Four-mode squeezed states: two-field quantum systems and the symplectic group $$\mathrm {Sp}(4,{\mathbb {R}})$$

Colas, Thomas; Grain, Julien; Vennin, Vincent

doi:10.1140/epjc/s10052-021-09922-y

Cited by 24 publications

(40 citation statements)

References 51 publications

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“…(A.11). However, note that the group of real 4×4 symplectic matrices, usually denoted Sp(4, R), is of dimension 10 [48]. Therefore, partition changes, that are described by 4 parameters, only correspond to a subgroup (isomorphic to U(2), and corresponding to the "rotation" generators of table 2 in Ref.…”

Section: Discussionmentioning

confidence: 99%

“…Therefore, partition changes, that are described by 4 parameters, only correspond to a subgroup (isomorphic to U(2), and corresponding to the "rotation" generators of table 2 in Ref. [48]) of the symplectic group.…”

Section: Discussionmentioning

confidence: 99%

“…which defines the group of symplectic matrices T [23,47,48]. In particular, any symplectic matrix has determinant 1 [47],…”

Section: A Partitionsmentioning

confidence: 99%

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Discord and Decoherence

Martin,

Micheli,

Vennin

2021

Preprint

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In quantum information theory, quantum discord has been proposed as a tool to characterise the presence of "quantum correlations" between the subparts of a given system. Whether a system behaves quantum-mechanically or classically is believed to be impacted by the phenomenon of decoherence, which originates from the unavoidable interaction between this system and an environment. Generically, decoherence is associated with a decrease of the state purity, i.e. a transition from a pure to a mixed state. In this paper, we investigate how quantum discord is modified by this quantum-to-classical transition. This study is carried out on systems described by quadratic Hamiltonians and Gaussian states, with generalised squeezing parameters. A generic parametrisation is also introduced to describe the way the system is partitioned into two subsystems. We find that the evolution of quantum discord in presence of an environment is a competition between the growth of the squeezing amplitude and the decrease of the state purity. In phase space, this corresponds to whether the semi-minor axis of the Wigner ellipse increases or decreases, which has a clear geometrical interpretation. Finally, these considerations are applied to primordial cosmological perturbations, thus allowing us to investigate how large-scale structures in our universe, which are believed to arise from quantum fluctuations, can exhibit classical properties.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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Discord and Decoherence

Martin,

Micheli,

Vennin

2021

Preprint

Self Cite

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“…Indeed, when restricting one's attention to the (coarse-grained) configurations of the field at locations x 1 and x 2 , one implicitly traces over its configuration at all other locations (to which the configurations at x 1 and x 2 are entangled), which leads to a non-pure bipartite system. In general, this effective "self-decoherence" can be measured with the purity parameter [13][14][15][16]…”

Section: Entanglement Entropy Of Gaussian Statesmentioning

confidence: 99%

Real-space entanglement of quantum fields

Martin,

Vennin

2021

Preprint

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We introduce a general method permitting the determination of entanglement entropy (and related quantities) between configurations of a quantum field, which is either free or in interaction with a classical source, at two distinct spatial locations. We show how such a setup can be described by a bipartite, continuous Gaussian system. This allows us to derive explicit and exact formulas for the entanglement entropy, the mutual information and the quantum discord, solely in terms of the Fourier-space power spectra of the field. As an example, we apply our formalism to massless fields in flat space, where exact expressions are derived that only involve the ratio between the size of the regions over which the field is coarse-grained, and the distance between these regions. In particular, we recover the well-known result that mutual information decays as the fourth power of this ratio at large distances, although our formula is exact and applies to arbitrary distances. We find that quantum discord identically vanishes (unless coarse-graining is performed over smeared spheres, in which case it obeys the same suppression at large distance as mutual information).

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“…Both sets of variables are related by a diagonal canonical transformation, which thus corresponds to a pure squeezing [53].…”

Section: Scalar Degrees Of Freedommentioning

confidence: 99%

Hamiltonian formalism for cosmological perturbations: the separate-universe approach

Artigas,

Grain,

Vennin

2021

Preprint

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The separate-universe approach provides an effective description of cosmological perturbations at large scales, where the universe can be described by an ensemble of independent, locally homogeneous and isotropic patches. By reducing the phase space to homogeneous and isotropic degrees of freedom, it greatly simplifies the analysis of large-scale fluctuations. It is also a prerequisite for the stochastic-inflation formalism. In this work, we formulate the separate-universe approach in the Hamiltonian formalism, which allows us to analyse the full phase-space structure of the perturbations. Such a phase-space description is indeed required in dynamical regimes which do not benefit from a background attractor, as well as to investigate quantum properties of cosmological perturbations. We find that the separate-universe approach always succeeds in reproducing the same phase-space dynamics for homogeneous and isotropic degrees of freedom as the full cosmological perturbation theory, provided that the wavelength of the modes under consideration are larger than some lower bound that we derive. We also compare the separate-universe approach and cosmological perturbation theory at the level of the gauge-matching procedure, where the agreement is not always guaranteed and requires specific matching prescriptions that we present.

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Four-mode squeezed states: two-field quantum systems and the symplectic group $$\mathrm {Sp}(4,{\mathbb {R}})$$

Cited by 24 publications

References 51 publications

Discord and Decoherence

Discord and Decoherence

Real-space entanglement of quantum fields

Hamiltonian formalism for cosmological perturbations: the separate-universe approach

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