commonly discussed in entanglement harvesting protocols.
We demonstrate the presence of multimode entanglement in the vacuum state of a free, massless scalar quantum field in four-dimensional flat spacetime between two sets of field modes, each contained within a spacetime region that is causally disconnected from the other. This is true despite the fact that entanglement between pairs of individual field modes is sparse and appears only when the two individual modes are carefully selected. Our results reveal that, while entanglement between individual modes is limited, bipartite multimode entanglement in quantum field theory is ubiquitous. We further argue that such multimode entanglement is operationally extractable, and it forms the basis of the entanglement commonly discussed in entanglement harvesting protocols.
We introduce a framework to identify where the total correlations and entanglement with a chosen degree of freedom reside within the rest of a system, in the context of bosonic many-body Gaussian quantum systems. Our results are organized into two main propositions. First, for pure Gaussian states, we show that every correlated mode possesses a unique single-degree-of-freedom partner that fully captures its correlations (consisting of entanglement), and we provide an explicit construction of this partner from the complex structure of the system's state. Second, for mixed Gaussian states, we constructively demonstrate that the notion of a partner subsystem splits into two: a correlation partner, which contains all classical and quantum correlations and need not correspond to a single degree of freedom, and an entanglement partner, which is always at most single-mode. Finally, we extend the construction of partners to multi-mode subsystems. Together, these results provide conceptual practical tools to study how bipartite correlations and entanglement are structured and where they can be found in complex Gaussian many-body systems.