Entanglement in relativistic quantum field theory

Abstract

I present some general ideas about quantum entanglement in relativistic quantum field theory, especially entanglement in the physical vacuum. Here, entanglement is defined between different single particle states (or modes), parameterized either by energy-momentum together with internal degrees of freedom, or by spacetime coordinate together with the component index in the case of a vector or spinor field. In this approach, the notion of entanglement between different spacetime points can be established. Some entanglement properties are obtained as constraints from symmetries, e.g., under Lorentz transformation, space inversion, time reverse and charge conjugation.Quantum entanglement is a notion about the structure of a quantum state of a composite system, referring to its non-factorization in terms of states of subsystems. It is regarded as an essential quantum characteristic [1,2]. Entanglement with environment is also crucial in decoherence, i.e. the emergence of classical phenomena in a quantum foundation, and may even be a possible explanation of superselection rules [3,4]. There has been a lot of activities on various aspects of entanglement, including some recent works which take into account relativity [5,6]. Investigations on entanglement in quantum field theories may provide useful perspectives on field theory issues. On the other hand, as the framework of fundamental physics, incorporating relativity, quantum field theory may be useful in deepening our understanding of entanglement. Besides, entanglement due to environmental perturbation may also be helpful in understanding spontaneous symmetry breaking. Most of the methods in field theory adopt Heisenberg or interaction picture, and do not need the explicit form of the underlying quantum state living in an infinite-dimensional Hilbert space. Nevertheless, in many circumstances, it is still important to know the nature of the quantum state, most notably the vacuum. In this paper, as an extension of some previous discussions on non-relativistic quantum field theories [7], I present some general ideas about the nature of entanglement in relativistic quantum field theory, and constraints from symmetries. Such investigations may offer useful insights on the structures of the vacua in quantum field theories on one hand, and on quantum information in relativistic regime on the other hand.First, I describe the basic method here of characterizing entanglement in quantum field theory. In quantum field theory, the dynamical variables are field operators (in real spacetime) or annihilation and creation operators (in energy-momentum space), in terms of which any observable can be expressed. Spacetime coordinate plus the component index in the case of a vector or spinor field, or energy-momentum plus internal degrees of freedoms (such as being particle or antiparticle, spin, polarization, etc.) are merely parameters. These parameters define the modes in either the real spacetime or the momentum space, and exactly provide the labels for the (distinguishable) subsys...

Unruh effect in quantum information beyond the single-mode approximation

“…The question of understanding entanglement in noninertial frames has been central to the development of the emerging field of relativistic quantum information [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. The main aim of this field is to incorporate relativistic effects to improve quantum information tasks (such as quantum teleportation) and to understand how such protocols would take place in curved space-times.…”

Spin-particles entanglement in Robertson-Walker spacetime

“…Their definition requires that in the distant past and in the far future spacetime tends to Minkowskian spacetime. Taking this approach it has been possible to learn about certain aspects of entanglement in curved spacetimes [11][12][13][14]. A suitable metric for a curved spacetime satisfying the above requirements turns out to be the (conformally flat) Robertson-Walker metric [10].…”

Pairwise mode entanglement in Schwinger production of particle-antiparticle pairs in an electric field

“…As usual [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37], the in vacuum can be expanded in terms of the out states of modes k and −k,…”

“…We treat the momentum modes as the subsystems [26], with each momentum mode k living in a four-dimensional Hilbert space with the basis states…”

“…It was proposed that in field theory, the proper approach of characterizing the quantum entanglement is to use modes, designated by the coordinates or momenta, as the subsystems [26]. Some concepts developed in quantum information theory have been applied to quantum states near a black hole [27][28][29][30][31][32] and in an expanding universe [33][34][35][36].…”